EMS Tracts in Mathematics 24
EMS Tracts in Mathematics Editorial Board: Carlos E. Kenig (The University of Chicago, USA) Andrew Ranicki (The University of Edinburgh, Great Britain) Michael Röckner (Universität Bielefeld, Germany, and Purdue University, USA) Vladimir Turaev (Indiana University, Bloomington, USA) Alexander Varchenko (The University of North Carolina at Chapel Hill, USA) This series includes advanced texts and monographs covering all fields in pure and applied mathematics. Tracts will give a reliable introduction and reference to special fields of current research. The books in the series will in most cases be authored monographs, although edited volumes may be published if appropriate. They are addressed to graduate students seeking access to research topics as well as to the experts in the field working at the frontier of research. For a complete listing see our homepage at www.ems-ph.org. 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
Sergio Albeverio et al., The Statistical Mechanics of Quantum Lattice Systems Gebhard Böckle and Richard Pink, Cohomological Theory of Crystals over Function Fields Vladimir Turaev, Homotopy Quantum Field Theory Hans Triebel, Bases in Function Spaces, Sampling, Discrepancy, Numerical Integration , Erich Novak and Henryk Wozniakowski, Tractability of Multivariate Problems. Volume II: Standard Information for Functionals Laurent Bessières et al., Geometrisation of 3-Manifolds Steffen Börm, Efficient Numerical Methods for Non-local Operators. 2-Matrix Compression, Algorithms and Analysis Ronald Brown, Philip J. Higgins and Rafael Sivera, Nonabelian Algebraic Topology. Filtered Spaces, Crossed Complexes, Cubical Homotopy Groupoids Marek Janicki and Peter Pflug, Separately Analytical Functions Anders Björn and Jana Björn, Nonlinear Potential Theory on Metric Spaces , Erich Novak and Henryk Wozniakowski, Tractability of Multivariate Problems. Volume III: Standard Information for Operators Bogdan Bojarski, Vladimir Gutlyanskii, Olli Martio and Vladimir Ryazanov, Infinitesimal Geometry of Quasiconformal and Bi-Lipschitz Mappings in the Plane Hans Triebel, Local Function Spaces, Heat and Navier–Stokes Equations Kaspar Nipp and Daniel Stoffer, Invariant Manifolds in Discrete and Continuous Dynamical Systems Patrick Dehornoy with François Digne, Eddy Godelle, Daan Kramer and Jean Michel, Foundations of Garside Theory Augusto C. Ponce, Elliptic PDEs, Measures and Capacities. From the Poisson Equation to Nonlinear Thomas–Fermi Problems
Hans Triebel
Hybrid Function Spaces, Heat and Navier-Stokes Equations
Author: Hans Triebel Friedrich-Schiller-Universität Jena Fakultät für Mathematik und Informatik Mathematisches Institut 07737 Jena Germany E-mail: hans.triebel @uni-jena.de
2010 Mathematical Subject Classification: 46-02, 46E35, 42B35, 42C40, 35K05, 35Q30, 76D03, 76D05 Key words: Function spaces, Morrey spaces, heat equations, Navier-Stokes equations
ISBN 978-3-03719-150-7 The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch. This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. © European Mathematical Society 2015 Contact address: European Mathematical Society Publishing House Seminar for Applied Mathematics ETH-Zentrum SEW A27 CH-8092 Zürich Switzerland Phone: +41 (0)44 632 34 36 Email: info @ems-ph.org Homepage: www.ems-ph.org Typeset using the author’s TEX files: le-tex publishing services GmbH, Leipzig, Germany Printing and binding: Beltz Bad Langensalza GmbH, Bad Langensalza, Germany ∞ Printed on acid free paper 987654321
Preface This book is the continuation of [T13]. Our aim is twofold. First we develop the theory of hybrid spaces LrAsp;q .Rn / which are between the nowadays well-known global spaces Asp;q .Rn / with A 2 fB; F g and their localization (or Morreyfication) LrAsp;q .Rn / as considered in detail in [T13]. Spaces Asp;q .Rn / cover (fractional) Sobolev spaces, (classical) Besov spaces and H¨older-Zygmund spaces, whereas local Morrey spaces Lrp .Rn / are special cases of the local spaces LrAsp;q .Rn /. In [T13] we applied the theory of spaces LrAsp;q .Rn / to nonlinear heat equations and NavierStokes equations. But this caused some problems which will be discussed in the Introduction (Chapter 1) below. It came out quite recently that it is more natural in this context to switch from local spaces LrAsp;q .Rn / to hybrid spaces LrAsp;q .Rn /. This again will be illuminated in the Introduction below. It is the second aim of this book to apply the theory of global spaces Asp;q .Rn / and hybrid spaces LrAsp;q .Rn / to the Navier-Stokes equations @t u C .u; r/u u C rP D 0 div u D 0 u.; 0/ D u0
in Rn .0; T /, in Rn .0; T /; in Rn ;
(0.1) (0.2) (0.3)
in the version of @t u u C P div .u ˝ u/ D 0 u.; 0/ D u0
in Rn .0; T /, in Rn ;
(0.4) (0.5)
reduced to the scalar nonlinear heat equations @t v D v 2 v D 0 v.; 0/ D v0
in Rn .0; T /; (0.6) n in R ; (0.7) 1 where 0 < T 1. Here u.x; t/ D u .x; t/; : : : ; un .x; t/ in (0.1)–(0.5) is the unknown velocity and P .x; t/ the unknown (scalar) pressure, whereas v.x; t/ in (0.6), (0.7) is a scalar function, 2 n 2 N. Recall @t D @=@t, @j D @=@xj if j D 1; : : : ; n; n X k uj @j uk ; .u; r/u D
k D 1; : : : ; n;
(0.8)
j D1
div u D
n X
@j uj ;
rP D .@1 P; : : : ; @n P /;
(0.9)
j D1
and by (0.2) .u; r/u D div .u ˝ u/;
div .u ˝ u/k D
n X j D1
@j .uj uk /:
(0.10)
vi
Preface
Furthermore, P is the Leray projector .Pf /k D f k C Rk
n X
Rj f j ;
k D 1; : : : ; n;
(0.11)
j D1
based on the (scalar) Riesz transforms Z _ yk k g.x y/ dy; b g .x/ D cn lim Rk g.x/ D i jj "#0 jyj" jyjnC1
x 2 Rn :
(0.12)
In (0.4), (0.5) there is no need to care about (0.2) any longer. But if, in addition, div u0 D 0 then div u D 0 in our context (mild solutions based on fixed point assertions). In the scalar equation (0.6) we used the abbreviation Df D
n X
@j f:
(0.13)
j D1
As mentioned above we dealt in [T13] with the above equations in the context of the global spaces Asp;q .Rn /. Rigorous reduction of (0.1)–(0.3) to (0.4), (0.5) and finally to (0.6), (0.7) requires a detailed study of the nonlinearity u 7! u2 and of boundedness of Riesz transforms in the underlying spaces. In [T13] we tried to extend this theory to some local spaces LrAsp;q .Rn /. But one needs some modifications, especially a replacement of the Riesz transforms by some truncated Riesz transforms. In the Introduction below we repeat the above considerations in greater details and discuss in particular this somewhat disturbing (but unavoidable) point. The hybrid spaces LrAsp;q .Rn / preserve many desirable properties of the local spaces LrAsp;q .Rn / but avoid the above-indicated shortcomings. They are between global spaces and local spaces, which may justify calling them hybrid spaces. They coin1 r n cide with the well-studied spaces As; p;q .R /, D p C n , including the global spaces s;0 Ap;q .Rn / D Asp;q .Rn / as special cases. Chapter 1 is the announced Introduction where we return to the above description in greater details and with some references. Chapter 2 deals with local and global Morrey spaces Lrp .Rn /, Lrp .Rn /, their duals and preduals and, in particular, with the question whether the Riesz transforms Rk in (0.12) are bounded maps in these spaces and what they look like. This chapter is self-contained and we hope that it is of interest for researchers in this field. In Chapter 3 we develop the theory of the hybrid spaces LrAsp;q .Rn / as needed for our above-outlined purposes. It comes out that many basic properties for the local spaces LrAsp;q .Rn / can be transferred easily from [T13] to the hybrid spaces LrAsp;q .Rn /. We concentrate on some new aspects which will be crucial in the context described above. Similarly we carry over and complement in Chapter 4 the theory of heat equations in the global spaces Asp;q .Rn / as developed in [T13] to the hybrid spaces LrAsp;q .Rn /. Chapter 5 deals with NavierStokes equations especially in the version (0.4), (0.5) in hybrid spaces. Then one is in a rather comfortable position, clipping together related assertions of the two
Preface
vii
preceding chapters. But again we add a few new aspects. In particular, if the admitted initial data are infrared-damped then the related local solutions of the Navier-Stokes equations can be extended globally in time. These considerations will be continued in Chapter 6 now specified to the global spaces Asp;q .Rn / and extended to the spaces r Sp;q A.Rn / with dominating mixed smoothness. We discuss conditions for the initial data in terms of Haar wavelets, Faber bases, and sampling in connection with the hyperbolic cross, ensuring solutions of the Navier-Stokes equations which are global in time. Furthermore we add some comments about the influence of large Reynolds numbers. This chapter is largely independent of the preceding considerations. We assume that the reader has a working knowledge about basic assertions for the spaces Asp;q .Rn /. But to make this book independently readable we provide related notation, facts, and detailed references. Formulae are numbered within chapters. Furthermore in each chapter all definitions, theorems, propositions, corollaries and remarks are jointly and consecutively numbered. References are ordered by names, not by labels, which roughly coincide, but may occasionally cause minor deviations. The bracketed numbers following the items in the Bibliography mark the page(s) where the corresponding entry is quoted. All unimportant positive constants will be denoted by c (with additional marks if there are several c’s in the same formula). To avoid any misunderstanding we fix our use of (equivalence) as follows. Let I be an arbitrary index set. Then ai bi for i 2 I (equivalence) (0.14) for two sets of positive numbers fai W i 2 I g and fbi W i 2 I g means that there are two positive numbers c1 and c2 such that c1 ai bi c2 ai
for all i 2 I :
Contents
1
Introduction
2
Morrey spaces 2.1 Introduction . . . . . . . . . . . . . 2.2 Definitions and preliminaries . . . . 2.2.1 Morrey spaces . . . . . . . . 2.2.2 Dual Morrey spaces . . . . . 2.2.3 Comments . . . . . . . . . . 2.3 Embeddings . . . . . . . . . . . . . 2.3.1 Besov spaces . . . . . . . . . 2.3.2 Main assertions . . . . . . . 2.3.3 Further embeddings . . . . . 2.3.4 Non-separability and density 2.4 Duality . . . . . . . . . . . . . . . . 2.4.1 Main assertions . . . . . . . 2.4.2 Complements . . . . . . . . 2.5 Calder´on-Zygmund operators . . . . 2.5.1 Preliminaries . . . . . . . . . 2.5.2 Main assertions . . . . . . . 2.5.3 Distinguished representations 2.6 Haar bases . . . . . . . . . . . . . . 2.6.1 Preliminaries and definitions 2.6.2 Main assertions . . . . . . . 2.6.3 Littlewood-Paley theorem . .
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6 6 7 7 8 10 15 15 16 20 23 24 24 29 31 31 32 36 39 39 42 43
Hybrid spaces 3.1 Introduction . . . . . . . . . . . . . . . . 3.2 Global spaces . . . . . . . . . . . . . . . 3.2.1 Definitions . . . . . . . . . . . . . 3.2.2 Atoms . . . . . . . . . . . . . . . 3.2.3 Wavelets . . . . . . . . . . . . . . 3.2.4 Multiplication algebras . . . . . . 3.2.5 Spaces on domains . . . . . . . . 3.3 Definitions and basic properties . . . . . . 3.3.1 Definitions . . . . . . . . . . . . . 3.3.2 Basic properties . . . . . . . . . . 3.4 Characterizations . . . . . . . . . . . . . 3.4.1 Wavelet characterizations . . . . . 3.4.2 Atomic characterizations . . . . . 3.4.3 Fourier-analytical characterizations
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45 45 46 46 49 51 54 55 56 56 60 63 63 66 68
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Contents
3.4.4 Haar wavelets . . . . . . . . . . . . . . . . . . . . 3.4.5 Morrey spaces, revisited . . . . . . . . . . . . . . . 3.4.6 Meyer wavelets . . . . . . . . . . . . . . . . . . . 3.5 Equivalent norms and Fourier multipliers . . . . . . . . . . 3.5.1 Equivalent norms . . . . . . . . . . . . . . . . . . 3.5.2 Fourier multipliers and Riesz transforms . . . . . . 3.6 Some properties . . . . . . . . . . . . . . . . . . . . . . . 3.6.1 Embeddings . . . . . . . . . . . . . . . . . . . . . 3.6.2 Multiplication algebras . . . . . . . . . . . . . . . 3.6.3 Morrey characterizations . . . . . . . . . . . . . . 3.6.4 Pointwise multipliers and diffeomorphisms . . . . . 3.6.5 Lifts . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.6 Thermic characterizations: comments and proposals 4 Heat equations 4.1 Preliminaries . . . . . . . . . . 4.2 Homogeneous heat equations . . 4.3 Inhomogeneous heat equations . 4.4 Nonlinear heat equations . . . . 4.4.1 Special cases . . . . . . . 4.4.2 General cases . . . . . . 4.4.3 Strong solutions . . . . . 4.4.4 Comments and examples
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5 Navier-Stokes equations in hybrid spaces 5.1 Preliminaries . . . . . . . . . . . . . 5.2 Special cases . . . . . . . . . . . . . 5.3 General cases . . . . . . . . . . . . . 5.4 Comments and examples . . . . . . . 5.5 Complements . . . . . . . . . . . . . 5.5.1 Supercritical spaces . . . . . . 5.5.2 Infrared-damped initial data . . 5.5.3 Global solutions . . . . . . . .
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6 Navier-Stokes equations in global spaces 6.1 Haar wavelets and Reynolds numbers . . . . . . . . . . . . . 6.1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . 6.1.2 Some embeddings . . . . . . . . . . . . . . . . . . . 6.1.3 Main assertions . . . . . . . . . . . . . . . . . . . . 6.1.4 Reynolds numbers . . . . . . . . . . . . . . . . . . . 6.1.5 Oscillation, persistency, lattice structure of initial data 6.2 Initial data in spaces with dominating mixed smoothness . . 6.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Definitions and basic properties . . . . . . . . . . . . 6.2.3 Sampling . . . . . . . . . . . . . . . . . . . . . . . .
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6.2.4 Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 167 6.2.5 Besov spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 6.2.6 Complements . . . . . . . . . . . . . . . . . . . . . . . . . . 171 Bibliography
175
Symbols
183
Index
185
Chapter 1
Introduction In [T13] we dealt with the Navier-Stokes equations in Rn .0; 1/, (1.1) @t u C .u; r/u u C rP D 0 n div u D 0 in R .0; 1/; (1.2) n in R ; (1.3) u.; 0/ D u0 where u.x; t/ D u1 .x; t/; : : : ; un .x; t/ is the unknown velocity and P .x; t/ the unknown (scalar) pressure, 2 n 2 N. Recall @t D @=@t, @j D @=@xj if j D 1; : : : ; n, and that the vector-function .u; r/u has the components n X k uj @j uk ; .u; r/u D
k D 1; : : : ; n;
(1.4)
j D1
whereas, as usual, div u D
n X
rP D .@1 P; : : : ; @n P /:
@j uj ;
(1.5)
j D1
By (1.2) one has .u; r/u D div .u ˝ u/;
div .u ˝ u/k D
n X
@j .uj uk /:
(1.6)
j D1
This reduces (1.1)–(1.3), now in the strip Rn .0; T / with T > 0, to @t u u C P div .u ˝ u/ D 0 u.; 0/ D u0
in Rn .0; T /, in Rn :
(1.7) (1.8)
Here P is the Leray projector, .Pf / D f C Rk k
k
n X
Rj f j ;
k D 1; : : : ; n;
(1.9)
j D1
based on the (scalar) Riesz transforms Rk , Z _ yk k b g .x/ D cn lim Rk g.x/ D i g.x y/ dy; jj "#0 jyj" jyjnC1
x 2 Rn :
(1.10)
2
1 Introduction
In (1.7), (1.8) there is no need to care about (1.2) any longer. But if in addition div u0 D 0 then div u D 0 in our context (mild solutions based on fixed point arguments). This well-known reduction of (1.1)–(1.3) to (1.7), (1.8) may also be found in [T13, Section 6.1.3, pp. 196-198]. The vector equation (1.7), (1.8) can be reduced to the nonlinear scalar heat equation @t u.x; t/ D u2 .x; t/ u.x; t/ D 0; u.x; 0/ D u0 .x/;
x 2 Rn ; 0 < t < T; x 2 Rn ;
(1.11) (1.12)
on the one hand and the mapping properties of Rj and P in the considered function spaces on the other hand. Here Df D
n X
@j f:
(1.13)
j D1
We dealt with the Cauchy problem (1.11), (1.12) in the context of local spaces LrAsp;q .Rn /, [T13, Theorem 5.24, p. 183], and of global spaces Asp;q .Rn /, [T13, Theorem 5.36, p. 189], under the crucial assumption that the underlying spaces LrAsp;q .Rn / and Asp;q .Rn / are multiplication algebras. This is ensured if s C r > 0 for local spaces and s > n=p (and some limiting spaces with s D n=p) for global spaces. The reduction of (1.7), (1.8) to (1.11), (1.12) requires in addition that the Riesz transforms Rj are linear and bounded maps in the underlying spaces. This applies to the global spaces Asp;q .Rn /;
1 < p < 1;
0 < q 1;
s 2 R;
(1.14)
[T13, Theorem 1.25, p. 17] where the additional restriction 1 < q < 1 for F -spaces mentioned there is not necessary (as a consequence of Theorem 3.52 below). Then one obtains satisfactory solutions for (1.7), (1.8) in the global spaces Asp;q .Rn /;
1 < p < 1;
1 q 1;
s > n=p;
(1.15)
(and some limiting cases with s D n=p). We refer the reader to [T13, Theorem 6.7, p. 203] (where 1 < q < 1 for F -spaces can be replaced by 1 q 1 as covered by Corollary 5.4 below). We could not find a counterpart in terms of the local spaces LrAsp;q .Rn / and replaced as a substitute the Leray projector P in (1.7) by the truncated Leray projector P 2 based on the truncated Riesz transforms R
;k f
Di
k b_ f ; jj
k D 1; : : : ; n;
(1.16)
where 2 C 1 .Rn /;
.x/ D 0 if jxj 1=2 and
.y/ D 1 if jyj 1;
(1.17)
[T13, pp. 199/200, Theorem 6.10, p. 205]. Hence, one removes the infrared (or low frequency) part of solutions of (1.7), (1.8). This point has also been discussed in
3
1 Introduction
[T13, p. 193, 199-201]. At that time we tried to find a way to deal with Navier-Stokes equations or with (1.7), (1.8) also in the context of the local spaces LrAsp;q .Rn /. But it came out quite recently that the Riesz transform (1.10) cannot be extended from D.Rn/ or S.Rn / to a linear and bounded operator acting in the local Morrey spaces Lrp .Rn / D LrLp .Rn /, 1 < p < 1, n=p r < 0, [RoT13, Theorem 1.1(i)]. We refer the reader also to Theorem 2.22 and Remark 2.23 below. On the one hand one can take this observation as a justification of the above truncation. But on the other hand one knows now that Rk are linear and bounded maps, Rk W
V rp .Rn / ,! LV rp .Rn / and Lrp .Rn / ,! Lrp .Rn /; L
(1.18)
1 < p < 1, n=p r < 0, in the global Morrey spaces Lrp .Rn / D Lr Lp .Rn / and in the completion of S.Rn / in Lrp .Rn /, denoted as LV rp .Rn /, [RoT13, Theorem 1.1], Theorem 2.22 and Remark 2.23 below. We refer the reader also to [RoT14]. It is crucial for us and the main motivation of this book that (1.18) can be extended to some hybrid spaces LrAsp;q .Rn / (being smaller than the local spaces LrAsp;q .Rn /). As far as properties are concerned these spaces are between local and global spaces. This may justify calling them hybrid spaces. In particular if 1 < p < 1;
0 < q 1;
s 2 R and
n=p r < 0;
(1.19)
LrAsp;q .Rn / ,! LrAsp;q .Rn /;
k D 1; : : : ; n;
(1.20)
then one has by Theorem 3.52 below Rk W
whereas the local spaces LrAsp;q .Rn / do not have this property. In addition LrAsp;q .Rn / are multiplication algebras if s C r > 0 (as their local counterparts LrAsp;q .Rn /). Then one can extend a corresponding theory for the nonlinear heat equations (1.11), (1.12), now in terms of the hybrid spaces LrAsp;q .Rn /, to the NavierStokes equations. We tried to find in [T13] related assertions in the context of the local spaces LrAsp;q .Rn /. Now it is clear that this is impossible, but it is also clear that one has a satisfactory theory with hybrid spaces LrAsp;q .Rn / in place of the local spaces LrAsp;q .Rn /. This extends corresponding assertions from Asp;q .Rn / D Ln=pAsp;q .Rn / to LrAsp;q .Rn /. Chapter 2 deals mainly with local and global Morrey spaces Lr .Rn /, LV r .Rn /, p
p
Lrp .Rn /, LV rp .Rn / and their (pre)duals. We follow closely [RoT13, RoT14] complemented by n=p < r < 0; (1.21) where L is the Lebesgue measure and w˛ .x/ D .1 C jxj2 /˛=2 with n < ˛ < n rp is a Muckenhoupt weight w˛ 2 Ap .Rn /. Then Rk g.x/ according to (1.10) is well-defined for x 2 Rn a.e., also in its integral version. Finally we characterize some of these spaces in terms of Haar wavelets. In Chapter 3 we introduce the hybrid spaces LrAsp;q .Rn / and collect some basic properties needed later on. This can be Lrp .Rn / ,! Lp .Rn ; ˛ /;
˛ D w˛ L ;
1 < p < 1;
4
1 Introduction
done largely in the same way as in [T13] for the local spaces LrAsp;q .Rn / mostly without additional efforts. Only occasionally we add a further argument. We observe that 1 r n LrAsp;q .Rn / D As; with D C (1.22) p;q .R / p n n for all admitted parameters s; p; q and n=p r < 1. The spaces As; p;q .R / have been studied in great detail in the book [YSY10], the survey [Sic12] and the underlying papers. There one finds many other properties which will not be repeated here. One may also consult [T13, pp. 38/39, Section 2.7.3, pp. 101-107]. There is one crucial exception needed to prove (1.20). Then we rely on
kf jLrAsp;q .Rn /k kf jLrAPsp;q .Rn /k C kf jLrp .Rn /k
(1.23)
1 < p < 1;
(1.24)
if 0 < q 1;
s > 0;
n=p r < 0:
Here LrAPsp;q .Rn / are homogeneous hybrid spaces (we do not need the homogeneous spaces themselves but only their homogeneous norms in the context of the inhomogeneous spaces LrAsp;q .Rn /). For these homogeneous spaces (or their norms) one has the Fourier multiplier assertion ˇ ˇ j˛j ˇ ˛ b/_ jLrAPs .Rn /k c ˇ kf jLrAPs .Rn /k (1.25) k.hf sup jxj h.x/ D p;q p;q j˛jk;x2Rn
of Michlin type with k 2 N sufficiently large (specified later on). This is essentially covered by [YaY10, Theorem 4.1, p. 3819]. We refer also to [YYZ12, Theorem 1.5, p. 6] and the recent survey [YaY13a]. This can be applied to Rk with h D k =jj. Then (1.20) with (1.19) follows essentially from (1.23) and (1.18), (1.25). This may be considered as the basic observation of what follows. Afterwards we return in Chapter 4 to the nonlinear heat equations (1.11), (1.12) and transfer assertions available so far in the context of the local spaces LrAsp;q .Rn / to their hybrid counterparts LrAsp;q .Rn / (again essentially without any additional efforts) complemented by some new observations. In Chapter 5 we deal with the Navier-Stokes equations (1.7), (1.8) in hybrid spaces LrAsp;q .Rn / extending a corresponding theory in [T13] for the spaces Asp;q .Rn / D Ln=p Asp;q .Rn / to LrAsp;q .Rn /. This extension applies not only to the obtained assertions, but also to the underlying technicalities. In particular (1.23) is the Morreyfied version of kf jAsp;q .Rn /k kf jAPsp;q .Rn /k C kf jLp .Rn /k if 0 < p < 1;
0 < q 1;
s > p D n
1 p
1
C
(1.26) ;
(1.27)
[T92, Theorem 2.3.3, p. 98]. Furthermore, (1.25) with APsp;q .Rn / D Ln=p APsp;q .Rn /
(1.28)
1 Introduction
5
is covered by [T83, Theorem 5.2.2, p. 241]. We refer the reader also to [T13, Theorem 1.25, p. 17]. The final Chapter 6 is to some extent independent of the main bulk of this book. It deals with Haar wavelets, Faber bases and sampling in the context of the hyperbolic cross and spaces with dominating mixed smoothness and their relations to solutions of Navier-Stokes equations, global in time, for large initial data.
Chapter 2
Morrey spaces
2.1 Introduction This chapter deals with local Morrey spaces Lrp .Rn / and global Morrey spaces Lrp .Rn / as well as their preduals H% Lp .Rn /, H % Lp .Rn / in the framework of tempered distributions S 0 .Rn /. This requires some restrictions for the parameters, typically 1 < p < 1. We are especially interested in duality properties and embeddings between these spaces and in relations to distinguished Besov spaces. It is our intention to present the material as self-contained as possible and to illuminate the somewhat tricky (topological) background to a larger extent than usually done in the literature. This requires that we include some basic material and a few so-called well-known properties for which we could not find proofs in the literature. A typical example is the claim that Morrey spaces are non-separable. We give a short proof. As a byproduct one obtains in one line the highly desirable (and well-known) assertion that neither D.Rn/, nor S.Rn /, nor distinguished Lebesgue spaces are dense in Morrey spaces. The second main aim of this chapter is the study of mapping properties of Calder´on-Zygmund operators in Morrey spaces and their preduals. The Riesz transforms (1.10) are distinguished cases and the mapping property (1.18) will be of great service for us in later chapters. It will be crucial to justify (1.20) based on (1.23), (1.25). There are apparently no books or up-to-date comprehensive surveys dealing with Morrey spaces and their (pre)duals especially in the limelight of Harmonic Analysis. Basic material about Morrey-Campanato spaces may be found in [KJF77, Chapter 4], taken over to the new edition [PKJF13, Chapter 5]. We do not deal with the numerous modifications and generalizations of (Campanato)-Morrey spaces. The interested reader may consult the overview [RSS13] where one finds also references to related classical and recent papers. In the last few years remarkable progress has been made to raise Morrey spaces in Harmonic Analysis to the same level as Lebesgue spaces. The most advanced paper in this direction is [AdX12], based on [AdX04], and the literature mentioned there spanning a period of several decades. This chapter may also be considered as a contribution to these recent developments providing some background material based on new proofs. In [T13, Chapter 3] we dealt with Morrey-Campanato spaces Lrp .Rn / in the larger context of so-called local spaces LrAsp;q .Rn /. This will now be complemented, identifying global Morrey spaces Lrp .Rn / with special hybrid spaces. As a consequence we characterize Morrey spaces in terms of Haar wavelets. In Section 2.2 we collect some definitions and comment on notation and the range of the admitted parameters. Section 2.3 deals with embeddings of Morrey spaces in the framework of S.Rn /; S 0 .Rn / , complemented by the non-separability of Lrp .Rn /
2.2 Definitions and preliminaries
7
and Lrp .Rn /. Duality will be treated in Section 2.4. Section 2.5 deals with mapping properties of Calder´on-Zygmund operators. Finally we characterize Morrey spaces in Section 2.6 in terms of Haar wavelets.
2.2 Definitions and preliminaries 2.2.1 Morrey spaces We use standard notation. Let N be the collection of all natural numbers and N0 D N [ f0g. Let Rn be Euclidean n-space, where n 2 N. Put R D R1 , whereas C is the complex plane. Let S.Rn / be the Schwartz space of all complex-valued rapidly decreasing infinitely differentiable functions on Rn and let S 0 .Rn / be the space of all tempered distributions on Rn . Let D.Rn / D C01 .Rn / be the collection of all functions f 2 S.Rn / with compact support in Rn . As usual D 0 .Rn / stands for the space of all distributions in Rn . Furthermore, Lp .Rn / with 0 < p 1, is the standard complex quasi-Banach space with respect to the Lebesgue measure, quasinormed by Z 1=p n kf jLp .R /k D jf .x/jp dx (2.1) Rn
with the usual modification if p D 1. Similarly Lp .M / where M is a Lebesguemeasurable subset of Rn . As usual Z is the collection of all integers; and Zn where n 2 N denotes the lattice of all points m D .m1 ; : : : ; mn / 2 Rn with mk 2 Z. Let Qj;m D 2j m C 2j .0; 1/n with j 2 Z and m 2 Zn be the usual dyadic cubes in Rn , n 2 N, with sides of length 2j parallel to the axes of coordinates and 2j m as n the lower left corner. As usual, Lloc p .R / collects all locally p-integrable functions f , hence f 2 Lp .M / for any bounded Lebesgue measurable set M in Rn . Definition 2.1. Let 1 < p < 1 and 0
Cr <
n p
n . p
n (i) Then Lrp .Rn / collects all f 2 Lloc p .R / such that
kf jLrp .Rn /k D
n
sup J 2N0 ;M 2Zn
2J. p Cr/ kf jLp .QJ;M /k
(2.2)
is finite. Furthermore, LV rp .Rn / is the completion of D.Rn / in Lrp .Rn /. n (ii) Then Lrp .Rn / collects all f 2 Lloc p .R / such that kf jLrp .Rn /k D
sup J 2Z;M 2Zn
n
2J. p Cr/ kf jLp .QJ;M /k
is finite. Furthermore, LV rp .Rn / is the completion of D.Rn / in Lrp .Rn /.
(2.3)
8
2 Morrey spaces
Remark 2.2. The above spaces are usually attributed to Morrey, [Mor38]. But Morrey himself was only interested in related integral inequalities in connection with smoothness properties (H¨older-continuity) of solutions of nonlinear elliptic and parabolic equations. The reformulation in terms of function spaces goes back to Campanato, Brudnyi and Peetre in the 1960s, [Bru71, Bru09, Cam63, Cam64, Pee66, Pee69] using quite similar notation as in the above Definition 2.1. Further references may be found in [T13, Chapter 3]. Let Lp .Rn ; w / with 0 < p 1 and w .x/ D .1 C jxj2 /=2 , 2 R, be the weighted Lebesgue spaces, quasi-normed by kf jLp .Rn ; w /k D kw f jLp .Rn /k:
(2.4)
As will be seen below in Theorem 2.8 one has the continuous embeddings Lu .Rn / ,! Lrp .Rn / ,! Lrp .Rn / ,! Lp .Rn ; w /
(2.5)
where ru D n, hence 1 < p u < 1, and < n=p. Then it follows by standard arguments, specified in Step 1 of the proof of Theorem 2.8, that all spaces introduced in Definition 2.1 are Banach spaces which can be treated within the dual pairing S.Rn /; S 0 .Rn / . We are not interested in limiting cases. This may explain why we excluded the reasonable case p D 1. Otherwise we discuss in Remark 2.5 below the range of the admitted parameters in the above definition. Furthermore as will be seen below in Proposition 2.16, both D.Rn / and S.Rn / are neither dense in Lrp .Rn / nor dense in Lrp .Rn /, 1 < p < 1, 0 < pn C r < pn . This applies also to
Ln=p .Rn / D Lp .Rn /; p normed by
1 < p < 1;
(2.6)
kf jLp .Rn /k D sup kf jLp .Q0;M /k;
(2.7)
M 2Zn
Remark 2.17, whereas (almost obviously, based on Fatou’s theorem and Lebesgue’s bounded convergence theorem) .Rn / D LV n=p .Rn /; Lp .Rn / D Ln=p p p
1 < p < 1:
(2.8)
This justifies introducing in the above definition the spaces LV rp .Rn / and LV rp .Rn / by completion of D.Rn / in the respective spaces. We call Lrp .Rn / local Morrey spaces (sometimes also denoted as inhomogeneous Morrey spaces) and Lrp .Rn / global Morrey spaces (sometimes also denoted as homogeneous Morrey spaces). But as far as further references and names of contributors are concerned one may consult the discussion in [T13, p. 110]. This applies in particular to the role of Yu. Brudnyi’s work.
2.2.2 Dual Morrey spaces It is one of the main aims of this chapter to study duals and preduals of the above spaces, always in the framework of the dual pairing S.Rn /; S 0 .Rn / . For this purpose we introduce the following spaces. Let again QJ;M D 2J M C 2J .0; 1/n be the above cubes, J 2 Z, M 2 Zn .
2.2 Definitions and preliminaries
Definition 2.3. Let 1 < p < 1 and n < % < n=p. (i) Then H% Lp .Rn / collects all h 2 S 0 .Rn / which can be represented as X hJ;M ; supp hJ;M QJ;M ; hD J 2N0
such that
9
(2.9)
;M 2Zn
X
n
2J. p C%/ khJ;M jLp .QJ;M /k < 1:
(2.10)
J 2N0 ;M 2Zn
Furthermore, X
kh jH% Lp .Rn /k D inf
J 2N0
n
2J. p C%/ khJ;M jLp .QJ;M /k
(2.11)
;M 2Zn
where the infimum is taken over all representations (2.9), (2.10). (ii) Then H % Lp .Rn / collects all h 2 S 0 .Rn / which can be represented as X hJ;M ; supp hJ;M QJ;M ; (2.12) hD J 2Z;M 2Zn
such that
X
n
2J. p C%/ khJ;M jLp .QJ;M /k < 1:
(2.13)
J 2Z;M 2Zn
Furthermore, kh jH % Lp .Rn /k D inf
X
n
2J. p C%/ khJ;M jLp .QJ;M /k
(2.14)
J 2Z;M 2Zn
where the infimum is taken over all representations (2.12), (2.13). Remark 2.4. One has to justify the above definition. Let again Lp .Rn ; w / be the weighted Lp -space normed by (2.4). As will be seen below in Theorem 2.8 one has the continuous embedding Lp .Rn ; w / ,! H% Lp .Rn / ,! H % Lp .Rn / ,! Lu .Rn /
(2.15)
where %u D n, hence 1 < u < p < 1, and > n=p0 with p1 C p10 D 1. Then it follows by standard arguments that all spaces introduced in Definition 2.3 are Banach spaces which can be treated within the dual pairing S.Rn /; S 0 .Rn / . More specific related arguments will be given in Step 1 of the proof of Theorem 2.8 below. In particular, (2.10) ensures that (2.9) converges unconditionally in Lu .Rn / and hence in S 0 .Rn /. Similarly (2.12), based on (2.13), converges unconditionally in Lu .Rn / and hence in S 0 .Rn /. In Section 2.2.3 we discuss not only the range of the parameters in the Definitions 2.1, 2.3 but also where the notations and names are coming from. In particular we took over from [Kal98] the notation dual Morrey spaces for the spaces introduced in Definition 2.3.
10
2 Morrey spaces
2.2.3 Comments First we wish to show that the restrictions for the parameters in the Definitions 2.1, 2.3 are natural. All spaces X.Rn/ introduced there should bedistributional Banach spaces in the framework of the dual pairing S.Rn /; S 0 .Rn / with the continuous embedding S.Rn / ,! X.Rn/ ,! S 0 .Rn / (2.16) as a minimal request. Secondly we comment on these spaces as far as notation, references and names are concerned. Finally we compare the above constructions with well-known extrapolation techniques, taking Zygmund spaces Lp .log L/a .Q/ as a distinguished example. Remark 2.5. Let 1 < p < 1. The restrictions for r in Definition 2.1, hence n=p r < 0, and for % in Definition 2.3, hence n < % < n=p, are natural. To substantiate this claim we extend temporarily Definition 2.1 to all r 2 R and Definition 2.3 to all % 2 R asking in specification of (2.16), S.Rn / ,! Lrp .Rn / ,! S 0 .Rn /;
S.Rn / ,! Lrp .Rn / ,! S 0 .Rn /
(2.17)
and S.Rn / ,! H % Lp .Rn / ,! S 0 .Rn /;
S.Rn / ,! H% Lp .Rn / ,! S 0 .Rn /: (2.18)
If r D 0 then it follows from a Lebesgue point argument that
L0p .Rn / D L0p .Rn / D L1 .Rn /:
(2.19)
If r > 0 then one obtains from n
2J p kf jLp .QJ;M /k c 2J r ! 0
if J ! 1
(2.20)
and again by a Lebesgue point argument that
Lrp .Rn / D Lrp .Rn / D f0g;
r > 0:
(2.21)
If r D n=p then one has by (2.6)–(2.8)
Ln=p .Rn / D Lp .Rn / p
and
Ln=p .Rn / D Lp .Rn /: p
(2.22)
If r < n=p then it follows from kf jLp .QJ;M /k ! 0 if J ! 1,
Lrp .Rn / D Lp .Rn / and Lrp .Rn / D f0g;
r < n=p:
(2.23)
This shows that the restriction n=p r < 0 in Definition 2.1 is natural (the limiting cases r D 0 and r D n=p will be not of interest for us in what follows). Let % D n=p in Definition 2.3. Let Lp .Rn /1 be normed by X kg jLp .Rn /1 k D kg jLp .Q0;M /k: (2.24) M 2Zn
2.2 Definitions and preliminaries
11
Then it follows from (2.9), (2.10) and the triangle inequality kh jHn=p Lp .Rn /k kh jLp .Rn /1 k kh jHn=p Lp .Rn /k:
(2.25)
Hence Hn=p Lp .Rn / D Lp .Rn /1 . Again by the triangle inequality one has kh jLp .Rn /k kh jH n=p Lp .Rn /k:
(2.26)
the converse. It is sufficient to deal with h 2 Lp .Rn / such that supp h ˚We prove x 2 Rn W xj 0 . Let QJ D QJ;0 D .0; 2J /n , J 2 N, be admitted cubes. In dependence on h there is a monotonically increasing sequence of natural numbers fJl g1 such that lD0 hD
1 X
hl ;
supp hl QJl ;
khl jLp .QJl /k 2l kh jLp .Rn /k
(2.27)
lD0
(convergence in Lp .Rn /). For suitable Jl one may choose h0 D hjQJ0 and hl D hjQJl n QJl1 if l 2 N. It is an admitted decomposition (2.12)–(2.14) (extended to % D n=p). This proves h 2 H n=p Lp .Rn / and the converse of (2.26) (with an additional factor 2). Then one obtains
Hn=p Lp .Rn / D Lp .Rn /1
H n=p Lp .Rn / D Lp .Rn /:
and
(2.28)
If % > n=p then one has again H% Lp .Rn / D Lp .Rn /1 . The spaces H % Lp .Rn / with % > n=p do not make sense in our context: If h 2 Lp .Rn / then one can argue similarly as in (2.27) where one has in (2.13), (2.14) the additional factors n 2J.%C p / ! 0 if J ! 1. One obtains kh jH % Lp .Rn /k D 0 for any h 2 Lp .Rn /, which contradicts our basis request (2.18). Hence
H% Lp .Rn / D Lp .Rn /1 ;
H % Lp .Rn / does not make sense; % > n=p: (2.29)
Let % D n and h 2 H n Lp .Rn / be optimally represented by (2.12), (2.13) (extended to % D n). Then one has with p1 C p10 D 1, kh jL1 .Rn /k
X
khJ;M jL1 .QJ;M /k
J 2Z;M 2Zn
X
2
J
n p0
khJ;M jLp .QJ;M /k
(2.30)
J 2Z;M 2Zn
c kh jH n Lp .Rn /k: Similarly for Hn Lp .Rn /. As for the converse we first remark that any h 2 L1 .Rn / can be decomposed as hD
1 X lD0
hl ;
khl jL1 .Rn /k 2l kh jL1 .Rn /k
(2.31)
12
2 Morrey spaces
with
X
hl D
J 2N0
lJ;M 2 C;
lJ;M J;M ;
(2.32)
;M 2Zn
where for fixed l only finitely many lJ;M are different from zero and the related cubes QJ;M are pairwise disjoint. Again J;M is the characteristic function of QJ;M . This follows from standard arguments using in particular that the Lebesgue measure is a Radon measure (regular measure). We refer to related textbooks, for example [Tri92, Sections 1.3.5, 1.3.6, Appendix A1, pp. 36–43, 459–463]. Clipping together (2.31), (2.32) one obtains hD
X
J;M J;M
J;M D
with
J 2N0 ;M 2Zn
1 X
lJ;M
(2.33)
lD0
and kh jHn Lp .Rn /k
1 X
X
2
J
lD0 J 2N0 ;M 2Zn 1 X khl jL1 .Rn /k
n p0
n
jlJ;M j 2J p (2.34)
2kh jL1 .Rn /k:
lD0
This proves the converse of (2.30) both for Hn Lp .Rn / and H n Lp .Rn /. Hence
Hn Lp .Rn / D H n Lp .Rn / D L1 .Rn /: Let % < % C " < n for some " > 0 and let X 2J.%C"/ J;MJ ; hK D
K 2 N;
(2.35)
(2.36)
KJ 2N
where J;MJ are characteristic functions of disjoint cubes QJ;MJ near the origin. Then one has for K 0 > K, X n n 0 0 khK hK jH % Lp .Rn /k khK hK jH% Lp .Rn /k 2J. p "/ 2J p D c" 2K" J >K
(2.37) and
khK jL1 .Rn /k D
X
2J.%C"Cn/ D c"0 2Kj%C"Cnj :
(2.38)
J K
Let ' 2 S.Rn / with '.0/ D 1 and the above cubes tending to the origin suitably chosen. Then Z hK .x/'.x/ dx ! 1 if K ! 1: (2.39) Rn K
On the other hand fh g is a fundamental sequence both in H % Lp .Rn / and H% Lp .Rn /. Then (2.39) contradicts our minimal request (2.18). In other words, with exception of some limiting cases the restrictions for r and % in Definitions 2.1 and 2.3 are natural.
2.2 Definitions and preliminaries
13
Remark 2.6. The notation Lrp .Rn / and Lrp .Rn / for the local and global Morrey spaces according to Definition 2.1 is very near to related proposals of the old masters, especially S. Campanato and J. Peetre, [Cam63, Cam64, Pee69]. Further (historical) references may be found in [T13, p. 110]. The situation is somewhat different as far as the notation H% Lp .Rn / and H % Lp .Rn / introduced in Definition 2.3 is concerned. The spaces Lrp .Rn / and H % Lp .Rn / have been characterized in [AdX04, AdX12] in terms of Hausdorff capacities and Muckenhoupt weights, including the suggestion to use the letter H in this context. We follow this proposal. The incorporation of Hausdorff capacities and related (Choquet) integrals into the theory of Morrey spaces has again some history. One may consult [Ada75, Ada88, Kal98]. Further references are % given in [AdX04]. There is surely a temptation to use Hp .Rn / instead of H% Lp .Rn / % n % n and Hp .R / instead of H Lp .R /. This would adapt the notation introduced in % [AdX04] to Definition 2.1 of the Morrey spaces. But Hp .Rn / denotes usually (fractional) Sobolev spaces and we wish to avoid any misunderstanding. On the other hand in the larger context of the theory of local function spaces LrAsp;q .Rn / as developed 0 .Rn / with in [T13] there are good reasons to replace Lrp .Rn / by LrLp .Rn / D LrFp;2 n=p r < 0, 1 < p < 1. We refer the reader to [T13, Theorem 3.13, p. 121]. The counterpart 0 Lrp .Rn / D Lr Lp .Rn / D Lr Fp;2 .Rn /;
n=p r < 0; (2.40) for the hybrid spaces may be found below in Remark 3.67. But we stick for the reasons just explained to the admittedly somewhat inconsistent notation Lrp .Rn /, Lrp .Rn / on the one hand and H% Lp .Rn /, H % Lp .Rn / on the other hand. The best proposal goes back to [Kal98] suggesting that we incorporate the H -spaces notationally into the L-scale, hence Lrp .Rn /;
n < r < 0;
1 < p < 1;
1 < p < 1;
(2.41)
with Lrp .Rn / D H r Lp .Rn / if n < r n=p. This is in good agreement with the restrictions for r and % in Definitions 2.1, 2.3 and also with .Rn / D H n=p Lp .Rn / D Lp .Rn /; Ln=p p
1 < p < 1;
(2.42)
according to (2.8), (2.28). Then all these spaces could be called Morrey spaces following the long-standing tradition in mathematics to extend the meaning of symbols and notation to wider ranges of parameters if one has good reasons to do so (one may think about the original and more recent use to speak about Besov spaces). The discussion in Remark 2.7 below shows how to relate the above spaces to extrapolation spaces. This would be a further good reason for the unification (2.41). However this convincing suggestion is unfortunately not in common use and we stick reluctantly to the notation introduced in Definitions 2.1, 2.3. Remark 2.7. The Morrey spaces Lrp .Rn /, Lrp .Rn / and the dual Morrey spaces H% Lp .Rn /, H % Lp .Rn / as introduced in Definitions 2.1, 2.3 fit in the larger scheme of extrapolation spaces. We will not rely on this technique later on. This may justify
14
2 Morrey spaces
that we illuminate the situation describing briefly as a distinguished example the extrapolation characterization of the Zygmund spaces Lp .log L/a .Q/ on the unit cube Q D .0; 1/n where 1 < p < 1 and a 2 R. These spaces can be quasi-normed by Z 1 1=p kf jLp .log L/a .Q/k D .1 C j log tj/ap f p .t/dt (2.43) 0
where f .t/ is the usual decreasing rearrangement of f . Let 1 1 1 D C 2j < 1 j p p n
and
1 1 1 D 2j > 0; j p n p
j0 j 2 N;
(2.44)
for some j0 2 N. If a < 0 then Lp .log L/a .Q/ collects all f 2 L1 .Q/ such that 1 X
2jap kf jLpj .Q/kp
1=p
<1
(2.45)
j Dj0
(equivalent norm). If a > 0 then Lp .log L/a .Q/ collects all f 2 L1 .Q/ which can be represented as 1 X fj ; fj 2 Lpj .Q/ (2.46) f D j Dj0
such that
1 X
2jap kfj jLpj .Q/kp
1=p
< 1:
(2.47)
j Dj0
Furthermore the infimum over all representations (2.46), (2.47) is an equivalent norm in Lp .log L/a .Q/. We refer the reader for details, explanations and proofs to [ET96, Section 2.6.2, pp. 69–75]. These typical extrapolation techniques have been extended afterwards in several directions. One may consult [KaM05, CFMM07] and the references given there. We inserted the above material to make clear that the spaces introduced in Definitions 2.1, 2.3 may be considered as extrapolation spaces. They are in good company with the above spaces Lp .log L/a .Q/. There are also breaking points. This is a D 0 for the spaces Lp .log L/a .Q/, hence Lp .log L/0 .Q/ D Lp .Q/ and its counterpart r D % D n=p according to (2.8), (2.28) and (2.42). In addition the two sides of these breaking points are related to each other by duality. In case of the Zygmund spaces one has Lp .log L/a .Q/0 D Lp0 .log L/a .Q/;
1 1 C 0 D 1; p p
a 2 R;
(2.48)
where again 1 < p < 1. One may consult [ET96, p. 68], the references to [BeS88] given there, and [EdE04, Theorem 3.1.15, Theorem 3.4.41, Corollary 3.4.44, pp. 69, 114, 117]. On the other hand, duality assertions for the spaces introduced in the Definitions 2.1, 2.3 will be considered in detail in Section 2.4 below. Then one has a counterpart of (2.48) in the context of extrapolation techniques. These similarities support the (not used) suggestion to unify the notation as indicated in (2.41).
2.3 Embeddings
15
2.3 Embeddings 2.3.1 Besov spaces We deal later on in detail with global, local and in particular hybrid spaces Asp;q .Rn /, LrAsp;q .Rn / and LrAsp;q .Rn /. Then the necessary definitions will be given. But to keep Chapter 2 independent we recall briefly the usual Fourier-analytical definition s of the global Besov spaces Bp;q .Rn /. We use standard notation. Otherwise in this Section 2.3 we follow closely [RoT14]. If ' 2 S.Rn / then Z n=2 eix '.x/ dx; 2 Rn ; (2.49) b ' ./ D .F '/./ D .2 / Rn
denotes the Fourier transform of '. As usual, F 1 ' and ' _ stand for the inverse Fourier transform, given by the right-hand side of (2.49) with i in place of i . Here x stands for the scalar product in Rn . Both F and F 1 are extended to S 0 .Rn / in the standard way. Let '0 2 S.Rn / with '0 .x/ D 1 if jxj 1 and let
and '0 .y/ D 0 if jyj 3=2;
'k .x/ D '0 .2k x/ '0 .2kC1 x/;
Since
1 X
'j .x/ D 1
x 2 Rn ;
k 2 N:
for x 2 Rn ;
(2.50) (2.51) (2.52)
j D0
f'j g1 j D0 _
forms a dyadic resolution of unity. The entire analytic functions ' D b .'j f / .x/ make sense pointwise in Rn for any f 2 S 0 .Rn /. We use (2.1) for 0 < p 1. Let 0 < p 1; 0 < q 1; s 2 R: (2.53) s Then Bp;q .Rn / is the collection of all f 2 S 0 .Rn / such that
kf
s .Rn /k' jBp;q
D
1 X
1=q b/_ jLp .Rn / q 2jsq .'j f
(2.54)
j D0
is finite (with the usual modification if q D 1/. As previously mentioned we return later on in greater detail to the global spaces Asp;q .Rn / with A 2 fB; F g. Otherwise we refer to [T83, T92, T06] where one finds the theory of these spaces and also their history. In particular these spaces are independent of admitted resolutions of unity ' according to (2.50)–(2.52) (equivalent quasi-norms). This justifies our omission of the subscript ' in (2.54). We denote by s C s .Rn / D B1;1 .Rn /;
the H¨older-Zygmund spaces.
s 2 R;
(2.55)
16
2 Morrey spaces
2.3.2 Main assertions If X and Y are Banach spaces then X ,! Y means continuous embedding. Recall that Lp .Rn ; w / with 1 p < 1 and w .x/ D .1 C jxj2 /=2 , 2 R, are the complex weighted Lebesgue spaces normed by (2.4). Let again Lp .Rn /1 with 1 p < 1 be the complex Banach spaces normed by (2.24). Then Lp .Rn /1 ,! Lp .Rn /;
1 p < 1;
(2.56)
is an immediate consequence of the triangle inequality. The other spaces have the s same meaning as in Definitions 2.1, 2.3. Let Bp;q .Rn / be the Besov spaces as intro1 1 duced in Section 2.3.1. Let p C p0 D 1 where 1 < p < 1. Let n 2 N. Recall our minimal request (2.16). We follow again [RoT14]. Theorem 2.8. (i) Let 1 < p < 1;
n=p < r < 0;
ru D n
and < n=p:
(2.57)
Then Lrp .Rn / and Lrp .Rn / are Banach spaces, 1 < p < u < 1 and n
n
p u .Rn / ,! Lu .Rn / ,! Lrp .Rn / ,! Lrp .Rn / ,! Lp .Rn ; w /: Bp;p
(2.58)
(ii) Let 1 < p < 1;
n < % < n=p;
%u D n
and > n=p0 :
(2.59)
Then H% Lp .Rn / and H % Lp .Rn / are Banach spaces, 1 < u < p and S.Rn / ,! Lp .Rn ; w / ,! Lp .Rn /1 ,! H% Lp .Rn / n
n
p u ,! H % Lp .Rn / ,! Lu .Rn / ,! Bp;p .Rn /:
(2.60)
Furthermore, D.Rn /, S.Rn /, Lp .Rn ; w / and Lp .Rn /1 are dense both in H% Lp .Rn / and H % Lp .Rn /. Proof. Step 1. The standard proof of the completeness of Lp . /, 1 p < 1, in domains in Rn as it may be found in any related textbook, for example [Tri92, Section 1.3.5, pp. 36–37], can be carried over to Lrp .Rn /, Lrp .Rn / and to H % Lp .Rn /, H% Lp .Rn / (taking for granted the embeddings (2.58), (2.60) which have nothing to do with completeness). Explicit proofs for the Morrey spaces may be found in [KJF77, Theorems 4.2.2, 4.4.1, pp. 210, 217], repeated in [PKJF13, Theorems 5.2.2, 5.4.1, pp. 173, 178]. But there is essentially no need to bother about the completeness: By Theorem 2.19 below all spaces Lrp .Rn /, Lrp .Rn / and H % Lp .Rn /, H% Lp .Rn / are duals of linear normed spaces. Then they are complete, [Rud91, Theorem 4.3, p. 94]. Step 2. We prove the remaining assertions of part (i) where 1 < p < u < 1 is obvious. Let f 2 Lrp .Rn /. Then the last embedding in (2.58) follows from <
17
2.3 Embeddings
n=p and X
kf jLp .Rn ; w /kp c
.1 C jM j/p kf jLp .Q0;M /kp
M 2Zn 0
(2.61)
c sup kf jLp .Q0;M /kp : M 2Zn
Let f 2 Lu .Rn / and p1 D u1 C v1 . Then 1=p Z n J. p Cr/ 2 jf .x/jp dx QJ;M
2
n J. p Cr n v/
Z
jf .x/ju dx
1=u
D
Z
QJ;M
jf .x/ju dx
(2.62)
1=u :
QJ;M
This proves the second inequality in (2.58). The third one is obvious by definition. The first assertion is a (sharp) embedding of the indicated Besov space into 0 .Rn / which can be found in [T06, p. 60] and the references given Lu .Rn / D Fu;2 n
n
p u .Rn / there, in particular to [SiT95]. (This can be improved by the larger space Bp;u n
n
p u .Rn /. A further improvement will be mentioned in (2.68) below.) in place of Bp;p Step 3. We prove (2.60) where 1 < u < p is obvious. The second embedding follows from p0 > n and
kf jLp .Rn /1 k c
X
.1 C jM j/p
0
1=p0
kf jLp .Rn ; w /k:
(2.63)
M 2Zn
The third embedding can be obtained from (2.9)–(2.11) and (2.24). The embedding of H% Lp .Rn / into H % Lp .Rn / is obvious by definition. We prove the last but one embedding and assume that h 2 H % Lp .Rn / is optimally represented according to (2.12)–(2.14). Let 1 1 1 D C ; u p q Then one has kh jLu .Rn /k
hence
X
1 1 % D : q p n
(2.64)
khJ;M jLu .QJ;M /k
J 2Z;M 2Zn
X
n
2J q khJ;M jLp .QJ;M /k
J 2Z;M 2Zn
X
(2.65)
n
2J. p C%/ khJ;M jLp .QJ;M /k:
J 2Z;M 2Zn
This proves the last but one inequality in (2.60). The last assertion is again a (sharp) n
n
n
n
p u p u 0 .Rn / into Bp;p .Rn / D Fp;p .Rn / covered by embedding of Lu .Rn / D Fu;2
18
2 Morrey spaces n
n
p u [T06, p. 60]. (This can be improved by the smaller space Bp;u .Rn / in place of n
n
p u .Rn /. A further improvement will be mentioned in (2.90) below.) Bp;p Step 4. We prove that D.Rn / is dense both in H% Lp .Rn / and H % Lp .Rn /. Let h be given by (2.12), (2.13) and let
hL D
X
hJ;M ;
L 2 N:
(2.66)
jJ jL;jM jL
Then kh hL jH % Lp .Rn /k ! 0
if L ! 1:
(2.67)
Any hJ;M with jJ j L, jM j L can be approximated in Lp .QJ;M / by functions belonging to D.QJ;M /. The sum of these functions approximates hL and also h. Hence D.Rn/ is dense in H % Lp .Rn /. One obtains by similar arguments that D.Rn / is dense in H% Lp .Rn /. Then it follows from (2.60) that also S.Rn /, Lp .Rn ; w / with > n=p0 and Lp .Rn /1 are dense both in H% Lp .Rn / and H % Lp .Rn /. Remark 2.9. The local Morrey spaces Lrp .Rn / have been considered in [T13, Chapter 3] in the wider context of local spaces of type LrAsp;q .Rn /. There one finds further embedding assertions and also relations to other types of spaces, including local Lorentz spaces, denoted by Lp;q .Rn /, and spaces of type Asp;q .Rn /. Restricted to the local spaces Lrp .Rn / one may consider the above Theorem 2.8 as a complement of the assertions obtained there. But this will not be repeated here and we do not try to extend results obtained in [T13] to other types of spaces included in the present book, in particular Lrp .Rn /, H% Lp .Rn / and H % Lp .Rn /. However mainly for historical reasons we insert the following assertion which complements (2.58). Let p; r; u be as in (2.57). Then n
Cr
p Bp;1 .Rn / ,! Lu;1 .Rn / ,! Lrp .Rn /;
(2.68)
where Lu;1 .Rn / are special Lorentz spaces (Marcinkiewicz spaces). Details and references about Lorentz spaces may be found in [T78, Sections 1.18.6, pp. 131– 135]. The first embedding in (2.68) can be proved by real interpolation. Let n=p < r0 < r < r1 < 0, r0 u0 D r1 u1 D n and r D .1 /r0 C r1 . Then it follows from (2.58) and well-known interpolation properties n Cr n n 0 p Cr p p Cr1 Bp;1 .Rn / D Bp;p .Rn /; Bp;p .Rn / ;1 n n ,! Lu0 .R /; Lu1 .R / ;1 D Lu;1 .Rn /:
(2.69)
We refer the reader to [T78, Sections 1.18.6, 2.4.1, pp. 131–135, 182]. The second embedding in (2.68) is covered by Step 1 of the proof of [T13, Theorem 3.7, p. 115].
19
2.3 Embeddings n
Cr
p The above embedding of Bp;1 .Rn / into Lrp .Rn / was observed by Nikol’skij in [Nik60] more than 50 years ago. One may also consult [BIN75,
27,28]. The assertion is sharp in the following sense. If for some 0 < q 1 and ~ 2 R, n
C~
p Bp;q .Rn / ,! Lrp .Rn /
n
C~
p then Bp;q .Rn / ,! C r .Rn /;
(2.70)
where the second embedding follows from Lrp .Rn / ,! Lrp .Rn / and [T13, (3.12), p. 111] or Corollary 2.14 below. But this is required by well-known embedding assertions ~ r. According to [Ros13, Proposition 2.4] the second embedding in (2.68) is strict (which means that the spaces do not coincide). We need a sharper embedding of Lrp .Rn / in Lp .Rn ; w / than stated so far in (2.58) with Lrp .Rn / in between. Let again 1 < p < 1. Then the Muckenhoupt class Ap .Rn / collects all weights w.x/ > 0, x 2 Rn , with h 1 Z ip1 1 w.x/ p1 dx : jBj B B B (2.71) The supremum is taken over all balls B in Rn , [Ste93, p. 194]. Then D w.x/L with the Lebesgue measure L is the related measure and Lp .Rn ; /, normed by sup MB < 1 where MB D
kf jLp .Rn ; /k D
Z
1 jBj
Z
w.x/dx
jf .x/jp .dx/
1=p
D
Rn
Z
jf .x/jp w.x/ dx
1=p ;
Rn
(2.72) the corresponding Lp -space. One should be aware that weighted Lp -spaces with respect to the Lebesgue measure L are normed by (2.4) whereas Lp -spaces based on a measure are normed by (2.72). Let again w˛ .x/ D .1 C jxj2 /˛=2 , ˛ 2 R. Then Theorem 2.8 can be complemented as follows. Proposition 2.10. Let 1 < p < 1;
n=p < r < 0;
ru D n;
n < ˛ < n rp:
(2.73)
Then 1 < p < u and w˛ 2 Ap .Rn /. Furthermore Lu .Rn / ,! Lrp .Rn / ,! Lp .Rn ; ˛ /;
˛ D w˛ L :
(2.74)
Proof. The first embedding in (2.74) is covered by (2.58). In Remark 2.11 below we prove that w˛ 2 Ap .Rn /
if, and only if,
n < ˛ < n.p 1/:
(2.75)
20
2 Morrey spaces
This covers in particular (2.73). Let f 2 Lrp .Rn /. Then the right-hand side of (2.74) follows from Z Z Z 1 X p p j˛ jf .x/j w˛ .x/ dx c jf .x/j dx C c 2 jf .x/jp dx Rn
jxj1
c
jf .x/jp dx C c jxj1 0
c kf
2j jxj2j C1
j D0
Z
1 X
2j.˛CnCrp/ kf jLrp .Rn /kp
j D0
jLrp .Rn /kp : (2.76)
Remark 2.11. We justify (2.75). Let R be the radius of the ball B. It is sufficient to deal with balls of radius R 1. If, in addition, dist .B; 0/ R then MB cR˛ R˛ D c:
(2.77)
If, in addition, dist .B; 0/ R (and again R 1) then MB c MB3R
with
B3R D fx 2 Rn W jxj 3Rg:
(2.78)
One has for ˛ 6D n and ˛ 6D n.p 1/, ˛
MB3R .Rn C R˛ /.Rn C R p1 /p1
(2.79)
.Rn C R˛ /.Rn.p1/ C R˛ / complemented by MB3R .Rn C Rn log R/.Rn.p1/ C R˛ /
if ˛ D n
(2.80)
and MB3R .Rn C R˛ /.Rn.p1/ C Rn.p1/ .log R/p1 /
˛ D n.p 1/: (2.81) If ˛ n or ˛ n.p 1/ then the right-hand sides of (2.79)–(2.81) tend to infinity if R ! 1. Otherwise one has MB3R c < 1 uniformly in R 1. This proves (2.75). One has the same conditions as for jxj˛ 2 Ap .Rn /, [Ste93, 6.4, p. 218]. With Lrp .Rn / in between one has only the weaker version (2.58) with ˛ D p < n. Then (2.75) is not satisfied. if
2.3.3 Further embeddings We introduced the spaces H% Lp .Rn / and H % Lp .Rn / in Definition 2.3 in terms of non-smooth atomic decompositions. This suggests that we compare these spaces with s atomic decompositions of other spaces, especially Besov spaces Bp;q .Rn / with, say,
2.3 Embeddings
21
s 1 p; q 1, s > 0. Of special interest are the spaces B1;1 .Rn /. Let d > 1 be fixed and let d QJ;M be the cube concentric with the above cube QJ;M D 2J M C 2J .0; 1/n where J 2 N0 , M 2 Zn , having side-length d 2J . Let K D 1 C Œs (smallest natural number larger than s > 0). Then smooth atoms aJ;M adapted s to B1;1 .Rn / with s > 0 are functions having classical derivatives up to order K inclusively such that
supp aJ;M d QJ;M ;
jD ˛ aJ;M .x/j 2J.ns/CJ j˛j
(2.82)
where J 2 N0 , M 2 Zn and j˛j K. We use again standard notation, hence @j D @=@xj if j D 1; : : : ; n and D ˛ D @˛1 1 @˛nn ;
˛ 2 Nn0 ;
n X
j˛j D
˛j ;
(2.83)
j D1
Nn0
where collects all points (multi-indices) m D .m1 ; : : : ; mn / 2 Rn with mj 2 s .Rn / if, and only if, it can be N0 D N [ f0g. Then f 2 S 0 .Rn / is an element of B1;1 represented by above atoms and J;M 2 C as X X f D J;M aJ;M ; jJ;M j < 1; (2.84) J 2N0 ;M 2Zn
J 2N0 ;M 2Zn
s convergence being in B1;1 .Rn /. Furthermore, s kf jB1;1 .Rn /k inf
X
J 2N0
jJ;M j
(2.85)
;M 2Zn
where the infimum is taken over all representations (2.84), (2.82). This is essentially a special case of [T92, Section 1.9.2, pp. 62/63] where one finds also some remarks about the history of the atomic decompositions of function spaces. As for a more recent version one may also consult [T13, Section 1.2.2, pp. 4–6] and the references given there. Corollary 2.12. Let 1 < p < 1, n < % < n=p and %u D n. Then 1 < u < p and %Cn
n
n
p u S.Rn / ,! B1;1 .Rn / ,! H% Lp .Rn / ,! H % Lp .Rn / ,! Lu .Rn / ,! Bp;p .Rn /: (2.86)
Proof. Compared with part (ii) of Theorem 2.8 it remains to prove that for some c > 0, %Cn %Cn h 2 B1;1 .Rn /: (2.87) kh jH% Lp .Rn /k c kh jB1;1 .Rn /k; %Cn
Let h 2 B1;1 .Rn / be optimally represented by (2.84), (2.85) where s D % C n > 0 and X hJ;M with hJ;M D J;M aJ;M ; J 2 N0 ; M 2 Zn : (2.88) hD J 2N0 ;M 2Zn
22
2 Morrey spaces
This can be taken as a representation (2.9) with the immaterial replacement of QJ;M by d QJ;M . One has X
X
n
2J. p C%/ khJ;M jLp .d QJ;M /k c
J 2N0 ;M 2Zn
jJ;M j:
(2.89)
J 2N0 ;M 2Zn
Then (2.87) follows from (2.85) and (2.11). %Cn
Remark 2.13. It has been observed in [Kal98] that B1;1 -norms are useful in connection with H % Lp .Rn /, called there dual Morrey spaces and denoted similarly as in (2.41). In this context it might be of interest that the embeddings in (2.60) and (2.86) can be strengthened by n
C%
p H % Lp .Rn / ,! Bp;1 .Rn /;
1 < p < 1;
% D n=u. This can be justified as follows. Let [T83, (12), p. 180] one has the duality ı
n 0 Cr
n
1 p
C
n < % < n=p; 1 p0
(2.90)
D 1 and r C % D n. By
C%
p B pp0 ;1 .Rn /0 D Bp;1 .Rn /;
(2.91)
ı
where B sp0 ;1 .Rn / is the completion of D.Rn / in Bps 0 ;1 .Rn /. Furthermore one obtains from (2.68) ı
n 0 Cr
B pp0 ;1 .Rn / ,! LV rp0 .Rn /:
(2.92)
Then (2.90) follows from (2.91), (2.92) and the duality according to (2.107) below. In the above remark we relied on some duality assertions with a reference to Theorem 2.19 below. We describe a second application of this theorem. By Theorem 2.8(ii) and well-known properties of Lebesgue and Besov spaces the Schwartz space S.Rn / is dense in all spaces in (2.86). Then duality in the framework of S.Rn /; S 0 .Rn / makes sense for all these spaces. In particular one has by Theorem 2.19 below
H% Lp .Rn /0 D Lrp0 .Rn / and H % Lp .Rn /0 D Lrp0 .Rn /
(2.93)
where 1 < p < 1, p1 C p10 D 1, n < % < n=p and % C n D r. Let r C r .Rn / D B1;1 .Rn / be the H¨older-Zygmund spaces according to (2.55). Corollary 2.14. Let Lrp .Rn / and Lrp .Rn / be the Morrey spaces according to Definition 2.1 where 1 < p < 1 and n=p < r < 0. Then Lrp .Rn / ,! Lrp .Rn / ,! C r .Rn /:
(2.94)
23
2.3 Embeddings
Proof. Let p1 C p10 D 1 and % D n r. Then n < % < n=p0 . Now it follows from the above remarks, (2.86) and (2.93) with p0 in place of p, r .Rn /0 D C r .Rn /; Lrp .Rn / ,! Lrp .Rn / D H% Lp0 .Rn /0 ,! B1;1
(2.95)
where the last assertion is covered by [T83, Theorem, Section 2.11.2, p. 178].
Remark 2.15. One may also consult [T13, Section 3.1.2, pp. 111–113] where one finds the second embedding in (2.94), further results and references as well as some discussions about the sharpness of assertions of this type. This is based, at least partly, on local means and wavelets. The above proof is much simpler as long as one accepts (2.93).
2.3.4 Non-separability and density By Theorem 2.8 the spaces D.Rn/ and S.Rn / are dense both in H% Lp .Rn / and H % Lp .Rn /. Nothing like this can be expected for the spaces Lrp .Rn / and Lrp .Rn / as introduced in Definition 2.1 (with exception of (2.8)). But one can say more. Recall that Lu .Rn / with u as in (2.57), (2.58) is continuously embedded in Lrp .Rn / and Lrp .Rn /. Proposition 2.16. Let 1 < p < 1;
n=p < r < 0
and ru D n:
(2.96)
Then the spaces Lrp .Rn / and Lrp .Rn / are non-separable. Furthermore, neither D.Rn/ nor S.Rn / nor Lu .Rn / is dense in Lrp .Rn / or dense in Lrp .Rn /. Proof. Step 1. First we prove the non-separability. Let QJl ;M l D 2Jl M l C 2Jl .0; 1/n;
l 2 N;
(2.97)
be disjoint cubes with QJl ;M l Q D .0; 1/n where Jl 2 N, J1 < J2 < : : : and M l 2 Zn . Let 1 X Jl ;Ml 2Jl r Jl ;M l (2.98) f D lD1
where Jl ;M l is the characteristic function of QJl ;M l and D fJl ;M l g1 lD1
with either Jl ;M l D 1 or Jl ;M l D 1:
(2.99)
Let J 2 Z and M 2 Zn . Then Z X X n n J. p Cr/p 2 jf .x/jp dx 2.Jl J /. p Cr/p C 2.J Jl /rp < 1 QJ;M
lWJl J
lWJl <J
(2.100)
24
2 Morrey spaces
where we used pn C r > 0 and r < 0. Hence f 2 Lrp .Rn / ,! Lrp .Rn /. If 1 and 2 are two different admitted sequences then one has 1J ;M l D 1 and 2J ;M l D 1 l l for some l 2 N and 1
2
1
2
kf f jLrp .Rn /k kf f jLrp .Rn /k 1=p Z ˇ ˇ 1 n ˇ.f f 2 /.x/ˇp dx 2Jl . p Cr/ D 2: QJ
l ;M
l
(2.101) But the set of all these admitted functions f is non-countable, having the cardinality of R. Then it follows from (2.101) that neither Lrp .Rn / nor Lrp .Rn / is separable. Step 2. The separable Lebesgue space Lu .Rn / is continuously embedded into the non-separable spaces Lrp .Rn / and Lrp .Rn /. This shows that this embedding cannot be dense. Remark 2.17. If r D n=p then one has (2.8). Hence D.Rn / and S.Rn / are dense in the separable space Lp .Rn /. On the other hand the above construction can be applied to Lp .Rn / according to (2.6), (2.7). In particular Lp .Rn / is non-separable. Furthermore, neither D.Rn/ nor S.Rn / nor Lp .Rn / is dense in Lp .Rn /. Remark 2.18. The assertion that Lrp .Rn / and Lrp .Rn / with 1 < p < 1 and n=p < r < 0 are non-separable seems to be something like mathematical folklore: no doubt, no proof. At least we have no explicit reference. On the other hand it is well known that, say, compactly supported bounded functions are not dense in Lrp .Rn / or Lrp .Rn /, 1 < p < 1, n=p < r < 0. This goes back to [Pic69] and has also been mentioned in [Zor86, Alv96]. Our own proof in [RoT13] is essentially a modification of corresponding arguments in [Zor86]. In any case D.Rn / is not dense in the above spaces which justifies introducing in Definition 2.1 the spaces LV rp .Rn / and LV rp .Rn /, in particular,
LV rp .Rn / 6D Lrp .Rn / and LV rp .Rn / 6D Lrp .Rn /
(2.102)
for 1 < p < 1 and n=p < r < 0.
2.4 Duality 2.4.1 Main assertions
Let G.Rn / be a Banach space embedded in the dual pairing S.Rn /; S 0 .Rn / , S.Rn / ,! G.Rn / ,! S 0 .Rn /:
(2.103)
Recall that ,! means continuous embedding. Let S.Rn / be dense in G.Rn /. Then the dual space G.Rn /0 can be identified with the collection of all g 0 2 S 0 .Rn / for
2.4 Duality
25
which there is a constant c 0 such that j.g 0 ; '/j c k' jG.Rn /k
for all
' 2 S.Rn /:
(2.104)
Then the dual pairing .g 0 ; g/ with g 2 G.Rn /, g 0 2 G.Rn /0 is defined by completion. If, in addition, S.Rn / ,! G.Rn /0 ,! S 0 .Rn /;
S.Rn / dense in G.Rn /0 ; (2.105) 0 n 00 / D G.Rn /0 makes sense in the then also the dual of G.Rn /0 , denoted as G.R context of the dual pairing S.Rn /; S 0 .Rn / . The duality assertions in the theorem below, but also in the next Section 2.4.2, must always be understood in this sense. We stick to the standard notation S 0 .Rn / (and D 0 . /) but in case of Banach spaces having indices or which are compounds of several symbols it might be better to use G.Rn /0 instead of G 0 .Rn / etc. In addition (and modification) we rely in the proofs on the standard duality for weighted Lp -spaces in the above cubes QJ;M D 2J M C 2J .0; 1/n, Lp QJ;M ; J ;
Lp0 QJ;M ; J ;
1 1 C 0 D 1; p p
1 < p < 1;
(2.106)
J 2 Z, M 2 Zn , where J .dx/ D 2J.nCrp/ dx. This must be reformulated after wards in the context of S.Rn /; S 0 .Rn / . Otherwise the spaces Lrp .Rn /, LV rp .Rn / and Lrp .Rn /, LV rp .Rn / as well as H% Lp .Rn /, H % Lp .Rn / have the same meaning as in the Definitions 2.1, 2.3. In this Section 2.4 we follow again closely [RoT14]. Theorem 2.19. (i) Let 1 < p < 1 and n=p < r < 0: Let r C % D n. Then .in the above interpretation/
1 p
LV rp .Rn /0 D H% Lp0 .Rn / and LV rp .Rn /0 D H % Lp0 .Rn /: (ii) Let 1 < p < 1 and n < % < n=p. Let Then .in the above interpretation/
1 p
C
1 p0
C
1 p0
D 1 and
(2.107)
D 1 and r C % D n.
H% Lp .Rn /0 D Lrp0 .Rn / and H % Lp .Rn /0 D Lrp0 .Rn /:
(2.108)
Proof. Step 1. It follows from % C r D n that all spaces in (2.107), (2.108) are covered by Definitions 2.1, 2.3. We can apply Theorem 2.8. In particular they are spaces of regular distributions sandwiched between (weighted) Lebesgue spaces to which both distributional and measure-theoretical arguments can be applied. It follows also from Theorem 2.8 that S.Rn / is dense in LV rp .Rn /, LV rp .Rn / with r; p as in part (i), and in H% Lp .Rn /, H % Lp .Rn / with %, p as in part (ii). Then the claimed duality is covered by the above considerations. Step 2. We prove part (i) in two steps and begin with a preparation. Let A D fAj W j 2 N0 g where Aj are complex Banach spaces. Let ˚ (2.109) c0 .A/ D a D faj W j 2 N0 g; aj 2 Aj ; aj ! 0 if j ! 1 ;
26
2 Morrey spaces
normed by
ka jc0 .A/k D sup kaj jAj k:
(2.110)
j 2N0
Let A0 D fA0j W j 2 N0 g be the collection of the (abstract) duals and let ˚ `1 .A0 / D a0 D faj0 W j 2 N0 g; aj0 2 A0j ; ka0 j`1 .A0 /k < 1 with ka0 j`1 .A0 /k D
1 X
kaj0 jA0j k:
(2.111)
(2.112)
j D0
Then one has c0 .A/0 D `1 .A0 /;
.a; a0/ D
1 X
.aj ; aj0 /:
(2.113)
j D0
This can be found in [ET96, pp. 73/74] with a reference to [T78, Lemma 1.11.1, p. 68]. For the proof of the second equality in (2.107) in the next step we apply this observation to AJ;M D Lp .QJ;M ; J /;
J 2 Z;
M 2 Zn ;
1 < p < 1;
(2.114)
and their duals according to (2.106), A0J;M D Lp0 .QJ;M ; J /;
1 1 C 0 D 1; p p
(2.115)
where J .dx/ D 2J.nCrp/ dx is the weighted Lebesgue measure in QJ;M . In particular if and G D fgJ;M g 2 `1 .fA0J;M g/
F D ffJ;M g 2 c0 .fAJ;M g/ then
Z
kF jc0 .fAJ;M g/k D sup
X Z
.F; G/ D
XZ J;M
;
(2.117)
0
jgJ;M .x/jp 2J.nCrp/ dx
1=p0 (2.118)
QJ;M
J;M
and the duality
1=p
QJ;M
J;M
kG j`1 .fA0J;M g/k D
jfJ;M .x/jp 2J.nCrp/dx
(2.116)
fJ;M .x/ gJ;M .x/ 2J.nCrp/ dx
(2.119)
QJ;M
in the context of Lp -spaces with respect to the measure J as explained above. Let f 2 S.Rn / and fJ;M D f J;M where J;M is the characteristic function of QJ;M . Then n 2J. p Cr/ kfJ;M jLp .QJ;M /k ! 0 if J ! 1 (2.120)
2.4 Duality
follows from
27
C r > 0 and
n p
n
2J. p Cr/ kfJ;M jLp .QJ;M /k ! 0
if J ! 1
(2.121)
follows from r < 0. Furthermore one has, for fixed J0 2 N and jJ j J0 , n
2J. p Cr/ kfJ;M jLp .QJ;M /k ! 0
if jM j ! 1:
(2.122)
Hence F D ffJ;M g 2 c0 .fAJ;M g/ and kf jLrp .Rn /k D kF jc0 .fAJ;M g/k:
(2.123)
This shows that LV rp .Rn / is isomorphic to a closed subspace of c0 .fAJ;M g/. More explicitly, we have a linear isometric map T W f ! ffJ;M g from LV rp .Rn / onto the ˚ closed subspace fJ;M jf 2 LV rp .Rn / of c0 fAJ;M g , hence ˚ V rp .Rn / D fJ;M jf 2 LV rp .Rn / ,! c0 fAJ;M g : TL
(2.124)
Step 3. After this preparation we come to the proof of the second equality in (2.107). Let ' 2 S.Rn / and let h 2 H % Lp0 .Rn / be optimally represented according to (2.12)–(2.14), X hJ;M ; supp hJ;M QJ;M ; (2.125) hD J 2Z;M 2Zn
X
2
J. pn0 C%/
khJ;M jLp0 .QJ;M /k 2 kh jH % Lp0 .Rn /k < 1:
(2.126)
J 2Z;M 2Zn
Recall pn0 C % C pn C r D 0. Furthermore by (2.60) the representation (2.125), (2.126) converges in some Lu .Rn / where 1 < u < p0 and hence the corresponding representation for 'h with ' 2 S.Rn / in L1 .Rn /. Then Lebesgue’s bounded convergence theorem ensures that integration and summation in Z X Z n J. n C%/ '.x/h.x/ dx D 2J. p Cr/ '.x/ 2 p0 hJ;M .x/dx (2.127) Rn
J 2Z;M 2Zn
QJ;M
can be interchanged. Now one obtains, by (2.3) and (2.126), Z ˇ ˇ ˇ ˇ ˇ ˇ.h; '/ˇ D ˇˇ '.x/ h.x/ dx ˇ c k' jLrp .Rn /k kh jH % Lp0 .Rn /k:
(2.128)
Rn
Then one has by the explanations at the beginning of this section h 2 LV rp .Rn /0 and kh jLV rp .Rn /0 k c kh jH % Lp0 .Rn /k;
h 2 H % Lp0 .Rn /:
(2.129)
28
2 Morrey spaces
V rp .Rn /0 . By (2.124) and the Hahn-Banach We prove the converse and assume h 2 L theorem, [Rud91, pp. 56–62], one has h 2 LV rp .Rn /0 if, and only if, h.'/ D .'; h/ can be represented for any ' 2 S.Rn / as X Z '.x/gJ;M .x/2J.nCpr/ dx (2.130) h.'/ D J 2Z;M 2Zn
QJ;M
with fgJ;M g 2 `1 fA0J;M g and
With
kh jLV rp .Rn /0 k D kfgJ;M g j`1 fA0J;M g k:
(2.131)
hJ;M .x/ D gJ;M .x/ J;M .x/ 2J.nCrp/
(2.132)
and r C % D n one has 1=p0 Z 0 J. pn0 C%/ jhJ;M .x/jp dx 2 QJ;M
D2 D
n J. p Cr/
Z
0
QJ;M
Z
0
jgJ;M .x/jp 2J.nCrp/ 2J.p 1/.nCrp/ 0
jgJ;M .x/jp 2J.nCpr/ dx
1=p0 (2.133)
1=p0 :
QJ;M
With e hD
P J 2Z;M 2Zn
hJ;M one has e h 2 H % Lp0 .Rn / and by (2.131)–(2.133) ke h jH % Lp0 .Rn /k kh jLV rp .Rn /0 k:
(2.134)
From (2.127), the preceding arguments ensuring that integration and summation can be interchanged, and (2.130) one obtains Z e (2.135) h.x/ '.x/ dx; ' 2 S.Rn /: h.'/ D Rn
Hence e h 2 H % Lp0 .Rn / is a representation of h 2 LV rp .Rn /0 and (2.134) is the converse of (2.129). This proves the second equality in (2.107) The proof of the first equality in (2.107) is the same replacing J 2 Z by J 2 N0 . Step 4. We prove the second assertion in (2.108). Let g 2 Lrp0 .Rn / and let h 2 H % Lp .Rn / be optimally represented according to (2.12)–(2.14). Then one has for the dual pairing .h; g/, ˇ ˇ X Z ˇ ˇ hJ;M .x/ g.x/ dx ˇ j.h; g/j D ˇ J 2Z;M 2Zn
X
QJ;M
J. n Cr/ n g jLp0 .QJ;M / 2J. p C%/ hJ;M jLp .QJ;M / 2 p0
J 2Z;M 2Zn
(2.136)
2.4 Duality
29
where we used n C % C r D 0 and p1 C p10 D 1. Again one has to justify that integration and summation can be interchanged. This is clear for functions hL as in (2.66), (2.67) which are dense in H % Lp .Rn /. The rest is afterwards a matter of completion as explained at the beginning of this section, where .h; g/ is defined by this procedure within the dual pairing S.Rn /; S 0 .Rn / . Then it follows from (2.136) and (2.3) (with p0 in place of p) in the above interpretation that g 2 H % Lp .Rn /0 and kg jH % Lp .Rn /0 k c kg jLrp0 .Rn /k:
(2.137)
We prove the converse. Let g 2 H % Lp .Rn /0 . According to Theorem 2.8(ii) the Schwartz space S.Rn / is dense in H % Lp .Rn /. Then .h; g/ with the dual pairing h 2 H % Lp .Rn / can be interpreted in the context of S.Rn /; S 0 .Rn / as explained at the beginning of this section. From (2.60) follows H % Lp .Rn /0 ,! Lp .Rn ; w /0 D Lp0 .Rn ; w / with > n=p0 . Hence one has for h 2 S.Rn /, ˇ ˇZ ˇ ˇ h.x/e g.x/dx ˇˇ kh jH % Lp .Rn /k kg jH % Lp .Rn /0 k j.h; g/j D ˇˇ
(2.138)
(2.139)
Rn
with the representative e g 2 Lp0 .Rn ; w / of g. Let ' 2 D.QJ;M /. Then n
k' jH % Lp .Rn /k 2J. p C%/ k' jLp .QJ;M /k: With
n p
ˇZ ˇ ˇ ˇ
(2.140)
C % D pn0 r one obtains ˇ ˇ J. n Cr/ '.x/e g .x/dx ˇˇ kg jH % Lp .Rn /0 k 2 p0 k' jLp .QJ;M /k: n
(2.141)
R
Then one has, by the density of D.QJ;M / in Lp .QJ;M / and the duality in Lp .QJ;M /, ke g jLp0 .QJ;M /k 2
J. pn0 Cr/
kg jH % Lp .Rn /0 k:
(2.142)
This proves e g 2 Lrp0 .Rn / and ke g jLrp0 .Rn /k kg jH % Lp .Rn /0 k:
(2.143)
Together with (2.137) one obtains the second assertion in (2.108). The proof of the first assertion is the same.
2.4.2 Complements Let again Lrp .Rn /, LV rp .Rn / and Lrp .Rn /, LV rp .Rn / be the spaces as introduced in Definition 2.1. As explained at the beginning of Section 2.4.1 the duality of Banach spaces G.Rn / satisfying (2.103) must always be understood in the context of the
30
2 Morrey spaces
dual pairing S.Rn /; S 0 .Rn / . There we fixed also our use of G.Rn /0 (dual space) if S.Rn / is dense in G.Rn / and of G.Rn /00 if S.Rn / is also dense in G.Rn /0 , bidual of G.Rn /, dual of G.Rn /0 . Otherwise our notation is quite standard. If X; Y; Z are Banach spaces with X 0 D Y and Y 0 D Z, then Y is the dual of X and Z is the bidual of X , whereas X is called the predual of Y . According to Theorem 2.19 all admitted spaces Lrp .Rn /, Lrp .Rn /, H%Lp .Rn /, H % Lp .Rn / have preduals. But not any Banach space is a dual Banach space (which means being isomorphic to the abstract dual of a Banach space). Let ˚ c0 D D fj g1 (2.144) j D1 ; j 2 C; jj j ! 0 if j ! 1 normed by k jc0 k D maxj 2N jj j. According to [AlK06, Corollary 2.5.6, p. 46] the space c0 is not a dual Banach space (in the above interpretation). This is formulated there for real spaces, but it applies also to complex spaces. In (2.124) we V rp .Rn / isomorphically onto a closed subspace of c0 fAJ;M g : Then the mapped L V rp .Rn / nor LV rp .Rn / with 1 < p < 1, above comments about c0 suggest that neither L n=p < r < 0 have preduals (although this is not a proof). We fix a useful complement of Theorem 2.19. Corollary 2.20. Let 1 < p < 1 and n=p < r < 0. Then
and
0 LV rp .Rn /00 D LV rp .Rn /0 D Lrp .Rn /
(2.145)
0 LV rp .Rn /00 D LV rp .Rn /0 D Lrp .Rn /:
(2.146)
Proof. It follows from Theorem 2.8 that S.Rn / is dense in H % Lp0 .Rn / and in H% Lp0 .Rn / where 1 < p0 < 1 and n < % < n=p0 . Then the above corollary is an immediate consequence of Theorem 2.19 and the explanation at the beginning of Section 2.4.1. Remark 2.21. Assertions of type (2.107), (2.108) for global spaces have some history. In particular the predual of Lrp .Rn / has been studied in [Alv96, Kal98, Zor86] by different means. It has been remarked in [Kal98] that homogeneous norms of %Cn spaces of type B1;1 .Rn / as considered in (2.86) play a role. This research has been complemented in [AdX04, AdX12] by powerful methods raising the Fourieranalytical theory of the Morrey spaces Lrp .Rn /, 1 < p < 1, n=p < r < 0, at the same level as for the Lebesgue spaces Lp .Rn /, 1 < p < 1. This has been based on [Ada75, Ada88]. A further method using so-called complementary Morrey spaces may be found in [GoM13]. As said above we followed here closely [RoT14]. There one finds also a more detailed discussion about the different methods underlying the above papers, including some comments about possible gaps. This will not be repeated here. It seems to be that the first self-contained complete proof of V rp .Rn /00 D H % Lp0 .Rn /0 D Lrp .Rn /; L
n=p < r < 0; r C % D n; (2.147)
31
2.5 Calder´on-Zygmund operators
1 < p < 1, p1 C p10 D 1, in the framework of S.Rn /; S 0 .Rn / has been given in [Ros13]. It uses some assertions of the above-mentioned papers and is different from the proof of Theorem 2.19 which we took over from [RoT14].
2.5 Calder´on-Zygmund operators 2.5.1 Preliminaries Let Sn1 D fx 2 Rn W jxj D 1g be the unit sphere in Rn , where n 2 N. Let T0 , Z .T0 f /.x/ D lim "#0
y2Rn ;jyj"
.y=jyj/ f .x y/ dy; jyjn
with
x 2 Rn ;
(2.148)
Z 2 L1 .Sn1 /;
. / d D 0;
(2.149)
Sn1
be the classical Calder´on-Zygmund operator where f 2 dom T0 D D.Rn/ D C01 .Rn / (domain of definition). If n D 1 then T0 refers to the Hilbert transform. Then T0 admits a unique linear and bounded extension from D.Rn / (or likewise S.Rn /) to Lp .Rn /, 1 < p < 1. This had been originally proved under some mild additional smoothness assumptions for , [CaZ52], [Ste70, Chapter II], [Ste93, Chapters VI, VII] and [Tor86, Chapter XI]. But according to [DuR86, Corollary 4.2, p. 552] and [Duo01, Theorem 8.38, p. 192] this assertion remains valid under the weaker natural condition (2.149). We are interested whether T0 has linear and bounded extensions to local and global Morrey spaces and what these extensions (if they exist) look like. The (unique) extension T of T0 to Lp .Rn /, 1 < p < 1 can be done by completion. But this does not mean immediately that T in Lp .Rn / can be represented analytically in the same way as T0 : The usual measure-theoretical arguments, convergence a.e., Fubini theorem and so on, do not apply directly because the integral in (2.148) is singular. But rescue comes (under mild additional continuity conditions for in (2.149)) from the so-called maximal Calder´on-Zygmund operators ensuring that the right-hand side of (2.148) with f 2 Lp .Rn /, 1 < p < 1, converges a.e. to .Tf /.x/. In connection with Proposition 2.10 it will be of interest for us that this assertion can be extended to Lp .Rn ; /, 1 < p < 1, according to (2.72) with the measure D w.x/L , where L is the Lebesgue measure in Rn and w 2 Ap .Rn / belongs to the Muckenhoupt class as introduced in (2.71). This applies both to the extension by completion of T0 with dom T0 D D.Rn / to T with dom T D Lp .Rn ; / and also its pointwise representation Z .Tf /.x/ D lim "#0
y2Rn ;jyj"
.y=jyj/ f .x y/ dy; jyjn
x 2 Rn ; a.e.
(2.150)
32
2 Morrey spaces
almost everywhere, again under some smoothness assumptions for , say, Z 1 n1 . / d D 0: 2 C .S /;
(2.151)
Sn1
Recall that D.Rn / is dense in Lp .Rn ; /. This has been mentioned explicitly in [Gra04, p. 714]. We refer the reader to [Tor86, Theorem 2.2, p. 331] as far as the extension by completion from D.Rn / to Lp .Rn ; / is concerned and to the proof of [Tor86, Theorem 6.4, p. 294] which applies also to a.e. pointwise representation (2.150) with f 2 Lp .Rn ; /. Likewise one can rely on [Gra04, Definitions 9.4.1, 9.4.2, Theorem 9.4.3, pp. 702/703, (proof of) Theorem 9.4.5, pp. 709, 712 and Theorem 2.1.14, p. 86]. We refer the reader also to [Ste93, Theorem 2, Corollary, p. 205]. We rely in this Section 2.5 mainly on [RoT13, RoT14], extending preceding assertions in [Ros13].
2.5.2 Main assertions Let X.Rn / be one of the spaces Lrp .Rn /, LV rp .Rn /, Lrp .Rn /, LV rp .Rn / and H% Lp .Rn /, H % Lp .Rn / as introduced in the Definitions 2.1 and 2.3 with 1 < p < 1;
n < % < n=p < r < 0:
(2.152)
A linear and bounded operator T acting in X.Rn /, hence T W X.Rn/ ,! X.Rn /, is called an extension of T0 according to (2.148), (2.149) if it coincides on dom T0 D D.Rn / with (2.148). Theorem 2.22. Let T0 be given by (2.148), (2.149) with dom T0 D D.Rn/. Let 1 < p < 1 and n < % < n=p < r < 0: (2.153) (i) Let, in addition, be continuous and non-trivial, this means k jL1 .Sn1 /k > 0. Then there are no linear and bounded extensions of T0 to any of the local spaces Lrp .Rn /, LV rp .Rn /, and H% Lp .Rn /. (ii) There is a .uniquely determined/ linear and bounded extension T of T0 to r V Lp .Rn /, (2.154) T W LV rp .Rn / ,! LV rp .Rn / and to H % Lp .Rn /,
T W
H % Lp .Rn / ,! H % Lp .Rn /:
(2.155)
(iii) Furthermore there are linear and bounded extensions T of T0 to Lrp .Rn /, T W
Lrp .Rn / ,! Lrp .Rn /:
(2.156)
Proof. Step 1. We prove part (i). We may assume that is real and .0 / > 0 for some 0 2 Sn1 . Let . / > 0 for all 2 Sn1 with j 0 j j1 0 j and some
33
2.5 Calder´on-Zygmund operators
1 2 Sn1 , 1 6D 0 . Let 'K 2 D.Rn / real with compact support in the sectoral domain ˚ y 2 Rn W y D jyj; 2j 0 j j1 0 j; jyj K1 ; (2.157) 0 'K 1, and ˚ y 2 Rn W y D jyj; 4j 0 j j1 0 j; K2 jyj K (2.158) where K1 , K2 and K are natural numbers, K1 < K2 < K. Then 'K 2 dom T0 . If K1 is chosen sufficiently large then one has by (2.148) for all x 2 Rn with jxj 1 and some c > 0, .T0 'K /.x/ c log K: (2.159) 'K .y/ D 1
if
Hence .T0 'K /.x/ ! 1 if K ! 1, whereas k'K jLrp .Rn /k is uniformly bounded as one checks easily. This proves that there is neither a linear and bounded extension of T0 to LV rp .Rn / nor to Lrp .Rn /. We assume that there is a linear and bounded extension T of T0 to H% Lp .Rn /. One has by Theorem 2.8 that D.Rn / D dom T0 is dense in H% Lp .Rn /. Recall that T0 is formally self-adjoint in D.Rn/. Then it follows from the duality assertions according to Theorem 2.19 that there is a linear and bounded extension of T0 to Lrp0 .Rn /, p1 C p10 D 1. But this is not the case. Step 2. We prove part (2.154). Both Lrp .Rn /, normed by (2.3), and T0 , given by (2.148), are translation-invariant. Hence to prove the requested estimate for T0 f in the counterpart of (2.3) it is sufficient to deal with the cubes QJ;0 or the model cubes QJ , centered at the origin with sides of length 2J parallel to the axes of coordinates, J 2 Z. In addition the spaces Lrp .Rn / have the following homogeneity property. Let > 0 and f 2 Lrp .Rn /. Then kf ./ jLrp .Rn /k D r
sup
D
2
sup
n J. p Cr/
J 2Z;M 2Zn
1 J 2
J 2Z;M 2Zn r kf jLrp .Rn /k;
n .p Cr/
Z
Z 0yl 2J Ml 2J
0yl 2J Ml 2J
jf .y/jp dy
jf .y/jp dy
1=p
1=p
(2.160) whereas T0 is homogeneous of degree zero. This shows that we may assume ˚ f 2 D.Rn /; supp f y 2 Rn ; jyj < 1 : (2.161) 2 Let J 2 Z and let f'j gJj D1 D.Rn / be a canonical dyadic resolution of unity in Rn with
'J 2 .x/ D 1 and
if
x 2 QJ 2 ;
supp 'j Qj 1 n Qj C1 ;
supp 'J 2 QJ 3 ;
(2.162)
j 2 Z;
(2.163)
j < J 2:
34
2 Morrey spaces
In particular, J 2 X
'j .x/ D 1;
x 2 Rn ;
(2.164)
j D1
and f D
J 2 X
J 3 X
'j f D fJ 2 C
j D1
fj :
(2.165)
j D1
Obviously, kT0 f jLp .QJ /k kT0 fJ 2 jLp .QJ /k C
J 3 X
kT0 fj jLp .QJ /k
(2.166)
j D1
(finite sum). Recall that the extension of T0 maps Lp .Rn / into itself. Then one has for the first term on the right-hand side of (2.166), n
n
2J. p Cr/ kT0 fJ 2 jLp .QJ /k c 2J. p Cr/ kfJ 2 jLp .Rn /k n
c 2J. p Cr/ kf jLp .QJ 3 /k c 0 kf jLrp .Rn /k:
(2.167)
If x 2 QJ and x y 2 Qj 1 n Qj C1 with j < J 2 then jyj 2j . Then one has for x 2 QJ by (2.148), (2.163) and H¨older’s inequality jn
jT0 fj .x/j c 2nj k'j f jL1 .Rn /k c 0 2 p kf jLp .Qj 1 /k c 00 2jr kf jLrp .Rn /k and
n
2J. p Cr/ kT0 fj jLp .QJ /k c 2.J j /r kf jLrp .Rn /k:
(2.168)
(2.169)
Since r < 0 one obtains, by (2.166)–(2.169), kT0 f jLrp .Rn /k c kf jLrp .Rn /k:
(2.170)
Hence T0 f 2 Lrp .Rn /. It remains to show that T0 f 2 LV rp .Rn /. Let f be as (2.161) and ' 2 S.Rn /. Then one has by the usual dual pairing and the Fubini-Lebesgue theorem Z Z .y=jyj/ j˛j ˛ .1/ T0 f; D ' D lim f .x y/.1/j˛j D ˛ '.x/ dx dy "#0 jyj" Rn jyjn D T0 D ˛ f; ' : (2.171)
35
2.5 Calder´on-Zygmund operators
Hence T0 f 2 Wpk .Rn / for all k 2 N. Then it follows by embedding that T0 f 2 Lrp .Rn / is a C 1 function. Furthermore one has by (2.148), ˇ ˇ ˇ T0 f /.x/ˇ c
Z Rn
jf .y/j dy; jx yjn
f 2 D.Rn/;
x 62 supp f:
(2.172)
Applied to f according to (2.161) one obtains j.T0 f /.x/j c jxjn
if
jxj 2:
(2.173)
Let R 2. Then one has for cubes QJ;M with QJ;M fx 2 Rn W jxj > Rg, n
2J. p Cr/ kT0 f jLp .QJ;M /k c 2J r Rn
if J 2 N0 :
(2.174)
Using 1 < p < 1 and again jT0 f .x/j c=jxjn one obtains n
n
1
2J. p Cr/ kT0 f jLp .QJ;M /k c 2J. p Cr/ Rn.1 p /
if
J 2 N:
(2.175)
Let R be a smooth cut-off function with R .x/ D 1 if jxj R. Then R T0 f 2 D.Rn/ and it follows from (2.174), (2.175) with 1 < p < 1 and 0 < pn C r < pn , lim kT0 f R T0 f jLrp .Rn /k D 0:
R!1
(2.176)
Hence T0 f 2 LV rp .Rn /. This proves the uniquely determined extension (2.154). Step 3. Let T with dom T D LV rp .Rn / be the extension of T0 according to (2.154) and the above Step 2. Then it follows from (2.107) and Banach space theory for the dual operator T 0 , T0 W
H % Lp0 .Rn / ,! H % Lp0 .Rn /;
1 1 C 0 D 1: p p
(2.177)
As for the abstract background one may consult [Yos80, pp. 112/113] and [Pie07, pp. 35/36]. But T is formally self-adjoint. Furthermore D.Rn/ is dense in H % Lp0 .Rn /, Theorem 2.8(ii). Both together shows that there is a uniquely determined extension of T0 with (2.155). Again by Banach space theory and (2.108) it follows that there are (not uniquely) determined extensions of T0 with (2.156). Remark 2.23. Part (ii) can be extended to r D n=p based on (2.8) and the mapping properties as described at the beginning of Section 2.5.1. By the same arguments as in Step 1 of the above proof one can extend part (i) to r D n=p, hence Lp .Rn / according to (2.6), (2.7). Otherwise our arguments rely on two observations. First there is the possibility of transferring mapping properties of operators of type (2.148) in Lp -spaces, 1 < p < 1, with the help of decay assertions of type (2.172) to more general spaces, in our case global Morrey spaces LV rp .Rn /. This observation goes back to [SoW94] and has been used afterwards by many authors inclusively related
36
2 Morrey spaces
decomposition techniques. We refer the reader to [RoT13, Remark 4.1, p. 9] where we collected related papers. The second basic ingredient in the above approach is the possibility according to Theorem 2.19 in the context of the of using the duality dual pairing S.Rn /; S 0 .Rn / . Then one can circumvent that D.Rn / is not dense in Lrp .Rn / as stated in Proposition 2.16. This abstract argument ensures the existence of an extension T of T0 according to part (iii) of the above theorem, but says nothing about uniqueness (which does not hold) or whether T can be represented according to (2.150) a.e.. We return to both questions in Section 2.5.3 below. Duality arguments in connection with Calder´on-Zygmund operators have been used in literature before. We refer the reader in particular to [Alv96] dealing in this context first with the predual H % Lp0 .Rn / of Lrp .Rn / according to (2.108) and that D.Rn / is dense in this space, Theorem 2.8(ii). We followed here essentially [RoT13, RoT14]. Compared with the literature we did in [RoT14] a further step back dealing first with LV rp .Rn /, the predual of H % Lp0 .Rn / and climbed afterwards the duality ladder to Lrp .Rn / with H % Lp0 .Rn / in between. This seems to be new (at least we have no reference). As we comment in Section 2.5.3 on some arising questions. Remark 2.24. In connection with the above discussion and Theorem 2.22(iii) one may ask whether any linear and bounded operator in some space admits linear and bounded extensions to larger spaces, similarly as for linear and bounded functionals ensured by the Hahn-Banach theorem. But this is not the case. According to [Pie07, p.133] a Banach space Y is said to have the extension property if every linear bounded operator T0 W X0 ,! Y defined on a closed subspace X0 of an arbitrary Banach space X admits a linear and bounded extension T W X ,! Y such that its restriction to X0 coincides with T0 , T jX0 D T0 . We refer the reader also to [Woj91, p.127]. Applied V rp .Rn /. to the above situation one may think about X D Y D Lrp .Rn / and X0 D L But there is no abstract assertion ensuring that T0 W LV rp .Rn / ,! Lrp .Rn / can be extended to T W Lrp .Rn / ,! Lrp .Rn /. According to [Pie07, p. 134] with a reference to [Gro53, p. 169] an infinite-dimensional Banach space with the extension property is necessarily non-separable. This applies to Lrp .Rn / as a non-separable target space, but not to LV rp .Rn / as an alternative separable target space. But sufficient conditions ensuring the extension property (for Lrp .Rn /) are apparently not known.
2.5.3 Distinguished representations In connection with Theorem 2.22 and the discussion in the preceding Section 2.5.2 one may ask several questions. The extensions T of T0 in (2.154), (2.155) are unique (because D.Rn / is dense in LV rp .Rn / and H % Lp .Rn /). The situation in (2.156) is different. Using the so-called Cotlar decomposition Peetre proved in [Pee66, Theorem 1.1, p. 296] that there are linear and bounded extensions T of T0 to Lrp .Rn /. But this covers neither uniqueness (which actually does not hold) nor a representation of
2.5 Calder´on-Zygmund operators
37
.Tf /.x/ as in (2.150) with f 2 Lrp .Rn / a.e..One can see quite easily that extensions T of T0 from D.Rn / to Lrp .Rn / are by no means unique. Let f1 2 Lrp .Rn / n LV rp .Rn /; and let Then
kf1 jLrp .Rn /k D 1;
(2.178)
˚ G D g D f0 C f1 W f0 2 LV rp .Rn /; 2 C :
(2.179)
kg jLrp .Rn /k kf0 jLrp .Rn /k C jj D kg jGk:
(2.180)
and k jGk. Of course G is a Banach space with respect to the norms k Then these norms are equivalent. This is a very simple case (which can also be proved in a few lines directly) of Banach’s famous observation that two norms on a Banach space (with respect to each of these two norms) are equivalent if, and only if, they are comparable. This version may be found in [Pie07, Section 2.5.5, p. 44] with a reference to Banach’s original paper [Ban29, II, p. 239]. One may also consult [Rud91, p. 50]. Then jLrp .Rn /k
L.g/ D ;
where g D f0 C f1 ;
f0 2 LV rp .Rn /;
(2.181)
is a non-trivial linear continuous functional on G which vanishes on LV rp .Rn /. By the Hahn-Banach theorem it can be extended to Lrp .Rn /, again denoted by L. If T is an extension of T0 according to (2.156) then T C L is also an extension. This shows that there are many extensions of T0 to Lrp .Rn /. The discussion in Section 2.5.1 supported by the references given there suggests that we ask whether some extensions T of T0 according to Theorem 2.22 can be represented as Z .y=jyj/ .Tf /.x/ D lim f .x y/ dy; x 2 Rn ; a.e.; (2.182) jyjn "#0 y2Rn ;jyj" with as in (2.151), hence Z 2 C .S 1
n1
. / d D 0:
/;
(2.183)
Sn1
Whereas the extensions in (2.154) and (2.155) are unique the above question asks in case of (2.156) for a distinguished extension. Proposition 2.25. Let T0 be given by (2.182), (2.183) with .T0 f /.x/ D .Tf /.x/, x 2 Rn , dom T0 D D.Rn /. Let 1 < p < 1 and n < % < n=p < r < 0: (i) Then the uniquely determined extensions T of T0 according to (2.154) with dom T D LV rp .Rn / and according to (2.155) with dom T D H % Lp .Rn / can be represented by (2.182). (ii) Furthermore (2.182) with dom T D Lrp .Rn / is a distinguished extension of T0 according to (2.156).
38
2 Morrey spaces
Proof. Step 1. Let D w.x/L with w 2 Ap .Rn /. As described at the end of Section 2.5.1 the uniquely determined extension T of T0 with dom T D Lp .Rn ; / can be represented by (2.182). The second embedding in (2.74) ensures that also T in (2.154) can be represented by (2.182). Similarly the last but one embedding in (2.60) with 1 < u < p shows that also T in (2.155) can be represented by (2.182). Step 2. The density argument does not work for extensions T in (2.156). But one can again rely on the second embedding in (2.74) which ensures that one has (2.182) for any f 2 Lrp .Rn /. Afterwards one can apply the same decomposition arguments as in (2.165)–(2.170) resulting in Tf 2 Lrp .Rn / and the counterpart of (2.170), kTf jLrp .Rn /k c kf jLrp .Rn /k;
f 2 Lrp .Rn /:
(2.184)
(The more restrictive assumptions (2.161) are mainly needed after (2.170) in (2.171)– (2.176) to justify (2.154).) Remark 2.26. In [RoT13, RoT14] we collected some recent papers dealing with mapping properties of Calder´on-Zygmund operators in generalized (weighted) Morrey spaces. Usually (2.182) is taken as a starting point. This is justified by the above considerations as long as the space in question is continuously embedded in some spaces Lp .Rn ; /, 2 Ap .Rn /, 1 < p < 1. But it surely requires some additional care if this is not guaranteed. Later on we apply the above assertions to the Riesz transforms Z _ yk k b f .x/ D cn lim f .x y/ dy; Rk f .x/ D i jj "#0 jyj" jyjnC1
x 2 Rn a.e.;
(2.185) k D 1; : : : ; n, as mentioned in (1.10). (As above a.e. stands for almost everywhere with respect to the Lebesgue measure in Rn ). To have a convenient reference we collect the outcome. Corollary 2.27. Let Rk;0 be given by (2.185) with dom Rk;0 D D.Rn/ where k D 1; : : : ; n. Let 1 < p < 1 and n < % < n=p < r < 0:
(2.186)
(i) There are .uniquely determined/ linear and bounded extensions Rk of Rk;0 to
LV rp .Rn /,
LV rp .Rn / ,! LV rp .Rn /
(2.187)
H % Lp .Rn / ,! H % Lp .Rn /:
(2.188)
Rk W and to H % Lp .Rn /, Rk W
Both extensions can be represented by (2.185).
2.6 Haar bases
39
(ii) There are linear and bounded extensions Rk of Rk;0 to Lrp .Rn /, Rk W
Lrp .Rn / ,! Lrp .Rn /:
(2.189)
Furthermore, Rk according to (2.185) with dom Rk D Lrp .Rn / is a distinguished extension. Proof. The above assertions follow from Theorem 2.22 and Proposition 2.25 with .y=jyj/ D yk =jyj. Remark 2.28. Recall that Theorem 2.22, Proposition 2.25 and Corollary 2.27 apply also to Lp .Rn / D Ln=p .Rn / D LV n=p .Rn / D H n=p Lp .Rn /; p p
1 < p < 1; (2.190) in the notation of (2.8) and (2.28). But according to Theorem 2.22(i) and Remark 2.23 there are no assertions of this type for the local spaces Lrp .Rn / with 1 < p < 1 and n=p r < 0. It is just this point which forced us in connection with local spaces to replace the Riesz transforms Rk by the truncated Riesz transforms according to (1.16), (1.17).
2.6 Haar bases 2.6.1 Preliminaries and definitions The Morrey spaces as introduced in Definition 2.1 are special local spaces LrAsp;q .Rn / and special hybrid spaces LrAsp;q .Rn /. In particular,
Lrp .Rn / D Lr Lp .Rn / and Lrp .Rn / D Lr Lp .Rn /;
1 < p < 1;
n=p < r < 0; (2.191)
based on Lp .Rn / D Hp0 .Rn /
s and Hps .Rn / D Fp;2 .Rn /;
s 2 R;
1 < p < 1; (2.192) where Hps .Rn / are the (fractional) Sobolev spaces. The first assertion in (2.191) may be found in [T13, Theorem 3.13, p. 121]. The hybrid counterpart is covered by (3.308). Later on we deal with characterizations of some spaces LrAsp;q .Rn / and LrAsp;q .Rn / in terms of Haar bases. This includes (2.191), (2.192). But there are also some specific assertions for the Morrey spaces which may be worth being fixed separately. We give now a complete description but postpone the proofs to Section 3.4.4. First we introduce some notation where we compromise between our later needs and the above specifications. Let, as usual in wavelet theory, G D .G1 ; : : : ; Gn / 2 G 0 D fF; M gn
(2.193)
40
2 Morrey spaces
which means that Gl is either F or M . Let G D .G1 ; : : : ; Gn / 2 G j D G D fF; M gn ;
j 2 N;
(2.194)
which means that Gl is either F or M where indicates that at least on the components of G must be an M . Hence G 0 has 2n elements, whereas G j D G , j 2 N, has 2n 1 elements. Let again QJ;M D 2J M C 2J .0; 1/n with J 2 Z and M 2 Zn be the same dyadic cubes as in Definition 2.1. Let J C D max.J; 0/ if J 2 Z. Then ˚ PJ;M D j J C ; G 2 G j ; m 2 Zn W Qj;m QJ;M ; J 2 Z; M 2 Zn ; (2.195) and ˚ PP J;M D j J; G 2 G ; m 2 Zn W Qj;m QJ;M ; J 2 Z; M 2 Zn : (2.196) Let j;m be the characteristic function of Qj;m . Definition 2.29. Let 0 < p < 1, s 2 R and n=p r < 1. (i) Let ˚ j n D j;G m 2 C W j 2 N0 ; G 2 G ; m 2 Z : Then
˚ Lrhsp .Rn / D W k jLrhsp .Rn /k < 1
(2.197) (2.198)
with k jLrhsp .Rn /k D
sup J 2N0 ;M 2Zn
n 2J. p Cr/
and
ˇ2 1=2 ˇ ˇ (2.199) ˇ ˇLp .Rn / 22js ˇj;G m j;m ./
X .j;G;m/2PJ;M
˚ Lr hsp .Rn / D W k jLr hsp .Rn /k < 1
(2.200)
with k jLr hsp .Rn /k D
sup J 2Z;M 2Zn
(ii) Let Then
n 2J. p Cr/
X
ˇ2 1=2 ˇ ˇ (2.201) n ˇ ˇ 22js ˇj;G ./ .R / L : j;m p m
.j;G;m/2PJ;M
˚ n D j;G m 2 C W j 2 Z; G 2 G ; m 2 Z :
(2.202)
˚ Lr hP sp .Rn / D W k jLr hP sp .Rn /k < 1
(2.203)
2.6 Haar bases
41
with k jLr hP sp .Rn /k D
sup J 2Z;M 2Zn
n 2J. p Cr/
X
2 22js jj;G m j;m ./j
1=2
jLp .Rn / : (2.204)
.j;G;m/2PP J;M
V r hP sp .Rn / of Lr hP sp .Rn / be the completion of all sequences in Let the subspace L Lr hP s .Rn / having only finitely many components different from zero. p
Remark 2.30. In this Section 2.6 we need the above sequence spaces for 1 < p < 1, s D 0 and n=p r < 0. But later on we use the above spaces in a more general context. We briefly recall what is meant by Haar bases in our context. Let for y 2 R, 8 ˆ if 0 < y < 1=2; <1 hM .y/ D 1 if 1=2 y < 1; (2.205) ˆ :0 if y 62 .0; 1/; and let hF .y/ D jhM .y/j be the characteristic function of the unit interval .0; 1/. Then n Y j HG;m .x/ D 2j n=2 hGl 2j xl ml ; j 2 N0 ; G 2 G j ; m 2 Zn ; lD1
(2.206) is the well-known orthonormal Haar basis in L2 .Rn /. One has in particular X X X j j n=2 j;G HG;m (2.207) f D m 2 j 2N0 G2G j m2Zn
with the Fourier coefficients
j j;G j n=2 j;G f; HG;m ; m D m .f / D 2
j 2 N0 ; G 2 G j ; m 2 Zn ;
(2.208)
(scalar product). We need the homogeneous counterpart of this assertion. Let G be as in (2.194). Then j
hG;m .x/ D 2j n=2
n Y
hGl 2j xl ml ;
j 2 Z;
G 2 G;
m 2 Zn ; (2.209)
lD1
is also an orthonormal basis in L2 .Rn /. Instead of (2.207), (2.208) one has now X X X j n=2 j j;G hG;m (2.210) f D m 2 j 2Z G2G m2Zn
with the Fourier coefficients
j;G j n=2 j;G f; hjG;m ; m D m .f / D 2
j 2 Z; G 2 G ; m 2 Zn :
(2.211)
42
2 Morrey spaces
2.6.2 Main assertions After these preparations we can now characterize the local Morrey spaces Lrp .Rn / and the global Morrey spaces Lrp .Rn / as introduced in Definition 2.1 in terms of the Haar expansion (2.206)–(2.208). If r D n=p then one has the usual Lebesgue spaces Lp .Rn / and their uniform modifications Lp .Rn / according to (2.6)–(2.8). We need now the sequence spaces Lr h0p .Rn / and Lr h0p .Rn / as introduced in Definition 2.29(i) with 1 < p < 1 and s D 0. Theorem 2.31. Let 1 < p < 1 and n=p r < 0. (i) Let f 2 S 0 .Rn /. Then f 2 Lrp .Rn / if, and only if, it can be represented by f D
1 X X X j D0
G2G j
j n=2 j j;G HG;m ; m 2
2 Lrh0p .Rn /;
(2.212)
m2Zn
unconditional convergence being in S 0 .Rn /. The representation (2.212) is unique with (2.208). Furthermore, I W
˚ j n f ! j;G m .f / W j 2 N0 ; G 2 G ; m 2 Z
(2.213)
is an isomorphic map of Lrp .Rn / onto Lr h0p .Rn /, kf jLrp .Rn /k k.f / jLr h0p .Rn /k:
(2.214)
(ii) Let f 2 S 0 .Rn /. Then f 2 Lrp .Rn / if, and only if, it can be represented by f D
1 X X X j D0 G2G j
j n=2 j j;G HG;m ; m 2
2 Lr h0p .Rn /;
(2.215)
m2Zn
unconditional convergence being in S 0 .Rn /. The representation (2.215) is unique with (2.208). Furthermore, I in (2.213) is an isomorphic map of Lrp .Rn / onto Lr h0p .Rn /, kf jLrp .Rn /k k.f / jLr h0p .Rn /k:
(2.216)
Remark 2.32. In Corollary 3.44 we prove a corresponding assertion for some hybrid Sobolev spaces Lr Hps .Rn / with 1 < p < 1. This covers in particular Lrp .Rn / D Lr Lp .Rn / D Lr Hp0 .Rn / and also the local counterparts Lrp .Rn / as mentioned in (2.191), (2.192). But it seems to be reasonable to incorporate this special case in this chapter about Morrey spaces and to shift the proof to the indicated later occasion. Part (i) coincides also with [Tri13, Corollary 4.1, p. 149].
2.6 Haar bases
43
2.6.3 Littlewood-Paley theorem The famous Littlewood-Paley characterization of Lp .Rn /, 1 < p < 1, in terms of the homogeneous Haar systems (2.209) is one of the cornerstones of Harmonic Analysis going back to J. Marcinkiewicz (1937), [Mar37], based on R.E.A.C. Paley (1932), [Pal32]. Details and further references may be found in [T10, p. 83]. For a new short proof we refer the reader to [T10, Section 2.2.5, pp. 86/87], n D 1. There is little hope that this assertion can be extended to Lrp .Rn / or Lrp .Rn /, n=p < r < 0, in Definition 2.1. According to Proposition 2.16 these spaces are non-separable and D.Rn/ is not dense. But the proof of the Littlewood-Paley characterization in terms of the homogeneous Haar basis in (2.209) relies essentially on the well-known assertion that D.Rn / is dense in Lp .Rn /, 1 < p < 1. This suggests that we ask for Littlewood-Paley characterizations for LV rp .Rn / introduced in Definition 2.1 as j the completion of D.Rn / in Lrp .Rn /. Let fhG;m g be the homogeneous Haar system according to (2.209) and let Lr hP 0p .Rn / and LV r hP 0p .Rn / be the sequence spaces as introduced in Definition 2.29(ii). Theorem 2.33. Let 1 < p < 1 and n=p r < 0. Then ˚ j hG;m W j 2 Z; G 2 G ; m 2 Zn
(2.217)
V rp .Rn / can be uniquely expanded by is an unconditional basis in LV rp .Rn /. Any f 2 L X X X j n=2 j f D j;G hG;m ; 2 LV r hP 0p .Rn /; (2.218) m 2 j 2Z G2G m2Zn
Z
with j;G j n=2 j;G m D m .f / D 2
Rn
f .x/ hjG;m .x/ dx;
j 2 Z, G 2 G , m 2 Zn . Furthermore, ˚ n I W f 7! j;G m .f / W j 2 Z; G 2 G ; m 2 Z
(2.219)
(2.220)
V r hP 0p .Rn /, is an isomorphic map of LV rp .Rn / onto L kf jLrp .Rn /k k.f / jLr hP 0p .Rn /k;
f 2 LV rp .Rn /:
(2.221)
Remark 2.34. This theorem coincides essentially with [Tri13, Theorem 4.3, p. 150]. The related proof will be repeated later on in Section 3.4.5. By (2.204) one can write (2.221) more explicitly as kf jLrp .Rn /k
sup J 2Z;M 2Zn
n 2J. p Cr/
X .j;G;m/2PP J;M
ˇ 1=2 ˇ j;G n ˇ .f / j;m./ˇ2 jL .R / ; (2.222) p m
44
2 Morrey spaces
f 2 LV rp .Rn /, with j;G m .f / as in (2.219). If r D n=p then one has by (2.8) kf jLp .Rn /k
X
ˇ 1=2 ˇ j;G n ˇ .f / j;m ./ˇ2 jL .R / ; p m
(2.223)
j 2Z;G2G m2Zn
f 2 Lp .Rn /, 1 < p < 1. This is the classical Littlewood-Paley theorem for the Lebesgue spaces Lp .Rn / with 1 < p < 1.
Chapter 3
Hybrid spaces
3.1 Introduction We dealt in [T13] with the local spaces LrAsp;q .Rn / which combine the global spaces Asp;q .Rn /;
s 2 R;
0 < p 1;
0 < q 1;
(3.1)
A 2 fB; F g, (p < 1 for F -spaces), with the typical localization or Morreyfication (2.2) for the local Morrey spaces Lrp .Rn / where n=p r < 1. Now we replace the localization (2.2) by (2.3) and call the outcome LrAsp;q .Rn / hybrid spaces. It follows immediately from the definitions below that LrAsp;q .Rn / ,! LrAsp;q .Rn /
(3.2)
for all admitted parameters. Furthermore Ln=p Asp;q .Rn / D Asp;q .Rn /
(3.3)
again for all admitted parameters. As far as properties are concerned the spaces LrAsp;q .Rn / are between the global spaces Asp;q .Rn / and the local spaces LrAsp;q .Rn / what may justify calling them hybrid spaces. As mentioned in (1.22) the hybrid spaces coincide with the spaces n r s n As; p;q .R / D L Ap;q .R /;
D
1 r C ; p n
(3.4)
n for all admitted parameters. The spaces As; p;q .R / have been studied in [YSY10, Sic12] and the underlying papers. After it has been established, (3.4) will be of some service for us. Otherwise we are in a comfortable position. Section 3.3 deals with definitions and basic properties of the hybrid spaces LrAsp;q .Rn /. In Section 3.4 we characterize these spaces in terms of atoms and wavelets and justify also (3.4) in Theorem 3.38. We rely for all spaces considered, global, local and now hybrid, on the same atoms, molecules and wavelets. In particular atomic and wavelet characterizations obtained in [T13] for local spaces LrAsp;q .Rn / can be carried over more or less immediately to the hybrid spaces LrAsp;q .Rn /. But we give detailed formulations and precise references. To make this chapter independently readable we collect in Section 3.2 the necessary assertions for the global spaces Asp;q .Rn / following again closely [T13]. Section 3.5 deals with distinguished equivalent norms and Fourier multipliers for the spaces LrAsp;q .Rn /. In contrast to the preceding sections the situation is now totally different. We are interested in equivalent norms as described in (1.23), (1.24) and resulting mapping properties for the Riesz transforms Rk according to (1.19),
46
3 Hybrid spaces
(1.20). There are no counterparts for the local spaces LrAsp;q .Rn /. Otherwise we refer the reader to Chapter 1 where we described in some detail our related motivations to switch from local spaces to hybrid spaces. Finally we collect in Section 3.6 some further properties where the Morrey characterizations of the hybrid spaces may be of self-contained interest. They shed some new light on these spaces.
3.2 Global spaces 3.2.1 Definitions First we recall and complement the notation introduced so far in Sections 2.2.1 and 2.3.1. Otherwise we take over from [T13, Section 1.2] some basic material ensuring (so we hope) that this Chapter 3 is independently readable. Let N be the collection of all natural numbers and N0 D N [ f0g. Let Rn be Euclidean n-space, where n 2 N. Put R D R1 , whereas C is the complex plane. Let S.Rn / be the Schwartz space of all complex-valued rapidly decreasing infinitely differentiable functions on Rn and let S 0 .Rn / be the space of all tempered distributions on Rn . As before Lp .Rn / with 0 < p 1, is the standard complex quasi-Banach space with respect to the Lebesgue measure in Rn , quasi-normed by Z 1=p n jf .x/jp dx (3.5) kf jLp .R /k D Rn
with the natural modification if p D 1. As usual Z is the collection of all integers; and Zn where n 2 N denotes the lattice of all points m D .m1 ; : : : ; mn / 2 Rn with mj 2 Z. Let Nn0 , where n 2 N, be the set of all multi-indices, ˛ D .˛1 ; : : : ; ˛n / with
˛ j 2 N0
and j˛j D
n X
˛j :
(3.6)
j D1
If x D .x1 ; : : : ; xn / 2 Rn and ˇ D .ˇ1 ; : : : ; ˇn / 2 Nn0 then we put x ˇ D x1ˇ1 xnˇn If ' 2 S.Rn / then b ' ./ D .F '/./ D .2 /
n=2
(monomials):
Z Rn
eix '.x/ dx;
(3.7)
2 Rn ;
(3.8)
denotes the Fourier transform of '. As usual, F 1 ' and ' _ stand for the inverse Fourier transform, given by the right-hand side of (3.8) with i in place of i . Here x stands for the scalar product in Rn . Both F and F 1 are extended to S 0 .Rn / in the standard way. Let '0 2 S.Rn / with '0 .x/ D 1 if jxj 1
and
'0 .y/ D 0 if jyj 3=2;
(3.9)
3.2 Global spaces
and let
'k .x/ D '0 2k x/ '0 2kC1 x ;
Since
1 X
'j .x/ D 1
x 2 Rn ;
k 2 N:
for x 2 Rn ;
47
(3.10)
(3.11)
j D0
b/_ .x/ the 'j form a dyadic resolution of unity. The entire analytic functions .'j f n 0 n make sense pointwise in R for any f 2 S .R /. Definition 3.1. Let ' D f'j g1 j D0 be the above dyadic resolution of unity. (i) Let 0 < p 1; 0 < q 1; s 2 R:
(3.12)
s Then Bp;q .Rn / is the collection of all f 2 S 0 .Rn / such that
s .Rn /k' D kf jBp;q
1 X
1=q b/_ jLp .Rn / q 2jsq .'j f <1
(3.13)
j D0
.with the usual modification if q D 1/. (ii) Let 0 < p < 1; 0 < q 1; Then
s .Rn / Fp;q
0
s 2 R:
(3.14)
is the collection of all f 2 S .R / such that n
1 X ˇ 1=q ˇ ˇ s b/_ ./ˇq ˇLp .Rn / .Rn /k' D 2jsq ˇ.'j f kf jFp;q <1
(3.15)
j D0
.with the usual modification if q D 1/. Remark 3.2. We call Asp;q .Rn / with A 2 fB; F g global spaces. The theory of these spaces and their history may be found in [T83, T92, T06]. In particular these spaces are independent of admitted resolutions of unity ' according to (3.9)–(3.11) (equivalent quasi-norms). This justifies our omission of the subscript ' in (3.13), (3.15) in the sequel. We remind the reader of a few special cases and properties which will be of some use later on. Greater details and explanations may be found in the above books, especially in [T06, Section 1.2]. (i) Let 1 < p < 1. Then 0 Lp .Rn / D Fp;2 .Rn /
(3.16)
is the well-known Littlewood-Paley theorem. (ii) Let 1 < p < 1 and k 2 N0 . Then k Wpk .Rn / D Fp;2 .Rn /
(3.17)
48
3 Hybrid spaces
are the classical Sobolev spaces usually equivalently normed by X 1=p D ˛ f jLp .Rn / p kf jWpk .Rn /k D :
(3.18)
j˛jk
This generalizes (3.16). (iii) Let 2 R. Then b _ I W f 7! hi f
with
hi D .1 C jj2 /1=2 ;
2 Rn ;
(3.19)
is a one-to-one map of S.Rn / onto itself and of S 0 .Rn / onto itself. Furthermore, I is a lift for the spaces Asp;q .Rn / with A 2 fB; F g and s 2 R, 0 < p 1 (p < 1 for F -spaces), 0 < q 1, n I Asp;q .Rn / D AsC p;q .R /
(3.20)
(equivalent quasi-norms). With
one has and
Hps .Rn / D Is Lp .Rn /;
s 2 R;
1 < p < 1;
(3.21)
s Hps .Rn / D Fp;2 .Rn /;
s 2 R;
1 < p < 1;
(3.22)
k 2 N0 ;
1 < p < 1:
(3.23)
Hpk .Rn / D Wpk .Rn /; Hps .Rn /
Nowadays one calls Sobolev spaces (sometimes fractional Sobolev spaces) with the classical Sobolev spaces Wpk .Rn / as special cases. (iv) We denote by s C s .Rn / D B1;1 .Rn /;
the H¨older-Zygmund spaces. Let 1 h f .x/ D f .x C h/ f .x/;
s 2 R;
(3.24)
lC1 h f .x/ D 1h lh f .x/;
(3.25)
where x 2 R , h 2 R , l 2 N, be the iterated differences in R . Let 0 < s < m 2 N. Then ˇ ˇ ˇ kf jC s .Rn /km D sup jf .x/j C sup jhjs ˇm (3.26) h f .x/ n
n
n
x2Rn
where the second supremum is taken over all x 2 Rn and h 2 Rn with 0 < jhj 1, are equivalent norms in C s .Rn / (for the continuous representative). s If 1 p; q 1 and s > 0, then Bp;q .Rn / are the classical Besov spaces f .x/. We refer the reader to [T92, which can again be normed with the help of m h Chapter 1] and [T06, Chapter 1] where one finds the history of these spaces, further special cases and classical assertions. Otherwise we assume that the reader is familiar with the theory of the spaces Asp;q .Rn /. But we collect some more recent properties of these spaces which which will be needed later on following again closely [T13, Section 1.2].
3.2 Global spaces
49
3.2.2 Atoms The theory of the local spaces LrAsp;q .Rn / and the hybrid spaces LrAsp;q .Rn / depends decisively on atomic representations and wavelet expansions. This relies on corresponding assertions for the global spaces Asp;q .Rn /. We deal first with atomic representations and shift the wavelet counterpart to Section 3.2.3 below. Otherwise we follow again [T13, Section 1.2.2] which in turn is based on [T08, Section 1.1.2, pp. 4,5]. As far as the history of atoms in function spaces is concerned we refer the reader to [T92, Section 1.9]. A few more recent comments may also be found in [T08, Remark 1.8, p. 5]. We follow the notation in [T13] adapted to our later purposes. We need some sequence spaces, both for atoms and in Section 3.2.3 for wavelets which differ by an additional summation over G 2 G j according to (2.193), (2.194) (repeated below) in case of wavelets. We denote the corresponding wavelet sequence s spaces by ap;q .Rn / and their atomic counterpart, introduced now, by asp;q .Rn /. Let again Qj;m D 2j m C 2j .0; 1/n; j 2 Z; m 2 Zn ; (3.27) be the usual dyadic cubes in Rn , n 2 N, with sides of length 2j parallel to the axes of coordinates and with 2j m as lower left corner. Let j;m be the characteristic function of Qj;m . If Q is a cube in Rn and d > 0 then dQ is the cube in Rn concentric with Q and with side-length d times of the side-length of Q. Definition 3.3. Let 0 < p; q 1 and s 2 R. Let ˚ D jm 2 C W j 2 N0 ; m 2 Zn : Then
˚ b sp;q .Rn / D W k jb sp;q .Rn /k < 1
with k jb sp;q .Rn /k D
1 X
n
2j.s p /q
j D0
and
(3.28)
X
jjm jp
(3.29)
q=p 1=q (3.30)
m2Zn
˚ f sp;q .Rn / D W k jf sp;q .Rn /k < 1
(3.31)
with k jf sp;q .Rn /k D
X
ˇq 1=q ˇ ˇ ˇLp .Rn / 2jsq ˇjm j;m ./ˇ
(3.32)
j 2N0 ;m2Zn
.usual modifications if max.p; q/ D 1/. Remark 3.4. Recall that b sp;p .Rn / D f sp;p .Rn / where s 2 R and 0 < p 1. If an assertion applies both to b sp;q .Rn / and f sp;q .Rn / then we write asp;q .Rn / with s s a 2 fb; f g. Similarly Asp;q .Rn / with A 2 fB; F g stands for Bp;q .Rn / and Fp;q .Rn /.
50
3 Hybrid spaces
Remark 3.5. We add a technical comment. The above definition coincides essentially with [T13, Definition 1.3, p. 5] where we preferred overlapping cubes with 2j m as left lower corners and of side-length 2j C1 . But this is immaterial. Let d > 0 and Ej;m d Qj;m
with
jEj;m j 2j n ;
j 2 N0 ;
m 2 Zn ;
(3.33)
for the Lebesgue measurable sets Ej;m with Lebesgue measure jEj;m j. Let Ej;m be the characteristic function of Ej;m . Then for 0 < p < 1, 0 < q 1, k jf sp;q .Rn /k
X
ˇq 1=q ˇ 2jsq ˇjm Ej;m ./ˇ jLp .Rn /
(3.34)
j 2N0 ;m2Zn
where the equivalence constants in (3.33) are independent of j; m and in (3.34) are independent of according to (3.28). One may consult [T06, Proposition 1.33, p. 19] and the literature mentioned there. Next we recall what is meant by L1 -normalized atoms in Rn , n 2 N. Definition 3.6. Let K 2 N0 , N 2 N0 and d > 1. Then L1 -functions aj;m W Rn 7! C with j 2 N0 , m 2 Zn are called atoms .more precisely .K; N; d /-atoms/ if supp aj;m d Qj;m ; jD ˛ aj;m .x/j 2j j˛j ; Z x ˇ aj;m .x/ dx D 0; Rn
j 2 N0 ;
j˛j K; jˇj < N;
m 2 Zn ;
j 2 N0 ; j 2 N;
m 2 Zn ; m 2 Zn :
(3.35) (3.36) (3.37)
Remark 3.7. If K D 0 then aj;m 2 L1 .Rn / need not to be continuous. If K 2 N then we assume that aj;m has classical continuous derivatives up to order K inclusively. The monomials x ˇ in (3.37) have the same meaning as in (3.7). No cancellation (3.37) for a0;m is required. Furthermore, if N D 0 then (3.37) is empty (no condition). In contrast to [T13, Definition 1.5, p. 5] where we dealt with overlapping cubes we need now d > 1 (excluding d D 1). Let for fixed n 2 N, p D n max p1 ; 1 1 ;
p;q D n max p1 ; q1 ; 1 1
(3.38)
where 0 < p; q 1. Theorem 3.8. (i) Let 0 < p 1, 0 < q 1, s 2 R. Let d > 1 and K 2 N0 , N 2 N0 with (3.39) K>s and N > p s:
51
3.2 Global spaces s Let f 2 S 0 .Rn /. Then f 2 Bp;q .Rn / if, and only if, it can be represented as
f D
1 X X j D0
jm aj;m ;
2 b sp;q .Rn /;
(3.40)
m2Zn
unconditional convergence being in S 0 .Rn /, where aj;m are .K; N; d /-atoms according to Definition 3.6. Furthermore, s kf jBp;q .Rn /k inf k jb sp;q .Rn /k
(3.41)
where the infimum is taken over all admissible representations (3.40). (ii) Let 0 < p < 1, 0 < q 1, s 2 R. Let d > 1 and K 2 N0 , N 2 N0 with K>s
and
N > p;q s:
(3.42)
s .Rn / if, and only if, it can be represented as Let f 2 S 0 .Rn /. Then f 2 Fp;q
f D
1 X X
jm aj;m ;
2 f sp;q .Rn /;
(3.43)
j D0 m2Zn
unconditional convergence being in S 0 .Rn /, where aj;m are .K; N; d /-atoms according to Definition 3.6. Furthermore, s .Rn /k inf k jf sp;q .Rn /k kf jFp;q
(3.44)
where the infimum is taken over all admissible representations (3.43). Remark 3.9. If 2 asp;q .Rn / then (3.40), (3.43) converge unconditionally at least in S 0 .Rn /. If max.p; q/ < 1 then (3.40), (3.43) converge unconditionally in Asp;q .Rn /. The above formulation coincides essentially with [T13, Theorem 1.7, p. 6]. In [T13, Remark 1.8, p. 6] one finds further comments and some references. In contrast to [T13] we deal now with the cubes Qj;m D 2j m C 2j .0; 1/n which do not overlap for fixed j 2 N0 . Then one needs d > 1 in the above theorem (in contrast to d 1 in [T13]). Otherwise one may consult our comments in Remark 3.5.
3.2.3 Wavelets We introduced the local spaces LrAsp;q .Rn / in [T13] in terms of wavelet approximations. This approach will be extended in Section 3.3 below to the hybrid spaces LrAsp;q .Rn /. Again we shall rely on wavelet characterizations of the global spaces Asp;q .Rn /. We give a brief description of what is needed in the sequel following closely [T13, Section 1.2.3, pp. 7–9] which in turn relies on [T10, Section 1.1.4, pp. 8–11]. We suppose that the reader is familiar with wavelets in Rn of Daubechies type and the related multiresolution analysis. The standard references are [Dau92, Mal99,
52
3 Hybrid spaces
Mey92, Woj97]. A short summary of what is needed may also be found in [T06, Section 1.7]. As usual, C u .R/ with u 2 N collects all bounded complex-valued continuous functions on R having continuous bounded derivatives up to order u inclusively. Let u u u 2 N; (3.45) F 2 C .R/; M 2 C .R/; be real compactly supported Daubechies wavelets with Z v for all v 2 N0 with v < u. M .x/ x dx D 0
(3.46)
R
Recall that F is called the scaling function (father wavelet) and M the associated wavelet (mother wavelet). We extend these wavelets from R to Rn by the usual multiresolution procedure. Let G D .G1 ; : : : ; Gn / 2 G 0 D fF; M gn
(3.47)
which means that Gr is either F or M . Let G D .G1 ; : : : ; Gn / 2 G D G j D fF; M gn ;
j 2 N;
(3.48)
which means that Gr is either F of M , where indicates that at least one of the components of G must be an M . Hence G 0 has 2n elements, whereas G j with j 2 N and G have 2n 1 elements. Let j .x/ ‰G;m
D2
j n=2
n Y Gr
j 2 x r mr ;
G 2 Gj ;
m 2 Zn ;
(3.49)
rD1
x 2 Rn , where (now) j 2 N0 . We always assume that F and L2 -norm 1. Then ˚ j W j 2 N0 ; G 2 G j ; m 2 Zn ‰ D ‰G;m
M
in (3.45) have (3.50)
is an orthonormal basis in L2 .Rn / (for any u 2 N) and f D
1 X X X j D0
with
G2G j
Z j;G m
D
j;G m .f
/D2
j n=2 Rn
j n=2 j;G ‰G;m m 2 j
(3.51)
m2Zn
j j f .x/ ‰G;m .x/ dx D 2j n=2 f; ‰G;m j
(3.52)
is the corresponding expansion, adapted to our needs, where 2j n=2 ‰G;m are uniformly bounded functions (with respect to j and m). In [T08], based on [HaT05, Tri04, T06, Tri08], we dealt in detail with an extension of the L2 -theory to spaces of type B and F , with and without weights, on the n-torus Tn , on smooth and rough
3.2 Global spaces
53
domains and on manifolds. In [T13] we based the wavelet theory for the local spaces LrAsp;q .Rn / on corresponding assertions for the global spaces Asp;q .Rn /. This will be extended below to the hybrid spaces LrAsp;q .Rn /. But first we describe the needed wavelet characterization of the global spaces Asp;q .Rn /. For this purpose we have to adapt the sequence spaces introduced in Definition 3.3 to the extra summation over G in (3.51). The characteristic functions j;m of the cubes Qj;m have the same meaning as there. Definition 3.10. Let 0 < p; q 1 and s 2 R. Let ˚ j n D j;G m 2 C W j 2 N0 ; G 2 G ; m 2 Z : Then
˚ s s bp;q .Rn / D W k jbp;q .Rn /k < 1
(3.53) (3.54)
with s .Rn /k D k jbp;q
1 X j D0
and
X X
n
2j.s p /q
G2G j
p jj;G m j
q=p 1=q (3.55)
m2Zn
˚ s s .Rn / D W k jfp;q .Rn /k < 1 fp;q
(3.56)
with s k jfp;q .Rn /k D
X
ˇq 1=q ˇ ˇ ˇ ˇLp .Rn / 2jsq ˇj;G m j;m ./
(3.57)
j 2N0 ;G2G j ; m2Zn
.usual modifications if max.p; q/ D 1/. Remark 3.11. Remark 3.5 applies also to the above definition, in particular to (3.57) with p < 1. s Let p and p;q be as in (3.38). Similarly as in Remark 3.4 we write ap;q .Rn / with a 2 fb; f g. Let ‰ be the wavelet system (3.50) based on (3.45), (3.46) with u 2 N.
Theorem 3.12. (i) Let 0 < p 1, 0 < q 1, s 2 R and u > max.s; p s/:
(3.58)
s .Rn / if, and only if, it can be represented as Let f 2 S 0 .Rn /. Then f 2 Bp;q
f D
X j 2N0 ;G2G j ; m2Zn
j n=2 j j;G ‰G;m ; m 2
s 2 bp;q .Rn /;
(3.59)
54
3 Hybrid spaces
unconditional convergence being in S 0 .Rn /. The representation (3.59) is unique, j j;G j n=2 f; ‰G;m j;G m D m .f / D 2 and
I W
˚ f 7! j;G m .f /
(3.60) (3.61)
s s is an isomorphic map of Bp;q .Rn / onto bp;q .Rn /. (ii) Let 0 < p < 1, 0 < q 1, s 2 R and
u > max.s; p;q s/:
(3.62)
s .Rn / if, and only if, it can be represented as Let f 2 S 0 .Rn /. Then f 2 Fp;q
f D
X
j n=2 j j;G ‰G;m ; m 2
s 2 fp;q .Rn /;
(3.63)
j 2N0 ;G2G j ; m2Zn
unconditional convergence being in S 0 .Rn /. The representation (3.63) is unique with s s (3.60). Furthermore I in (3.61) is an isomorphic map of Fp;q .Rn / onto fp;q .Rn /. Remark 3.13. This is the wavelet counterpart of Theorem 3.8. It coincides essentially with [T13, Theorem 1.10, pp. 8,9], which, in turn, is based on [T10, Theorem 1.18, pp. 10,11]. This theorem and in particular the sharp restrictions for u in (3.58), (3.62) have some history. One may consult [T13, Remark 1.11, p. 9]. There one finds also references about some technical points concerning of the convergence j s n (3.59), (3.63) and how to understand the dual pairing f; ‰G;m if f 2 Ap;q .R / with A 2 fB; F g. This will not be repeated here.
3.2.4 Multiplication algebras Recall that Asp;q .Rn / is called a multiplication algebra if f1 f2 2 Asp;q .Rn / for any f1 2 Asp;q .Rn /, f2 2 Asp;q .Rn / and if there is a constant c > 0 such that kf1 f2 jAsp;q .Rn /k c kf1 jAsp;q .Rn /k kf2 jAsp;q .Rn /k
(3.64)
for all f1 2 Asp;q .Rn /, f2 2 Asp;q .Rn /. Of course one has to say what is meant by the product f1 f2 in the context of S 0 .Rn /. All this has been done with great care spanning a period of several decades beginning with the late 1970s and is connected with what is called paramultiplication. This will not be repeated here. One may consult [T13, Section 1.2.5, pp. 12, 13]. There we commented on this sophisticated question supported by related references. We take over the final outcome without further discussions, mainly to have an easy reference later on. Theorem 3.14. Let 0 < p; q 1 .p < 1 for the F -spaces/ and s 2 R. Then the following assertions are pairwise equivalent:
3.2 Global spaces
(a) Asp;q .Rn / is a multiplication algebra. (b) s > 0 and Asp;q .Rn / ,! L1 .Rn /. (c) Either ( s > n=p where 0 < p; q 1; s n s n Ap;q .R / D Bp;q .R / with s D n=p where 0 < p < 1, 0 < q 1,
55
(3.65)
or Asp;q .Rn /
D
s Fp;q .Rn /
( s > n=p with s D n=p
where 0 < p < 1, 0 < q 1; (3.66) where 0 < p 1, 0 < q 1:
Remark 3.15. This final version goes back to [SiT95]. It coincides with [T13, Section 1.2.5, pp. 12, 13]. There one finds also further references and comments, including in particular [RuS96, pp. 221, 222, 258] for a more detailed description of the history of this problem.
3.2.5 Spaces on domains In [T13] we dealt with the global spaces Asp;q .Rn / and the local spaces LrAsp;q .Rn /. We described in the above Chapter 1, Introduction, our motivations to extend these considerations now to hybrid spaces LrAsp;q .Rn /. All these spaces are defined on Rn . But both LrAsp;q .Rn / and LrAsp;q .Rn / combine global aspects with localizations or Morreyfications in the spirit of Definition 2.1 where we introduced the Morrey spaces Lrp .Rn / and Lrp .Rn /. Roughly speaking we replace Lp .QJ;M / in (2.2), (2.3) by Asp;q .QJ;M / combined with some shift space approximations in terms of wavelets according to (3.50). In any case one needs some information about the restrictions of Asp;q .Rn / to domains in Rn . We have nothing new to say about global spaces on domains and take over some material from [T13, pp. 10, 11] which in turn is based on the references given there, in particular to [T08, pp. 28, 29]. Let be a domain in Rn , n 2 N. Domain means open set. As usual, D. / D 1 C0 . / stands for the collection of all complex-valued infinitely differentiable functions in Rn with compact support in . Let D 0 . / be the dual spaces of all distributions in . We assume that the reader is familiar with the theory of distributions in the usual extent as it may be found, for example, in [HT08]. Let g 2 S 0 .Rn /. Then we denote by gj its restriction to , gj 2 D 0 . / W
.gj /.'/ D g.'/ for ' 2 D. /:
(3.67)
With A D B or A D F the spaces Asp;q .Rn / have the same meaning as in Definition 3.1. Definition 3.16. Let be an arbitrary domain in Rn with 6D Rn and let 0 < p 1, 0 < q 1, s 2 R with p < 1 for the F -spaces.
56
3 Hybrid spaces
(i) Then ˚ Asp;q . / D f 2 D 0 . / W f D gj for some g 2 Asp;q .Rn / ; kf
jAsp;q . /k
D
inf kg jAsp;q .Rn /k;
where the infimum is taken over all g 2 Asp;q .Rn / with gj D f . (ii) Let ˚ e Asp;q . / D f 2 Asp;q .Rn / W supp f :
(3.68) (3.69)
(3.70)
Then ˚ e Asp;q . / ; Asp;q . / D f 2 D 0 . / W f D gj for some g 2 e kf je Asp;q . /k D inf kg jAsp;q .Rn /k;
(3.71) (3.72)
where the infimum is taken over all g 2 e Asp;q . / with gj D f . Remark 3.17. Part (i) is the usual definition of Asp;q . / by restriction. The spaces e Asp;q . / are closed subspaces of Asp;q .Rn /. One has a one-to-one correspondence between the subspace e Asp;q . / of S 0 .Rn / and the subspace e Asp;q . / of D 0 . /, written in a somewhat sloppy way as ˚ e Asp;q . /; if, and only if, h 2 Asp;q .Rn / W supp h @ D f0g: Asp;q . / D e (3.73) Here @ is the boundary of , hence @ D n . This is the case if is a bounded Lipschitz domain, 0 < p; q 1 and s > p according to (3.38) since j@ j D 0 n and Asp;q .Rn / Lloc 1 .R /. If is not a Lipschitz domain then some curiosities may happen which we discussed in [T08, p.29]. In our context we need the spaces Asp;q .QJ;M / with J 2 Z (hybrid spaces) or J 2 N0 (local spaces) and M 2 Zn where again QJ;M D 2J M C 2J .0; 1/n are the above dyadic cubes. But we are in a comfortable position. The behaviour of an element f of LrAsp;q .Rn / or of LrAsp;q .Rn / in a given cube QJ;M is totally determined by the restriction of f to neighbouring cubes QJ;M 0 with jM M 0 j c (independent of J ). In particular there is no need to look at such elements in isolated cubes QJ;M for the spaces introduced in the above definition.
3.3 Definitions and basic properties 3.3.1 Definitions We come to the main topic of Chapter 3, the study of the hybrid spaces LrAsp;q .Rn /. As a preparation we collected so far in Section 3.2 some assertions about the global spaces Asp;q .Rn / needed later on. On the other hand some parts of the theory of the local spaces LrAsp;q .Rn / as developed in [T13] can be carried over to the hybrid
57
3.3 Definitions and basic properties
spaces LrAsp;q .Rn / essentially without any additional efforts. This applies to the present Section 3.3 and to some characterizations of these spaces in terms of atoms and wavelets considered in Section 3.4 below. Otherwise one may consult Section 3.1, Introduction, where we described the situation and the plan of this Chapter 3. First we repeat, complement and adapt what had been said so far in Section 3.2.3 about wavelets. There one finds also the standard references about wavelet theory underlying what follows. Recall that C u .R/ with u 2 N collects all bounded complexvalued continuous functions on R having continuous bounded derivatives up to order u inclusively. Let 2 C u .R/;
F
M
2 C u .R/;
be real compactly supported Daubechies wavelets with Z v for all v 2 N0 M .x/ x dx D 0
u 2 N;
v < u;
with
(3.74)
(3.75)
R
and
k
Let and
jL2 .R/k D 1:
(3.76)
G D .G1 ; : : : ; Gn / 2 G 0 D fF; M gn
(3.77)
F
jL2 .R/k D k
M
G D .G1 ; : : : ; Gn / 2 G D G j D fF; M gn ;
j 2 N;
be as in (3.47), (3.48). Then ˚ j W j 2 N0 ; G 2 G j ; m 2 Zn ‰ D ‰G;m
(3.78)
(3.79)
with j .x/ ‰G;m
D2
j n=2
n Y
Gr
2j xr mr /;
j 2 N0 ;
G 2 Gj ;
m 2 Zn ;
rD1
(3.80)
is an orthonormal basis in L2 .Rn /. Let J 2 N0 and J ‰m .x/
D2
J n=2
n Y F
J 2 x r mr ;
m 2 Zn ;
(3.81)
rD1
be based on the scaled orthonormal father wavelets F D F ;u according to (3.74) indicating now u 2 N. If J 2 Z then J C D max.J; 0/. It follows from the multiresolution analysis of wavelet theory that for any J 2 Z, ˚ JC ˚ j ‰m W m 2 Zn ˚ ‰G;m W J C j 2 N0 ; G 2 G ; m 2 Zn (3.82) is an orthonormal basis in L2 .Rn /. Here G has the same meaning as in (3.78). Let ˚ J ‰ ‰ D span ‰m W m 2 Zn if J 2 N0 ; VJ;u D f0g if J 2 N ; (3.83) VJ;u
58
3 Hybrid spaces
with N D N D fJ 2 Z W J 2 Ng where ‰ based on F D F ;u refers to (3.79). Again it follows from the multiresolution analysis for the above wavelets that ‰ VJ;u VJ‰C1;u ;
J 2 Z:
(3.84)
Let Pk with k 2 N0 be the collection of all polynomials in Rn of degree less than or ‰ D f0g. Then equal to k. Let P1 D f0g and VJ;0 ‰ Pu1 VJ;u ;
u 2 N0 ;
J 2 N0 ;
follows from the polynomial reproducing formula X J J J 2 N0 ; P; ‰m ‰m .x/; P .x/ D
(3.85)
x 2 Rn ;
(3.86)
m2Zn
P 2 Pu1 , with pointwise convergence or, better, convergence in some weighted s -type. We refer the reader to [T08, Section 6.5.1, Theorem 6.83, spaces of Bp;q p. 237]. This observation will be of some service for us later on. Recall that aC D max.a; 0/ if a 2 R and for 0 < p; q 1, p D n max p1 ; 1 1 and p;q D n max p1 ; q1 ; 1 1 : (3.87) Let Asp;q . / be the spaces as introduced in Definition 3.16. Let QJ;M D 2J M C 2J .0; 1/n, J 2 Z, M 2 Zn , be the same dyadic cubes as in (3.27) and let d QJ;M be as explained there. Definition 3.18. Let 0 < p; q 1 .p < 1 for F -spaces/; s 2 R and n=p r < ‰ with k 2 N0 be as in (3.83) with u D k C 1, based on F ;kC1 . Let 1. Let VJ;kC1 ( kC1>
B-spaces; max.s C r C ; p s/; max.s C r C ; p;q s/; F -spaces:
(3.88)
Then LrAsp;q .Rn /‰ collects all f 2 S 0 .Rn / such that kf jLrAsp;q .Rn /‰ k D
sup
n
2J. p Cr/
J 2Z;M 2Zn
inf
‰ g2VJ;kC1
kf g jAsp;q .2QJ;M /k (3.89)
is finite. Let LV rAsp;q .Rn /‰ be the completion of D.Rn / in LrAsp;q .Rn /‰ . Remark 3.19. First we remark that (3.89) makes sense for fixed ‰ (and hence fixed k 2 N0 ): If J 2 N0 , M D 0 and J ! 1 then it follows from (3.84) that the second factor in n 2J. p Cr/ inf kf g jAsp;q .2QJ;0 /k (3.90) ‰ g2VJ;kC1
59
3.3 Definitions and basic properties
decreases whereas the first factor increases. Similarly if J 2 N (hence J 2 N), M D 0 and J ! 1 then the second factor in n
2J. p Cr/ kf jAsp;q .2QJ;0 /k
(3.91)
increases whereas the first factor decreases. If 1 < p < 1, n=p r < 0 and 0 Asp;q .Rn / D Fp;2 .Rn / D Lp .Rn / then one can compare Lr Lp .Rn / with the Morrey r n space Lp .R / as introduced in Definition 2.1. In particular, kf jLr Lp .Rn /‰ k kf jLrp .Rn /k;
(3.92)
where any k 2 N0 in (3.88) can be admitted. As indicated in (2.40) one has in (3.92) equivalence, hence 0 Lrp .Rn / D Lr Lp .Rn / D Lr Fp;2 .Rn /;
n=p r < 0: (3.93) But this is not covered by the above definition even if one accepts that the spaces LrAsp;q .Rn /‰ are independent of admitted ‰ (as will be justified in Section 3.3.2 below). However it is an easy consequence of a corresponding assertion for related local spaces and may also be found in Remark 3.67 below. But in any case it is now quite clear that one can call LrAsp;q .Rn /‰ a Morreyfication of Asp;q .Rn /. Furthermore one can compare the above approximation by admitted wavelet systems ‰ with polynomial approximations. Let again Pk with k 2 N0 be the collection of all polynomials in Rn of degree less than or equal to k, complemented by P1 D f0g. Let N1 D N0 [ f1g and as before N D N. Let temporarily LrAsp;q .Rn /k with k 2 N1 and the other parameters as in the above definition be the collection of all f 2 S 0 .Rn / such that kf jLrAsp;q .Rn /k k D
sup J 2N0 ;M 2Zn
C
1 < p < 1;
n
2J. p Cr/ inf kf P jAsp;q .2QJ;M /k
sup
P 2Pk
2
n J. p Cr/
J 2N ;M 2Zn
(3.94)
kf jAsp;q .2QJ;M /k
is finite. We discussed in [T13, pp. 41, 42] polynomial approximations of this type in the context of local spaces. This can be transferred to hybrid spaces. However this will not yet be done here in detail. But we return to this point in Step 1 of the proof of Theorem 3.41 below. The spaces LrAsp;q .Rn /k are monotone with respect to k 2 N1 . Then it follows from (3.85) and suitably chosen ‰ that kf jLrAsp;q .Rn /‰ k c kf jLrAsp;q .Rn /k k; Using (3.93) and (2.3) (this means k D 1 in (3.94) with obtains Lrp .Rn / D Lr Lp .Rn /k ;
1 < p < 1;
k 2 N1 : Asp;q .Rn /
n=p r < 0;
(3.95)
D Lp .R /) one n
k 2 N1 : (3.96)
In other words, additional polynomial approximation gives the same Morrey spaces Lrp .Rn / (equivalent norms). This observation is some 50 years old and goes back to S. Campanato, [Cam64].
60
3 Hybrid spaces
3.3.2 Basic properties Let p; q; s; r and also ‰ with k 2 N0 be restricted by (3.88) as in Definition 3.18. Then the local space LrAsp;q .Rn /‰ collects all f 2 S 0 .Rn / such that kf jLrAsp;q .Rn /‰ k D sup kf jAsp;q .2Q0;M /k M 2Zn
C
sup
n
2J. p Cr/
J 2N;M 2Zn
inf
‰ g2VJ;kC1
kf g jAsp;q .2QJ;M /k
(3.97) if finite. This coincides essentially with [T13, Definition 1.26, p. 20], where we used the cubes 2J M C 2J C1 .0; 1/n to guarantee the necessary overlap for fixed J 2 N0 instead of 2QJ;M . But this is immaterial. We add a further comment about the replacement of 2QJ;M by 2J M C2J C1 .0; 1/n and even 2J M C2J CJ0 .0; 1/n for some J0 2 N later on in connection with wavelet characterizations of LrAsp;q .Rn /‰ in Section 3.4.1 below. One has LrAsp;q .Rn /‰ ,! LrAsp;q .Rn /‰ ; kf jLrAsp;q .Rn /‰ k kf jLrAsp;q .Rn /‰ kC
sup J 2N ;M 2Zn
(3.98) n
2J. p Cr/ kf jAsp;q .2QJ;M /k:
(3.99) The minor modification as far as the terms with 2Q0;M are concerned is again immaterial. But (3.99) makes clear that some basic properties for the local spaces can be transferred to the hybrid spaces without any additional efforts. This applies in particular to the independence of these spaces of ‰ (and hence of k), the completeness, some embeddings, but also to wavelet and atomic characterizations. Whereas wavelet and atomic representations are shifted to Section 3.4 we collect now some basic properties. We stick to our minimal request (2.16). In other words, all local and hybrid spaces X.Rn / introduced above quasi-Banach spaces in the should be distributional framework of the dual pairing S.Rn /; S 0 .Rn / with the continuous embedding S.Rn / ,! X.Rn / ,! S 0 .Rn /:
(3.100)
Let again C s .Rn /, s 2 R, be the H¨older-Zygmund spaces according to (2.55). Recall that ,! means continuous embedding. Theorem 3.20. (i) Let 0 < p; q 1 .p < 1 for the F -spaces/; s 2 R and n=p r < 1. Then the spaces LrAsp;q .Rn /‰ according to Definition 3.18 are independent of the wavelet system ‰ in (3.79), based on (3.74)–(3.78) with u D k C1 as in (3.88) .equivalent quasi-norms/. They are quasi-Banach spaces. Furthermore S.Rn / ,! LrAsp;q .Rn /‰ ,! LrAsp;q .Rn /‰ ,! C sCr .Rn / ,! S 0 .Rn /:
(3.101)
(ii) Let 0 < p; q 1 .p < 1 for the F -spaces/ and s 2 R. Then Ln=p Asp;q .Rn /‰ D Asp;q .Rn /:
(3.102)
3.3 Definitions and basic properties
61
Proof. Step 1. According to [T13, Corollary 1.38, p. 32] and the comments in [T13, p. 21] the corresponding local spaces LrAsp;q .Rn /‰ are quasi-Banach spaces which are independent of admitted wavelet systems ‰. Then it follows from (3.99) that also the hybrid spaces LrAsp;q .Rn /‰ are quasi-Banach spaces which are independent of admitted wavelet systems ‰. The completeness of LrAsp;q .Rn /‰ is also a consequence of Theorem 3.26 below where we prove the LrAsp;q .Rn /‰ is isomorphic to some quasi-Banach sequence space. Step 2. The second embedding in (3.101) follows again from (3.99) whereas the subsequent embeddings for the local spaces are covered by [T13, Theorem 2.1, p. 45]. We prove S.Rn / ,! LrAsp;q .Rn /‰ (3.103) which we apparently forgot to state explicitly in [T13] although it is an easy consequence of some properties of the local spaces LrAsp;q .Rn /‰ derived there. Let be a domain in Rn and let C. / be the restriction of the space C.Rn / of all bounded continuous functions on Rn to . If s < 0 and 0 < p; q 1 (p < 1 for the F -spaces) then C.2QJ;M / ,! C 0 .2QJ;M / ,! Asp;q .2QJ;M /;
J 2 N0 ;
M 2 Zn ; (3.104)
uniformly in J and M , [T83, Theorem 3.3.1, pp. 196/197] and elementary embeddings. (Uniformly means that the related embedding constants can be chosen independently of J 2 N0 and M 2 Zn ). Let ' 2 S.Rn /. Then it follows from the Taylor expansion of ' at the off-point 2J M and a suitable choice of the polynomial P 2 Pk that k' P jAsp;q .2QJ;M /k c
sup
j'.x/ P .x/j c 0 2J.kC1/ ;
(3.105)
x22QJ;M
where c and c 0 can be estimated from above by c 00 supx2Rn ;0j˛jkC1 jD ˛ '.x/j independently of J 2 N0 and M 2 Zn . One has by (3.85) n
2J. p Cr/
inf
‰ g2VJ;kC1
n
k'g jAsp;q .2QJ;M /k c 2J. p Crk1/
sup
jD ˛ '.x/j
x2Rn ;0j˛jkC1
(3.106) where c is independent of J , M , and '. One can choose k 2 N sufficiently large. This proves (3.103). Let Iı , ı 2 R, b _; f 2 S 0 .Rn /; (3.107) Iı f D .1 C jj2 /ı=2 f be the well-known lift, mapping S.Rn / onto itself and S 0 .Rn / onto itself. According to [T13, Theorem 2.39, p. 88] one has the isomorphic map n Iı LrAsp;q .Rn /‰ D LrAsCı p;q .R /‰
(3.108)
for all admitted parameters. This shows that one can extend (3.103) from s < 0 to all s 2 R by lifting. Then the first embedding in (3.101) follows from (3.103), (3.99) and n sup 2J. p Cr/ k' jAsp;q .2QJ;M /k ck' jAsp;q .Rn /k (3.109) J 2N ;M 2Zn
62
3 Hybrid spaces
using J < 0 and pn C r 0. A similar argument may be found in [T13, p. 43]. Step 3. We prove (3.102). If f 2 Asp;q .Rn / then one has kf jLn=p Asp;q .Rn /‰ k
sup J 2Z;M 2Zn
kf jAsp;q .2QJ;M /k kf jAsp;q .Rn /k:
Conversely, if f 2 Ln=p Asp;q .Rn / then f 2 Asp;q .Rn / follows from kf jAsp;q .Rn /k c sup kf jAsp;q .2QJ;0 k;
(3.110) (3.111)
J 2N
where we used the Fatou property of the spaces Asp;q .Rn / which we recall just now in Remark 3.21 below. s n Remark 3.21. Let fgj g1 j D1 be a bounded sequence in Ap;q .R / with
gj ! g
in
S 0 .Rn /
if j ! 1:
(3.112)
Then g 2 Asp;q .Rn / and there is a positive constant c (depending only on Asp;q .Rn / and the chosen quasi-norm) such that kg jAsp;q .Rn /k c sup kgj jAsp;q .Rn /k:
(3.113)
j 2N
This is the so-called Fatou property of the spaces Asp;q .Rn /. The notation Fatou property goes back to [Fra86] in a wider context. One may also consult [RuS96, p. 15]. We justify (3.111). Let 2 S.Rn / be a cut-off function, say, .x/ D 1 if jxj 1 and .y/ D 0 if jyj 2. Let f 2 Ln=p Asp;q .Rn /. Then .2j /f 2 Asp;q .Rn / and by pointwise multiplier properties k .2j /f jAsp;q .Rn /k c kf jAsp;q .2QJ;0 /k if J is sufficiently large. Since (3.113). This justifies (3.111).
(3.114)
.2j /f ! f in S 0 .Rn / one can apply (3.112),
Notation 3.22. Both the local spaces LrAsp;q .Rn /‰ and by Theorem 3.20 the hybrid spaces LrAsp;q .Rn /‰ are independent of ‰. This justifies denoting these spaces in the sequel as
LrAsp;q .Rn / and LrAsp;q .Rn /;
0 < p; q 1;
s 2 R;
n=p r < 1 (3.115) (p < 1 for the F -spaces). This extends [T13, Notation 1.39, p. 32] from local spaces to hybrid spaces. Remark 3.23. The smoothness assumptions (3.88) for the underlying wavelets in Definition 3.18 are the same as the natural restrictions for the global spaces Asp;q .Rn / in Theorem 3.12 with u D k C 1 if r 0. If r > 0 then one has the additional term r in (3.88). But this extra smoothness is again natural. We discussed this point in [T13, Remark 1.28, p. 21].
63
3.4 Characterizations
3.4 Characterizations 3.4.1 Wavelet characterizations In [T13, Section 1.3.2, pp. 22–26] we characterized the local spaces LrAsp;q .Rn / as introduced in (3.97) and Notation 3.22 in terms of wavelets. Using (3.99) one can extend these assertions more or less immediately to the hybrid spaces LrAsp;q .Rn / according to Definition 3.18, Theorem 3.20 and Notation 3.22. We prefer now a few minor technical modifications which will be mentioned and discussed below. First we adapt some sequence spaces underlying wavelet characterizations for local spaces to the hybrid situation. Let again Qj;m D 2j m C 2j .0; 1/n be the dyadic cubes according to (3.27), where j 2 Z and m 2 Zn . Let j;m be the characteristic function of Qj;m . Let G j with j 2 N0 be as in (3.47), (3.48). Let J C D max.J; 0/ if J 2 Z, ˚ J 2 Z; M 2 Zn ; PJ;M D j J C; G 2 G j ; m 2 Zn W Qj;m QJ;M ; (3.116) and ˚ PJ;M D j J C; m 2 Zn W Qj;m QJ;M ; J 2 Z; M 2 Zn : (3.117) Of course, j 2 N0 in (3.116), (3.117). Definition 3.24. Let 0 < p; q 1, s 2 R and r n=p. Let ˚ j n D j;G m 2 C W j 2 N0 ; G 2 G ; m 2 Z : Then
˚ s s .Rn / D W k jLr bp;q .Rn /k < 1 Lr bp;q
(3.118) (3.119)
with s k jLr bp;q .Rn /k
D
n
2J. p Cr/
sup
X 1
J 2Z;M 2Zn
X
j DJ C G2G j
n
2j.s p /q
X
p jj;G m j
q=p 1=q
mW.j;m/2PJ;M
(3.120) and
˚ s s Lrfp;q .Rn / D W k jLrfp;q .Rn /k < 1
(3.121)
with s .Rn /k k jLrfp;q
D
sup J 2Z;M 2Zn
n 2J. p Cr/
X .j;G;m/2PJ;M
.usual modification if max.p; q/ D 1/.
ˇq 1=q ˇ ˇ n (3.122) ˇ ˇ 2jsq ˇj;G ./ .R / L j;m p m
64
3 Hybrid spaces
s s Remark 3.25. As usual Lr ap;q .Rn / with a 2 fb; f g stands for Lr bp;q .Rn / and s Lrfp;q .Rn /. In [T13, Definition 1.30, pp. 22/23] we introduced the local counterparts s Lrap;q .Rn / for the same sequences as in (3.118) but with J 2 N0 instead of J 2 Z in (3.120), (3.122), hence s .Rn /k k jLrbp;q
D
2
sup
n J. p Cr/
X 1 X
J 2N0 ;M 2Zn
2
n j.s p /q
j DJ G2G j
X
p jj;G m j
q=p 1=q
mW.j;m/2PJ;M
(3.123) and s .Rn /k k jLrfp;q
D
sup J 2N0 ;M 2Zn
n 2J. p Cr/
ˇq 1=q ˇ ˇ (3.124) ˇ ˇLp .Rn / 2jsq ˇj;G ./ j;m m
X .j;G;m/2PJ;M
(usual modification if max.p; q/ D 1). In [T13] we preferred in (3.124) characteristic functions of overlapping cubes, typically of type 2J M C 2J C1 .0; 1/n or even 2J M C 2J CJ0 .0; 1/n for some J0 2 N. But this is immaterial as has been mentioned in Remark 3.5 if p < 1 (which will always be assumed in case of F -spaces, at least in the context of the above sequence spaces). s Let LrAsp;q .Rn / with A 2 fB; F g and Lr ap;q .Rn / with a 2 fb; f g be the spaces as introduced in Definitions 3.18, 3.24 and Notation 3.22.
Theorem 3.26. Let 0 < p; q 1 .p < 1 for the F -spaces/; s 2 R and n=p r < 1. Let ‰ be the wavelet system (3.79) based on (3.74)–(3.78) with u D k C 1 as in (3.88). Let f 2 S 0 .Rn /. Then f 2 LrAsp;q .Rn / if, and only if, it can be represented as f D
X
j n=2 j;G ‰G;m ; m 2 j
s 2 Lr ap;q .Rn /;
(3.125)
j 2N0 ;G2G j ; m2Zn
unconditional convergence being in S 0 .Rn /. The representation (3.125) is unique, j j;G j n=2 j;G f; ‰G;m m D m .f / D 2
(3.126)
and I W is an isomorphic map of
LrAsp;q .Rn /
f 7! fj;G m .f /g onto
(3.127)
s Lr ap;q .Rn /,
s kf jLrAsp;q .Rn /k k.f / jLr ap;q .Rn /k:
(3.128)
3.4 Characterizations
65
Proof. This is the hybrid counterpart of [T13, Theorem 1.32, p. 24] where we proved a corresponding assertion for the local spaces LrAsp;q .Rn /. This covers the first terms on the right-hand side of (3.99), in particular the second terms on the right-hand side of (3.97). At the end of Step 1 of the proof in [T13, p. 25] we remarked that the first terms on the right-hand side of (3.97) can be incorporated having in mind that ‰ V0;kC1 D f0g (in the notation preferred there, which differs from (3.83) just by terms with J D 0). But this applies also to the second terms in (3.99). Then one obtains the counterpart of [T13, (1.162), p. 25], s k.f / jLr ap;q .Rn /k
sup
2
X
n J. p Cr/
J 2Z;M 2Zn
j;G m .f
/2
j n=2
j ‰G;m jAsp;q .Rn /
(3.129)
.j;G;m/2PJ;M
c kf jLrAsp;q .Rn /k: The proof of the converse is essentially the same as in [T13, pp. 25, 26]. We add a technical comment about the cubes used in [T13] and above. For any wavelet system ‰ there is a number J0 2 N such that j supp ‰G;m 2J0 QJ;M
if .j; G; m/ 2 PJ;M :
(3.130)
But this is the substitute of [T13, (1.137), p. 22] which ensures (3.129).
Remark 3.27. We add a further technical comment. If f 2 LrAsp;q .Rn / then it follows from (3.101) that f admits at least a wavelet expansion according to (3.59) sCr with 2 b1;1 .Rn / and u > js C rj in (3.58). But this is not really satisfactory as far as the minimal smoothness assumptions (3.88) for the underlying wavelets are concerned. One obtains more natural minimal restrictions for the smoothness of the wavelets involved if one embeds LrAsp;q .Rn /, and hence also LrAsp;q .Rn /, into weighted Besov spaces Bp;p .Rn ; w / with < s and some weights w .x/ D 2 =2 .1 C jxj / , 2 R. We discussed this point in some detail in [T13, pp. 23, 24] with a reference to [HaT05], [T06, Chapter 6] and [T08, Section 1.2.3]. We do not go into further detail. It is useful to fix (3.129) and its inverse. Corollary 3.28. Let 0 < p; q 1 .p < 1 for the F -spaces/; s 2 R and n=p r < 1. Let ‰ be the wavelet system (3.79) based on (3.74)–(3.78) with u D k C 1 as in (3.88). Let j;G m .f / be as in (3.126). Then kf jLrAsp;q .Rn /k
sup J 2Z;M 2Zn
2
X
n J. p Cr/
.j;G;m/2PJ;M
are equivalent quasi-norms.
j;G m .f
/2
j n=2
j ‰G;m jAsp;q .Rn /
(3.131)
66
3 Hybrid spaces
Proof. This follows from (3.129) and its converse, the hybrid counterpart of [T13, (1.166), p. 26]. Remark 3.29. This is essentially the hybrid version of [T13, Corollary 1.33, p. 26]. Remark 3.30. We fix the local counterpart of Theorem 3.26 and Corollary 3.28. Let s Lrap;q .Rn / be the local sequence spaces quasi-normed by (3.123), (3.124). Then s Theorem 3.26 with LrAsp;q .Rn / in place of LrAsp;q .Rn / and Lrap;q .Rn / in place of r s n L ap;q .R / remains valid for the same parameters p; q; s; r and u D k C 1 as there. This coincides essentially with [T13, Theorem 1.32, p. 24] and the indicated technical modifications as far as the underlying cubes are concerned. Similarly Corollary 3.28 with LrAsp;q .Rn / in place of LrAsp;q .Rn / and J 2 N0 in place of J 2 Z in (3.131). This coincides essentially with [T13, Corollary 1.33, p. 26].
3.4.2 Atomic characterizations We extend the atomic representation Theorem 3.8 for the global spaces Asp;q .Rn / to the hybrid spaces LrAsp;q .Rn /. We rely on corresponding assertions for the local spaces LrAsp;q .Rn / according to [T13, Section 1.3.3, pp. 27–32]. One needs the hybrid version of the sequence spaces asp;q .Rn / as introduced in Definition 3.3. Let again j;m be the characteristic function of the cube Qj;m D 2j mC2j .0; 1/n with j 2 Z and m 2 Zn . Let PJ;M be as in (3.117). Recall J C D max.J; 0/ if J 2 Z. Definition 3.31. Let 0 < p; q 1, s 2 R and r n=p. Let
Then
D fjm 2 C W j 2 N0 ; m 2 Zn g:
(3.132)
Lr b sp;q .Rn / D f W k jLr b sp;q .Rn /k < 1g
(3.133)
with k jLr b sp;q .Rn /k D
sup
2
n J. p Cr/
J 2Z;M 2Zn
and
X 1
2
n j.s p /q
j DJ C
X
jjm jp
q=p 1=q
(3.134)
mW.j;m/2PJ;M
Lr f sp;q .Rn / D f W k jLr f sp;q .Rn /k < 1g
(3.135)
with k jLr f sp;q .Rn /k D
sup J 2Z;M 2Zn
n 2J. p Cr/
X .j;m/2PJ;M
.usual modification if max.p; q/ D 1/.
2jsq jjm j;m ./jq
1=q
jLp .Rn /
(3.136)
67
3.4 Characterizations
Remark 3.32. As usual Lr asp;q .Rn / with a 2 fb; f g stands for Lr b sp;q .Rn / and Lr f sp;q .Rn /. In [T13, Definition 1.35, p. 27] we introduced the local counterpart Lr asp;q .Rn / for the same sequences as in (3.132) but with J 2 N0 instead of J 2 Z in (3.134) and (3.136), hence k jLr b sp;q .Rn /k D
2
sup
n J. p Cr/
J 2N0 ;M 2Zn
X 1
2
n j.s p /q
j DJ
X
jjm jp
q=p 1=q
(3.137)
mW.j;m/2PJ;M
and k jLr f sp;q .Rn /k D
sup J 2N0
;M 2Zn
n 2J. p Cr/
X
2jsq jjm j;m ./jq
1=q
jLp .Rn / :
(3.138)
.j;m/2PJ;M
In [T13] we dealt with overlapping cubes. But this is immaterial as has been discussed in Remark 3.5 (with p < 1 in case of F -spaces what will always be assumed, at least in the context of the above sequence spaces). We rely on .K; N; d /-atoms as introduced in Definition 3.6. Recall again r C D max.r; 0/, r 2 R, and for 0 < p; q 1, and p;q D n max p1 ; q1 ; 1 1 : (3.139) p D n max p1 ; 1 1 Theorem 3.33. Let 0 < p; q 1 .p < 1 for the F -spaces/; s 2 R, and n=p r < 1. Let K 2 N0 , N 2 N0 , d > 1 and ( p s; B-spaces; C N > (3.140) K >sCr ; p;q s; F -spaces: Let f 2 S 0 .Rn /. Then f 2 LrAsp;q .Rn / if, and only if, it can be represented as f D
1 X X
jm aj;m;
2 Lr asp;q .Rn /;
(3.141)
j D0 m2Zn
where aj;m are .K; N; d /-atoms, unconditional convergence being in S 0 .Rn /. Furthermore, kf jLrAsp;q .Rn /k inf k jLr asp;q .Rn /k (3.142) .equivalent quasi-norms/ where the infimum is taken over all admissible representations (3.28).
68
3 Hybrid spaces
Proof. This is the hybrid counterpart of the atomic representation of the local spaces LrAsp;q .Rn / according to [T13, Theorem 1.37, pp. 28–32]. The proof given there does not only apply to the first terms on the right-hand side of (3.99), but essentially also to the second terms on the right-hand side of (3.99) (in obvious modifications of corresponding arguments related to the first term of the right-hand side of (3.97)). Remark 3.34. We fix the local counterpart of the above theorem. Let Lr asp;q .Rn / be the local sequence spaces quasi-normed by (3.137), (3.138). Then Theorem 3.33 with LrAsp;q .Rn / in place of LrAsp;q .Rn / and Lr asp;q .Rn / in place of Lr asp;q .Rn / remains valid for the same parameters p; q; s; r as there. This coincides essentially with [T13, Theorem 1.37, p. 28]. Remark 3.35. After the atomic representation theorem for the local spaces LrAsp;q .Rn / had been established we switched in [T13, Notation 1.39, p. 32] notationally from LrAsp;q .Rn /‰ to LrAsp;q .Rn / (because it is ensured that LrAsp;q .Rn /‰ is independent of ‰). One could do the same now for the hybrid spaces LrAsp;q .Rn /‰ . But we justified this simplification already in Notation 3.22 above.
3.4.3 Fourier-analytical characterizations We have already mentioned in (3.4) that the hybrid spaces LrAsp;q .Rn / coincide with n the Fourier-analytically defined spaces As; p;q .R /. We have now a closer look at this relation and its local counterpart. We use the same notation as introduced in Section 3.2.1. Let again '0 2 S.Rn / with '0 .x/ D 1 if jxj 1
and '0 .y/ D 0 if jyj 3=2;
(3.143)
and let 'k .x/ D '0 .2k x/ '0 .2kC1 x/;
x 2 Rn ;
k 2 N:
(3.144)
Since 1 X
'j .x/ D 1
for x 2 Rn ;
(3.145)
j D0
the 'j form a dyadic resolution of unity. The global spaces Asp;q .Rn /, A 2 fB; F g, are introduced in Definition 3.1 in terms of the Fourier-analytical decompositions (3.13), (3.15). This will now be localized by the typical Morrey procedure according to Definition 2.1. Let again QJ;M D 2J M C 2J .0; 1/n, J 2 Z, M 2 Zn . Let as before J C D max.J; 0/ if J 2 Z. Definition 3.36. Let ' D f'j g1 j D0 be the above resolution of unity. Let 0 < p; q 1 .p < 1 for F -spaces/; s 2 R and 0 < 1.
69
3.4 Characterizations s; (i) Then Bp;q .Rn / is the collection of all f 2 S 0 .Rn / such that X 1=q s; b/_ jLp .QJ;M /kq kf jBp;q .Rn /k' D sup 2J n 2jsq k.'j f J 2Z;M 2Zn
j J C
(3.146) s; .Rn / is the collection of all f 2 S 0 .Rn / such that is finite and Fp;q X 1=q J n jsq _ q b f jF s; .Rn / D sup 2 2 j.' ./j jL .Q / f / j p J;M p;q ' J 2Z;M 2Zn
j J C
(3.147) is finite. s; .Rn / is the collection of all f 2 S 0 .Rn / such that (ii) Then Bp;q X 1=q s; b/_ jLp .QJ;M / q kf jBp;q .Rn /k' D sup 2J n 2jsq .'j f J 2N0 ;M 2Zn
j J
(3.148) s; is finite and Fp;q .Rn / is the collection of all f 2 S 0 .Rn / such that X ˇ 1=q ˇ s; b/_ ./ˇq kf jFp;q .Rn /k' D sup 2J n 2jsq ˇ.'j f jLp .QJ;M / J 2N0 ;M 2Zn
j J
(3.149) is finite. s; s; n n n Remark 3.37. Again As; p;q .R / with A 2 fB; F g means Bp;q .R / or Fp;q .R /. s; n s n Similarly Ap;q .R /. Compared with Ap;q .R / according to Definition 3.1 and in n agreement with the above notation As; p;q .R / is the hybrid (Fourier-analytical) vers; s n n sion of Ap;q .R /, and Ap;q .R / is the related local (Fourier-analytical) version. As their global ancestors Asp;q .Rn / they are independent of ' (already tacitly used in the above definition), equivalent quasi-norms. The hybrid inhomogeneous spaces n Ps; n As; p;q .R / and their homogeneous counterparts Ap;q .R /, defined later on, having been introduced in [YaY08, YaY10, YSY10]. They attracted a lot of attention. Further references about these homogeneous and inhomogeneous spaces may be found in [T13, Remark 1.40, p. 39], the book [YSY10] and the survey [Sic12]. This will not be repeated here with exception of some more recent papers needed to establish n (1.22). The local spaces As; p;q .R / according to part (ii) of the above definition go back to [T13, Definition 2.59, p. 102]. In [T13, Section 2.7.3, pp. 101–107] one finds n also a detailed discussion how the local spaces As; p;q .R / are related to other weighted and unweighted, hybrid and global spaces. We refer the reader in this connection also to [YSY13].
Theorem 3.38. Let 0 < p; q 1 .p < 1 for the F -spaces/; s 2 R, n=p r < n 1 and D p1 C nr . Let LrAsp;q .Rn / and As; p;q .R / be the spaces according to the Definitions 3.18 and 3.36(i). Then n LrAsp;q .Rn / D As; p;q .R /:
(3.150)
70
3 Hybrid spaces
Proof. We rely on wavelet characterizations, this means Theorem 3.26 for the spaces n LrAsp;q .Rn /. The characterization of As; p;q .R / in terms of wavelets developed over the time and reached finally the same level as for the spaces LrAsp;q .Rn / (no additional restrictions for the parameters involved). First assertions go back to [YSY10, Section 4.2.3, pp. 96–100]. One may also consult [Sic12, I, Theorem 3.4, p. 133]. n A corresponding characterization covering all spaces As; p;q .R / in a more general context has been obtained recently in [LYYSU13, Theorem 4.12, p. 42]. One may also consult [LSUYY12, Section 6]. A specification of these rather general assern tions to orthonormal wavelets as considered here and to As; p;q .R / has been described in [YHSY14, Proposition 3.3, Remark 3.4]. This gives the possibility to compare the natural smoothness assumptions according to (3.88), hence ( B-spaces; max.s C r C ; p s/; j ‰G;m 2 C kC1 .Rn /; k C 1 > (3.151) C max.s C r ; p;q s/; F -spaces; with the corresponding restrictions in [YHSY14] in specification of [LYYSU13]. In n any case, if k in (3.151) is chosen sufficiently large, then LrAsp;q .Rn / and As; p;q .R / have the same wavelet characterizations. This proves (3.150). Instead of wavelets one can use atoms. On the one hand one has the atomic representation Theorem 3.33 n for the spaces LrAsp;q .Rn /. On the other hand atomic representations for As; p;q .R / may be found in [YSY10, Theorem 3.3, p. 59] and more recently in [Dri13]. However the comparison of these atomic representations requires some additional efforts. We refer the reader to [T13, pp. 104, 105]. There is a counterpart of the above theorem for local spaces. Corollary 3.39. Let 0 < p; q 1 .p < 1 for the F -spaces/; s 2 R, n=p n r < 1 and D p1 C nr . Let LrAsp;q .Rn / and As; p;q .R / be the spaces according to Notation 3.22, based on (3.97) and Definition 3.36(ii). Then n LrAsp;q .Rn / D As; p;q .R /:
(3.152)
Proof. This assertion is essentially covered by [T13, Theorem 2.63, Remark 2.65, pp. 104, 105]. Remark 3.40. Both Theorem 3.38 and Corollary 3.39 rely on rather complicated n wavelet and atomic characterizations for LrAsp;q .Rn / and As; p;q .R /, and their local counterparts. More direct proofs would be desirable. Some further assertions in case of local spaces may be found in [T13, Sections 2.7.2, 2.7.3, pp. 100–107]. One may also consult [YSY13].
3.4.4 Haar wavelets The requested smoothness of the underlying wavelets in Definition 3.18 and (3.151) is natural. But it excludes Haar wavelets which have for ages attracted a lot of attention in the theory of function spaces. In [T10, Chapter 2] we dealt with Haar bases in
71
3.4 Characterizations
some global spaces Asp;q .Rn / and extended these considerations in [Tri13] to related local spaces LrAsp;q .Rn /. These assertions can be transferred to corresponding hybrid spaces LrAsp;q .Rn / without any additional efforts. Haar expansions for global spaces Asp;q .Rn / will be used in Chapter 6 in connection with global solutions for NavierStokes equations. A possible extension of these applications to hybrid spaces has not yet been done. But the topic is of interest for its own sake. Some consequences for the Morrey spaces Lrp .Rn /, Lrp .Rn / and LV rp .Rn / have already been described in Section 2.6 on a somewhat preliminary basis about Haar wavelets and related sequence spaces. This will now be recalled and extended in a more systematic way. We briefly recall what is meant by Haar bases in our context. Let again for y 2 R, 8 ˆ if 0 < y < 1=2; <1 hM .y/ D 1 if 1=2 y < 1; (3.153) ˆ :0 if y 62 .0; 1/; and let hF .y/ D jhM .y/j be the characteristic function of the unit interval .0; 1/. Let as in Section 3.3.1 and (3.47), (3.48), G D .G1 ; : : : ; Gn / 2 G 0 D fF; M gn and
G D .G1 ; : : : ; Gn / 2 G D G j 2 fF; M gn ;
(3.154)
j 2 N;
(3.155)
with the explanations given there. Then j .x/ D 2j n=2 HG;m
n Y
hGl 2j xl ml ;
j 2 N0 ;
G 2 Gj ;
m 2 Zn ;
lD1
(3.156) is the well-known orthonormal Haar basis in L2 .Rn /. One has in particular X X X j j n=2 f D j;G HG;m (3.157) m 2 j 2N0 G2G j m2Zn
with the Fourier coefficients j j;G j n=2 f; HG;m ; j;G m D m .f / D 2
j 2 N0 ;
G 2 Gj ;
m 2 Zn ; (3.158)
(scalar product). We are looking for a counterpart of Theorem 3.26 which requires some restrictions for the parameters involved. First we recall what is known in case of the global spaces Asp;q .Rn / and, in particular, their restrictions to the unit cube Q D Q0;0 D .0; 1/n, hence Asp;q .Q/ according to Definition 3.16(i). Let be the characteristic function of Q and let ˚ j 2 N0 : (3.159) Pj D m 2 Zn W 0 ml 2j 1I l D 1; : : : ; n ; Then
˚
; HG;m W j 2 N0 ; G 2 G ; m 2 Pj j
(3.160)
72
3 Hybrid spaces
is an orthonormal basis in L2 .Q/. One has f D 0 C
1 X X X j D0 G2G
with
Z
j;G m
D
j;G m .f
/D2
j n=2 Q
j n=2 j;G HG;m m 2 j
(3.161)
m2Pj
j f .x/ HG;m .x/ dx;
j 2 N0 ; G 2 G ; m 2 Pj ;
(3.162) R complemented by 0 D .f; / D Q f .x/ dx. This restricts (3.157), (3.158) from Rn to Q. It is now convenient for us to isolate the starting term with . Otherwise we rely on the representation theorem for some spaces Asp;q .Q/ in terms of Haar bases according to [T10, Section 2.3.3, pp. 95–98] adapted to the above notation. s s For a description we need the counterpart ap;q .Q/ of the sequence spaces ap;q .Rn / according to Definition 3.10. Let 0 < p; q 1, s 2 R, and ˚ D 0 2 C; j;G (3.163) m 2 C W j 2 N0 ; G 2 G ; m 2 Pj : Let again j;m be the characteristic function of the cube Qj;m D 2j m C 2j .0; 1/n. Then ˚ s s bp;q .Q/ D W k jbp;q .Q/k < 1 (3.164) with s .Q/k k jbp;q
D j j C 0
1 X j D0
and
n
2j.s p /q
X X G2G
p jj;G m j
q=p 1=q (3.165)
m2Pj
˚ s s .Q/ D W k jfp;q .Q/k < 1 fp;q
(3.166)
with kf
s .Q/k jfp;q
1 X X X ˇq 1=q ˇ ˇ D j j C 2jsq ˇj;G ./ jL .Q/ : j;m p m 0
j D0 G2G m2Pj
(3.167) s s s As before ap;q .Q/ with a 2 fb; f g stands for bp;q .Q/ and fp;q .Q/. Otherwise s kf jap;q .Q/k is the term with J D 0 and M D 0 in (3.120), (3.122), hence .j; G; m/ 2 P0;0 . We recall now the Haar wavelet expansion according to [T10, Theorem 2.26, p. 96]. Let 0 < p; q 1 and max n p1 1 ; p1 1 < s < min p1 ; 1 ; (3.168) s .Q/ if, and only if, it can on the left in Figure 3.1. Then f 2 D 0 .Q/ belongs to Bp;q be represented as
f D 0 C
1 X X X j D0 G2G m2Pj
j n=2 j;G HG;m ; m 2 j
s 2 bp;q .Q/:
(3.169)
73
3.4 Characterizations
s
s
1
s D n. p1 1/
s D n. p1 1/
1 2
1 1C n1
1 p
12
1 2
1 1 1C 2n
1 p
1
1 B-spaces
H -spaces Figure 3.1. Haar bases, n 2 N
The representation (3.169) is unique, Z j j;G j n=2 D .f / D 2 f .x/ HG;m .x/ dx; j;G m m
j 2 N0 ; G 2 G ; m 2 Pj ;
Q
(3.170) s s .Q/ onto bp;q .Q/. There 0 D .f; /, and f 7! .f / is an isomorphic map of Bp;q s is a counterpart for Fp;q .Q/ with 8 1 1 ˆ ; ; 1 1 < s < min p1 ; q1 ; 1 ; 0 < p < 1; 0 < q < 1; n max < p q ˆ1 < p < 1; 1 < q < 1; s D 0; : 1 < p < 1; 1 < q 1; max p1 ; q1 1 < s < 0; (3.171) s s .Q/ in (3.169) in place of 2 bp;q .Q/. The right-hand side of Figand 2 fp;q s .Q/. ure 3.1, shows the restrictions in case of the Sobolev spaces Hps .Q/ D Fp;2 We add a technical comment. In Definition 3.18 of the hybrid spaces LrAsp;q .Rn / and its local counterparts in (3.97) we relied on overlapping cubes 2QJ;M where QJ;M D 2J M C 2J .0; 1/n, J 2 Z and M 2 Zn . This is indispensable in general. But if s; p (and q) are restricted by (3.168), covering also (3.171), then one can replace 2QJ;M in (3.89) and its local counterpart (3.97) by QJ;M D 2J M C 2J .0; 1/n. This can be justified as follows. First we remark that the characteristic function C of the half-space ˚ (3.172) RnC D x D .x1 ; : : : ; xn / 2 Rn W xn > 0 is a pointwise multiplier for Asp;q .Rn / if, and only if, max n p1 1 ; p1 1 < s < p1 ; 0 < p 1;
0 < q 1;
(3.173)
74
3 Hybrid spaces
(p < 1 for F -spaces). We quoted and discussed this observation in [T08, p. 169] with a reference to [RuS96, pp. 208, 258]. Let Q D .0; 1/n or any other cube in Rn and let Asp;q .Q/, e Asp;q .Q/ and e Asp;q .Q/ be as in Definition 3.16 with D Q. According to [T08, Propositions 6.12, 6.13, pp. 182–184] one has Asp;q .Q/ D e Asp;q .Q/ Asp;q .Q/ D e
(3.174)
with s; p; q as in (3.173) (and p < 1 for F -spaces). In particular the somewhat sloppy second equality in (3.174) relies on (3.73). A detailed proof may be found in [T08, pp. 183/184]. Now it can be seen easily that one can replace 2QJ;M in (3.89) by QJ;M D 2J M C 2J .0; 1/n where J 2 Z and M 2 Zn if s; p; q are restricted by (3.173) (p < 1 for the F -spaces). Similarly for (3.97). j For the Haar wavelets HG;m according to (3.156) one has supp HG;m D Qj;m D 2j m C 2j Œ0; 1n; j
m 2 Zn : (3.175) s Let Lr ap;q .Rn / with a 2 fb; f g be the sequence spaces as introduced in Definition 3.24. After these preparations we can now complement Theorem 3.26 as follows. Theorem 3.41. Let
max n p1 1 ;
0 < p; q 1;
1 p
j 2 N0 ;
G 2 Gj ;
1 < s < min p1 ; 1
(3.176)
for the B-spaces, on the left-hand side of Figure 3.1, and 8 1 1 1 1 ˆ <0 < p < 1; 0 < q < 1; n max p ; q ; 1 1 < s < min p ; q ; 1 ; 1 < p < 1; 1 < q < 1; ˆ : 1 < p < 1; 1 < q 1;
s D 0; max p1 ; q1 1 < s < 0;
(3.177) for the F -spaces, on the right-hand side of Figure 3.1 for q D 2. Let n=p r < s. Let f 2 S 0 .Rn /. Then f 2 LrAsp;q .Rn / if, and only if, it can be represented by X j j n=2 s j;G HG;m ; 2 Lr ap;q .Rn /; (3.178) f D m 2 j 2N0 ;G2G j ; m2Zn
unconditional convergence being in S 0 .Rn /. The representation (3.178) is unique, j j;G j n=2 f; HG;m (3.179) j;G m D m .f / D 2 and
I W
˚ f 7! j;G m .f /
LrAsp;q .Rn /
(3.180)
s Lr ap;q .Rn /,
is an isomorphic map of onto f jLrAs .Rn / .f / jLr as .Rn / : p;q p;q
(3.181)
75
3.4 Characterizations
Proof. Step 1. If s; p; q are restricted by (3.176), which covers also (3.177), then it follows from the above discussion that one can replace 2QJ;M in (3.89), (3.97) (overlapping cubes) by QJ;M D 2J M C 2J .0; 1/n. Let Pk with k 2 N0 be the collection of all polynomials in Rn of degree less than or equal to k, complemented by P1 D f0g. We replaced in (3.94) the wavelet approximation by polynomial approximations. One has always (3.95). But the above restrictions of s; p; q and s C r < 0, n=p r ensure equivalence now with QJ;M in place of 2QJ;M , hence kf jLrAsp;q .Rn /k C
n
sup J 2N0 ;M 2Zn
2J. p Cr/ inf kf P jAsp;q .QJ;M /k P 2Pk
2
sup J 2N ;M 2Zn
n J. p Cr/
kf jAsp;q .QJ;M /k;
(3.182)
where k 2 N1 D N0 [ f1g. This may be found in [T13, (1.261)-(1.263), Remark 2.30, pp. 42, 77]. We return to this point later on in Section 3.6.3. Step 2. If J 2 N D N then the desired expansions by Haar wavelets for Asp;q .Rn /, Asp;q .QJ;M / and Asp;q .Q/ with Q D .0; 1/n are covered by [T10, Sections 2.3.2, 2.3.3, pp. 92–98] and the above considerations. One has to be careful about the terms with J 2 N0 in (3.182). This reduces by (3.97), (3.99) the above theorem to its counterpart for the local spaces LrAsp;q .Rn /. Then one can follow [Tri13]. Of interest for us is the case k D 0 in (3.182), hence best approximations of f 2 Asp;q .QJ;M / by constants. We deal first with a model case, say, Q D .0; 1/n. The conditions (3.176)=(3.168) and (3.177)=(3.171) show that we can apply (3.169) s and its counterpart with fp;q .Q/, hence f c D .0 c/ C
1 X X X
j n=2 j;G HG;m ; m 2 j
s 2 ap;q .Q/; (3.183)
j D0 G2G m2Pj j;G
with m as in (3.162), c 2 C and 0 D .f; /. Then one has by (3.165), (3.167), inf kf c jAsp;q .Q/k D kf .f; / jAsp;q .Q/k;
c2C
(3.184)
which eliminates the starting term in (3.169). This reduces (3.165), (3.167) (now with 0 D 0) to corresponding terms with P0;0 in Definition 3.24 with G 2 G D fF; M gn . Let now QJ D 2J .0; 1/n, J 2 N0 . We rely on the local homogeneity n kf c jAsp;q .QJ /k 2J.s p / kf 2J c jAsp;q .Q/k (3.185) with Q D .0; 1/n, where the equivalence constants are independent of J 2 N0 . This substantial property is covered by [T13, Theorem 1.4, p. 11], based on [T08, Sections 2.2.1, 3.3.2, pp. 33–35, 92–94], and (3.174). Pulling back one has in QJ the optimal decomposition Z X X X j Jn j n=2 f 2 f .x/ dx D j;G HG;m ; (3.186) m 2 QJ
j J G2G m2Pj J
76
3 Hybrid spaces
R where QJ f .x/ dx in case of s 0 must be understood as the dual pairing of f 2 Asp;q .Rn / with the characteristic function of QJ , and Pj J is given by (3.159). There is a counterpart for QJ;M D 2J M C 2J .0; 1/n, J 2 N0 , M 2 Zn , cutting out j j in (3.178) the Haar functions HG;m with j J 2 N0 , G 2 G and supp HG;m QJ;M . The theorem follows from (3.182) with k D 0 (approximation by constants). As far as the underlying homogeneity of sequence spaces is concerned one may also consult Remark 3.51 below. Remark 3.42. Characteristic functions of half-spaces or cubes are pointwise multipliers in Asp;q .Rn / if, and only if, one has (3.173). Furthermore according to [T10, Proposition 2.24, p. 94] with a reference to [Tri78] one cannot expect that Haar syss s tems are bases in Bp;q .Rn / or Bp;q .Q/ if s > 1. Both together shows that the s condition (3.176) is natural, Figure 3.1. The situation for the spaces Fp;q .Rn / is less favourable. As indicated above we took over the condition (3.177)=(3.171) from [T10]. The step from Haar systems in Asp;q .Rn / to Haar systems in LrAsp;q .Rn / requires the additional assumption s C r < 0 which is natural. This follows from Proposition 3.54 below. Remark 3.43. As previously mentioned we followed in the above proof closely [Tri13] where we dealt with corresponding assertions for the local spaces LrAsp;q .Rn / which we introduced at the beginning of Section 3.3.2 with a reference to [T13, Defs inition 1.26, p. 20]. Let Lrap;q .Rn / according to (3.123), (3.124) with a 2 fb; f g be s the local counterparts of the sequence spaces Lr ap;q .Rn / as introduced in Definition 3.24. Then the local counterpart of Theorem 3.41 reads as follows: Let p; q; s and r be as in Theorem 3.41. Let f 2 S 0 .Rn /. Then f 2 LrAsp;q .Rn / if, and only if, it can be represented by X j j n=2 s j;G HG;m ; 2 Lrap;q .Rn /; (3.187) f D m 2 j 2N0 ;G2G j ; m2Zn
unconditional convergence being in S 0 .Rn /. The representation is unique with j;G m as in (3.179). Furthermore, I in (3.180) is an isomorphic map of LrAsp;q .Rn / onto s Lrap;q .Rn /, f jLrAs .Rn / .f / jLras .Rn / : (3.188) p;q p;q This coincides with [Tri13, Theorem 3.3, p. 145]. The proof of Theorem 3.41 covers also the above assertion and repeats essentially corresponding arguments in [Tri13]. s .Rn / with 1 < p < Some special cases might be of interest. Let Hps .Rn / D Fp;2 1, s 2 R, be the usual (fractional) Sobolev spaces according to (3.21)–(3.23) and let Lr Hps .Rn /, n=p r < 1 be the related hybrid spaces. Recall that bmo.Rn / collects all complex-valued locally Lebesgue-integrable functions f in Rn such that Z Z n 1 kf jbmo.R /k D sup jf .y/j dx C sup jQj jf .x/ fQ j dx (3.189) jQjD1
Q
jQj1
Q
3.4 Characterizations
77
R is finite. Here Q stands for cubes in Rn and fQ D jQj1 Q f .y/ dy is the related mean value. As will be justified below in (3.308) and (3.295) one has Lrp .Rn / D Lr Lp .Rn /;
1 < p < 1;
n=p r < 0;
(3.190)
for the Morrey spaces introduced in Definition 2.1(ii) and bmo.Rn / D L0 Lp .Rn /;
2 p < 1:
(3.191)
The lifting (3.19), (3.20) for global spaces Asp;q .Rn / can be extended to local spaces LrAsp;q .Rn /, [T13, Theorem 2.39, p. 88], and to hybrid spaces LrAsp;q .Rn / according to Section 3.6.5 below. In particular, Is Lrp .Rn / D LrHps .Rn /;
1 < p < 1;
n=p r < 0;
s 2 R;
(3.192)
and bmos .Rn / D Is bmo.Rn / D L0 Hps .Rn /;
2 p < 1;
s 2 R:
(3.193)
s .Rn / to s 2 R, 0 < p < 1, We extend the above notation LrHps .Rn / D LrFp;2 r s n r s n n=p r < 1. Let L hp .R / D L fp;2 .R / be the corresponding sequence spaces according to (3.121), (3.122) and (2.200), (2.201).
Corollary 3.44. Let 8 1 1 ˆ <2 p < 1; 2 < s < p ; 1 1 < s < 12 ; 1 p < 2; p ˆ :p < 1; n. p1 1/ < s < 12 ;
(3.194)
right-hand side of Figure 3.1. Let n=p r < s. Let f 2 S 0 .Rn /. Then f 2 LrHps .Rn / if, and only if, it can be represented by f D
X
j n=2 j j;G HG;m ; m 2
2 Lrhsp .Rn /;
(3.195)
j 2N0 ;G2G j ; m2Zn
unconditional convergence being in S 0 .Rn /. The representation (3.195) is unique, j j;G j n=2 f; HG;m (3.196) j;G m D m .f / D 2 and
I W
˚ f 7! j;G m .f /
(3.197)
is an isomorphic map of LrHps .Rn / onto Lrhsp .Rn /, kf jLrHps .Rn /k k.f / jLr hsp .Rn /k:
(3.198)
78
3 Hybrid spaces
Proof. This is a special case of Theorem 3.41 where (3.194) comes from (3.177) with q D 2. Remark 3.45. The interest in this corollary comes from the above examples. In particular the Morrey spaces Lrp .Rn / in (3.190) can be characterized by Haar wavelets with s D 0 in (3.195), (3.198). This has been stated explicitly in Theorem 2.31 both for Lrp .Rn / and Lrp .Rn /. But it applies also to the (lifted inhomogeneous) bmospaces bmos .Rn / according to (3.193) with 1=2 < s < 0, and its sequence counterparts L0 hs2 .Rn / in (2.201), kf jbmos .Rn /k k.f / jL0hs2 .Rn /k Z Jn 2 sup Rn
J 2Z;M 2Zn
sup
2J n
Z
J 2N0 ;M 2Zn
ˇ2 1=2 ˇ ˇ 22js ˇj;G m .f / j;m .x/ dx
X .j;G;m/2PJ;M
Rn
X
ˇ2 1=2 ˇ ˇ 22js ˇj;G m .f / j;m .x/ dx
.j;G;m/2PJ;M
(3.199) j;G with m .f / as in (3.196). As far as the last equivalence is concerned we remark that one can complement (3.191) and (3.193) by it local counterparts, hence bmo.Rn / D L0 Lp .Rn / D L0 Lp .Rn /;
2 p < 1;
(3.200)
and bmos .Rn / D L0 Hps .Rn / D L0Hps .Rn /;
2 p < 1;
s 2 R;
(3.201)
covered by [T13, (3.83), p. 122] and the indicated lifting. Then the last equivalence in s .Rn / D L0hsp .Rn /. But it can (3.199) follows from (3.188) and (2.199) with L0ap;2 also be seen by direct arguments similarly as in (3.347) below. By [T10, Theorem s s 2.21, p. 92] or the above Theorem 3.41 with L0 B1;q .Rn / D B1;q .Rn / according s n to (3.102) the spaces B1;q .R /, can be characterized in terms of Haar wavelets if 1 < s < 0. For the near-by spaces bmos .Rn / one has the restriction 1=2 < s < 0. It is not clear what happens if 1 < s 1=2. The spaces bmo.Rn / and bmo1 .Rn / surely do not admit Haar expansions. From (3.193) and Theorem 3.26 follows that one has wavelet expansions (3.125) if F ; M 2 C u .R/ with u > jsj. This means u 1 for bmo.Rn / and u 2 for bmo1 .Rn /. There is a connection of these questions to Navier-Stokes equations to which we return in Chapter 6 at the end of this book.
3.4.5 Morrey spaces, revisited We prove Theorem 2.33. All notation has the same meaning as there. In particular fhjG;m g in (2.217) is the homogeneous orthonormal Haar basis in L2 .Rn / according to (2.209)–(2.211).
3.4 Characterizations
Step 1. Recall that D.Rn/ D C01 .Rn /. First we prove that Z ˚ D.Rn /ı D ' 2 D.Rn / W '.x/ dx D 0
79
(3.202)
Rn
V rp .Rn /. It is sufficient to approximate 2 D.Rn / in Lrp .Rn / by funcis dense in L tions belonging to D.Rn /ı . Let Z ! 2 D.Rn/; !.y/ dy D 1; (3.203) Rn
and !J .x/ D 2J n !.2J x/, x 2 Rn , J 2 Z. Then homogeneity (2.160) one has k!J jLrp .Rn /k 2J nCJ r k! jLrp .Rn /k ! 0
R Rn
if
!J .y/ dy D 1. By the
J ! 1; R D !J Rn
(3.204)
> 0. Furthermore J .y/ dy 2 n ı r n V D.R / . Then the density of D.R / in Lp .R / follows from ˇZ ˇ ˇ ˇ .y/ dy ˇ k!J jLrp .Rn /k ! 0 if J ! 1: (3.205) k J jLrp .Rn /k D ˇ where we used n C r n
n p
n ı
Rn
R Step 2. Let ' 2 D.Q/ D C01 .Q/ with Q '.y/ dy D 0 where again Q D .0; 1/n. We expand ' according to (3.161), (3.162) where the starting term is zero, hence 'D
1 X X X
j n=2 j;G hG;m m 2 j
(3.206)
j D0 G2G m2Pj
Z
with j;G m
D
j;G m .'/
D2
j n=2 Rn
'.x/ hjG;m .x/ dx:
(3.207)
R Let ' 2 D.Rn / with Rn '.y/ dy D 0 and 'K .x/ D '.2K x/. Then one has for some K 2 Z and a possible immaterial translation x 7! x C M.1; ; 1/ for some M 2 N that Z 'K 2 D.Q/; 'K .y/ dy D 0: (3.208) Q
We apply (3.206), (3.207) to 'K in place of '. By (2.216) we have (2.222) with 'K in place of f (where the integral is restricted to Q). Then the expansion (2.218) with 2 Lr hP 0p .Rn /, (2.219) and (2.221) specified by (2.222) with ' in place of f follows from the homogeneity (2.160) and its sequence counterpart explicitly menV r hP 0p .Rn / follows from 2 Lr hP 0p .Rn / tioned below in (3.253). The assertion 2 L and that the expansions by Haar functions, say, for 'K above, work according to the beginning of Section 3.4.4 and the right-hand side of Figure 3.1, not only for Lp .Q/ but also for Hps .Q/ with 1 < p < 1 and some s > 0. This gives by (3.167) and
80
3 Hybrid spaces
q D 2 some extra factors 2js , j 2 N0 , and shows that one has even 2 LV r hP 0p .Rn /. Then I in (2.220) is the indicated isomorphic map. The standard unconditional basis for LV r hP 0p .Rn / is mapped isomorphically onto fhjG;m g according to (2.217), which, V rp .Rn /. Our minimal request (2.16) follows from hence, is an unconditional basis in L the embedding (2.58) or, better, (2.74). We add a related comment in the following remark. Remark 3.46. By (2.8) Theorem 2.33 covers the classical Haar expansion for the Lebesgue spaces Lp .Rn /, 1 < p < 1, with the Littlewood-Paley equivalence (2.223). Historical comments may be found in [T10, Section 2.1, pp. 63–71] and also at the beginning of Section 2.6.3. It is of interest how Theorem 2.33 is related to other expansions in terms of (homogeneous and inhomogeneous) Haar wavelets. In connection with distinguished representations of the Calder´on-Zygmund operators (2.150), (2.151) according to Proposition 2.25 we relied on corresponding assertions in Lp .Rn ; /, 1 < p < 1 where D w.x/L , w 2 Ap .Rn / is a Muckenhoupt weight, (2.71), (2.72). Corresponding references have been given at the end of Section 2.5.1. The theory of Muckenhoupt weights and also related applications to Calder´on-Zygmund operators goes back to the late 1970s and early 1980s. We refer the reader to the note sections in [Tor86, Gra04, Ste93] in connection with assertions described at the end of Section 2.5.1. But a satisfactory counterpart of Haar expansions and Littlewood-Paley assertions for weighted Lp -spaces needed apparently quite a long time to appear. Finally it came out that there is a full counterpart of Theorem 2.33 with Lp .Rn ; /, D w.x/L , w 2 Ap .Rn / in place of LV rp .Rn / (or in place of Lp .Rn /). This may be found in [ABM03, Theorem 6, Corollary 3, pp. 502, 503]. Furthermore, D w.x/L with w 2 Ap .Rn / is not only sufficient but also (almost) necessary. We refer the reader for details but also for the history of this problem to [ABM03]. In particular it follows from (2.74) that any f 2 Lrp .Rn /, and hence any f 2 LV rp .Rn /, can be expanded by homogeneous Haar wavelets, convergence being at least in Lp .Rn ; ˛ /. Theorem 2.33 improves this assertion substantially if f 2 LV rp .Rn /. Remark 3.47. For weighted spaces on Rn there is a decisive difference between exj pansions in terms of homogeneous Haar wavelets fhG;m g in (2.217) and their inhomoj geneous counterpart fHG;m g according to (3.156). Let again w .x/ D .1 C jxj2 /=2 , x 2 Rn , 2 R, be smooth weights of polynomial type. Let Asp;q .Rn / with 0 < p; q 1 (p < 1 for F -spaces) be the spaces as introduced in Definition 3.1. Then Asp;q .Rn ; w / is the collection of all f 2 S 0 .Rn / such that
kf jAsp;q .Rn ; w /k D kw f jAsp;q .Rn /k
(3.209)
is finite. The theory of these spaces attracted some attention in the last two decades. One may consult [T06, Chapter 6] and the references given there. One has in particular 0 Lp .Rn ; w / D Fp;2 .Rn ; w /; 1 < p < 1; (3.210)
3.4 Characterizations
81
with Lp .Rn ; w / as in (2.4). There is a full counterpart of the wavelet expansion Theorem 3.12 for all 2 R. The final version may be found in [T08, Theorem 1.26, pp. 18/19], but it goes back to [HaT05] and [T06, Section 6.2, pp. 268–273]. This applies to all 2 R. There is little doubt that one can replace the smooth Daubechies j wavelets by the inhomogeneous Haar wavelets fHG;m g under the same restrictions for p; q; s as in (3.168), (3.171) (which coincides with the corresponding restrictions in Theorem 3.41). This has been done in the one-dimensional case in [T12, Theos rem 3.3, p. 47] for the spaces Bp;q .R; w /. If is a Muckenhoupt weight according to (2.71) then one can introduce corresponding spaces Asp;q .Rn ; / replacing Lp .Rn / in Definition 3.1 by Lp .Rn ; / according to (2.72). Again one can ask for atomic and wavelet expansions. First results may be found in [HaS08, HaP08] in terms of smooth atoms and wavelets. One may ask for corresponding assertions for (inhomogeneous) Haar wavelets.
3.4.6 Meyer wavelets Definition 3.18 of the hybrid spaces LrAsp;q .Rn / relies on (sufficiently smooth) Daubechies wavelets having, in particular, compact supports. We added in Section 3.4.4 some assertions about Haar wavelets to which we return in Chapter 6 at the end of this book where we discuss infrared-damped initial data for Navier-Stokes equations. For the same reason we add now a brief comment about Meyer wavelets. The Fourier transform of the underlying one-dimensional scaling function F 2 S.R/ (father wavelet) and the related (mother) wavelet M 2 S.R/ have now compact supports, j j
b./j > 0 F
b ./j > 0 M
jj < 4 =3;
(3.211)
2 =3 < jj < 8 =3:
(3.212)
if, and only if,
if, and only if,
Otherwise one has again (inhomogeneous) orthonormal bases in L2 .Rn / as in (3.79), (3.80) and related homogeneous counterparts as in (3.219), (3.220) below. Standard references for Meyer wavelets are [Mey92, Dau92, Woj97, Mal99]. For an explicit construction of Meyer wavelets we refer the reader to [Woj97, Section 3.2, Exercise 3.2, pp. 49–51, 71]. Further (historical) references may also be found in [T06, Sections 1.7.3, 3.1.5, pp. 30–35, 159–160]. There is little doubt (but not yet done in detail) that one can replace the Daubechies wavelets in Definition 3.18 by the Meyer wavelets with the same outcome for the hybrid spaces LrAsp;q .Rn /. Similarly for the local spaces LrAsp;q .Rn /. This claim can be justified by using molecular characterizations of the spaces in question. In case of the local spaces LrAsp;q .Rn / we refer the reader to [T13, Proposition 2.35, Remark 2.36, pp. 85/86]. One can extend the sketchy arguments given there to the hybrid spaces LrAsp;q .Rn /. Corresponding assertions are also covered by [YSY10, Chapter 3] combined with Theorem 3.38. We rely in this book on compactly supported Daubechies wavelets. But the above comments about Meyer wavelets will illuminate in Remark 5.12 below what is meant by infrared-damped initial data for Navier-Stokes equations.
82
3 Hybrid spaces
3.5 Equivalent norms and Fourier multipliers 3.5.1 Equivalent norms As indicated in Chapter 1, Introduction, it is one of the main aims of this book to justify (1.23), (1.24) and to apply this observation to Navier-Stokes equations. As previously established (1.23), (1.24) is the Morreyfied version of (1.26), (1.27) which we describe first in some detail and which will be also useful later on. We follow [T13, p. 17] which, in turn, is based on [T92, Section 2.3.3, Theorem, p. 98]. Let ' 2 S.Rn / with '.x/ D 1 if jxj 1 and '.y/ D 0 if jyj 3=2;
(3.213)
and let ' j .x/ D '.2j x/ '.2j C1 x/; Then
X
x 2 Rn ;
j 2 Z:
' j .x/ D 1 for x 2 Rn n f0g:
(3.214) (3.215)
j 2Z
Let s 2 R and 0 < p; q 1 (with p < 1 for the F -spaces). Then 8 1=q P ˆ jsq j b _ n q < f / 2 jL .R / ; .' p j 2Z kf jAPsp;q .Rn /k' D P 1=q ˇ ˇ ˆ jsq ˇ j b _ ˇq n : f / 2 ./ jL .R / .' ; p j 2Z
B-spaces;
F -spaces; (3.216) (with the usual modification if q D 1) are the homogeneous counterparts of (3.13), (3.15) in Definition 3.1 where we introduced the inhomogeneous spaces Asp;q .Rn /. We are not interested in the homogeneous spaces APsp;q .Rn / themselves as discussed in [T83, Chapter 5, pp. 237–244], but only in the homogeneous quasi-norms (3.216) applied to elements f 2 Asp;q .Rn / or simply to smooth functions.Then different admitted basic functions ' according to (3.213) produce equivalent quasi-norms (3.216). This justifies dropping the subscript ' in (3.216) in the sequel. If 0 < p; q 1; s > p D n max p1 ; 1 1 (3.217) (p < 1 for the F -spaces) then kf jAsp;q .Rn /k kf jAPsp;q .Rn /k C kf jLp .Rn /k;
f 2 Asp;q .Rn /; (3.218)
are equivalent quasi-norms. This may be found in [T92, Section 2.3.3, Theorem, p. 98], repeated in [T13, Proposition 1.23, p. 17]. This was the basis to prove in [T13, Theorem 1.25, p. 17–18] that the Fourier-analytical version of the Riesz transform Rk in (1.10) generates a linear and bounded map in Asp;q .Rn / if s 2 R, 1 < p < 1, (0 < q 1 for B-spaces, 1 < q < 1 for F -spaces). We are interested in a corresponding
83
3.5 Equivalent norms and Fourier multipliers
observation for some hybrid spaces LrAsp;q .Rn /. But instead of Fourier-analytical counterparts of (3.216) we rely on the homogeneous wavelet system ˚ P D ‰ j W j 2 Z; G 2 G ; m 2 Zn ; (3.219) ‰ G;m where again G D fF; M gn and n Y
j ‰G;m .x/ D 2j n=2
Gl .2
j
xl ml /;
j 2 Z;
G 2 G;
m 2 Zn ; (3.220)
lD1
in modification of (3.78), (3.80), again based on (3.74)–(3.76). This is again an orthonormal wavelet basis in L2 .Rn /. It differs from the orthonormal wavelet basis ‰ ‰ in (3.79) by the terms with j 0 as a replacement of V0;u according to (3.83), hence ‰m .x/ D
n Y
F .xl
ml /;
m 2 Zn :
(3.221)
lD1
We do not really need the homogeneous spaces LrAPsp;q .Rn / themselves but the homogeneous counterparts of the related sequence spaces according to Definition 3.24. Let ˚ PP J;M D j J; G 2 G ; m 2 Zn W Qj;m QJ;M ; J 2 Z; M 2 Zn ; (3.222) and ˚ PP J;M D j J; m 2 Zn W Qj;m QJ;M ; J 2 Z; M 2 Zn ; (3.223) be the homogeneous versions of (3.116), (3.117) where Qj;m D 2j m C 2j .0; 1/n are the same dyadic cubes as there. Of course j 2 Z in (3.222), (3.223). Let j;m be the characteristic functions of Qj;m . Definition 3.48. Let 0 < p; q 1, s 2 R and r n=p. Let ˚ n D j;G m 2 C W j 2 Z; G 2 G ; m 2 Z : Then
Lr bP sp;q .Rn / D f W k jLr bP sp;q .Rn /k < 1g
(3.224) (3.225)
with k jLr bP sp;q .Rn /k D
sup J 2Z;M 2Zn
2
n J. p Cr/
X 1 X j DJ G2G
2
n j.s p /q
X
p jj;G m j
q=p 1=q
mW.j;m/2PP J;M
(3.226)
84
3 Hybrid spaces
and
Lr fPsp;q .Rn / D f W k jLr fPsp;q .Rn /k < 1g
(3.227)
with k jLr fPsp;q .Rn /k D
sup J 2Z;M 2Zn
n 2J. p Cr/
X
q 2jsq jj;G m j;m ./j
1=q
jLp .Rn / (3.228)
.j;G;m/2PP J;M
.usual modification if max.p; q/ D 1/. Remark 3.49. This is the homogeneous counterpart of Definition 3.24. As above Lr aP sp;q .Rn / with a 2 fb; f g stands for Lr bP sp;q .Rn / and LrfPsp;q .Rn /. As far as the use of the characteristic function j;m of non-overlapping cubes Qj;m for fixed j 2 Z and m 2 Zn is concerned we refer the reader to the Remarks 3.5 and 3.25. If f 2 LrAsp;q .Rn / then we complement (3.125), (3.126), hence X X j n=2 j f D m .f /‰m C j;G ‰G;m m .f / 2 j 2N0 ;G2G ; m2Zn
m2Zn
(3.229)
D f0 C f C ; j as in (3.221), (3.220) by with ‰m , ‰G;m
f D
X
j n=2 j j;G ‰G;m m .f / 2
(3.230)
j 2N ;G2G ; m2Zn
where again N D N D fJ 2 Z W J 2 Ng and j j n=2 m .f / D f; ‰m ; j;G f; ‰G;m ; m .f / D 2
(3.231)
j 2 Z, G 2 G , m 2 Zn . We borrow now some equivalence assertions obtained in the framework of the homogeneous counterparts n Lr APsp;q .Rn / D APs; p;q .R /;
D
1 r C ; p n
(3.232)
of Theorem 3.38 with 0 < p; q 1 (p < 1 for F -spaces), s 2 R, n=p r < 1 as in Definition 3.18. We refer the reader to [YSY10, Definition 8.1, p. 253] for the Fourier-analytical definition. We return to this point in Section 3.5.2 below in greater detail. Characterizations of these spaces in terms of the homogeneous P in (3.219) maybe found in [YSY10, Theorem 8.3, p. 260]. But there wavelet system ‰ are some restrictions for the parameters, especially s > 0 and p < 1 also for Bspaces. Although this would be sufficient for our later purposes we wish to emphasize
85
3.5 Equivalent norms and Fourier multipliers
that wavelet characterizations of all spaces in (3.232) without any restrictions for the parameters have been obtained recently in [LSUYY12, Theorem 6.3, pp. 1106/1107] (under more restrictive smoothness assumptions for the underlying wavelets than in (3.88) = (3.151)). Later on we use (3.232) and the indicated references to ensure the equivalence s kf jLr APsp;q .Rn /k k.f / jLr aP p;q .Rn /k (3.233) s .Rn / as in Definition 3.48, Remark 3.49 for sufficiently smooth wavelets with Lr aP p;q j;G
and f D f0 C f C D f C C f 2 LrAsp;q .Rn / with m .f / and m .f / as in (3.231). At this moment and in the theorem below one can interpret the left-hand side of (3.233) as another writing of the right-hand side. The above-described Fourieranalytical background will be explained in detail when needed in Section 3.5.2 below dealing with Fourier multipliers. Recall that Lrp .Rn / are the Morrey spaces according to Definition 2.1. Theorem 3.50. Let 1 < p < 1, 0 < q 1, s > 0 and n=p r < 0. Then kf jLrAsp;q .Rn /k kf jLrAPsp;q .Rn /k C kf jLrp .Rn /k
(3.234)
are equivalent quasi-norms on LrAsp;q .Rn /. Proof. Step 1. First we prove kf jLrAsp;q .Rn /k c kf jLrAPsp;q .Rn /k C c kf jLrp .Rn /k
(3.235)
for some c > 0 and all f 2 LrAsp;q .Rn / where the first term on the right-hand side must be interpreted as the related sequence space according to (3.233) based on (3.226), (3.228). With f D f0 C f C as in (3.229) one has kf C jLrAsp;q .Rn /k c kf jLr APsp;q .Rn /k
(3.236)
as an immediate consequence of the related sequence spaces in Definitions 3.24, 3.48 P and Theorem 3.26. One obtains by Definition 3.18 applied to f0 D m2Zn m .f /‰m , kf0 jLrAsp;q .Rn /k
n
sup J 2N ;M 2Zn
2J. p Cr/ kf0 jAsp;q .2QJ;M /k:
(3.237)
Recall Wpk .Rn / ,! Asp;q .Rn / for the classical Sobolev spaces Wpk .Rn / with k > s. Then one obtains from (3.231) and fixed J 2 N , M 2 Zn , kf0 jAsp;q .2QJ;M /kp c kf0 jWpk .2QJ;M /kp Z X 0 c j.f; ‰m /jp m .x/ dx Rn
(3.238)
Q0;m CQJ;M
where C D C.‰/ > 0 is a constant reflecting the overlap of ‰m , and m is the characteristic function of CQ0;m . We insert j.f; ‰m /j c kf jLp .CQ0;m /k. Then one has kf0 jAsp;q .2QJ;M /kp c kf jLp .C 0 QJ;M /kp (3.239)
86
3 Hybrid spaces
for some C 0 > 0 (independent of J 2 N and M 2 Zn ). Then kf0 jLrAsp;q .Rn /k c kf jLrp .Rn /k
(3.240)
follows from (3.237) and (2.3). Together with (3.236) one obtains (3.235). Step 2. We prove the converse of (3.235) interpreting the first term on the righthand side as the related sequence space according to (3.233). First we remark that one has by [T13, (3.81), p. 121], 0 Lrp .Rn / D LrLp .Rn / D LrFp;2 .Rn /:
(3.241)
By Definition 2.1 and (3.99) one has a hybrid counterpart with Lrp .Rn / in place of 0 0 Lrp .Rn / and Lr Fp;2 .Rn / in place of Lr Fp;2 .Rn / as stated explicitly in (3.308). We return to equivalences of this type later on in Section 3.6.3 more systematically. Using s > 0 one obtains kf jLrp .Rn /k c kf jLrAsp;q .Rn /k:
(3.242)
With f C as in (3.229) one has by Theorem 3.26 and (3.233) based on Definition 3.48, kf C jLrAPsp;q .Rn /k kf C jLrAsp;q .Rn /k c kf jLrAsp;q .Rn /k:
(3.243)
We prove kf jLrAPsp;q .Rn /k c kf jLrp .Rn /k c 0 kf jLrAsp;q .Rn /k
(3.244)
where the second inequality is covered by (3.242). This means in case of the Bspaces that one has to estimate the corresponding terms in (3.226) with J 2 N . Let j 2 N , m 2 Zn and G 2 G . Then it follows from (3.229), (3.231) and the orthogonality of the wavelets in the homogeneous system (3.219), jn j;G m .f / D 2
X
L .f /‰L ; 2j n=2 ‰G;m j
(3.245)
L2Zn
and for some C D C.‰/ > 0, reflecting the overlap of ‰L , X
jn jj;G m .f /j c 2
jL .f /j
Q0;L CQj;m
c02
j n jpn0
jn
c0 2 p
X
jL .f /jp
Q0;L CQj;m
X Q0;L CQj;m
jL .f /jp
1=p (3.246)
1=p :
3.5 Equivalent norms and Fourier multipliers
87
For fixed G 2 G , J 2 N , M 2 Zn and q < 1 (modification if q D 1) one obtains 1 q=p X X n p 2j.s p /q jj;G .f /j m j DJ
mW.j;m/2PP J;M 1 X
c
2sj q
j DJ
X
jL .f /jp
(3.247)
q=p :
Q0;L CQJ;M
Recall that L .f / is given by (3.231). Then one has Z X X jL .f /jp c Q0;L CQJ;M
Q0;L CQJ;M
c
0
Z
jf .x/jp dx
CQ0;L
(3.248)
jf .x/j dx: p
C 0 QJ;M
From (3.247), (3.248) and s > 0 follows n
2J. p Cr/
X 1 X j DJ G2G
n
2j.s p /q
X
p jj;G m .f /j
mW.j;m/2PP J;M
q=p 1=q (3.249)
n
c 2J. p Cr/ kf jLp .CQJ;M /k: In (3.226) applied to f there are no terms with J 2 N0 and j 2 N0 . Then one has by (3.249), (3.233) (so far as a notation) with f and j;G m .f / and (2.3) s kf jLr BP p;q .Rn /k c kf jLrp .Rn /k:
(3.250)
The corresponding assertion for the F -spaces follows by embedding of the related sequence spaces (the right-hand side of (3.250) is independent of q). Then (3.242)– (3.244) prove the converse of (3.235). Remark 3.51. Theorem 3.50 is the Morreyfied counterpart of (3.218) with (3.217). The possibility of having equivalent quasi-norms in (inhomogeneous) spaces consisting of two homogeneous quasi-norms is very useful for several purposes. We used (3.218) in [T13, Theorem 1.25, p. 17] to prove that (the Fourier-analytical version of) the Riesz transform Rk in (1.10) generates a linear and bounded map in Asp;q .Rn / if 1 < p < 1. This will be extended in Section 3.5.2 below to some space LrAsp;q .Rn / based on (3.234). Observations of type (3.218) and now also of type (3.234) are also useful to study Gagliardo-Nirenberg inequalities, [Tri14] and [T13, Chapter 4]. But this will not be done here. However it seems to be of interest to justify that the two terms on the right-hand side of (3.234) are homogeneous quasi-norms (which will not be needed later on explicitly). It is sufficient to deal with dyadic dilations % D 2k ,
88
3 Hybrid spaces
k 2 Z. Let j 2 Z, G 2 G and m 2 Zn . Then one has by (3.231) and (3.220) Z n Y j j;G k jn m f .2 / D 2 f .2k x/ Gl 2 xl ml dx Rn
D 2.j k/n
lD1 n Y
Z
f .x/ Rn
D
jmk;G .f
Gl
2j k xl ml dx
(3.251)
lD1
/:
We insert (3.251) in (3.226) and use Qj k;m QJ k;M if Qj;m QJ;M . Then one has n n s s k f .2k / jLr bPp;q .Rn /k D 2k. p Cr/ 2k.s p / k.f / jLr bPp;q .Rn /k: (3.252) Similarly for the F -spaces using j;m .x/ D j k;m .2k x/. Hence, s s k f .2k / jLr aP p;q .Rn /k D 2k.sCr/ k.f / jLr aP p;q .Rn /k
(3.253)
and by (3.233) kf .%/ jLr APsp;q .Rn /k %sCr kf jLr APsp;q .Rn /k
(3.254)
for all admitted parameters s; p; q; r and % > 0. If 1 < p < 1, n=p r < 0 and % > 0 then one has by (2.160), kf .%/ jLrp .Rn /k %r kf jLrp .Rn /k;
f 2 Lrp .Rn /:
(3.255)
Hence the two terms on the right-hand side of (3.234) are homogeneous with the indicated factors. We will not directly use this observation. But it is the underlying property also for what follows.
3.5.2 Fourier multipliers and Riesz transforms We rely mainly on (3.234) in connection with the (inhomogeneous) spaces LrAsp;q .Rn /. This may justify that we recall the original Fourier-analytical definition of the homogeneous spaces n LrAPsp;q .Rn / D APs; p;q .R /;
D
1 r C ; p n
(3.256)
in analogy to (3.216) without further discussions of the topological background. Let '; ' j be as in (3.213), (3.214), 0 < p 1 (p < 1 for F -spaces), 0 < q 1, s 2 R and n=p r < 1. Then X 1=q n s b/_ jLp .QJ;M / q kf jLrBP p;q .Rn /k D sup 2J. p Cr/ 2jsq .' j f J 2Z;M 2Zn
j J
(3.257)
3.5 Equivalent norms and Fourier multipliers
89
and s .Rn /k D kf jLr FPp;q
X ˇ 1=q ˇ n b/_ ./ˇq 2J. p Cr/ 2jsq ˇ.' j f jLp .QJ;M / :
sup J 2Z;M 2Zn
j J
(3.258) This is the homogeneous counterpart of Definition 3.36(i). These homogeneous spaces have been considered in [YSY10, Chapter 8] (where one finds further references) and [LSUYY12]. In particular one has (3.233) for all admitted parameters, [LSUYY12, Theorem 6.3, pp. 1106/1107]. Fourier multipliers for these spaces have been studied in [YaY10, Theorem 4.1, p. 3819], [YYZ12, Theorem 1.5, p. 6] and [YaY13a]. In particular one has for all admitted parameters b/_ jLrAPs .Rn / c .hf p;q where k 2 N with
sup x2Rn ;j˛jk
jxjj˛j jD ˛ h.x/j kf jLrAPsp;q .Rn /k (3.259)
( n ; n k> C p 2 n max p1 ; q1 ;
B-spaces; F -spaces:
(3.260)
This is the extension of the Michlin version of Fourier multiplier theorems for APsp;q .Rn / according to [Pee75, Corollary 5.1, p. 129] and [T83, Theorem 5.2.2, p. 241] to LrAPsp;q .Rn /. The corresponding assertion in [YaY10, Theorem 4.1, p. 3819] is formulated in terms of the sharper Peetre version Z ˇ _ ˇ _ r Ps n b .1Cjja /ˇ ./h.2j / ./ˇ dkf jLrAPsp;q .Rn /k k.hf / jL Ap;q .R /k c sup j 2Z Rn
where
.x/ D ' 0 .x/ D '.x/ '.2x/ as in (3.213), (3.214) and ( a>
n ; p
B-spaces; n max p ; ; F -spaces: 1
1 q
(3.261)
(3.262)
Conditions of this type appear for the first time by Peetre in [Pee75] in connection s .Rn /. Although well known we justify that (3.259) follows from with the spaces FPp;q (3.261). One has for fixed j 2 Z, Z ˇ _ ˇ .1 C jja /ˇ ./h.2j / ./ˇ d Rn 1=2 X nZ ˇ ˇ ˇ ./h.2j / _ ./ˇ2 d 2l 2 c (3.263) jj2l l0 1=2 Z X ˇ ˇ n ˇ ./h.2j / _ ./ˇ2 d Cc 2l.aC 2 / : l>0
jj2l
90
3 Hybrid spaces
Here jj 2l means 2l1 jj 2l . The terms with l 0 can be estimated from above by Z 1=2 j .x/ h.2j x/j2 dx c sup jh.2j x/j: (3.264) Rn
jxj1
This is covered by ˛ D 0 on the right-hand side in (3.259). Let l > 0 and " > 0. Then the second term on the right-hand side of (3.263) can be estimated from above by Z 1=2 X ˇ ˇ n ˇ ./h.2j / _ ./ˇ2 d 22l.aC"C 2 / : (3.265) c jj2l
l>0
Recall .D ˇ g/_ ./ D .i /jˇ j ˇ g _ ./. Let a C n2 < k 2 N. Then (3.265) can be estimated from above similarly as in (3.264) by 1=2 X X Z ˇ ˇ ˇ ˇ ˇD ˛ .x/ h.2j x/ ˇ2 dx c c0 2j jˇ j sup ˇ D ˇ h .2j x/ˇ: j˛jDk
Rn
jxj1
jˇ jk
(3.266) This is again covered by the right-hand side of (3.259). Then (3.259), (3.260) follows from (3.261)–(3.263) and the above estimates. Instead of the Michlin version in terms of L1 -norms one obtains by the above arguments also the H¨ormander version in terms of L2 -norms (which is between the Peetre version and the Michlin version). We refer the reader in this context again to [YYZ12, Theorem 1.5, p. 6]. Let _ k Rk g.x/ D i k D 1; : : : ; n; (3.267) b g .x/; x 2 Rn ; jj be the Riesz transforms according to (1.10) = (2.185) in their Fourier-analytical version. Let LrAsp;q .Rn / and LV rAsp;q .Rn / be the hybrid spaces as introduced in Definition 3.18, Notation 3.22. Recall that ,! means linear and bounded embedding. Theorem 3.52. Let 1 < p < 1, 0 < q 1, s 2 R and n=p r < 0. Then
and
Rk W
LrAsp;q .Rn / ,! LrAsp;q .Rn /
(3.268)
Rk W
V rAsp;q .Rn /: LV rAsp;q .Rn / ,! L
(3.269)
Proof. Step 1. Let, in addition, s > 0. Then it follows from (3.234), (3.259) with h D k =jj and Corollary 2.27(ii) (applied to the integral version of Rk ) Rk W
LrAsp;q .Rn / ,! LrAsp;q .Rn /;
s > 0:
(3.270)
ı 2 R;
(3.271)
Step 2. Let Iı , b _; Iı f D .1 C jj2 /ı=2 f
f 2 S 0 .Rn /;
3.5 Equivalent norms and Fourier multipliers
91
be the well-known lifts in Asp;q .Rn /, recalled in (3.19), (3.20). According to [T13, Theorem 2.39, p. 88] these are also lifts in the local spaces LrAsp;q .Rn /. The proof given there applies also to the hybrid spaces LrAsp;q .Rn /, hence n Iı LrAp;q .Rn / D LrACı p;q .R /;
ı 2 R;
2 R:
(3.272)
We formalize this observation later on in Section 3.6.5. This is also covered by Theorem 3.38 and [YSY10, Proposition 5.1, p. 142] (some restrictions for r) and [Sic12, I, Corollary 3.2, p. 131, Proposition 3.5, p. 129]. Then Rk D Iı ı Rk ı Iı W
LrAp;q .Rn / ,! LrAp;q .Rn /;
(3.273)
2 R, follows from (3.270), (3.272) with C ı D s > 0. This proves (3.268) in all cases. Step 3. We prove (3.269). Let f 2 D.Rn /. Then it follows from (2.171), (2.173) that ˇ ˇ c˛ Rk f 2 C 1 .Rn / and ˇD ˛ Rk f .x/ˇ ; jxjn
jxj 1;
(3.274)
˛ 2 Nn0 . Let WpK .Rn /, K 2 N, 1 < p < 1, be the usual Sobolev spaces, % > 1 and QJ;M fx 2 Rn W jxj > %g. Then n 2J. p Cr/ Rk f jWpK .QJ;M / c 2J r %n
if J 2 N0
(3.275)
and n n 1 2J. p Cr/ Rk f jWpK .QJ;M / c 2J. p Cr/ %n.1 p /
if J 2 N ;
(3.276)
where c is independent of J and M . Let % 2 D.Rn / be a cut-off function with n % .x/ D 1 if jxj %. Then % Rk f 2 D.R / and it follows from (3.275), (3.276) n n with 1 < p < 1, K 2 N and 0 p C r < p , lim Rk f
%!1
% Rk f
jLr WpK .Rn / D 0:
(3.277)
Hence Rk f 2 LV r WpK .Rn /. This covers also (3.269) using WpK .Rn / ,! Asp;q .Rn / if K > s. Remark 3.53. Compared with Definition 3.18 we did not use in Step 3 of the above ‰ if J 2 N0 . proof approximations by VJ;u
92
3 Hybrid spaces
3.6 Some properties 3.6.1 Embeddings We studied in [T13] properties of local spaces LrAsp;q .Rn / in some details. This will not be carried over systematically to the hybrid spaces LrAsp;q .Rn / based on (3.99) and Notation 3.22. But it is quite clear by (3.99) that LrAsp;q .Rn / are hybrid versions between the local spaces LrAsp;q .Rn / and the global spaces Asp;q .Rn /. We collect in Section 3.6 some properties which are useful in our context. A few of them have been already mentioned in the preceding sections with a reference to what follows. First we ask for some embeddings. Recall that C .Rn /, 2 R, are H¨older-Zygmund spaces according to (3.24). Proposition 3.54. Let 0 < p; q 1 .p < 1 for F -spaces/; s 2 R and n=p r < 1. Then (3.278) LrAsp;q .Rn / ,! C sCr .Rn /: If, in addition, r > 0 then LrAsp;q .Rn / D C sCr .Rn /:
(3.279)
Proof. This is the counterpart of a corresponding assertion for the local spaces LrAsp;q .Rn / according to [T13, Theorem 2.1, p. 45]. In particular, (3.278) follows from (3.99) and [T13, (2.5), p. 45], LrAsp;q .Rn / ,! LrAsp;q .Rn / ,! C sCr .Rn /:
(3.280)
Let r > 0 and f 2 C sCr .Rn /. Then one has by (3.99) and [T13, (2.6), p. 45], kf jLrAsp;q .Rn /k kf jC sCr .Rn /k C
n
sup J 2N ;M 2Zn
2J. p Cr/ kf jAsp;q .2QJ;M /k:
(3.281) According to Theorem 3.26 and (3.120) it remains to show in case of the B-spaces n
2J. p Cr/
X 1 X j D0 G2G j
n
2j.s p /q
X
p jj;G m .f /j
q=p 1=q
mW.j;m/2PJ;M
(3.282)
c kf jC sCr .Rn /k for some c > 0, all J 2 N D N, M 2 Zn (and all f 2 C sCr .Rn /). The j;G F -spaces can be incorporated afterwards by embedding. Here m .f / is given by (3.126)=(3.60). Recall kf jC sCr .Rn /k sup 2j.sCr/ jj;G m .f /j; j;G;m
(3.283)
3.6 Some properties
93
Theorem 3.12 and the references given there. By embedding we may assume q < 1 in (3.282). Then one has 1 X X j D0 G2G j
n
2j.s p /q
X
p jj;G m .f /j
q=p
mW.j;m/2PJ;M
c kf jC sCr .Rn /kq
1 X
n
n
2j.rC p /q 2.j J / p q
(3.284)
j D0
c2
n p Jq
kf jC
sCr
.Rn /kq ;
where we used r > 0. Then (3.282) follows from (3.284) and J r < 0.
Remark 3.55. Both assertions of the above proposition are known, (3.278) may be found in [YSY10, Proposition 2.6, p. 45] and (3.279) is covered by [YaY13, Theorem 2, p. 560], based on Theorem 3.38. The embedding (3.278) is strict for all spaces LrAsp;q .Rn / with r < 0 (which means that the two spaces involved do not coincide). This follows from (3.280) and a corresponding assertion for the local spaces LrAsp;q .Rn / in [T13, Remark 2.3, pp. 47/48]. Remark 3.56. We dealt in [T13, Sections 2.1.2, 2.1.3, pp. 48–53] with embeddings between local spaces LrAsp;q .Rn /. One may ask for corresponding assertions in terms of the hybrid spaces spaces LrAsp;q .Rn /. This will not be done in detail. Some assertions, partly in terms of necessary and sufficient conditions, have been obtained recently in [YHSY14]. Reformulating the restrictions in [YHSY14] for the parameters according to Theorem 3.38 then it is again clear that limiting embeddings both for hybrid spaces and local spaces with p0 into corresponding spaces with p1 where p1 > p0 are governed by s1 C r1 D s0 C r0 and
(differential dimension invariance)
r1 p1 D r0 p0 n
(slope invariance):
(3.285) (3.286)
For hybrid spaces the second terms on the right-hand side of (3.99) must be taken into account. One has 2J r1 2J r0 , J 2 N , if r1 r0 . But it is not so clear whether this monotonicity for r is really helpful for embeddings in hybrid spaces. Even worse, if r0 < 0, r1 < 0 then Lr0 Asp;q .Rn / is embedded in Lr1 Asp;q .Rn / if, and only if, r0 D r1 . This assertion is covered by [YSY10, Proposition 2.2, p. 40] reformulated according to Theorem 3.38. However there is the following monotonicity with respect to p. Proposition 3.57. Let 0 < p1 p0 1 .p0 < 1 for F -spaces/; 0 < q 1, s 2 R and n=p0 r < 1. Then
Lr Bps 0 ;q .Rn / ,! Lr Bps 1 ;q .Rn /;
Lr Bps 0 ;q .Rn / ,! Lr Bps 1 ;q .Rn /
(3.287)
94
3 Hybrid spaces
and
Lr Fps0 ;q .Rn / ,! Lr Fps1 ;q .Rn /;
Lr Fps0 ;q .Rn / ,! Lr Fps1 ;q .Rn /:
(3.288)
Proof. We rely on Theorem 3.26. Then (3.287) follows from (3.120), (3.123) and 1=p1 1=p0 X X n.j J /. p1 p1 / p1 j;G p0 1 0 jj;G j c 2 j j : m m mW.j;m/2PJ;M
mW.j;m/2PJ;M
(3.289) Similarly one obtains (3.288) from (3.122), (3.124) and X ˇq 1=q ˇ ˇ 2jsq ˇj;G jLp1 .Rn / m j;m ./ .j;G;m/2PJ;M
c2
nJ. p1 p1 / 1
0
ˇq 1=q ˇ n ˇ 2jsq ˇj;G ./ jL .R / : j;m p0 m
X
(3.290)
.j;G;m/2PJ;M
Remark 3.58. More general assertions of this type for the local spaces LrAsp;q .Rn / may be found in [T13, Section 2.1.2, pp. 48/49]. This can be extended to the hybrid spaces LrAsp;q .Rn /. Furthermore, if 0 < q 1 and s 2 R then Asp0 ;q .Rn / ,! Ln=p0 Asp1 ;q .Rn /;
0 < p1 p0 1;
(3.291)
(p0 < 1 for F -spaces) where we used in (3.102). There is no embedding of Asp0 ;q .Rn / into Asp1 ;q .Rn / if p1 < p0 . Hence (3.291) may be considered as a substitute. Remark 3.59. It might be of some interest that s s L0 Fp;q .Rn / D L0 Fp;q .Rn /;
0 < p < 1;
0 < q 1;
s 2 R:
(3.292)
This can be justified as follows. One has by (3.99) s s kf jL0 Fp;q .Rn /k kf jL0 Fp;q .Rn /k C
n
sup J 2N ;M 2Zn
s 2J p kf jFp;q .2QJ;M /k:
(3.293) From the localization principle for F -spaces according to [T92, Theorem 2.4.7, p. 124] (nowadays also an easy consequence of wavelet characterizations) follows for J 2 N , n
n
n
s .2QJ;M /k c 2J p 2J p 2J p kf jFp;q
sup 2Q0;L 4QJ;M
s kf jFp;q .2Q0;L /k: (3.294)
This shows that the second terms in (3.293) can be incorporated in the first ones. This 0 proves (3.292). Then one has by [T13, (3.83), p. 122] with Lp .Rn / D Fp;2 .Rn /, L0 Lp .Rn / D L0 Lp .Rn / D bmo.Rn /; n
This justified (3.191) with bmo.R / as in (3.189).
2 p < 1:
(3.295)
95
3.6 Some properties
3.6.2 Multiplication algebras In Section 3.2.4 we described under which conditions the global space Asp;q .Rn / is a multiplication algebra. We refer the reader also to [T13, Section 1.2.5, pp. 12, 13]. There one finds the necessary references covering in particular the sophisticated question which is meant by f1 f2 if, say, f1 2 Asp;q .Rn / and f2 2 Asp;q .Rn /. This will not be repeated here. In [T13, Section 2.5, pp. 89–95] we dealt with multiplication algebras for the local spaces LrAsp;q .Rn /. We are again in the comfortable position that the (rather sketchy) arguments given there apply also to the hybrid spaces LrAsp;q .Rn /. We do not repeat the technicalities. As there we ask for spaces LrAsp;q .Rn / such that kf1 f2 jLrAsp;q .Rn /k c kf1 jLrAsp;q .Rn /k kf2 jLrAsp;q .Rn /k
(3.296)
for some c > 0 and all f1 ; f2 2 LrAsp;q .Rn /. We assume from the very beginning n LrAsp;q .Rn / Lloc 1 .R /:
(3.297)
Then one can interpret f1 .x/ f2 .x/ first pointwise almost everywhere and ask whether this product belongs in addition to LrAsp;q .Rn / with (3.296) if both f1 and f2 are elements of LrAsp;q .Rn /. Theorem 3.60. Let 0 < p; q 1 .p < 1 for F -spaces/; s 2 R and n=p r < 1. Let n LrAsp;q .Rn / Lloc (3.298) 1 .R /: (i) If LrAsp;q .Rn / is a multiplication algebra then s C r 0. (ii) If s C r > 0 then LrAsp;q .Rn / is a multiplication algebra. Proof. This is the counterpart of [T13, Theorem 2.43, p. 91] with the same assertions as for the spaces LrAsp;q .Rn /. The outlined proof, based on paramultiplication for wavelets, works also for the spaces LrAsp;q .Rn /, where we use now (3.278) with s C r > 0. Remark 3.61. If LrAsp;q .Rn / is a multiplication algebra then one has LrAsp;q .Rn / ,! L1 .Rn /;
(3.299)
by the same arguments as in [T13, Proposition 2.41, p. 90]. By (3.278) this is the case if s C r > 0. But this assertion cannot be extended to s C r D 0. If, in addition, r > 0 then one has by (3.279) LrAsp;q .Rn / D C 0 .Rn /;
s C r D 0;
r > 0:
(3.300)
0 .Rn /, which is larger than bmo.Rn /, is not embedded in But C 0 .Rn / D B1;1 n L1 .R /, [RuS96, p. 30], based on [SiT95]. We refer the reader in this context also to [YSY10, Corollary 6.1, p. 157].
96
3 Hybrid spaces
Corollary 3.62. Theorem 3.60 remains valid if one replaces LrAsp;q .Rn / by LV r Asp;q .Rn /. Proof. This follows immediately from Theorem 3.60 and Definition 3.18.
Problem 3.63. In connection with Theorem 3.60 one has the following two problems. First it would be of interest to clarify which spaces LrAsp;q .Rn / with s C r D 0 are multiplication algebras. In case of the global spaces Asp;q .Rn / we recalled in Theorem 3.14 the final answer. Furthermore one may ask whether (3.296) with, say, s C r > 0 can be strengthened by kf1 f2 jLrAsp;q .Rn /k c kf1 jLrAsp;q .Rn /k kf2 jL1 .Rn /k C c kf1 jL1 .Rn /k kf2 jLrAsp;q .Rn /k:
(3.301)
In case of the global spaces Ln=p Asp;q .Rn / D Asp;q .Rn / we refer to [RuS96, Theorem 2, p. 222]. This has been extended in [YSY10, Theorem 6.3, p. 156] to some 1 r n r s n spaces As; p;q .R / D L Ap;q .R /, D p C n , according to Theorem 3.38. We refer the reader also to [Sic12, I, Theorem 3.10.3, p. 141]. Such assertions are not only of interest for its own sake but also very useful for so-called persistency properties of (unique global) solutions of Navier-Stokes equations. We return to questions of this type in Remark 5.18 and and Chapter 6. Related greater details and the role of inequalities of type (3.301) may be found in [T13, pp. 212–214] and the references given there.
3.6.3 Morrey characterizations We introduced in Definition 2.1 the local Morrey spaces Lrp .Rn / and their global counterparts Lrp .Rn /, restricted by 1 < p < 1 and n=p r < 0. It is well known that the restriction for r can be extended to n=p r < 1 at the expense of polynomial approximations in the same way as in (3.94) with Lp .QJ;M / in place of Asp;q .2QJ;M /. These are the Morrey-Campanato spaces with (3.96) as a special assertion. We dealt in [T13, Chapter 3] in detail with local Morrey-Campanato spaces Lrp .Rn / and their relations to other distinguished spaces, in particular global spaces Asp;q .Rn / and local spaces LrAsp;q .Rn /. We do not try to transfer all these assertions to global Morrey-Campanato spaces Lrp .Rn / and respective hybrid spaces LrAsp;q .Rn /. There is surely a temptation to replace Lp .QJ;M / and the indicated polynomial approximations by other spaces, similarly as in (3.94). But this has apparently not yet done systematically. The only reference which we mentioned and discussed in [T13] is [KiK13]. In what follows we transfer some distinguished assertions for the local spaces LrAsp;q .Rn / to their hybrid counterparts LrAsp;q .Rn /. This can be done without substantial additional efforts. The differential dimension s C r plays a crucial role in the theory of the spaces LrAsp;q .Rn / and LrAsp;q .Rn /. We refer the reader to Section 3.6.1, especially Remark 3.56, [YHSY14] (after reformulations based on Theorem 3.38 switching notationally n r s n from As; p;q .R / to L Ap;q .R /), and also to Theorem 3.60 which shows that sCr D 0
3.6 Some properties
97
is something like a watershed. If s C r > 0 then both LrAsp;q .Rn / and LrAsp;q .Rn / are multiplication algebras. If s C r < 0 then LrAsp;q .Rn / and LrAsp;q .Rn / can be described similarly as in Definition 2.1 which may justify speaking about Morrey characterizations. What follows depends decisively on the homogeneity in the small for the spaces Asp;q .Rn / which we described in [T13, Section 2.3]. This will not be repeated here. Theorem 3.64. 8 ˆ <0 < p 1
Let
< 1; 0 < q 1; n max p1 ; 1 1 < s < 1=p; < 1; 0 < q 1; s D 0; 1; 0 < q 1; s < 0;
for B-spaces and 8 ˆ <0 < p < 1; 0 < q 1; 1 < p < 1; 1 q < 1; ˆ : 0 < p < 1; 0 < q 1;
(3.302)
n max p1 ; q1 ; 1 1 < s < 1=p; s D 0; s < 0;
(3.303)
for F -spaces, n=p r < 1 and s C r < 0. Let f 2 S 0 .Rn /. Then f 2 LrAsp;q .Rn / if, and only if, n
sup J 2Z;M 2Zn
2J. p Cr/ kf jAsp;q .2QJ;M /k
(3.304)
is finite .equivalent quasi-norms/. Proof. This follows immediately from a corresponding assertion for the local spaces LrAsp;q .Rn / according to [T13, Theorem 2.29, p. 75] and (3.99). We return to the above discussion about polynomial approximation. Let again Pk with k 2 N0 be the collection of all polynomials in Rn of degree less than or equal to k, complemented by P1 D f0g. Recall N D N D fJ 2 Z W J 2 Ng. Corollary 3.65. Let p; q; s and r be as in Theorem 3.64. Let k 2 N1 . Let f 2 S 0 .Rn /. Then f 2 LrAsp;q .Rn / if, and only if, sup J 2N0
;M 2Zn
n
2J. p Cr/ inf kf P jAsp;q .2QJ;M /k P 2Pk
C
sup J 2N ;M 2Zn
n
2J. p Cr/ kf jAsp;q .2QJ;M /k
(3.305)
is finite .equivalent quasi-norms/. Proof. This follows immediately from Theorem 3.64 and (3.94), (3.95). One may also consult [T13, Remark 2.30, pp. 42/77] where we justified a corresponding assertion for the local spaces LrAsp;q .Rn /.
98
3 Hybrid spaces
One can extend the above observations to all spaces LrAsp;q .Rn /. Let again r C D max.r; 0/, r 2 R. Corollary 3.66. Let 0 < p; q 1 .p < 1 for F -spaces/; s 2 R and n=p r < 1. Let m 2 N0 with s C r C < m. Let f 2 S 0 .Rn /. Then f 2 LrAsp;q .Rn / if, and only if, n sup 2J. p Cr/ D ˛ f jAsm (3.306) p;q .2QJ;M / J 2Z;M 2Zn ; 0j˛jm
is finite .equivalent quasi-norms/. Proof. The corresponding assertion for the first terms on the right-hand side of (3.99), hence for the local spaces LrAsp;q .Rn /, is covered by [T13, Corollary 2.31, p. 78]. As for the second terms on the right-hand side of (3.99) we recall X kf jAsp;q .2QJ;M /k kD ˛ f jAsm (3.307) p;q .2QJ;M /k j˛jm
uniformly in J 2 N and M 2 Z , [T08, Proposition 4.21, p. 113] and its proof. n
0 Remark 3.67. Recall that Lp .Rn / D Fp;2 .Rn /, 1 < p < 1, as remarked in (3.16). Then one has by (3.304) and (2.3) for the Morrey spaces Lrp .Rn /, 0 Lrp .Rn / D Lr Lp .Rn / D Lr Fp;2 .Rn /;
1 < p < 1;
n=p r < 0: (3.308) It is the counterpart of Lrp .Rn / D Lr Lp .Rn / according to [T13, (3.81), p. 121] where again 1 < p < 1, n=p r < 0. Corollary 3.65 applied to (3.308) and its local counterpart have a long history and go back essentially to Campanato, [Cam63, Cam64]. We dealt in [T13, Chapter 3] in detail with Morrey-Campanato spaces, also in the larger context of measurable functions admitting 0 < p 1 in Definition 2.1 (where spaces with p < 1 cannot be interpreted in the framework of S 0 .Rn //. There one finds also further references, some of them have already been mentioned in Remark 2.2. Assertion (3.308) may also be found in [YSY10, Corollary 3.3, pp. 63, 64] with related references to [SaT07, Saw08, SYY10], which can be complemented by [Ros12]. m Remark 3.68. Let Wpm .Rn / D Fp;2 .Rn / with m 2 N0 , 1 < p < 1, be the classical Sobolev spaces according to (3.18). Then the corresponding hybrid (or Morreyfied) spaces
Lr Wpm .Rn /;
m 2 N0 ;
1 < p < 1;
n=p r < 0;
(3.309)
2J. p Cr/ kD ˛ f jLp .QJ;M /k:
(3.310)
can be equivalently normed by X kf jLr Wpm .Rn /k kD ˛ f jLrp .Rn /k j˛jm
sup J 2Z;M 2Zn ; 0j˛jm
n
3.6 Some properties
99
This looks reasonable but it is not immediately covered by (3.306). Here Lrp .Rn / are again the Morrey spaces identified in (3.308) with Lr Lp .Rn / D Lr Wp0 .Rn /:
(3.311)
The first equivalence in (3.310) follows again from (3.99), the local version according to [T13, (2.239), p. 78] with s D m, complemented by (3.307) and based on (3.308). Then one obtains the second equivalence from (2.3). Sometimes Lr Wpm .Rn / are called Morrey-Sobolev spaces with the first equivalence in (3.310) as definition (extended to p D 1). These spaces attracted a lot of attention and the case m D 1 and n D 2 goes essentially back to Morrey, [Mor38]. His celebrated assertion in R2 , called a lemma, can be formulated in the above context (now in Rn with 2 n 2 N) as follows: If f 2 Lrp .Rn / and jrf j 2 Lrp .Rn / with 1 < p < 1 and max pn ; 1 < r < 0 then f 2 C 1Cr .Rn /:
(3.312)
This is a special case of (3.278), Lr Wpm .Rn / ,! C mCr .Rn /;
m 2 N0 ;
s Lr Hps .Rn / D Lr Fp;2 .Rn /;
s 2 R;
1 < p < 1;
n=p r < 0 (3.313) (and (3.279) if r > 0). In this context we wish also to mention the (fractional) Morrey-Sobolev spaces 1 < p < 1;
n=p r < 1; (3.314) (especially s > 0 and n=p r < 0) which have been studied in detail in [Ada75, AdX12]. One may also consult [AdH96, p. 79]. Of special interest are sharp embeddings Lr0 Hps00 .Rn / ,! Lr1 Lp1 .Rn / D Lrp11 .Rn /: (3.315) A detailed description may be found in [T13, pp. 122/123]. Corresponding sharp embeddings Lr0 Asp00 ;q0 .Rn / ,! Lr1 Asp11 ;q1 .Rn / (3.316) have been proved quite recently in [YHSY14] with references to [HaS12, HaS14] in n terms of the spaces As; p;q .R /. After reformulation according to Theorem 3.38 the conditions obtained there look more natural and handsome as indicated in Remark 3.56. Then they can also be compared with [T13, Sections 2.1.3, 3.3.2, pp. 50–53, 122–123] (replacing the local spaces LrAsp;q .Rn / by the hybrid spaces LrAsp;q .Rn /).
3.6.4 Pointwise multipliers and diffeomorphisms By Theorem 3.64 and Corollary 3.66 one can transfer some properties from the global spaces Asp;q .Rn / to the hybrid spaces LrAsp;q .Rn /. This applies in particular to pointwise multipliers and diffeomorphisms. The same has been done in [T13, Section 2.7.1, pp. 98–100] with respect to the local spaces LrAsp;q .Rn /. We follow now the
100
3 Hybrid spaces
representation given there closely. Recall that C k .Rn / with k 2 N0 is the collection of all complex-valued continuous functions g in Rn having classical continuous derivatives up to order k inclusively with kg jC k .Rn /k D
jD ˛ g.x/j < 1:
sup
(3.317)
j˛jk;x2Rn
As usual C 0 .Rn / D C.Rn /. Let A.Rn / be a distributional quasi-Banach space in Rn with the continuous embeddding S.Rn / ,! A.Rn / ,! S 0 .Rn /
(3.318)
in analogy to (2.16). Then g 2 L1 .Rn / is said to be a pointwise multiplier for A.Rn / if f 7! g f generates a bounded map in A.Rn /: (3.319) Of course one has to say what this multiplication means. But here we rely on related previous considerations. If A.Rn / D Asp;q .Rn / then one finds careful discussions and rather final assertions in [RuS96, Chapter 4], [T83, Section 2.8], [T92, Section 4.2], [T06, Section 2.3] and most recently [Scha13]. In particular for any given space Asp;q .Rn / there is a sufficiently large number k 2 N (depending on s; p; q and A D B or A D F ) and a constant c > 0 such that kgf jAsp;q .Rn /k c kg jC k .Rn /k kf jAsp;q .Rn /k
(3.320)
for all g 2 C k .Rn / and f 2 Asp;q .Rn /. This theory has been extended in [T13, Section 2.7.1, pp. 98–100] to the local spaces LrAsp;q .Rn /. This will now be transferred to the hybrid spaces LrAsp;q .Rn / without essential changes. Let k 2 N. A continuous one-to-one map y D .x/of Rn onto Rn is called an k-diffeomorphism if the components j .x/ of .x/ D 1 .x/; : : : ; n .x/ have ˛ n classical ˇ derivatives ˇ up to order k inclusively with D j n2 C.R / for 0 < j˛j k and if ˇdet .x/ˇ c > 0 for some c and all x 2 R . Here stands for the Jacobian matrix. Recall that the inverse 1 of an k-diffeomorphism is also a kdiffeomorphism. According to [T92, Section 4.3] for given Asp;q .Rn / and sufficiently large k 2 N (depending on s; p; q and A D B or A D F ) (3.321) D f D f ı D f ./ makes sense for any f 2 Asp;q .Rn / and any k-diffeomorphism D W
Asp;q .Rn / ,! Asp;q .Rn /
. Furthermore
is an isomorphic map.
(3.322)
We refer the reader to [Scha13] where one finds a careful discussion about the smoothness of the components of 1 for k-diffeomorphisms and how large k must be to ensure (3.322). The described pointwise multiplier properties and diffeomorphic maps can be transferred from Asp;q .Rn / to LrAsp;q .Rn / quite easily. We use (3.319)–(3.322) with LrAsp;q .Rn / in place of Asp;q .Rn /.
3.6 Some properties
101
Theorem 3.69. Let LrAsp;q .Rn / with 0 < p; q 1 .p < 1 for F -spaces/; s 2 R and n=p r < 1 be the spaces according to Definition 3.18 and Notation 3.22. (i) There is a number k 2 N and a constant c > 0 such that kg f jLrAsp;q .Rn /k c kg jC k .Rn /k kf jLrAsp;q .Rn /k
(3.323)
for all g 2 C k .Rn / and f 2 LrAsp;q .Rn /. (ii) There is a number k 2 N such that D W
LrAsp;q .Rn / ,! LrAsp;q .Rn /
for any k-diffeomorphism
is an isomorphic map
(3.324)
in Rn .
Proof. This follows from the above-described pointwise multiplier assertions and diffeomorphisms for the spaces Asp;q .Rn / applied to Corollary 3.66. In part (ii) one uses .QJ;M / C QJ;M 0
for all J 2 Z;
M 2 Zn ;
some C > 0 and M 0 D M 0 .J; M / 2 Zn .
(3.325)
Remark 3.70. One has by (3.279) LrAsp;q .Rn / D C sCr .Rn /;
s 2 R;
r > 0:
(3.326)
Then the above theorem is covered by corresponding sharp assertions in [T92, Corollary 4.2.2, p. 205] and [Scha13], even with some improvements in terms of the spaces C % .Rn /, % > 0, in place of C k .Rn /. If r < 0 and s < 0 then one can apply Theorem 3.64. This shows that sharp assertions about k 2 N in C k .Rn / or even % > 0 in C % .Rn / in the above theorem can be transferred from [T92, Sections 4.2, 4.3, pp. 201– 211] and in particular [Scha13] from Asp;q .Rn / to LrAsp;q .Rn /. Afterwards one can apply Corollary 3.66 which gives at least more or less natural (maybe not optimal) estimates for k 2 N and % > 0 in the other cases. Remark 3.71. As previously noted we followed closely [T13, Section 2.7.1, pp. 98– 100]. There one finds counterparts of the above theorem for the local spaces LrAsp;q .Rn /. The properties themselves are known, based on Theorem 3.38. We refer the reader to [YSY10, Sections 6.1, 6.2, pp. 147–162].
3.6.5 Lifts As mentioned in (3.19), (3.20) the classical lifts b _ with hi D .1 C jj2 /1=2 ; Iı W f 7! hiı f
2 Rn ;
ı 2 R; (3.327)
n map the global spaces Asp;q .Rn / isomorphically onto AsCı p;q .R /, n Iı Asp;q .Rn / D AsCı p;q .R /;
(3.328)
102
3 Hybrid spaces
for all admitted parameters 0 < p 1 (p < 1 for F -spaces), 0 < q 1, s 2 R, A 2 fB; F g, [T83, Section 2.3.8, pp. 58/59]. We extended this assertion in [T13, Section 2.4.3, pp. 88/89] to the local spaces LrAsp;q .Rn /, hence n Iı LrAsp;q .Rn / D LrAsCı p;q .R /
(3.329)
for all s; p; q, A 2 fB; F g as above and n=p r < 1. The proof given there applies also to the hybrid spaces LrAsp;q .Rn /. Theorem 3.72. Let LrAsp;q .Rn / with 0 < p; q 1 .p < 1 for F -spaces/; s 2 R and n=p r < 1 be the spaces according to Definition 3.18 and Notation 3.22. Let Iı be as in (3.327) with ı 2 R. Then Iı maps LrAsp;q .Rn / isomorphically onto n LrAsCı p;q .R /, n (3.330) Iı LrAsp;q .Rn / D LrAsCı p;q .R /: Proof. The proof of (3.329) in [T13, pp. 88/89] applies also to the hybrid spaces LrAsp;q .Rn /. Remark 3.73. Assertions of type (3.330) may be found in [YSY10, Proposition 5.1, p. 142] based on Theorem 3.38 (some restrictions of the parameters).
3.6.6 Thermic characterizations: comments and proposals One can take the Morrey characterizations according to Theorem 3.64 and Corollary 3.66, but also Morrey’s original observation (3.312), to ask for similar assertions in terms of other distinguished means. In connection with heat and Navier-Stokes equations characterizations based on the Gauss-Weierstrass semi-group Z _ jxyj2 1 2 Wt w.x/ D e 4t w.y/ dy D et jj w b./ .x/; (3.331) n=2 .4 t/ Rn x 2 Rn , t > 0, are of special interest. We have no final answers but some proposals, formulated as problems. Nothing will be used later on in connection with heat and Navier-Stokes equations but we indicate how closely these (possible) characterizations are related to some current assertions about Navier-Stokes equations. First we recall thermic characterizations of the global spaces Asp;q .Rn / according to [T92, Section 2.6.4, pp. 151–155] based on [Tri88] with [T78, Section 2.5.2, pp. 190–192] and [Tri82] as a forerunners. Let s<0
and 0 < p; q 1
Then kf
s jBp;q .Rn /k
and kf
s jFp;q .Rn /k
Z
1
0
1
t
(3.332)
dt 1=q t
(3.333)
sq
t 2 kWt f jLp .Rn /kq
0
Z
(with p < 1 for F -spaces):
sq 2
1=q ˇ ˇ ˇWt f ./ˇq dt jLp .Rn / t
(3.334)
3.6 Some properties
103
(usual modification if q D 1) are equivalent quasi-norms. More precisely: f 2 s S 0 .Rn / belongs to Bp;q .Rn / if, and only if, the right-hand of (3.333) is finite. Sims n ilarly for Fp;q .R /. This coincides essentially with [T92, Theorem, p. 152]. But we b/_ jLp .Rn /k add a technical comment. In [T92] we have the additional term k.'0 f n with '0 2 D.R /, '0 .0/ 6D 0. It follows from the arguments given there, based on [T92, Theorems 2.4.1, 2.5.1, pp. 100/101, 132, Corollaries 2.4.1/1, 2.5.1/1, pp. 108, 2 134] that one can also choose '0 .x/ D ejxj . But this term can be incorporated in the integrals on the right-hand sides of (3.333), (3.334) (with t 1). In recent times the homogeneous counterparts of these quasi-norms play some role in the theory of Navier-Stokes equations. But they are treated sometimes as mathematical folklore: no doubt, no proof, vague citations, if any. This may justify adding here related comments and proper references. Recall the standard Fourieranalytical definition of the homogeneous spaces APsp;q .Rn / according to (3.216). But the topological background of these spaces requires some care. We refer the reader to [T83, Chapter 5]. In their Fourier-analytical version one has to deal with these spaces modulo polynomials. However if s; p; q are restricted by (3.332) (this means s < 0) then one can avoid this ambiguity replacing (3.216) by Z 1 sq dt 1=q s n s n P P t 2 kWt f jLp .Rn /kq kf jBp;q .R /k kf jBp;q .R /kW D t 0 (3.335) and Z 1 sq ˇ ˇq dt 1=q s n s n P P kf jFp;q .R /k kf jFp;q .R /kW D t 2 ˇWt f ./ˇ jLp .Rn / t 0 (3.336) (usual modification if q D 1). This is covered by [Tri88, Corollary 16, p. 201] with [Tri82, Corollary 1, p. 283] as a forerunner. (In [T92] we took over the related material from [Tri88] for the inhomogeneous spaces Asp;q .Rn / but not for the homogeneous spaces APsp;q .Rn /). From a technical point of view the proof of (3.335), (3.336) with s; p; q as in (3.332) is simpler than the corresponding assertions (3.333), (3.334). One can rely on the global homogeneity kf ./ jAPsp;q .Rn /k s p kf jAPsp;q .Rn /k; n
> 0:
(3.337)
From a topological point of view the inhomogeneous spaces Asp;q .Rn / are simpler than their homogeneous counterparts. But if one relies on (3.335), (3.336) then there is no longer any ambiguity modulo polynomials because Wt P 62 Lp .Rn / for polynomials P 6D 0 (if p D 1 and P D c 6D 0 a constant then one has to rely on s < 0). We wish to emphasize that s < 0 is crucial. Nothing like (3.335), (3.336) can be s expected if s 0. One can go one step further and define BP p;q .Rn / with s; p; q as in 0 n (3.332) as the collection of all f 2 S .R / such that the right-hand side of (3.335) is s finite. Similarly for FPp;q .Rn /, hence ˚ APsp;q .Rn / D f 2 S 0 .Rn / W kf jAPsp;q .Rn /kW < 1
(3.338)
104
3 Hybrid spaces
with kf jAPsp;q .Rn /kW as in (3.335), (3.336). This makes sense because Wt f is a C 1 function for any f 2 S 0 .Rn /, [T78, p. 152], [T06, pp. 5,6]. For the inhomogeneous spaces Asp;q .Rn / we dealt in detail with characterizations of type (3.338). We refer the reader to [T92, Theorem 2.6.4, p. 152]. This applies also to the homogeneous spaces APsp;q .Rn /, based on [Tri88]. The proof in [T92] that the right-hand sides of (3.333), (3.334) are not only equivalent quasi-norms but characterizations is somewhat tricky. The final assertion also for more general means may be found in [T06, Sections 1.3, 1.4, pp. 4–12] (covering also the history of this sophisticated question). Now one has by, (3.333), (3.334), APsp;q .Rn / ,! Asp;q .Rn / ,! S 0 .Rn /;
s < 0;
0 < p; q 1;
(3.339)
(p < 1 for F -spaces) in the framework of S 0 .Rn /. Based on the above references it comes out afterwards that the right-hand sides of (3.216) for given APsp;q .Rn / are equivalent quasi-norms in APsp;q .Rn /. One may compare this type of assertions with the Littlewood-Paley theorems for Lp .Rn / and LV rp .Rn / according to Theorem 2.33 and Remark 2.34. In other words, if s < 0 then the homogeneous spaces APsp;q .Rn / are well-defined quasi-Banach spaces in the framework of S 0 .Rn / enjoying the global homogeneity (3.337). One may ask whether there are counterparts of (3.333) and (3.334) for the local spaces LrAsp;q .Rn / and the hybrid spaces LrAsp;q .Rn /. We formulate a proposal (conjecture, problem) for the hybrid spaces. Let again J C D max.J; 0/, J 2 Z. Problem 3.74. Let s; p; q be as in (3.332). Let n=p r < 1 and s C r < 0. Let s .Rn / if, and only if, f 2 S 0 .Rn /. Then f 2 LrBp;q 2
sup
n J. p Cr/
Z
J 2Z;M 2Zn
22J
C
t
sq 2
kWt f jLp .QJ;M /kq
0
dt 1=q t
(3.340)
s .Rn / if, and only if, is finite, and f 2 LrFp;q
sup J 2Z;M 2Zn
2
Z
n J. p Cr/
0
22J
C
t
sq 2
1=q ˇ ˇ ˇWt f ./ˇq dt jLp .QJ;M / t
(3.341)
is finite (equivalent quasi-norms, usual modification for q D 1). Remark 3.75. In case of the local spaces LrAsp;q .Rn / one has to replace J 2 Z in (3.340), (3.341) by J 2 N0 . The justification of the above proposal comes from (3.332)–(3.334) on the one hand, (3.304) on the other hand, and the Fourieranalytical characterization of LrAsp;q .Rn / according to Theorem 3.38 based on Definition p 3.36(i). According to the arguments in [T92, p. 153, (14)] one has to compare '. t/ with '.2j /, hence t 22j . This may explain that the correct replacement of the summation j J C in the Fourier-analytical version of LrAsp;q .Rn / is
105
3.6 Some properties
the integration over 0 < t < 22J if J 2 N0 and 0 < t < 1 if J 2 N D N. We mention in this context also [Maz03, (1.13), (2.57), Proposition 2.26, pp. 1301, 1318] where one finds corresponding characterizations for so-called Besov-Morrey spaces s; s which are near to (but not always the same as) Bp;q .Rn / D Lr Bp;q .Rn /, D p1 C nr . In [Maz03] assertions of this type have been used to study Navier-Stokes equations extending earlier results in [KoY94]. In [YSY10, Chapter 1] one finds discussions s; of how Besov-Morrey spaces are related to Bp;q .Rn /. One may also consult [Sic12] and [T13, Section 1.3.4, pp. 32–39] and the references within. Taking the assertions in Problem 3.74 for granted one can apply Corollary 3.66 with the following outcome (again formulated as a problem although it is a corollary to Problem 3.74). Let again r C D max.r; 0/, r 2 R. The derivatives D ˛ apply to the space variables. Problem 3.76. Let 0 < p; q 1 (p < 1 for F -spaces), s 2 R and n=p r < s .Rn / if, and 1. Let m 2 N0 with s C r C < m. Let f 2 S 0 .Rn /. Then f 2 LrBp;q only if, sup
2
n J. p Cr/
Z
J 2Z;M 2Zn ; 0j˛jm
22J
0
C
q dt 1=q q t .ms/ 2 D ˛ Wt f jLp .QJ;M / t
(3.342)
s is finite, and f 2 LrFp;q .Rn / if, and only if,
sup J 2Z;M 2Zn; 0j˛jm
2
Z
n J. p Cr/
0
22J
C
ˇq dt 1=q ˇ q ˇ ˇLp .QJ;M / t .ms/ 2 ˇD ˛ Wt f ./ˇ t (3.343)
is finite (equivalent quasi-norms, usual modification for q D 1).
Remark 3.77. As previously stated, under the hypothesis that the assertions in Problem 3.74 are valid it follows from Corollary 3.66 and D ˛ Wt D Wt D ˛ that (3.342), (3.343) are characterizing equivalent quasi-norms in the hybrid spaces LrAsp;q .Rn /. Similarly for the local spaces LrAsp;q .Rn / with J 2 N0 in (3.342), (3.343) in place of J 2 Z. Recall that @t Wt f D Wt f , where is the Laplacian and @t D @=@t. One may ask whether one can replace D ˛ Wt f , 0 j˛j 2m in (3.342), (3.343) by s n @m t Wt f as in case of the global spaces Ap;q .R / according to [T92, Theorem 2.6.4, p. 152]. Of interest are some special cases and their relations to already known assertions of this type. According to (3.295) and [T83, Theorem 2, p. 93] one has 0 L0Lp .Rn / D L0Lp .Rn / D bmo.Rn / D F1;2 .Rn /;
2 p < 1:
(3.344)
Application of the lifts in Theorem 3.72, (3.329), (3.330) gives 1 .Rn /; L0Hp1 .Rn / D L0Hp1 .Rn / D bmo1 .Rn / D F1;2
2 p < 1; (3.345)
106
3 Hybrid spaces
s where Hps .Rn / D Fp;2 .Rn / are (fractional) Sobolev spaces according to (3.21). Taking (3.341) = (3.340) with r D 0, s D 1, p D q D 2 for granted one obtains
1
kf jbmo
Z .R /k n
2
2
sup J 2Z;M 2Zn
22J
C
Z
ˇ ˇ ˇWt f .x/ˇ2 dx dt
Jn 0
Z
22J
Z
QJ;M
ˇ ˇ ˇWt f .x/ˇ2 dx dt
2J n
sup J 2N0 ;M 2Zn
0
(3.346)
QJ;M
for the lifted (by 1) inhomogeneous space bmo.Rn /. We justify the second equivalence. The cube QJ;M D 2J M C 2J .0; 1/n with J 2 N D N can be decomposed into 2J n D 2jJ jn cubes of side-length 1. Then one has for these cubes Z
1
Z
Z jWt f .x/j2 dx dt
2J n 0
1Z
jWt f .x/j2 dx dt:
sup Q0;M 0 QJ;M
QJ;M
0
Q0;M 0
(3.347) This proves the second equivalence in (3.346) if the first one holds. Formally one obtains the characterizations (3.335), (3.336) for the homogeneous spaces APsp;q .Rn / R1 with (3.332) if one replaces 0 dt in the corresponding R 1 characterizations (3.333), (3.334) for the inhomogeneous spaces Asp;q .Rn / by 0 dt. We take for granted that this substitute applies also to the homogeneous spaces BMO1 .Rn / compared with the characterization (3.346) of its inhomogeneous counterpart bmo1 .Rn /, hence kf jBMO
1
Z .R /k n
2
2
sup
22J
Z
ˇ ˇ ˇWt f .x/ˇ2 dxdt:
Jn
J 2Z;M 2Zn
0
(3.348)
QJ;M
Then (3.346) and (3.348) coincide with related definitions of BMO1 .Rn / and n bmo1 .Rn / D BMO1 1 .R / in [KoT01]. Furthermore one can apply (3.343) = (3.342) with s D r D 0, m D 1, and p D q D 2 to (3.344). Then one has Z kf jbmo.R /k n
2
sup
2
J 2Z;M 2Zn ; 0j˛j1
sup
sup J 2N0 ;M 2Zn ; 0j˛j1
Z
C
0
ˇ ˇ ˛ ˇD Wt f .x/ˇ2 dx dt
QJ;M
Z
22J
Z
ˇ ˇ ˛ ˇD Wt f .x/ˇ2 dx dt
2J n
J 2N0 ;M 2Zn ; 0j˛j1
22J
Jn
0
Z 2
QJ;M 2J
Z
Jn 0
QJ;M
(3.349)
ˇ ˇ2 t ˇD ˛ Wt 2 f .x/ˇ dx dt:
The second equivalence follows from the first one and a related counterpart of (3.347). The step from (3.346) to (3.348) suggests now for the homogeneous counterpart
3.6 Some properties
107
BMO.Rn / of bmo.Rn / characterized by (3.349), Z kf jBMO.R /k n
2
sup
2
J 2Z;M 2Zn ; j˛jD1
sup J 2Z;M 2Zn ; j˛jD1
22J
Z
0
Z 2
ˇ ˇ ˛ ˇD Wt f .x/ˇ2 dx dt
Jn QJ;M 2J
Z
Jn 0
QJ;M
ˇ ˇ2 t ˇD ˛ Wt 2 f .x/ˇ dx dt:
(3.350)
The above characterizations of bmo.Rn /; BMO.Rn /
and bmo1 .Rn /; BMO1 .Rn /
(3.351)
are known. In particular, (3.346), (3.348) and the last equivalences in (3.349), (3.350) may be found in [KoT01]. Here the second version of (3.350) is a special case of the Carleson measure characterization of BMO.Rn / as it may be found in [Ste93]. Direct proofs of (3.346), (3.348) and the first equivalences of (3.349), (3.350) are given in [Lem02, pp. 92–95, 100, 159–162]. Later on it will be of some use for us to complement (3.344), (3.345) by the homogeneous assertions 0 BMO.Rn / D FP1;2 .Rn /
1 and BMO1 .Rn / D FP1;2 .Rn /:
(3.352)
s .Rn / are the homogeWe refer the reader also to [Can04, pp. 180/181]. Here FP1;2 neous spaces as introduced in [T83, Section 5.1.4, pp. 239/240], where (3.352) is covered by [T83, Theorem, p. 244]. Greater details about homogeneous spaces APsp;q .Rn / including BMO.Rn / may be found in [Tr78, Chapter 3].
Remark 3.78. In other words, if the assertions in Problems 3.74, 3.76 (and their homogeneous counterparts) are correct then one has new proofs of the already known equivalence relations (3.346), (3.348)–(3.350). These spaces and the above representations play a role in the recent theory of Navier-Stokes equations. We return to this point in Chapters 5 and 6 below. Remark 3.79. Instead of thermic characterizations of the hybrid spaces LrAsp;q .Rn / and their local counterparts LrAsp;q .Rn / one can also ask for harmonic characterizations based on Cauchy-Poisson semi-groups. The corresponding theorem for the global spaces Asp;q .Rn / may be found in [T78, Section 2.5.3, pp. 192–196] and, more general, [T92, Section 2.6.4, pp. 151–155]. This might be of some interest for its own sake, but it has apparently no relevance for heat and Navier-Stokes equations. Remark 3.80. As previously noted the recent interest in homogeneous spaces APsp;q .Rn / with (3.332) and normed by (3.335), (3.336) comes from the Navier-Stokes equations. This may justify adding a few further properties (which we could not find in the literature). But we are very sketchy and shift more detailed discussions (and
108
3 Hybrid spaces
proofs) to later occasions. Nothing of what follows will be needed later on. First we note that (3.339) can be strengthened by S.Rn / ,! APsp;q .Rn / ,! Asp;q .Rn / ,! S 0 .Rn / if 1 < p 1;
0 < q 1;
n
1 p
1 <s <0
(3.353)
(3.354)
(p < 1 for F -spaces). This can be proved rather directly. Let D.Rn/ı be as in (3.202). If 1 1 < p < 1; 0 < q < 1; n 1 <s <0 (3.355) p then D.Rn/ı and, hence by (3.353) also D.Rn/ and S.Rn /, are dense in APsp;q .Rn /. We outline the main arguments. Let j .x/ D '.2j x/ with ' as in (3.213). Then Z 1 sq 1=q 2 t jj2 b _ n q dt b /_ jBP s .Rn /k D t ./e jL .R / k. j f f j p p;q t 0 ! 0 if j ! 1 (3.356) as a consequence of the equivalent Fourier-analytical norm (3.216). Similarly for s .Rn /. This annihilates the infrared terms and reduces the question to the inFPp;q homogeneous spaces Asp;q .Rn / where one already knows that S.Rn / and D.Rn / are dense. The assertion that even D.Rn/ı is dense follows from the homogeneity (3.337) in the same way as in Step 1 of the proof of Theorem 2.33 in Section 3.4.5. These observations have some interesting consequences. The usual duality procedure in the framework of the dual pairing S.Rn /; S 0 .Rn / and (3.353) show that n s n 0 n 0 n 0 n Ps Ps S.Rn / ,! As p 0 ;q 0 .R / D Ap;q .R / ,! Ap;q .R / D Ap 0 ;q 0 .R / ,! S .R / (3.357)
if 1 < p; q < 1;
1 1 1 1 C 0 D C 0 D 1; p p q q
n
1 p
1 < s < 0:
(3.358)
Hence one has distinguished realizations of the homogeneous spaces APsp;q .Rn / if
1 < p; q < 1;
n
1
n 1 <s < ; p p
s 6D 0:
(3.359)
Afterwards one can ask for further equivalent norms in the now fixed spaces APsp;q .Rn / (which are usually not characterizing in S 0 .Rn /). This applies to (3.216) but also to diverse other norms in the above references to [T92, T06] and, in particular, [Tri88]. The density of D.Rn /ı in the spaces APsp;q .Rn / with (3.355) gives the possibility to complement the famous Littlewood-Paley assertion for Lp .Rn /, 1 < p < 1, and
3.6 Some properties
109
its extension to the Morrey spaces LV rp .Rn / according to Theorem 2.33. By the same proof as in Section 3.4.5 and a duality argument (and interpolation as far as s D 0 is concerned) one obtains the following assertion: The homogeneous Haar system ˚ j hG;m W j 2 Z; G 2 G ; m 2 Zn (3.360) according to (2.209), (2.217) is an unconditional basis in the homogeneous Besov spaces 1 1 s .Rn / if 1 < p; q < 1; BP p;q 1<s < ; (3.361) p p and in the homogeneous Sobolev spaces ( 2 p < 1; 12 < s < p1 ; s n s n HP p .R / D FPp;2 .R / if (3.362) 1 1 < s < 12 : 1 p < 2; p In addition to the above references one has to rely on HP p0 .Rn / D Lp .Rn / and Section 3.4.4, Figure 3.1 on page 73. A further comment might be of interest. If s < 0 then Wt w according to (3.331) may be considered as distinguished local means with nonnegative kernels eliminating polynomials. But this observation applies to all nonnegative kernels of local means. We refer the reader to [T06, Section 1.4, pp. 9–12] covering characterizations of the inhomogeneous spaces Asp;q .Rn / with s < 0 by local means based on non-negative kernels. In other words, there is apparently a wide range of distinguished realizations of homogeneous spaces APsp;q .Rn / with (3.339) in the framework of S 0 .Rn / eliminating the usual ambiguity of polynomials. It seems to be of interest to elaborate the above sketchy comments and to compare the outcome with the already existing literature about this topic. We refer the reader in particular to the related work of Bourdaud in [Bou88, Bou11, Bou13]. But we wish to indicate how distinguished norms of homogeneous spaces APsp;q .Rn / coming out by the duality (3.357), (3.358) in the framework of S.Rn /; S 0 .Rn / annihilating polynomials look. Recall @t D @=@t. Let 1 < p; q < 1;
0<s<
n ; p
n n Ds r p
(3.363)
and s=2 < m 2 N (1 < q r for B-spaces). Then 1=q Z 1 s s n n q dt P kf jBp;q .R /kW;m D t .m 2 /q @m C kf jLr .Rn /k t Wt f jLp .R / t 0 (3.364) and Z 1 ˇq dt 1=q ˇ s s n ˇ .Rn /kW;m D t .m 2 /q ˇ@m W f ./ jL .R / kf jFPp;q Ckf jLr .Rn /k t p t t 0 (3.365) are equivalent norms of the homogeneous spaces APsp;q .Rn / satisfying in particular (3.337). We indicate how this claim can be justified. According to [T92, Theorem,
110
3 Hybrid spaces
p. 152, Remark, p. 155] one has for the related inhomogeneous spaces the equivalent norms 1=q Z 1 s s n q dt kf jBp;q .Rn /kW;m D t .m 2 /q @m W f jL .R / C kf jLp .Rn /k t p t t 0 (3.366) and Z 1 ˇq dt 1=q ˇ s s n ˇ kf jFp;q .R /kW;m D t .m 2 /q ˇ@m jLp .Rn / t Wt f ./ t (3.367) 0 C kf jLp .Rn /k: Furthermore, kf jLr .Rn /k c kf jAsp;q .Rn /k;
(3.368)
[T01, Theorem, p. 170] where one finds explanations and references, in particular to [SiT95]. This is the point where the additional restriction q r for the B-spaces comes in. If f 2 Asp;q .Rn / and supp f fy W jyj "g D K" then (for small " > 0) kf jLr .R /k c" n
Z 0
1
1=q s n q dt t .m 2 /q @m W f jL .R / t p t t
(3.369)
and an F -counterpart. This follows from H¨older’s inequality applied to Lr .K" / ,! Lp .K" / with p < r < 1. Afterwards (3.369) (and its F -counterpart) can be extended to all f 2 APsp;q .Rn / by homogeneity using (3.337) and (3.363). As for the duality (3.357) with (now) s; p; q as in (3.358) we first mention that Z 1 Z 1 t jj2 dt dt 2 2 et jj t m @m e2t jj .1/m .tjj2 /m e D D C 6D 0; t t t 0 0 (3.370) independently of 2 Rn . With, say, '; 2 S.Rn / real, one has by (3.331) for the dual pairing Z Z '.x/ .x/ dx D b ' ./ b./ d Rn Rn Z Z 1 t jj2 2 b./ d dt et jj b ' ./ t m @m Dc (3.371) t e n t ZR Z0 1 dt s s D c0 t 2 Wt '.x/ t .mC 2 / @m dx: t Wt .x/ t Rn 0 From the usual duality for Lp .`q / and `q .Lp / with 1 < p; q < 1 applied to (3.335), (3.336) and (3.357) with (3.358) follows Z 1 q 0 dt 1=q 0 s 0 s n P t .mC 2 /q @m jLp0 .Rn / (3.372) k jBp0 ;q 0 .R /k t Wt t 0
3.6 Some properties
111
and an F -counterpart. After replacing s; p0 ; q 0 by s; p; q one obtains the righthand side of (3.364) from (3.369) and (3.372). Similarly for the F -spaces. To repeat our point of view: All happens within the dual pairing S.Rn /; S 0 .Rn / . After a particular space is fixed one may ask for equivalent norms (or bases) within this given space (which neednot to be and often will not be characterizing in the framework of S.Rn /; S 0 .Rn / ). Examples are the right-hand side of (3.369), the related F -counterpart, and (3.216), but also bases in terms of the homogeneous Haar system (3.360) with the famous Littlewood-Paley theorem for Lp .Rn /, 1 < p < 1, as the first assertion of this type. This point of view applies at least to the spaces APsp;q .Rn / according to (3.359). It covers not only the so-called critical spaces for Navier-Stokes equations, hence s pn D 1 (1 < q n for B-spaces, some problems with s D 0), but also the related supercritical spaces s pn D nr , n < r < 1. It might be of interest that kf jLr .Rn /k in (3.365) can be replaced by 1=p Z 1=p Z n p p dx sp p r jxj jf .x/j D jxj jf .x/j dx : (3.373) jxjn Rn Rn This is a sharp Hardy inequality according to [T01, Theorem 16.3, p. 238]. There is also a (more complicated) B-counterpart (again with q r). But preference should be given to (3.364) and (3.365). These norms reflect the translation-invariance of the respective spaces.
Chapter 4
Heat equations
4.1 Preliminaries In [T13, Chapter 5] we dealt with the Cauchy problem for linear and nonlinear heat equations of type (1.11)–(1.13) in the global spaces Asp;q .Rn / as recalled in Definition 3.1, Remark 3.2 and in the local spaces LrAsp;q .Rn / according to (3.97), Notation 3.22. This will be complemented now by corresponding assertions in the hybrid spaces LrAsp;q .Rn /, Definition 3.18, Notation 3.22. Since Ln=p Asp;q .Rn / D Asp;q .Rn /;
(4.1)
(3.102), one could also say that we extend the theory developed in [T13] in the global spaces Asp;q .Rn / to the hybrid spaces LrAsp;q .Rn /, n=p r < 1. There is essentially no difference with the local spaces LrAsp;q .Rn / as far as proofs are concerned. But we give complete selfcontained formulations and add also a few new aspects. As described in Chapter 1, Introduction, the nonlinear heat equations (1.11)–(1.13) are the scalar case of the Navier-Stokes equations (1.1)–(1.3) or (1.7), (1.8). In contrast to the local spaces LrAsp;q .Rn / we have now the mapping properties of the Riesz transforms Rk according to Theorem 3.52 and hence of the Leray projector P in (1.9). This gives the possibility to extend the theory for the Navier-Stokes equations as developed in [T13] from Asp;q .Rn / to LrAsp;q .Rn /. There is no counterpart for the spaces LrAsp;q .Rn /. In Chapter 1, Introduction, we gave a description of the related situation. We use the same standard notation as introduced in Sections 2.2.1 and 3.2.1. Otherwise we take over some basic material from [T13, Section 5.1.1, pp. 159–161] where one finds additional explanations. Let again Rn be Euclidean n-space, n 2 N. Let ˚ D .x; t/ 2 RnC1 W x 2 Rn ; t > 0 (4.2) RnC1 C be its closure, hence the set of all .x; t/ 2 RnC1 with x 2 Rn , t 0. and let RnC1 C As usual D.Rn / D C01 .Rn / collects all complex-valued C 1 functions in Rn with compact support. Its dual space D 0 .Rn / consists of all complex-valued distributions. Let w 2 D 0 .Rn /. Then w ˝ ı D w.x/ ˝ ı.t/ 2 D 0 .RnC1 / is the usual tensor product of w 2 D 0 .Rn / and the ı-distribution in R. Let again @j D @=@xj ; j D 1; : : : ; n; (4.3) @t D @=@t; P and D x D njD1 @2j , the Laplacian with respect to the space variables x 2 Rn . 0 nC1 / with supp W Let f 2 D 0 .RnC1 / with supp f RnC1 C . Then W 2 D .R
4.1 Preliminaries
RnC1 and C
@t W W D f C w.x/ ˝ ı.t/
113
(4.4)
is called a (weak) solution of the Cauchy problem in RnC1 , n 2 N, with the initial data w. In particular W .; t/ tends to w./ if t # 0 distributional. But in Sections 4.4.3, 4.4.4 below we discuss under which circumstances one can say more (strong solutions). This is desirable especially when it comes to the Navier-Stokes equations. Let w 2 S 0 .Rn /. Then Z 2 2 1 1 jxyj jxj 4t 4t w; e ; t > 0; e w.y/ dy D Wt w.x/ D .4 t/n=2 Rn .4 t/n=2 (4.5) x 2 Rn , is the well-known Gauss-Weierstrass semi-group which can be written on the Fourier side as
b
2
Wt w./ D et jj w b./;
2 Rn ;
t > 0:
(4.6)
The Fourier transform is taken with respect to the space variables x 2 Rn . Of course, both (4.5), (4.6) must be interpreted in the context of tempered distributions (distributional for short). But we recall that (4.5) makes sense pointwise: It is the convolution of w 2 S 0 .Rn / and gt .y/ D .4 t/n=2 e w gt 2 C 1 .Rn /;
jyj2 4t
2 S.Rn / and hence
N=2 j.w gt /.x/j ct 1 C jxj2 ;
x 2 Rn ;
(4.7)
for some ct > 0 and, say, N 2 N. Details and references may be found in [T78, Section 2.2.1, p. 152] and [T06, pp. 5,6]. If w and f are regular distributions, subject to some restrictions, then W .x; t/ Z
1 e w.y/ dy C .4 /n=2 Rn Z t D Wt w.x/ C Wt f d .x/; 1 D .4 t/n=2
2 jxyj 4t
Z tZ 0
Rn
jxyj2
e 4.t / f .y; / dyd .t /n=2
0
(4.8) x 2 Rn , t > 0, with f .y/ D f .y; / is the well-known unique solution of the classical Cauchy problem @t W .x; t/ W .x; t/ D f .x; t/; W .x; 0/ D w.x/;
x 2 Rn ; x 2 Rn :
t > 0;
(4.9) (4.10)
Using (4.6) then (4.8) can be written on the Fourier side in terms of the Duhamel formula Z t 2 t jj2 b w b./ C e.t /jj fb ./ d; (4.11) W .; t/ D e 0
114
4 Heat equations
2 Rn , t > 0. Details and references may be found in [Tri92, Sections 3.3.4– 3.3.6, pp. 169–172], [T78, Section 2.5.2, pp. 190–192] and [T13, p. 160]. Let X D LrAsp;q .Rn / be the hybrid spaces according to Definition 3.18, Notation 3.22, T > 0, b 2 R and 1 v 1. Then we rely in what follows on the weighted spaces Lv .0; T /; b; X normed by 1=v Z T kf jLv .0; T /; b; X k D t bv kf .; t/ jX kv dt ; (4.12) 0
usual modification if v D 1. Some explanations and related references about these vector-valued Banach spaces may be found in [T13, p. 177]. In our context one has in addition to ensure that these spaces make sense in the framework of D 0 .RnC1 / as a minimal request. This will be done below in preparation of Theorem 4.6. Otherwise we transfer those parts of the theory of the heat equation in the local spaces LrAsp;q .Rn / and the global spaces Asp;q .Rn / as developed in [T13] to the hybrid spaces which will be needed later on.
4.2 Homogeneous heat equations First we transfer [T13, Theorem 5.12, p. 171] from the local spaces to the hybrid spaces. Theorem 4.1. Let 1 p; q 1 .p < 1 for F -spaces/; s 2 R. Let d 0 and n=p r < 1. Then there is a constant c > 0 such that for all t with 0 < t 1 and all w 2 LrAsp;q .Rn /, n r s n (4.13) t d=2 Wt w jLrAsCd p;q .R / c kw jL Ap;q .R /k: Proof. The proof is the same as in [T13, pp. 172/173] where one now uses the lift property according to Theorem 3.72 and the wavelet characterizations in Theorem 3.12. Remark 4.2. The case r D n=p, hence (4.1), has been considered separately in [T13, Theorem 5.30, p. 187]. Remark 4.3. Similarly one can transfer other properties and examples of type (4.13) for the local spaces LrAsp;q .Rn / in [T13, Section 5.2.3, pp. 171–174] to the hybrid spaces LrAsp;q .Rn /.
4.3 Inhomogeneous heat equations The second equality in (4.8), justified by (4.7), is the solution of (4.4). The Cauchy problem for nonlinear heat equations considered below will be based on some in-
115
4.3 Inhomogeneous heat equations
equalities related to inhomogeneous heat equations in terms of the spaces introduced in (4.12). We deal first (as a model case) with L1 .0; T /; 0; X D L1 .0; T /; X , kf jL1 .0; T /; X k D sup kf .; t/ jX k: (4.14) t 2.0;T /
Proposition 4.4. Let 1 p; q 1 .p < 1 for F -spaces/, s 2 R. Let 0 d < 2 and n=p r < 1. Let T > 0. Let w 2 LrAsp;q .Rn / and f 2 L1 .0; T /; LrAsp;q .Rn / . Then there is a constant c > 0 such that for all w; f and t with 0 < t T , W .; t/ jLrAsCd .Rn / p;q
n 1 d 2 c Wt w jLrAsCd p;q .R / C c t
sup f ./ jLrAsp;q .Rn /
2.0;t /
d c t d=2 w jLrAsp;q .Rn / C c t 1 2
sup f ./ jLrAsp;q .Rn / :
2.0;t /
(4.15) n Proof. We apply the triangle inequality for the Banach spaces LrAsCd p;q .R / to the second equality in (4.8). One has by (4.13), Z t Z t r sCd n Wt f jLr AsCd .Rn / d Wt f d ./ jL Ap;q .R / p;q 0 0 Z t c .t /d=2 f jLrAsp;q .Rn / d 0
d
c 0 t 1 2
sup f jLrAsp;q .Rn / :
0<
(4.16) Then the first estimate in (4.15) follows from (4.8). The second estimate is again a consequence of (4.13). Remark 4.5. We inserted the above short proof although it is essentially a copy of a corresponding proof in [T13, Proposition 5.16, pp. 177/178] for the local spaces LrAsp;q .Rn /. But it illuminates the role of the diverse parameters. We used (4.13) with 0 < t instead of 0 < t 1. But this is immaterial. We deal with estimates of type (4.15) in terms of L1 -spaces as a (distinguished) model case. But we are mainly interested in corresponding assertions based on weighted Lv -spaces according to (4.12) where one has first to ensure (4.17) Lv .0; T /; a; LrAsp;q .Rn / S 0 .RnC1 / in the usual interpretation (indicated below in (4.21)). Let f .x; t/ 2 Lv .0; T /; a; LrAsp;q .Rn / and '.x; t/ 2 S.RnC1 /:
(4.18)
116
4 Heat equations
We extend f from Rn .0; T / to RnC1 by zero. Let k'kK D
sup
2 K jxj C t 2 jD ˛ '.x; t/j;
x2Rn ;t 2R; j˛jK
K 2 N;
(4.19)
be the norms generating the topology of S.Rn / with D ˛ similarly as in (2.83). By sr .Rn /0 D C sCr .Rn / according to [T83, Theorem 2.11.2, (3.278) and the duality B1;1 p. 178] one has for fixed t with 0 < t T (or t 2 R after the indicated extension by zero) ˇ ˇ ˇ f .; t/; '.; t/ ˇ c kf .; t/ jC sCr .Rn /k k'.; t/ jB sr .Rn /k 1;1 (4.20) c 0 kf .; t/ jLrAsp;q .Rn /k k'kK for some K 2 N, where c 0 is independent of t. In what follows we always assume 1 < v 1 and av 0 < 1 where v1 C v10 D 1. Then one obtains (4.17) from (4.12), (4.20) and H¨older’s inequality, Z ˇ ˇ ˇ ˇ.f; '/ˇ D ˇˇ
T
0
c
Z
ˇˇ f .; t/; '.; t/ dt ˇ
T
t 0
av 0
1=v0 f jLv .0; T /; a; LrAs .Rn / k'kK : dt p;q
(4.21)
In other words, all spaces we are dealing with make sense in D 0 .RnC1 / and even in S 0 .RnC1 /. But we will not stress this point in the sequel. We modified (and improved) corresponding arguments in [T13, p. 183]. The L1 -spaces in (4.14) are covered by (4.17) with a D 0 and v D 1. Let W be as in the second line of (4.8). Theorem 4.6. Let 1 p; q 1 .p < 1 for F -spaces/; n=p r < 1 and s 2 R. Let T > 0 and 1 < v 1;
1 1 < a < 1 ; v
Let n w 2 LrAsCg p;q .R /
1 0d <2 1 ; v
1 < g d: (4.22)
and f 2 Lv .0; T /; a; LrAsp;q .Rn / :
(4.23)
Then there is a constant c > 0 such that for all w; f and t with 0 < t T , n W .; t/ jLrAsCd .Rn / c t d g 2 kw jLrAsCg p;q p;q .R /k Z t v 1=v 1 d av f ./ jLrAsp;q .Rn / d C c t 1 v 2 a 0
.with the usual modification if v D 1/.
(4.24)
4.4 Nonlinear heat equations
117
Proof. We follow closely the proof of [T13, Theorem 5.18, p. 178]. We apply the triangle inequality for Banach spaces to (4.8). The first term on the right-hand side of (4.24) follows again from (4.13). As for the second term we combine the arguments resulting in (4.16), (4.21) with 1v C v10 D 1, Z t r sCd n W f d ./ jL A .R / t p;q 0 Z t (4.25) .t /d=2 a a f jLrAsp;q .Rn / d c 0 Z Z 1=v0 t t v 1=v 0 0 .t /dv =2 av d av f jLrAsp;q .Rn / d : c 0
0
Then (4.24) follows from (4.25) and the assumptions (4.22).
Remark 4.7. This is the counterpart of [T13, Theorem 5.18, p. 178] and its proof. The special case v D 1 and a D g D 0 reduces Theorem 4.6 to Proposition 4.4. Nevertheless we stick at least partly in what follows to this special case to provide a more transparent understanding of what is going on in the sequel.
4.4 Nonlinear heat equations 4.4.1 Special cases Let L1 .0; 1/; X with X D LrAsp;q .Rn / or X D LV rAsp;q .Rn / according to Definition 3.18, Notation3.22, p 1, q 1, be the Banach spaces normed by (4.14) complemented by C Œ0; 1/; X (continuity for 0 t < 1 normed by (4.14)). Let Pn Df D jP D1 @j f with @j D @=@xj , j D 1; : : : ; n. Recall @t D @=@t and D x D njD1 @2j . Theorem 4.8. Let 1 p; q 1 .p < 1 for F -spaces/; s 2 R, n=p r < 1 and s C r > 0. Then there is a number ı, ı > 0, such that @t u.x; t/ D u2 .x; t/ u.x; t/ D 0; u.x; 0/ D u0 .x/;
x 2 Rn ; 0 < t 1; x 2 Rn ;
(4.26) (4.27)
has for any (4.28) u0 2 LrAsp;q .Rn / with ku0 jLrAsp;q .Rn /k ı a solution u 2 L1 .0; 1/; LrAsp;q .Rn / . If, in addition, u0 2 LV rAsp;q .Rn / then V rAsp;q .Rn / . u 2 C Œ0; 1/; L
118
4 Heat equations
Proof. Step 1. This is the counterpart of [T13, Theorem 5.20, pp. 180/181] where one has to replace the local spaces LrAsp;q .Rn / by the hybrid spaces LrAsp;q .Rn /. As there one can reduce the proof to a fixed point problem in L1 .0; 1/; LrAsp;q .Rn / for the operator Tu0 u.x; t/ D Wt u0 .x/ C
Z
t
Wt Du2 .; / d .x/
(4.29)
0
based on (4.8). The main ingredients are u; v 7! uv W
LrAsp;q .Rn / LrAsp;q .Rn / ,! LrAsp;q .Rn /;
(4.30)
multiplication algebra according to Theorem 3.60(ii) and u 7! Du W
n LrAsp;q .Rn / ,! LrAs1 p;q .R /
(4.31)
as a consequence of the atomic characterization as described in Theorem 3.33. Otherwise the arguments given in [T13, p. 181] can be taken over using Proposition 4.4 and u 7! Wt u W LrAsp;q .Rn / ,! LrAsp;q .Rn /; 0 < t 1; (4.32) according to Theorem 4.1. Step 2. If u; v 2 S.Rn / then uv and Du belong also to S.Rn /. This applies also to Wt u as follows from (4.6). Hence one can replace LrAsp;q .Rn / in the above arguments by LV rAsp;q .Rn / if u0 2 LV rAsp;q .Rn /. The proof of Theorem 4.14 below, in par V rAsp;q .Rn / ticular (4.50) with a D 0, g D d D 1 and v D 1 ensures u 2 C.Œ0; 1/; L if u0 2 LV rAsp;q .Rn /. Remark 4.9. One can replace 0 on the right-hand side of (4.26) by h 2 L1 .0; 1/; V rAsp;q .Rn / . This has been done in [T13, Corollary LrAsp;q .Rn / or h 2 L1 .0; 1/; L 5.22, p. 182] in the context of the local spaces LrAsp;q .Rn / which can be transferred to the above cases. We do not go into detail.
4.4.2 General cases Theorem 4.8 is a model case to provide a better understanding of what is going on. We wish to weaken the assumptions for the initial data u0 from u0 2 LrAsp;q .Rn / .Rn / with 0 < g 1. For this to u0 2 LrAs1Cg p;q purpose we replace the spaces L1 .0; 1/; X by their weighted counterparts Lv .0; T /; b; X according to (4.12) where X D LrAsp;q .Rn / or X D LV rAsp;q .Rn /. Our minimal request (4.17) is ensured if av 0 < 1 as justified by (4.21). Otherwise we follow [T13, Section 5.4.3, pp. 182– 187] where one finds further discussions and explanations.
4.4 Nonlinear heat equations
119
Theorem 4.10. Let 1 p; q 1 .p < 1 for F -spaces/; s 2 R, n=p r < 1 and s C r > 0. Let 2 < v 1; 0 < g 1
and a D 1
1 v
~g with 0 < ~ < 1:
(4.33)
.Rn /. Then there is a number T , T > 0, such that Let u0 2 LrAs1Cg p;q @t u.x; t/ D u2 .x; t/ u.x; t/ D 0; u.x; 0/ D u0 .x/;
x 2 Rn ; 0 < t < T; x 2 Rn ;
(4.34) (4.35)
has a unique solution u 2 L2v .0; T /; a=2; LrAsp;q .Rn / :
(4.36)
.Rn / then Furthermore u 2 C 1 Rn .0; T / . If, in addition, u0 2 LV rAs1Cg p;q V rAsp;q .Rn / : u 2 L2v .0; T /; a=2; L Proof. Since
1 2v
<1
a 2
one has in modification of (4.17)–(4.21) L2v .0; T /; a=2; LrAsp;q .Rn / S 0 .RnC1 /:
(4.37)
Otherwise one can follow the arguments in the proof of [T13, Theorem 5.24, pp. 184– 186] based on the same ingredients as in Step 1 of the indicated proof of Theorem 4.8. In particular one relies 4.6 with d D 1 which requires v > 2. We again on Theorem added here u 2 C 1 Rn .0; T / which is covered by [T13, Step 2, Remark 5.25, .Rn / then it follows as in the proof of pp. 185–186]. Furthermore if u0 2 LV rAs1Cg p;q r s V Ap;q .Rn / . Theorem 4.8 that u 2 L2v .0; T /; a=2; L Remark 4.11. Recall that Asp;q .Rn / D Ln=p Asp;q .Rn /, (3.102). Then the above theorem with s > n=p coincides essentially with [T13, Theorem 5.36, p. 189]. Hence Theorem 4.10 extends assertions about the Cauchy problem for nonlinear heat equations from the global spaces Asp;q .Rn / to the hybrid spaces LrAsp;q .Rn / and V rAsp;q .Rn /. Any g with 0 < g 1 can be admitted. One has to choose a D 1 1 " L v with 0 < " < g 1 in (4.33). Next we transfer the examples discussed in [T13, pp. 186/187] from the local spaces LrAsp;q .Rn / to the hybrid spaces LrAsp;q .Rn /. Example 4.12. The lifted bmo-space bmos .Rn / D L0Hps .Rn / with 2 p < 1 and s > 0 according to (3.193) is a special case covered by the above theorem. The preceding remark shows that any initial data u0 2 bmo .Rn / with > 1 can be allowed. By (3.279) with r > 0 and s C r > 0 it follows that any initial data u0 2 C .Rn / with > 1 can also be admitted. In particular (4.34), (4.35) has for any u0 2 C .Rn / with > 1 a unique solution u 2 L1 .0; T /; a=2; C s .Rn / \ C 1 Rn .0; T / (4.38)
120
4 Heat equations
for some T > 0 with v D 1 in (4.33), hence 0 < s D C 1 g;
0 s D 1 g < a < 1:
(4.39)
In other words, for given u0 2 C .Rn / with > 1, one has a unique solution u of (4.34), (4.35) according to (4.38) for any s with 0 < s < C 1 and any a with 0 s < a < 1. Example 4.13. According to (3.308) and (3.314) one has Lrp .Rn / D LrLp .Rn / D LrHp0 .Rn /
1 < p < 1;
n=p r < 0;
(4.40)
where Lrp .Rn / are the classical (global) Morrey spaces as introduced in (2.3). If u0 2 Lrp .Rn / and v D 1 then one has by (4.39) with D 0 that u.; t/ 2 LrHps .Rn / for any s with 0 < s < 1. To ensure that LrHps .Rn / is a multiplication algebra one needs in addition s C r > 0, hence r < s < 1. In other words, for given u0 2 Lrp .Rn / with 1 < p < 1, n=p r and 1 < r < 0 one has a unique solution u of (4.34), (4.35) according to (4.41) L1 .0; T /; a=2; Lr Hps .Rn / \ C 1 Rn .0; T / for any s with r < s < 1 and any a with s < a < 1. .Rn / and This applies in particular to the Lebesgue spaces Lp .Rn / D Ln=p p s n n=p s n Hp .R / in (4.41) with n < p < 1. Some the Sobolev spaces Hp .R / D L comments and references may be found in [T13, Remark 5.37, p. 190].
4.4.3 Strong solutions In connection with (nonlinear) heat equations and Navier-Stokes equations one calls solutions u.x; t/, say, of (4.34), (4.35) weak if for t # 0 the solution u.; t/ approaches the initial data u0 distributional. This is our point of view so far as indicated in Section 4.1. Solutions u of (nonlinear) heat equations or Navier-Stokes equations are called mild if they originate from fixed point theorems. This applies to the above considerations asking for solutions in the spaces according to (4.36). One has always (4.37) which says a little bit more about how the solution u.; t/ approaches u0 if t # 0. But especially for Navier-Stokes equations (caused by their physical background) one is interested to find conditions ensuring u 2 C Œ0; T /; X.Rn/ (4.42) where C.Œ0; T /; X.Rn / is the usual space of continuous functions on Œ0; T / with values in the Banach space X.Rn/ obviously normed as in (4.14). Such solutions are called strong. We discussed this point in [T13, Section 6.2.5, pp. 209, 210]. There one finds also related references which will not be repeated here. It comes out that our approach can also be used to obtain assertions of type (4.42) in generalization of Theorem 4.8. For this purpose we strengthen (4.33) somewhat and deal with the spaces LV rAsp;q .Rn / as introduced in Definition 3.18, Notation 3.22. Otherwise we use the same notation as before and explained in Section 4.4.1.
4.4 Nonlinear heat equations
121
Theorem 4.14. Let 1 p; q 1 .p < 1 for F -spaces/; s 2 R, n=p r < 1 and s C r > 0. Let 2 < v 1; 0 < g 1
and a D 1
1 v
~g with 1=2 ~ < 1
(4.43)
.Rn /. Then there is a number T , .1=2 < ~ < 1 if v D 1/: Let u0 2 LV rAs1Cg p;q T > 0, such that @t u.x; t/ D u2 .x; t/ u.x; t/ D 0; x 2 Rn ; 0 < t < T; u.x; 0/ D u0 .x/; x 2 Rn ; has a unique solution in L2v .0; T /; a=2; LrAsp;q .Rn / . Furthermore,
(4.44) (4.45)
V rAs1Cg V rAsp;q .Rn / \ C Œ0; T /; L u 2 L2v .0; T /; a=2; L .Rn / \ C 1 Rn .0; T / : p;q (4.46) Proof. Compared with Theorem 4.10 it remains to prove V rAs1Cg .Rn / : u 2 C Œ0; T /; L p;q
(4.47)
By (4.29) one has u.x; t/ u0 .x/ D Wt u0 .x/ u0 .x/ C
Z
t
Wt f ./ d .x/
(4.48)
0
where f .x/ D D u2 .x; /. By (4.31), (4.30) one has n r s n 2 kf jLrAs1 p;q .R /k c ku jL Ap;q .R /k :
(4.49)
We rely on (4.25) with s 1 in place of s, 0 < d D g 1 and v > 2. Together with (4.49) one obtains Z t r s1Cg n W f ./ d jL A .R / t p;q 0 Z t 1=v 1 g 1 v 2 a n v (4.50) av kf jLr As1 ct p;q .R /k 0 Z t 1=v 1 g c 0 t 1 v 2 a av ku jLrAsp;q .Rn /k2v d : 0
We already know that u belongs to the first space on the right-hand side of (4.46). Then it follows from (4.43) that the right-hand side of (4.50) tends to zero if t # 0. We apply the norm of LrAs1Cg .Rn / to (4.48). By (4.50) it remains to estimate p;q Z 2 1 jxyj 4t e u0 .y/ dy u0 .x/ Wt u0 .x/ u0 .x/ D .4 t/n=2 Rn Z (4.51) p 1 jzj2 e u0 .x C 2 t z/ u0 .x/ dz: D n=2
Rn
122
4 Heat equations
For given " > 0 and M D M."; u0 / > 0 chosen sufficiently large one has for all t > 0, kWt u0 u0 jLrAs1Cg .Rn /k p;q Z p 2 ejzj u0 . C 2 tz/ u0 ./ jLrAs1Cg .Rn / dz: "Cc p;q
(4.52)
jzjM
Let u0 2 S.Rn /. Then it follows from the first embedding in (3.101) that the integral in (4.52) tends to zero if t # 0. This can be extended by completion to u0 2 LV rAs1Cg .Rn /. p;q
4.4.4 Comments and examples So far we have described the distinguished Examples 4.12, 4.13 of Theorem 4.10. Now we collect some special cases of Theorems 4.8 and 4.14. Recall that C .Rn / D .Rn /, 2 R, are the H¨older-Zygmund spaces according to (3.24). As above B1;1 CV .Rn / is the completion of S.Rn / in C .Rn /. Corollary 4.15. Let > 0. Then there is number ı, ı > 0, such that @t u.x; t/ D u2 .x; t/ u.x; t/ D 0; u.x; 0/ D u0 .x/;
x 2 Rn ; 0 < t 1; x 2 Rn ;
(4.53) (4.54)
has for any u0 2 CV .Rn / a solution u 2 C Œ0; 1/; CV .Rn / .
with
ku0 jC .Rn /k ı
(4.55)
Proof. Let 0 < r < and D s C r. Then the above assertion follows from Theorem 4.8 and (3.279). Remark 4.16. One may ask whether Theorem 4.8 and Corollary 4.15 can be reformulated similarly as the Theorems 4.10, 4.14 covering in particular uniqueness assertions. But the corresponding proof of Theorem 4.10 and its references to [T13, Theorem 5.24, pp. 183–186] cannot be carried over immediately. In particular it is not so clear whether the solutions u.x; t/ in Theorem 4.8 and Corollary 4.15 are unique. On the other hand the in Theorem 4.14. But then one asks case v D 1 is admitted for solutions in L1 .0; T /; a=2; LrAsp;q .Rn / where a is restricted by (4.43), hence a > 0 which excludes a D 0. In Example 4.13 we dealt with solutions of (4.34), (4.35) if u0 belongs to some Morrey spaces Lrp .Rn / according to (4.40). Now we ask what can be said using Theorem 4.14 instead of Theorem 4.10. Let 0 LV rp .Rn / D LV rLp .Rn / D LV rFp;2 .Rn / D LV rHp0 .Rn /;
1 < p < 1;
n=p r < 0;
(4.56)
4.4 Nonlinear heat equations
123
be the Morrey spaces according to Definition 2.1, (3.308), (3.314) where again H .Rn / D Fp;2 .Rn /, 1 < p < 1, 2 R, are the Sobolev spaces, (3.22). In particular V n=p Hp .Rn /; Hp .Rn / D Ln=p Hp .Rn / D L
2 R;
1 < p < 1; (4.57)
according to Theorem 3.20. Corollary 4.17. Let 1 < p < 1, n=p r and 1 < r < 0. Let 2 < v 1, 0 < g < 1 C r and a D1
1 ~g v
for some 1=2 < ~ < 1:
(4.58)
Let u0 2 LV rp .Rn /. Then there is a number T , T > 0, such that x 2 Rn ; 0 < t < T; @t u.x; t/ D u2 .x; t/ u.x; t/ D 0; u.x; 0/ D u0 .x/; x 2 Rn ; has a unique solution in L2v .0; T /; a=2; LrHp1g .Rn / . Furthermore,
(4.59) (4.60)
V rp .Rn / \ C 1 Rn .0; T / : V rHp1g .Rn / \ C Œ0; T /; L u 2 L2v .0; T /; a=2; L (4.61) Proof. We apply Theorem 4.14 where (4.43) is covered by the above assumptions. Let s D 1 g and LV rAs1Cg .Rn / D LV rp .Rn /: (4.62) p;q Then s C r > s C g 1 D 0. This show that (4.61) follows from (4.46).
Remark 4.18. If n < p < 1 then one can choose r D n=p and u0 2 Lp .Rn / based on (4.56), (4.57) with D 0, hence .Rn / D Ln=p .Rn / D Lp .Rn /; LV n=p p p
1 < p < 1:
(4.63)
By (4.57) and (4.61) one has for the above unique solution u, u 2 L2v .0; T /; a=2; Hp1g .Rn / \ C Œ0; T /; Lp .Rn / \ C 1 .Rn .0; T / ; (4.64) where g is now restricted by 0 < g < 1
n p
and a is given by (4.58).
Remark 4.19. Small values of g > 0 in (4.61), (4.64) seem to be desirable. But one must be well aware that there is an interplay between the smoothness 1 g and the weight t a=2 in Z 1=2v T av u jL2v .0; T /; a=2; L V rHp1g .Rn / D t ku.; / jLrHp1g .Rn /k2v d 0
(4.65)
124
4 Heat equations
with a as in (4.58). Small values of g require large values of a and, hence, strong decay of t av at t D 0. On the other hand, with ~g D 1Cr" one has a D v1 CjrjC" for any " > 0. This means that any a > 1=2 is possible if jrj is small. By (3.278) one has always LV rHp1g .Rn / ,! CVrC1g .Rn /
where 0 < r C 1 g < 1:
(4.66)
Hence u0 2 LV rp .Rn / ensures some global H¨older-continuity of the solution u in the space variables. Similarly for u0 2 Lrp .Rn / in Example 4.13. Remark 4.20. According to Theorem 3.20 one has always Ln=p Asp;q .Rn / D Asp;q .Rn /:
(4.67)
Furthermore Asp;q .Rn / D AVsp;q .Rn / if p < 1, q < 1, [T83, Theorem 2.3.3, p. 48]. The assumption s C r > 0 in Theorem 4.14, hence s > n=p, ensures that the global space Asp;q .Rn / is a multiplication algebra. But in case of these global spaces one has a sharper final assertion ensuring that Asp;q .Rn / is a multiplication algebra which we formulated in Theorem 3.14. In particular Theorems 4.10 and 4.14 applied to (4.67) with r D n=p and (3.65), (3.66) in place of s pn > 0 improve and complement [T13, Theorem 5.36, Section 6.2.5, pp. 189, 209/210].
Chapter 5
Navier-Stokes equations in hybrid spaces
5.1 Preliminaries The Navier-Stokes equations (1.1)–(1.3) can be reformulated as (1.7), (1.8), hence @t u u C P div .u ˝ u/ D 0 in Rn .0; T /, in Rn ; u.; 0/ D u0 2 n 2 N, where u.x; t/ D u1 .x; t/; : : : ; un .x; t/ , div .u ˝ u/k D
n X
@j .uj uk /;
(5.1) (5.2)
k D 1; : : : ; n;
(5.3)
k D 1; : : : ; n;
(5.4)
j D1
with the Leray projector P, n X
.Pf /k D f k C Rk
Rj f j ;
j D1
f D .f 1 ; : : : ; f n /, based on the (scalar) Riesz transforms Z _ yk k g.x y/ dy; b g .x/ D cn lim Rk g.x/ D i jj "#0 jyj" jyjnC1
x 2 Rn : (5.5)
As before @t D @=@t and @j D @=@xj if j D 1; : : : ; n. Detailed explanations may be found in [T13, Sections 6.1.1–6.1.3, pp. 191–198], including a few historical comments and references to the recent (book) literature. This will not be repeated here. From (5.1), (5.2) and the vector-valued version of the Duhamel formula (4.11) follows Z t ^ 2 t jj2 b u.; t/ D e ub0 ./ e.t /jj P div .u ˝ u/ .; / d; (5.6) 0
2 Rn , t > 0. If the initial data u0 are divergence-free, hence div u0 D 0 in Rn
or, equivalently
n X
c j j u0 ./ D 0;
2 Rn ;
(5.7)
j D1
then it follows from (5.6) (on which we rely in what follows) that u is also divergence free. Hence the property to be divergence-free is inherited from the initial data u0 to
126
5 Navier-Stokes equations in hybrid spaces
the solution u. But we will not care about this point in the sequel. We do not assume that u0 is divergence-free. Otherwise we refer the reader again to [T13, Section 6.1.3, pp. 196–198] how (1.1)–(1.3) on the one hand and (1.7), (1.8), hence (5.1), (5.2) are related. Without P the Cauchy problem (5.1), (5.2) is the vector-valued version of the nonlinear heat equation (4.26), (4.27) and one can transfer Theorems 4.8, 4.10 and 4.14. So far we could incorporate P only in the context of global spaces, hence Ln=pAsp;q .Rn / D Asp;q .Rn /;
1 < p < 1;
(5.8)
(3.102), which are in addition multiplication algebras. This has been done in [T13, Section 6.2.1, pp. 201–205]. We could not incorporate P in the context of the local spaces LrAsp;q .Rn /. A brief description and related references have been given in Chapter 1, Introduction, especially in connection with (1.16), (1.17). In particular Theorem 2.22(i) shows that one cannot expect that P, based on the Riesz transforms Rk according to (5.4), (5.5), is a linear bounded map in the (vector-valued) local spaces LrAsp;q .Rn /. The situation is now much better, replacing the local spaces LrAsp;q .Rn / by the hybrid spaces LrAsp;q .Rn / which cover (5.8) as a special case. Then one has (1.19), (1.20) and Theorem 3.52. It remains to clip together these ingredients. We need the vector-valued versions of some of the above notation. Let LrAsp;q .Rn / V rAsp;q .Rn / be the spaces introduced in Definition 3.18, Notation 3.22. Then and L n Y
LrAsp;q .Rn /n D
LrAsp;q .Rn /
(5.9)
j D1
collects all f D .f 1 ; : : : ; f n /, n 2 N, such that f j 2 LrAsp;q .Rn / and kf jLrAsp;q .Rn /n k D
n X
kf j jLrAsp;q .Rn /k:
(5.10)
j D1
Similarly LV rAsp;q .Rn /n . Let X D LrAsp;q .Rn / and let Xn be as in (5.9). Let T > 0, b 2 R and 1 v 1. Then L v j .0; T /; b; Xn / collects all f .; t/ 2 Xn , hence 1 n f .; t/ D f .; t/; : : : ; f .; t/ , f .; t/ 2 X; such that n Z X f jLv .0; T /; b; Xn D j D1
T
v 1=v t bv f j .; t/ jX dt
(5.11)
0
is finite (usual modification if v D 1). This is the vector-valued counterpart of (4.12). We are interested in the counterpart of (4.17), hence Lv .0; T /; a; Xn S 0 .RnC1 /n (5.12) in obvious notation. This can be ensured in the same way as in (4.18)–(4.21) if av 0 < 1 as there.
5.2 Special cases
127
5.2 Special cases After these preparations we combine first version of Theorem 4.8 the vector-valued with Theorem 3.52. Let L1 .0; 1/; Xn D L1 .0; 1/; 0; Xn according to (5.11) which can be equivalently normed by (5.13) kf jL1 .0; 1/; Xn k D sup f j .; t/ jX : 0
Theorem 5.1. Let 1 < p < 1, 1 q 1, s 2 R, n=p r < 0 and s C r > 0. Then there is a number ı, ı > 0, such that @t u u C P div .u ˝ u/ D 0 u.; 0/ D u0
in Rn .0; 1/, in Rn ;
(5.14) (5.15)
ku0 jLrAsp;q .Rn /n k ı
(5.16)
has for any u0 2 LrAsp;q .Rn /n
with
V rAsp;q .Rn /n then a solution u 2 L1 .0; 1/; LrAsp;q .Rn /n . If, in addition, u0 2 L V rAsp;q .Rn /n . u 2 C Œ0; 1/; L Proof. The vector-valued version of Theorem 4.8 is quite obvious. This applies also the (5.14), (5.15) without the Leray projector P. Again s C r > 0 ensures that LrAsp;q .Rn / is a multiplication algebra according to (4.30). Instead of (4.30), (4.31) one has now (5.3) with the same (vector-valued) outcome. The counterpart of (4.29) can now be formulated as Z t Wt P div .u ˝ u/.; / d .x/: (5.17) Tu0 u.x; t/ D Wt u0 .x/ 0
The additional restrictions for p and r, hence 1 < p < 1, n=p r < 0, ensure that Theorem 3.52 can be applied. By (5.4) one can extend this assertion to P. Then (5.14), (5.15) can be reduced to a fixed point problem for Tu0 in L1 .0; 1/; LrAsp;q .Rn /n with (5.6) as the desired outcome (on the Fourier side). This can be treated in the same way as the scalar case in [T13, p. 181] with the technical modification in the vector-valued case as in [T13, (6.73), (6.74), p. 203] (v D 1). This proves the main assertion of the theorem. If, in addition, u0 2 LV rAsp;q .Rn /n then one can argue as in Step 2 of the proof of Theorem 4.8 and the proof of Theorem 4.14, especially (4.50) adapted to the above situation (a D 0, v D 1, g D d D 1). Here one needs (3.269).
128
5 Navier-Stokes equations in hybrid spaces
5.3 General cases We transfer Theorems 4.10 and 4.14 from nonlinear heat equations to Navier-Stokes equations. Again some additional restrictions for the parameters are needed such that Theorem 3.52 can be applied. Theorem 5.2. Let 1 < p < 1, 1 q 1, s 2 R, n=p r < 0 and s C r > 0. (i) Let 2 < v 1; 0 < g 1 and a D 1
1 v
~g with 0 < ~ < 1:
(5.18)
.Rn /n . Then there is a number T , T > 0, such that Let u0 2 LrAs1Cg p;q @t u u C P div .u ˝ u/ D 0 u.; 0/ D u0
in Rn .0; T /, in Rn ;
(5.19) (5.20)
has a unique solution in L2v .0; T /; a=2; LrAsp;q .Rn /n :
(5.21)
.Rn /n then Furthermore u 2 C 1 Rn .0; T / n . If, in addition, u0 2 LV rAs1Cg p;q V rAsp;q .Rn /n . u 2 L2v .0; T /; a=2; L (ii) Let 2 < v 1; 0 < g 1
and a D 1
1 v
~g with 1=2 ~ < 1
(5.22)
.Rn /n . Then .1=2 < ~ < 1 if v D 1/: Let u0 2 LV rAs1Cg p;q there is a number T , T > 0, such that (5.19), (5.20) has a unique solution in L2v .0; T /; a=2; LrAsp;q .Rn /n . Furthermore, V rAsp;q .Rn /n \ C Œ0; T /; L V rAs1Cg u 2 L2v .0; T /; a=2; L .Rn /n p;q (5.23) \ C 1 Rn .0; T / n : Proof. We are in the same position as in the proof of Theorem 5.1 with the same references to [T13] about some minor technicalities. In particular part (i) is now the vector-valued version of Theorem 4.10 combined with Theorem 3.52 and (5.4). Similarly, part (ii) follows from Theorem 4.14 combined again with Theorem 3.52 and (5.4). Remark 5.3. If r D n=p then (5.8) and its vector-valued counterpart reduce Theorems 5.1 and 5.2 to corresponding assertions in the framework of the global spaces Asp;q .Rn /. This is essentially covered by [T13, Sections 6.2.1, 6.2.5, pp. 201–205, 209–211]. We tried to extend this theory in [T13] to the local spaces LrAsp;q .Rn /. But this is not possible. We discussed this situation in the Introduction, Chapter 1, in connection with (1.16), (1.17). Whether the replacement of the Riesz transforms by
5.4 Comments and examples
129
their truncated modification (1.16), (1.17) is of interest or even of physical relevance (ignoring infrared frequencies) is not known to us. But now one has a satisfactory extension of the theory of Navier-Stokes equations in the global spaces Asp;q .Rn / (excluding limiting cases) to the hybrid spaces LrAsp;q .Rn / and LV rAsp;q .Rn /. In case of global spaces one has AVsp;q .Rn / D LV n=p Asp;q .Rn / D Asp;q .Rn /
if p < 1, q < 1;
(5.24)
Theorem 3.20, Notation 3.22 and [T83, Theorem 2.3.3, p. 48]. But the situation for the spaces LrAsp;q .Rn / with n=p < r < 1 is totally different. They are even non-separable. We refer the reader to Proposition 2.16 and (2.102) where we dealt with this question in case of the Morrey spaces Lrp .Rn / D Lr Lp .Rn /, 1 < p < 1, n=p < r < 0, (3.308). The above assumption s > n=p and (5.8) ensure that Asp;q .Rn / is a multiplication algebra. This can be strengthened by Theorem 3.14. We fix the outcome. Corollary 5.4. The assertions of Theorems 5.1 and 5.2 remain valid for the global spaces Ln=pAsp;q .Rn / D Asp;q .Rn / with Asp;q .Rn /
D
s Bp;q .Rn /
s .Rn / Asp;q .Rn / D Fp;q
and
( 1 < p < 1; 1 q 1; if 1 < p < 1; q D 1; if
AVsp;1 .Rn /;
s > n=p; s D n=p;
(5.25)
1 < p < 1; 1 q 1; s > n=p;
(5.26)
1 < p < 1;
s > n=p:
(5.27)
Proof. As remarked above, s > n=p can be replaced by (3.65), (3.66). By (5.24) only the spaces AVsp;q .Rn / with q D 1 are not yet covered by (5.26), (5.27).
5.4 Comments and examples Basically we combined assertions about nonlinear heat equations according to Section 4.4 with Theorem 3.52, (5.4), complemented by some technical references to [T13]. We established the outcome in the above Theorems 5.1 and 5.2 whereas Corollary 5.4 may be considered as a specification to the global spaces Asp;q .Rn /. One may ask to what extent examples and corollaries as considered in Sections 4.4.2 and 4.4.4 can be transferred from nonlinear heat equations to Navier-Stokes equations. There is no direct counterpart of Example 4.12. We used there (3.279) based on r > 0. Now one needs r < 0. Then one has (3.278) and the embedding is strict as mentioned in Remark 3.55. On the other hand Example 4.13 can be transferred to
130
5 Navier-Stokes equations in hybrid spaces
Navier-Stokes equations as a special case of Theorem 5.2(i). We return to this point briefly in Remark 5.9 below. At this moment we are mainly interested in strong solutions asking for counterparts of the Corollaries 4.15, 4.17 and some consequences of the Theorems 5.1 and 5.2. Recall that C .Rn / D B1;1 .Rn / are the H¨older-Zygmund spaces which can be equivalently normed by (3.26) if > 0. Corollary 5.5. Let 0 < < s D r with n=p r < 0 for some 1 < p < 1. Let 1 q 1. Then there is a number ı, ı > 0, such that @t u u C P div .u ˝ u/ D 0 u.; 0/ D u0
in Rn .0; 1/, in Rn ;
(5.28) (5.29)
has for any initial data u0 2 LV rAsp;q .Rn /n with ku0 jLrAsp;q .Rn /n k ı a solution u 2 C Œ0; 1/; CV .Rn /n .
Proof. This follows from Theorem 5.1 and (3.278).
V r Asp;q .Rn /n which means Remark 5.6. By Theorem 5.1 one has u 2 C Œ0; 1/; L that u is a strong solution. We have no uniqueness assertion as mentioned in Remark 4.16 with respect to Corollary 4.15. Otherwise the above assertion is the substitute of Corollary 4.15. There we used D s C r with r > 0. Then one can use (3.279). This is no longer possible because Theorem 5.1 requires r < 0 and one has only the embedding (3.278) which is strict as mentioned in Remark 3.55. One can specify Asp;q .Rn / in the above corollary by some distinguished Sobolev and Besov spaces, s .Rn / D Hps .Rn / Asp;q .Rn / D Fp;2
s or Asp;q .Rn / D Bp;p .Rn /:
(5.30)
One arrives always at the global H¨older continuity in the space variables of the solution u. Next we ask for a counterpart of Corollary 4.17 specifying part (ii) of Theorem 5.2. We rely on the vector-valued counterpart of the Morrey and Morrey-Sobolev spaces according to (4.56) assuming that the initial data u0 belong to LV rp .Rn /n D LV rLp .Rn /n D LV rHp0 .Rn /n ;
1 < p < 1;
n=p r < 0; (5.31)
s .Rn /, asking for solutions u.; t/ in LrHps .Rn /n and in with Hps .Rn / D Fp;2 LV rHps .Rn /n . If r D n=p then one has obvious vector-valued counterparts of (4.57),
Hp .Rn /n D Ln=p Hp .Rn /n D LV n=p Hp .Rn /n ; based again on Theorem 3.20.
2 R;
1 < p < 1; (5.32)
5.4 Comments and examples
131
Corollary 5.7. Let 1 < p < 1, n=p r and 1 < r < 0. Let 2 < v 1, 0 < g < 1 C r and a D1
1 ~g v
for some 1=2 < ~ < 1:
(5.33)
Let u0 2 LV rp .Rn /n . Then there is a number T , T > 0, such that @t u u C P div .u ˝ u/ D 0 u.; 0/ D u0
in Rn .0; T /, in Rn ;
(5.34) (5.35)
has a unique solution in L2v .0; T /; a=2; LrHp1g .Rn /n . Furthermore, V rHp1g .Rn /n \ C Œ0; T /; L V rp .Rn /n \ C 1 Rn .0; T / : u 2 L2v .0; T /; a=2; L n (5.36) Proof. We apply part (ii) of Theorem 5.2, where (5.22) is covered by the above assumptions. Let s D 1 g. Then one has u0 2 LV rp .Rn /n D LV rHps1Cg .Rn /n and s C r D 1 g C r > 0. Then (5.36) follows from (5.23).
(5.37)
Remark 5.8. If n < p < 1 then one can choose r D n=p and u0 2 Lp .Rn /n according to (4.63). Then (5.34), (5.35) with 2 < v 1, 0 < g < 1 pn and a as in (5.33) has a unique solution u, u 2 L2v .0; T /; a=2; Hp1g .Rn /n \ C.Œ0; T /; Lp .Rn /n \ C 1 Rn .0; T / n : (5.38) This follows from the above corollary and (4.56), (4.57). It is the counterpart of Remark 4.18. Remark 5.9. One can also carry over Remark 4.19 to the above vector-valued situation. In particular if u0 2 LV rp .Rn /n then it follows from (4.66) that the solution u according to (5.36) is globally H¨older-continuous with respect to the space variables, u 2 L2v .0; T /; a=2; CVrC1g .Rn /n
where 0 < r C 1 g < 1:
(5.39)
But this is just in the spirit of [Mor38] to enhance u0 2 Lp .Rn / by u0 2 Lrp .Rn / in order to ensure H¨older-continuity of solutions of nonlinear elliptic and parabolic problems. Similarly one can transfer Example 4.13, based on Theorem 4.10. This corresponds now to part (i) of Theorem 5.2 with 0 < ~ < 1 in (5.18). Then one obtains by the same arguments as there the following assertion:
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5 Navier-Stokes equations in hybrid spaces
For given u0 2 Lrp .Rn /n with 1 < p < 1, n=p r and 1 < r < 0 one has for some T , T > 0, a unique solution u of (5.34), (5.35) according to L1 .0; T /; a=2; Lr Hps .Rn /n \ C 1 Rn .0; T / n (5.40) for any s with r < s < 1 and any a with s < a < 1. In contrast to (5.36) it is not clear whether u is a strong solution. This applies in particular to the Lebesgue spaces Lp .Rn / D Ln=p Lp .Rn / and the Sobolev spaces Hps .Rn / D Ln=p Hps .Rn / in (5.40) with n < p < 1.
5.5 Complements 5.5.1 Supercritical spaces In recent times there have appeared several books and surveys, as well as numerous papers dealing with Navier-Stokes equations in function spaces. This includes Morrey spaces, Morrey-Besov spaces and some (homogeneous and inhomogeneous) spaces of type Asp;q .Rn / where A 2 fB; F g (with a preference of B-spaces). Our approach is based on the hybrid spaces LrAsp;q .Rn / generalizing (localizing or Morreyfying) the spaces Asp;q .Rn /. We indicate a few relations of our results and methods to other assertions in the recent theory of Navier-Stokes equations. However this will not be done systematically (which might be not so easy because of the large variety of different aspects and notation). As far as the recent literature is concerned we refer the reader to [BCD11, BaG13, BoP08, BoF13, CPZ14, Can95, Can04, DeY13, Fef06, FLZZ00, Ger06, GGS10, Gui13, KeK11, KoT01, Lem02, Lem07, Lem12, Lem12a, Mey01a, Mey06, Tsu11, Yon10, Zhe14]. Further (historical) references may be found in [T13, Sections 6.1.1, 6.2.5, pp. 191–194, 209–214]. In the theory of Navier-Stokes equations it plays a decisive role whether the space X.Rn / for the initial data u0 2 X.Rn /n is critical, subcritical or supercritical. We add a corresponding remark in the framework of the hybrid spaces LrAsp;q .Rn /. Closely related is the infrared problem to which we return below in Sections 5.5.2, 5.5.3. According to [Can04, Section 3, pp. 188–204] (adapted to inhomogeneous spaces) one calls X.Rn / D Ap;q .Rn / with u0 2 X.Rn /n for the initial data critical if n D 1. Here D n=p is the breaking condition for Ap;q .Rn / in order to be p a multiplication algebra, Theorem 3.14, and the desired minimal smoothness of the initial data u0 drops by 1 compared with the global smoothness of the solution. In particular, the spaces 0 .Rn /; Ln .Rn / D Fn;2
1 1 bmo1 .Rn / D F1;2 .Rn / and B1;q .Rn /
(5.41)
1 .Rn / are critical (for Navier-Stokes equa0 < q 1, with C 1 .Rn / D B1;1 n tions). As far as bmo .R / is concerned one may consult (3.189), (3.193), (3.201) and (3.346). Spaces Ap;q .Rn / with pn > 1 are called supercritical and with
133
5.5 Complements
pn < 1 subcritical. In particular the spaces Lp .Rn / are supercritical if n < p 1 and subcritical if 1 p < n. Subcritical spaces Ap;q .Rn / with < pn 1 are not very well adapted for initial data of Navier-Stokes equations. This applies in particular to Lp .Rn /, 2 p < n where n 3, [Can04, Theorem 19, p. 233] with a reference to [Mey01]. The critical spaces, including the examples in (5.41) attracted a lot of attention. Then one has solutions local in time, but also global in time if the initial data in the indicated spaces are small. In the supercritical case one has solutions local in time. If the initial data in the supercritical case are small and highly oscillating then there are solutions global in time. A detailed discussion may be found in [Can04, Chapter 4]. We return to this problem in our context below. In case of the hybrid spaces LrAp;q .Rn / one has Cr D 0 as the breaking condition for multiplication algebras, Theorem 3.60. Following the above notation one would call spaces for initial data u0 2 LrAp;q .Rn /n supercritical if C r > 1. This applies to all spaces considered above in the Theorems 5.1, 5.2 and Corollary 5.5 where u0 2 LrAp;q .Rn /n
with
C r D s C r C g 1 > 1;
(5.42)
but also to the Morrey spaces Lrp .Rn / in Corollary 5.7 where D 0 and r > 1. Also the specification to the global spaces Asp;q .Rn / in Corollary 5.4, based on (5.24), Theorem 3.20, Notation 3.22, fits in this scheme. This covers in particular the corresponding well-known assertions for the supercritical Lebesgue spaces Lp .Rn / D Ln=p Lp .Rn /, n < p < 1, as mentioned in Remarks 5.8, 5.9.
5.5.2 Infrared-damped initial data As indicated in Section 5.5.1 if the initial data u0 2 Lp .Rn /n in the supercritical case n < p 1 are small and oscillating then one has solutions of the NavierStokes equations global in time. One may ask whether there is a counterpart in the above more general context for the hybrid spaces LrAp;q .Rn /;
D s 1 C g;
n=p r < 0;
(5.43)
in Theorem 5.2 and for the Morrey spaces Lrp .Rn /;
1 < p < 1;
n=p r;
1 < r < 0;
(5.44)
in Corollary 5.7. We offer a wavelet version, replacing oscillating data by infrareddamped data. We rely again on the inhomogeneous wavelet system ˚ j ‰ D ‰G;m W j 2 N0 ; G 2 G j ; m 2 Zn
(5.45)
as introduced at the beginning of Section 3.2.3 and its homogeneous counterpart ˚ P D ‰ j W j 2 Z; G 2 G ; m 2 Zn ; ‰ G;m
(5.46)
134
5 Navier-Stokes equations in hybrid spaces
according to (3.219). In particular, j .x/ D 2j n=2 ‰G;m
n Y
Gl .2
j
x l ml /
(5.47)
lD1
with j 2 N0 , G 2 G j , m 2 Zn , in case of the inhomogeneous system ‰ and j 2 Z, P Otherwise G j and G have G 2 G , m 2 Zn , in case of the homogeneous system ‰. the same meaning as in (3.47), (3.48) whereas F is the father wavelet and M the P is an orthonormal basis in mother wavelet according to (3.45), (3.46). Both ‰ and ‰ L2 .Rn /. Let ‰m .x/ D
n Y
F .xl
ml /;
x 2 Rn ;
m 2 Zn ;
(5.48)
lD1
as in (3.221). Then the wavelet expansion of f 2 LrAsp;q .Rn / in Theorem 3.26 can also be written as in (3.229), hence X j n=2 j f D j;G ‰G;m m .f / 2 j 2N0 ;G2G j ; m2Zn
D
X
X
m .f /‰m C
j n=2 j j;G ‰G;m m .f / 2
(5.49)
j 2N0 ;G2G ; m2Zn
m2Zn
D f0 C f C ; where
m .f / D f; ‰m ;
j j n=2 j;G f; ‰G;m ; m .f / D 2
(5.50)
m 2 Zn , j 2 N0 , G 2 G j (recall G j D G if j 2 N). The formal counterpart of P in (5.46) is given by (5.49) in terms of the homogeneous wavelet system ‰ X j n=2 j f D j;G ‰G;m (5.51) m .f / 2 j 2Z;G2G ; m2Zn
where again j j n=2 f; ‰G;m ; j;G m .f / D 2
j 2 Z;
G 2 G;
m 2 Zn :
(5.52)
Now the additional terms according to f in (3.230) are coming in. One may consult the discussion in Remark 3.49. So far we have not really used the wavelet expansion (5.51) for the homogeneous spaces Lr APsp;q .Rn / but relied on the equivalence (3.233) with the related sequence spaces. If u0 D .u10 ; : : : ; un0 / 2 LrAp;q .Rn /n then we agree that .u0 ; ‰m / D 0
means .ul0 ; ‰m / D 0;
l D 1; : : : ; n:
(5.53)
5.5 Complements
135
j Similarly for u0 ; ‰m .2k / and u0 ; ‰G;m . We always assume that the underlying smoothness of the wavelets according to (3.88) is sufficiently large such that (5.53) (and similarly for the other wavelets) makes sense as dual pairings. Definition 5.10. Let 1 < p < 1, 1 q 1, 2 R and n=p r < 0. Let u0 2 LrAp;q .Rn /n . Let ‰ be a fixed wavelet system according to (5.45). (i) Then u0 is said to be infrared-damped if .u0 ; ‰m / D 0;
for all m 2 Zn :
(5.54)
(ii) Let k 2 N. Then u0 is said to be k-infrared-damped if u0 ; ‰m .2k / D 0;
for all m 2 Zn :
(5.55)
Remark 5.11. With k D 0 one can incorporate (5.54) in (5.55). But we prefer the above version. Furthermore, (5.55) can be reformulated as
j D 0; u0 ; ‰G;m
j 2 N0 ; j < k; G 2 G j ; m 2 Zn :
(5.56)
This follows from the multiresolution analysis of wavelets. A short description and related references may be found in [T06, Section 1.7.1, pp. 26–28]. One can take (5.54)–(5.56) as a wavelet version to express high oscillation. This can be seen from the wavelet decomposition (5.49), (5.50) and the well-known cancellation properties j with j 2 N0 , G 2 G , m 2 Zn , according to (5.47) based of the wavelets ‰G;m on (3.46). If u0 is infrared-damped then it can be expanded as in (5.49) without the P according to (5.46), starting terms ‰m and hence also by the homogeneous system ‰ X X X j n=2 j j;G ‰G;m ; (5.57) u0 D m .u0 / 2 j 2Z G2G m2Zn
with (5.52) and with vanishing infrared tail (the terms with j < 0). Conversely, let j;G u0 be given by (5.57) with m .u0 / D 0 if j < 0. Expanded by the inhomogeneous system ‰ in (5.45) one obtains by the well-known orthogonality relations of the wavelets that m .u0 / D 0, m 2 Zn , which means that u0 is infrared-damped according to (5.54). Hence infrared-damped initial data u0 can be expanded likewise by the inhomogeneous system ‰ similarly as in (5.49) without the starting terms ‰m P in (5.46) with vanishing infrared tail or as in (5.57) by the homogeneous system ‰ (terms with j < 0). j Remark 5.12. Let ‰G;m with j 2 N0 , G 2 G , m 2 Zn , given by (5.47), be based on F ; M 2 C kC1 .R/, k 2 N0 , as in Definition 3.18. Then one has by (3.45), (3.46) with u D k C 1 2 N,
1 ./ˇˇ c 2
ˇ j ˇ‰
G;m
ˇ 2
j n=2 ˇ j
ˇkC1 ˇ ;
0 jj 1;
(5.58)
136
5 Navier-Stokes equations in hybrid spaces
where c is independent of j; G; m. If u0 satisfies (5.54) then one has f D f C in j (5.49) and (5.58) applies to all admitted ‰G;m with j 2 N0 , G 2 G and m 2
1
j at the origin may justify calling u0 subject to (5.54) Zn . The strong decay of ‰G;m infrared-damped. If one replaces the compactly supported Daubechies wavelets by the Meyer wavelets as discussed briefly in Section 3.4.6 then one has for some c > 0 j and the corresponding wavelets ‰G;m ,
1
j ‰G;m ./ D 0 if
jj c 2j ; j 2 N0 ; G 2 G ; m 2 Zn :
(5.59)
This means that there are no low or infrared frequencies at all. One may also consult [Can04, pp. 208/209] for corresponding discussions. A modified way to avoid an infrared disaster is the assumption b _!0 Sj f D '0 .2j /f
if
j ! 1
(5.60)
with '0 as in (3.9), say, for the initial data f D u0 . We refer the reader to [BCD11, Definition 1.26, p. 22/23] and the recent papers [CPZ14, Zhe14]. It annihilates at least polynomials. This relates (5.60) to our comments in Remark 3.80 and the work of Bourdaud in [Bou88, Bou11, Bou13] mentioned there. Remark 5.13. We add a comment which illuminates the role of (5.54). Let 0 < p; q 1, (p < 1 for F -spaces), s 2 R, d 0, s C d 0 and n=p r < 1. Let f 2 LrAsp;q .Rn / be expanded by (5.49) = (3.229). Then n r s n t d=2 Wt f C jLrAsCd p;q .R / c kf jL Ap;q .R /k;
t 1;
(5.61)
where c is independent of f 2 LrAsp;q .Rn / and t 1. Here Wt is the GaussWeierstrass semi-group as introduced in Section 4.1 and (5.61) is the counterpart of Theorem 4.1 for large t. This can be justified if one transfers a corresponding assertion for local spaces LrAsp;q .Rn / according to [T13, Theorem 5.1, (5.42), pp. 164– 165] to the hybrid spaces LrAsp;q .Rn / what can be done without any additional efforts. The smoothness of the wavelets underlying the justification of (5.61) must be sufficiently large as in (3.88). This restricts d > 0 in (5.61). Greater details are given in [T13, Theorem 5.1, p. 164]. In any case in dependence on the smoothness of the wavelets one has for infrared-damped initial data u0 according to (5.54) an extra decay of Wt u0 if t tends to infinity. But we will not use this observation below. There is a second point which illustrates assumptions of type (5.54) and (5.55). Recall that X ‰m .x/; x 2 Rn ; (5.62) 1D m2Zn
is a distinguished resolution of unity in Rn , where the compactly supported functions ‰m are given by (5.48). This is a well-known property of scaling functions which may also be found in [T08, (6.365), p. 238]. By Definition 3.18 the scaling function (father
5.5 Complements
137
wavelet) F must be sufficiently smooth in dependence on the space LrAsp;q .Rn /n considered. Then one has by (5.54), Z X X u0 D u0;m D ‰ m u0 ; u0;m .x/ dx D 0; m 2 Zn ; (5.63) m2Zn
m2Zn
Rn
where the integral must be understood as the dual pairing .u0 ; ‰m /. Similarly for ‰m .2k / in place of ‰m if one assumes (5.55). This again makes clear that infrareddamped might be considered as a wavelet substitute of oscillation.
5.5.3 Global solutions By (3.191) = (3.344) one has bmo .Rn / D L0Lp .Rn / D L0Lp .Rn /;
2 p < 1;
(5.64)
where bmo.Rn / has the usual meaning normed by (3.189). Hence bmo.Rn / fits in our scheme. Infrared-damped initial data u0 as introduced in Definition 5.10 and discussed above can be expanded likewise by the inhomogeneous system ‰ in (5.45) P according to (5.46) with vanishing with (5.54) or by the homogeneous system ‰ infrared tail (terms with j < 0 in (5.57)). The expansion of BMO.Rn /, the homogeP hence (5.51), (5.52), may be found in [Mey92, neous counterpart of bmo.Rn /, by ‰, Theorem 4, p. 154]. This can be reformulated in the context of the homogeneous hybrid spaces LrAPsp;q .Rn / according to (3.232), (3.233) as BMO.Rn / D L0 LP 2 .Rn /;
(5.65)
where the related condition [Mey92, (6.7), p. 154] for the wavelet coefficients coincides with 0 kf jL0 LP 2 .Rn /k k.f / jL0 b2;2 .Rn /k < 1 (5.66) j;G
according to (3.226). Here .f / D fm .f /g are the coefficients (5.52) in the expansion (5.51). Otherwise we refer the reader to Remark 3.77 where we discussed several properties of bmo.Rn /, BMO.Rn / and related spaces in the context of thermic characterizations. Let IPı , ı 2 R, b _ (5.67) IPı f D jjı f be the homogeneous counterpart of the lift Iı in (3.327). Then it follows from the homogeneous version of Theorem 3.72 and (3.352) that BMO1 .Rn / D IP1 BMO.Rn / D L0 HP 1 .Rn /;
(5.68)
1 where HP 1 .Rn / D HP 21 .Rn / D FP2;2 .Rn / is the homogeneous Sobolev space of smoothness order 1. This fits again in our scheme of (homogeneous) hybrid spaces. In particular it follows from (3.233) that BMO1 .Rn / can be expanded by the hoP according to (5.51), (5.52) in the expected way. We refer the mogeneous system ‰
138
5 Navier-Stokes equations in hybrid spaces
reader also to [Mey06, Theorem 19.2, (19.4), p. 181] where the related condition for the wavelet coefficients coincides with the counterpart of (5.66), 1 kf jL0 HP 1 .Rn /k k.f / jL0 b2;2 .Rn /k < 1:
(5.69)
We switch (temporarily for sake of simplicity) to the scalar case and assume, in addition, u0 2 LrAp;q .Rn / with D s 1 C g and, hence, r C > 1 as in (5.42), the scalar case of the initial data in Theorem 5.2. By (3.278) one has 1 LrAp;q .Rn / ,! C rC .Rn / ,! bmo1 .Rn / D F1;2 .Rn /;
(5.70)
as the lifted consequence of the well-known embedding for inhomogeneous spaces " 0 C " .Rn / D B1;1 .Rn / ,! B1;1 .Rn / ,! bmo .Rn /;
" > 0:
(5.71)
As far as the last embedding is concerned we add (as a digression) the sharp embedding if, and only if, 0 < u 2, v D 1; (5.72) which seems to be again a piece of mathematical folklore. The left-hand side is covered by [SiT95, Theorem 3.3.2, Corollary 3.3.1, pp. 114/115] whereas the righthand side is a matter of duality as follows. By [T83, Theorem 2.11.2, p. 178] one has 0 0 .Rn / ,! bmo.Rn / ,! B1;v .Rn / B1;u
0 0 0 0 bmo.Rn / D F1;2 .Rn / D F1;2 .Rn /0 ,! B1;1 .Rn /0 D B1;1 .Rn /:
(5.73)
0 .Rn / be the comLet cmo .Rn / be the completion of D.Rn / in bmo.Rn / and BV 1;q n 0 n pletion of D.R / in B1;q .R /, 0 < q 1. If the last embedding in (5.72) is valid for some v with 1 v 1 then it follows again by duality with v1 C v10 D 1, 0 n n 0 n 0 0 n V0 B1;v 0 .R / D B1;v .R / ,! cmo .R / D F1;2 .R /:
(5.74)
The first duality is covered by [T83, (12), p. 180] whereas the second duality is due to [Daf02]. But this embedding requires v 0 D 1, hence v D 1, [ET96, Section 2.3.3, (1), p. 44] with a reference to [SiT95, Theorem 3.1.1, p. 112]. After this digression we return to (5.70). One has by the above considerations for infrared-damped functions u0 2 LrAp;q .Rn /, ku0 jBMO1 .Rn /k c ku0 jLrAp;q .Rn /k:
(5.75)
As far as the spaces bmo.Rn /, bmo1 .Rn /, BMO.Rn / and BMO1 .Rn / are concerned one may also consult Section 3.6.6. According to [KoT01, Theorem 2, p.24] there is a number ı > 0 such that the Navier-Stokes equations (1.1)–(1.3), reformulated as (5.1), (5.2) with T D 1, have for any initial data u0 2 BMO1 .Rn /n with div u0 D 0 and ku0 jBMO1 .Rn /n k ı (5.76)
5.5 Complements
139
a unique solution u.x; t/ in Rn .0; 1/ in the Banach space E.RnC1 C / normed by ku jE.RnC1 C /k Z tZ h j 2 n=2 D sup t ju .x; t/j C t x2Rn ;t >0; j D1;:::;n
0
p jxyj t
juj .y; /j2 dy d
i1=2 :
(5.77)
We refer the reader also to [Lem02, Theorem 16.2, p.166] and [BCD11, Section 5.5] for further assertions. In particular it follows from the arguments in [Lem02, Proposition 11.1, Lemma 16.3, Theorems 16.1, 16.2, pp. 107, 162–166] that this assertion applies also to (5.1), (5.2) without the additional assumption div u0 D 0. The arguments in [Lem02, Proposition 11.1, p. 107] rely on so-called Oseen kernels extending corresponding assertions in [Ose27, pp. 12, 28, 53] from n D 3 to n 2 N. Then Theorem 5.2 can be complemented for infrared-damped initial data according to Definition 5.10(i) as follows. Proposition 5.14. Let 1 < p < 1, 1 q 1, s 2 R, n=p r < 0 and s C r > 0. Let 2 < v 1; 0 < g 1
and a D 1
1 v
~g with 0 < ~ < 1:
(5.78)
Then there are numbers ı > 0 and T > 0 such that @t u u C P div .u ˝ u/ D 0 u.; 0/ D u0
in Rn .0; 1/, in Rn ;
(5.79) (5.80)
has for any infrared-damped initial data u0 2 LrAs1Cg .Rn /n p;q
with
ku0 jLrAs1Cg .Rn /n k ı p;q
(5.81)
a unique solution r s n u 2 E.RnC1 C / \ L2v .0; T /; a=2; L Ap;q .R /n :
(5.82)
.Rn /n and 1=2 ~ < 1 .1=2 < ~ < 1 in case of If, in addition, u0 2 LV r As1Cg p;q v D 1/ then n Vr s u 2E.RnC1 C / \ L2v .0; T /; a=2; L Ap;q .R /n V rAs1Cg .Rn /n \ C 1 Rn .0; T / n : \ C Œ0; T /; L p;q
(5.83)
Proof. The additional assumption (5.81) for the infrared-damped initial data u0 (compared with Theorem 5.2) ensures (5.75) and hence (5.76). Then the proposition follows from Theorem 5.2 and the above considerations.
140
5 Navier-Stokes equations in hybrid spaces
Remark 5.15. There arise several questions. First one may ask how the two spaces in (5.82) are related to each other and whether the additional smoothness expressed by the second spaces can be preserved if T tends to infinity. Furthermore one may ask whether the requested smallness of u0 2 Lr As1Cg .Rn /n in (5.81) is necessary p;q (in contrast to Theorem 5.2). First we compare the two spaces in (5.82). We switch again (for sake of simplicity) to the scalar case. From s C r > 0 and (3.278) follows LrAsp;q .Rn / ,! C sCr .Rn / ,! L1 .Rn /: Based on (4.12) one obtains by H¨older’s inequality, av 0 < 1 and tju.x; t/j C t 2
n=2
(5.84) 1 v
C
1 v0
D 1,
Z tZ p jxyj t
0
ju.y; /j2 dy d Z
t
a a ku.; /jLrAsp;q .Rn /k2 d 0 Z t 1=v aC v10 r s n 2 av ku.; / jLrAsp;q .Rn /k2v : c t ku.; t/ jL Ap;q .R /k C c t
ct
ku.; t/ jLrAsp;q .Rn /k2
Cc
0
(5.85) In particular if v D 1 (and hence 0 < a < 1 by (5.78)) then one has for 0 < t T < 1, Z tZ 2 tju.x; t/j2 C t n=2 p ju.y; /j dy d 0
jxyj t
c t ku.; t/ jLrAsp;q .Rn /k2 C c t 1a sup a ku.; / jLrAsp;q .Rn /k2
(5.86)
0< t
cT 1a ku jL1 .0; T /; a=2; LrAsp;q .Rn / k2 : This shows that the two spaces in (5.82) with v D 1 coincides locally in time with the second space. But this says nothing about what happens if T tends to infinity. We remove the assumption that u0 in (5.81) is small in the indicated space and require for this purpose that u0 is k-infrared-damped for a sufficiently large k 2 N. Recall that we assumed in the related Definition 5.10 that the underlying wavelet system ‰ is fixed. Corollary 5.16. Let 1 < p < 1, 1 q 1, s 2 R, n=p r < 0 and s C r > 0. Let 2 < v 1; 0 < g 1 and a D 1
1 v
~g with 0 < ~ < 1:
(5.87)
Let C > 0. Then there are numbers T > 0 and k 2 N such that @t u u C P div .u ˝ u/ D 0 u.; 0/ D u0
in Rn .0; 1/, in Rn ;
(5.88) (5.89)
5.5 Complements
141
has for any k-infrared-damped initial data .Rn /n u0 2 LrAs1Cg p;q
ku0 jLrAs1Cg .Rn /n k C p;q
with
(5.90)
a unique solution r s n u 2 E.RnC1 C / \ L2v .0; T /; a=2; L Ap;q .R /n :
(5.91)
If, in addition, u0 2 LV r As1Cg .Rn /n and 1=2 ~ < 1 .1=2 < ~ < 1 in case of p;q v D 1/ then n Vr s u 2E.RnC1 C / \ L2v .0; T /; a=2; L Ap;q .R /n V rAs1Cg \ C Œ0; T /; L .Rn /n \ C 1 Rn .0; T / n : p;q
(5.92)
Proof. Let " > 0 such that r C > 1 C " where again D s 1 C g. Then it follows from the same arguments as in (5.70), (5.75) (in the context of inhomogeneous spaces) ku0 jBMO1 .Rn /n k c ku0 jC 1C" .Rn /n k c 0 ku0 jC rC .Rn /n k c 00 ku0 jLrAp;q .Rn /n k:
(5.93)
We expand u0 by the wavelet system ‰ underlying Definition 5.10 and use % D r C C 1 " > 0. Then one obtains by (3.283) and (5.56) ˇ ˇ ku0 jC 1C" .Rn /n k sup 2j."1/ ˇj;G .u0 /ˇ m
j k;G;m
c 2%k
sup j k;G;m
0
c 2
ˇ ˇ ˇ 2j.rC/ ˇj;G m .u0 /
(5.94)
%k
c0C 2
ku0 jLrAp;q .Rn /n k %k :
Then (5.76) follows from (5.93) if k is chosen sufficiently large. Now the corollary can be obtained in the same way as in the proof of Proposition 5.14 and the references given there. Remark 5.17. As mentioned in Section 5.5.2 one may consider the assumption that u0 is infrared-damped as a wavelet version of highly oscillating (small) initial data. We refer the reader again to [Can04, Chapter 4] and [Mey01a, Mey06]. In [Mey01a, pp. 71, 72, 84] and also in [Mey06] initial data u0 are called oscillating if ku0 jAPsp;q .Rn /n k is small in some critical homogeneous spaces APsp;q .Rn / with s s pn D 1 and s < 0, especially in case of related Besov spaces BP p;q .Rn / and BMO1 .Rn /. Recall that APsp;q .Rn / is normed by (3.216).
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5 Navier-Stokes equations in hybrid spaces
Remark 5.18. The question arises whether the unique global solution u 2 E.RnC1 C / in (5.82) and (5.91) belongs globally in time to some other spaces, say, u 2 L2v .0; 1/; a=2; LrAsp;q .Rn /n (5.95) at the best, or whether this additional smoothness breaks down if time tends to infinity. The case of preference might be v D 1 for which one has (5.86) local in time. Problems of this type have been considered: If the initial data u0 2 X.Rn /n satisfy also u 2 E.RnC1 (5.76) then one has for some spaces X.Rn / that C / the global solution n according to (5.77) belongs in addition to L1 .0; 1/; X.R /n . This remarkable observation, called persistency, goes back to [FLZZ00]. We refer also to [Lem02, Theorem 19.3, pp. 194/195], [Mey01a, Theorems 29, 32, pp. 85, 87] and [Can04, Theorem 10, pp. 209–211]. This observation covers Lebesgue spaces Lp .Rn /, Morrey spaces Lrp .Rn /, LV rp .Rn /, Lorentz spaces Lp;q .Rn /, but also some (homogeneous and inhomogeneous) Besov spaces. Further discussions and references may be found in [T13, pp. 212–214]. Of special interest in this connection are assertions of type (3.301). But it remains to be seen to what extent one can employ this method for the above purposes. Remark 5.19. The crucial estimate (5.94) suggests that one can relax (5.55) or, equivalently (5.56), by j j u0 ; ‰G;m j small; j 2 N0 ; j < k; G 2 G j ; m 2 Zn : (5.96) One has to say what this means quantitatively. This will not be done at this moment. But we return to this question in greater detail in connection with Haar wavelets in Section 6.1.5 below.
Chapter 6
Navier-Stokes equations in global spaces
6.1 Haar wavelets and Reynolds numbers 6.1.1 Preliminaries Corollary 5.16 deals with solutions of Navier-Stokes equations, global in time, for large initial data u0 , provided that they are k-infrared damped. This means by (5.55) and (5.62) that u0 can be decomposed as Z X u0 ‰m .2k / with u0 .x/‰m .2k x/ dx D 0; m 2 Zn ; (6.1) u0 D Rn
m2Zn
which extends (5.63). Recall again that the integrals must be understood as the dual pairing u0 ; ‰m .2k / based on sufficiently smooth wavelets such that Theorem 3.26 can be applied. Daubechies wavelets are powerful but rather complicated functions constructed with some efforts. One may ask for simpler conditions. Let again Qm D m C .0; 1/n and Qk;m D 2k m C 2k .0; 1/n, k 2 N0 and m 2 Zn be the above dyadic cubes. Then Z u0 .x/ dx D 0; for all m 2 Zn ; (6.2) Qm
and for some k 2 N,
Z u0 .x/ dx D 0;
for all m 2 Zn ;
(6.3)
Qk;m
are the counterparts of (5.54), (5.55) with characteristic functions of cubes in place of ‰m in (5.62) and (6.1). This suggests that we replace in Sections 5.5.2, 5.5.3 the P in (5.45), (5.46) by their Haar counterparts (2.206) smooth wavelet systems ‰ and ‰ and (2.209) briefly recalled below in (6.8), (6.11). Then one has so far expansions by Haar wavelets for the spaces covered by Theorem 3.41 and Corollary 3.44, but also for some special cases, including bmos .Rn /, 1=2 < s < 0, with (3.199). Unfortunately this does not apply to bmo1 .Rn /. This can By be seen as follows: (5.74) and lifting one has in the context of the dual pairing S.Rn /; S 0 .Rn / , 1 .Rn /: cmo1 .Rn /0 D F1;2
(6.4)
If bmo1 .Rn / and, hence, cmo1 .Rn / could be characterized by Haar expansions, then it would follow from (6.4) that the Haar system (2.206) = (6.8) is a basis in
144
6 Navier-Stokes equations in global spaces
1 F1;2 .Rn /. Then characteristic functions of cubes must be pointwise multipliers in 1 F1;2 .Rn /. But this contradicts the related criterion in (3.173). Otherwise we refer the reader to our discussion in Remark 3.45. As indicated there the situation is better if one deals with smooth wavelets. This was also mentioned at the beginning of Section 5.5.3 with references to [Mey92, Mey06] and used afterwards in a decisive way (preferably for near-by B-spaces). This type of argument cannot be carried over to Haar bases. But one can circumvent this shortcoming by some sharp embeddings in the context of the global spaces Asp;q .Rn / as introduced in Definition 3.1. Recall that Asp;q .Rn / D Ln=p Asp;q .Rn / are special hybrid spaces, Theorem 3.20, Notation 3.22. But the extension of what follows from the global spaces Asp;q .Rn / to more general hybrid spaces LrAsp;q .Rn / is not so clear. This may justify our dealing in this Chapter 6 exclusively with global spaces. First we repeat what is meant by Haar expansions, adapted to our later needs. Let for y 2 R, 8 ˆ if 0 < y < 1=2; <1 hM .y/ D 1 if 1=2 y < 1; (6.5) ˆ :0 if y 62 .0; 1/;
and let hF .y/ D jhM .y/j be the characteristic function of the unit interval .0; 1/. Let G D .G1 ; : : : ; Gn / 2 G 0 D fF; M gn
(6.6)
which means that Gr is either F or M . Let G D .G1 ; : : : ; Gn / 2 G D G j D fF; M gn ;
j 2 N;
(6.7)
which means that Gr is either F or M , where indicates that at least one of the components of G must be an M . Then j HG;m .x/
D2
j n=2
n Y
hGl 2j xl ml ;
G 2 Gj ;
j 2 N0 ;
m 2 Zn ; (6.8)
lD1
is the well-known orthonormal Haar basis in L2 .Rn /. One has in particular X j j n=2 j;G HG;m f D m 2
(6.9)
j;G;m
with the Fourier coefficients j;G j n=2 j;G m D m .f / D 2
Z
j
Rn
f .x/ HG;m .x/ dx:
(6.10)
Let hjG;m .x/ D 2j n=2
n Y lD1
hGl 2j xl ml /;
j 2 Z;
G 2 G;
m 2 Zn ; (6.11)
6.1 Haar wavelets and Reynolds numbers
145
be the homogeneous counterpart of the inhomogeneous system in (6.8). It is also an orthonormal basis in L2 .Rn / with obvious counterparts of (6.9), (6.10). So far we have dealt several times with Haar expansions in function spaces, Sections 2.6 and 3.4.4. We need now expansions of type (6.9) for some spaces Asp;q .Rn / according to [T10, Section 2.3.2, pp. 92–95]. This is also covered by Theorem 3.41 applied to Asp;q .Rn / D Ln=p Asp;q .Rn /. But an explicit formulation is desirable, restricted to what is really needed below. Let ˚ j n D j;G (6.12) m 2 C W j 2 N0 ; G 2 G ; m 2 Z and let 0 < p; q 1, s 2 R. Then ˚ s s bp;q .Rn / D W k jbp;q .Rn /k < 1
(6.13)
with s .Rn /k D k jbp;q
1 X
n
2j.s p /q
j D0
X X G2G j
p jj;G m j
q=p 1=q (6.14)
m2Zn
(standard modification if max.p; q/ D 1). We recall the main part of [T10, Theorem 2.21, p. 92] for B-spaces. Proposition 6.1. Let n 2 N, 0 < p 1, 0 < q 1 and 1 1 1 max n 1 ; 1 < s < min ; 1 ; p p p
(6.15)
s .Rn / if, and only left side of Figure 3.1 on page 73. Let f 2 S 0 .Rn /. Then f 2 Bp;q if, it can be represented as
f D
1 X X X
j n=2 j;G HG;m ; m 2 j
s 2 bp;q .Rn /;
(6.16)
j D0 G2G j m2Zn
unconditional convergence being in S 0 .Rn /. The representation (6.16) is unique with j;G j;G m D m .f / as in (6.10). Furthermore J W
f 7! .f /
(6.17)
s s is an isomorphic map of Bp;q .Rn / onto bp;q .Rn /. If, in addition, p < 1, q < 1, then n o jn n j 2j.s p / 2 HG;m W j 2 N0 ; G 2 G j ; m 2 Zn (6.18) s .Rn /. is an unconditional normalized basis in Bp;q
146
6 Navier-Stokes equations in global spaces
Remark 6.2. We refer the reader for proofs and explanations to [T10, Section 2.3.2]. s There is also a counterpart for Fp;q .Rn /, but this will not be needed here. Later on s we use the above proposition for Bp;q .Rn / with 1 < p; q < 1. Of special interest will be expansions (6.16) without the starting terms 0 .x/; m .x/ D HG;m
G D fF; : : : ; F g;
m 2 Zn ;
(6.19)
where m is the characteristic function of Qm D m C .0; 1/n. This means that Z 0;G f .x/ dx D 0; G D fF; : : : ; F g; m 2 Zn ; (6.20) m .f / D Qm
for the corresponding coefficients in (6.10). It follows by mollification that such elements f (without these starting terms and 1 < p; q < 1) can be approximated in s Bp;q .Rn / by functions Z n ' 2 D.R /; '.x/ dx D 0: (6.21) Rn
6.1.2 Some embeddings By Propositions 6.1 and (6.20) it is now quite clear that (6.2), (6.3) is the counterpart of (5.54), (5.55) with the Haar system according to (6.8) in place of the (smooth) wavelet system ‰ in Definition 5.10. The arguments in Sections 5.5.2, 5.5.3 rely on an interplay between expansions by inhomogeneous wavelet systems without starting terms and their interpretations as expansions by related homogeneous wavelet systems without infrared tail. Now we are doing the same with respect to the Haar systems in (6.8) and (6.11). But there are no counterparts of the crucial estimates (5.93), (5.94). As a substitute we rely on some peculiar (more or less known) embeddings which may also be of interest for their own sake. This applies in particular to the critical homogeneous and inhomogeneous spaces n
n
p 1 p 1 Bp;q .Rn / and BP p;q .Rn /;
1 n 1 < p 1;
0 < q 1; (6.22)
as well as 1 .Rn / bmo1 .Rn / D F1;2
1 and BMO1 .Rn / D FP1;2 .Rn /:
(6.23)
We have dealt so far several times not only with the global inhomogeneous spaces Asp;q .Rn / as introduced in Definition 3.1 but also with their homogeneous counterparts APsp;q .Rn / and their quasi-norms according to (3.216) (in the context of inhomogeneous spaces). The resulting homogeneity in (3.337), kf ./ jAPsp;q .Rn /k s p kf jAPsp;q .Rn /k; n
> 0;
(6.24)
will be of some use for us later on. We refer the reader also to Section 3.6.6 where we discussed in (3.335), (3.336) thermic characterizations of homogeneous spaces
6.1 Haar wavelets and Reynolds numbers
147
APsp;q .Rn / if s < 0 (which is essentially sufficient for what follows). We extended these characterizations in (3.363)–(3.365) to some spaces APsp;q .Rn / with s > 0. As far as the spaces in (6.23) and also bmo.Rn /, BMO.Rn / are concerned one may consult (3.189) (and an obvious homogeneous counterpart where the related seminorm coincides with the second term on the right-hand side of (3.189) extended to all cubes Q), (3.344), (3.345), (3.352) with the references and explanations given there. In particular we mentioned some thermic characterizations of the spaces in (3.351). Recall again that we are not so much interested in the homogeneous spaces themselves but in their norms, for example (3.216), applied to elements belonging to the related inhomogeneous spaces. What follows should be understood in this interpretation. Proposition 6.3. Let n 2 N. (i) Let 0 < q 1 and 0 < p0 p1 1. Then n p
1
n p
1
n
1
n
1
1 Bp00;q .Rn / ,! Bp11;q .Rn / ,! B1;q .Rn /
and
(6.25)
p p 1 BP p00;q .Rn / ,! BP p11;q .Rn / ,! BP 1;q .Rn /:
(6.26)
1 1 B1;2 .Rn / ,! bmo1 .Rn / ,! B1;1 .Rn /
(6.27)
1 1 .Rn / ,! BMO1 .Rn / ,! BP 1;1 .Rn /: BP 1;2
(6.28)
(ii) Furthermore
and
Proof. Step 1. Both (6.25) and (6.26) follow from (3.13), (3.216) and Nikol’skij’s inequality j _ 1 1 b jLp .Rn / c 2j n. p0 p1 / ' j f b _ jLp .Rn / ; ' f 1 0
j 2 Z;
(6.29)
which may be found in [T83, Remark 1, p. 18, Remark 4, p. 23] where c > 0 is independent of j (and f ). Step 2. The embedding (6.27) follows by duality from 1 1 1 B1;1 .Rn / ,! F1;2 .Rn / ,! B1;2 .Rn /:
(6.30)
This is covered by [T83, (9), p. 47, Theorem 2.11.2, p. 178, Proposition 2, p. 51] and (6.23). The homogeneous counterpart of (6.30) can be proved in the same way, [T83, p. 47]. This has also been mentioned explicitly in [T83, Section 5.2.5, p. 244]. There we remarked also that the duality theory according to [T83, Section 2.11] has a homogeneous counterpart. Then (6.28) follows from (6.23).
148
6 Navier-Stokes equations in global spaces
Remark 6.4. Although not needed we remark that the above embeddings can be complemented by n
1
p .Rn / ,! bmo1 .Rn /; Bp;1
n
p 1 BP p;1 .Rn / ,! BMO1 .Rn /;
0 < p < 1: (6.31) This follows from [T01, (13.88), p. 217] with a reference to [Mar95, Lemma 16, p. 253] (inhomogeneous case, combined with lifting and related arguments for the homogeneous case). As mentioned at the beginning of Section 6.1.1 the arguments in Sections 5.5.2, 5.5.3 cannot be extended directly from smooth wavelets to Haar wavelets because the distinguished spaces bmo1 .Rn / and BMO1 .Rn / do not admit Haar expansions. We circumvent this shortcoming by a combination of Propositions 6.1 and 6.3. In connection with the assumption that the initial data u0 are k-infrared damped according to (5.55) or that they satisfy the Haar counterparts (6.3) raises the question whether these rigid conditions can be relaxed assuming that the absolute value of the corresponding expressions are small but not necessarily zero. In case of smooth wavelets we mentioned this problem in Remark 5.19. Its Haar counterpart will be discussed in greater detail in Section 6.1.5 below. But this requires (again in contrast to smooth wavelets) some additional efforts. One must be sure that characteristic functions of cubes are elements of the homogeneous spaces involved. (Recall that the usual monotonicity with respect to the smoothness parameter s for the inhomogeneous spaces Asp;q .Rn / is no longer valid for the homogeneous spaces APsp;q .Rn /.) Let be the characteristic function of the cube Q D .1; 1/n. Using the obvious monotonicity 0 < p; q 1;
s1 s0 .Rn / ,! Bp;q .Rn /; Bp;q
1 < s0 s1 < 1;
(6.32)
of the inhomogeneous spaces one obtains as a by-product of Proposition 6.1, s 2 Bp;q .Rn / if
1 p 1; 0 < q 1; 1 < s < 1=p:
(6.33)
s s A final answer under which conditions belongs to Bp;q .R/ and Fp;q .R/ may be found in [T10, Remark 6.19, p. 263]. There is no counterpart of (6.32) for homogeneous spaces and nothing like (6.33) can be expected. For our later discussions of the above question the following assertion will be helpful.
Proposition 6.5. Let n 2 N and 1 p 1. Then ( s 2 BP p;q .Rn /
if
n. p1 1/ < s < n. p1 1/ D s;
1 ; p
0 < q 1; q D 1:
(6.34)
6.1 Haar wavelets and Reynolds numbers
149
Proof. By (6.33) it is sufficient to deal with the terms j 0 for the B-spaces in (3.216). One has with % D ' 0 in (3.214) n Y sin k _ /_ jLp .Rn /k ' j ./ ./ jLp .Rn / k.' j b k kD1 j _ %.2 / ./ jLp .Rn / 1 2j n %_ .2j / jLp .Rn / 2j n.1 p / :
(6.35)
In the second equivalence we used Fourier multiplier assertions according to [T83, Theorem 1.5.2, p. 26] with n Y sin k k
n Y
k sin k
(6.36)
where 2 S.R / is a cut-off function with ./ D 1 if jj 3=2 and jj 2. Then (6.34) follows from the terms with j 0 in (3.216).
./ D 0 if
M./ D
./
and M./ D
kD1
./
kD1
n
Remark 6.6. Instead of (3.216) one can alternatively use (3.335) to prove (6.34) in case of s < 0 and 1 < p 1. Corollary 6.7. Let n 2 and let m be the characteristic functions of the cubes Qm D m C .0; 1/n, m 2 Zn . Let X X f D m m ; m 2 C; jm j < 1: (6.37) m2Zn
m2Zn
Then f 2 BMO1 .Rn / and for some c > 0, X kf jBMO1 .Rn /k c jm j D c kf jL1 .Rn /k:
(6.38)
m2Zn
Proof. This follows from (6.28) and (6.34) with p D 1, the translation invariance of BMO1 .Rn / and the triangle inequality.
6.1.3 Main assertions After the above preparations we come to the main assertions of Section 6.1. We wish to replace the assumption that the initial data u0 in Corollary 5.16 are k-infrared damped by its Haar counterpart Z u0 .x/ dx D 0 for all m 2 Zn : (6.39) Qk;m
Recall that Qk;m D 2k m C 2k .0; 1/n with k 2 N0 and m 2 Zn are the above dyadic cubes. In contrast to Chapter 5 we deal now exclusively with global spaces
150
6 Navier-Stokes equations in global spaces
Asp;q .Rn /. This means r D n=p in Corollary 5.16, covered by Theorem 3.20, Notation 3.22. Then one has s > n=p in the respective spaces Asp;q .Rn /. The related .Rn / for the initial data u0 cover all supercritical spaces according to spaces As1Cg p;q Section 5.5.1, Ap;q .Rn /;
1 < p < 1;
1 q 1;
n > 1; p
(6.40)
0 .Rn /, n < p < 1. If pn 1 < 0 then (6.39) must in particular Lp .Rn / D Fp;2 be understood as the dual pairing of u0 with the characteristic function k;m of Qk;m , Z n u0 .x/ dx D u0 ; k;m ; (6.41) k;m 2 Bp 0 ;1 .R /; Qk;m
n 0 n n where p1 C p10 D 1 and Bp 0 ;1 .R / D Bp;1 .R / within the dual pairing S.R /; S 0 .Rn / , [T83, Theorem 2.11.2, p. 178]. Proposition 6.1 ensures that k;m belongs n to Bp 0 ;1 .R /. Otherwise we use the same notation as in Chapter 5, especially in connection with Proposition 5.14 and Corollary 5.16, now specified to the global spaces Asp;q .Rn / as introduced in Definition 3.1. But we repeat what we need to keep this Chapter 6 largely selfcontained, at least as far as notation and assertions are concerned. Let AVsp;q .Rn / be the completion of D.Rn / in Asp;q .Rn /. Recall that D.Rn / is dense in Asp;q .Rn / if p < 1, q < 1, hence AVsp;q .Rn / D Asp;q .Rn / if p < 1, q < 1. Let X be a Banachspaces with X S 0 .Rn /, 0 < T < 1, b 2 R and 1 v 1. Then Lv .0; T /; b; X collects all f .; t/ 2 X , 0 < t T , such that kf jLv .0; T /; b; X k D
Z
T
t bv kf .; t/ jX kv dt
1=v (6.42)
0
is finite in the understanding of vector-valued Lv -spaces (Bochner integral) with the usual modification if v D 1. In what follows we specify X to the Banach spaces Asp;q .Rn / with 1 p; q 1. Then kf .; t/ jX k are isomorphic to t-dependent sequence spaces according to the wavelet characterizations in Theorem 3.12. Afterwards weighted Lv -norms with respect to t as in (6.42) do not cause any problems. Let 1 < b 1 v1 and X D Asp;q .Rn /, 1 p; q 1. Then one has always Lv .0; T /; b; Asp;q .Rn / S 0 .RnC1 /
(6.43)
after extending functions of this space from Rn .0; T / to RnC1 by zero. This has been justified in (4.17)–(4.21). Q With X D Asp;q .Rn / and Xn D njD1 X let Asp;q .Rn /n D
n Y j D1
Asp;q .Rn /
(6.44)
151
6.1 Haar wavelets and Reynolds numbers
be the collection of all f D .f 1 ; : : : ; f n / such that f j 2 Asp;q .Rn / and n X
kf jAsp;q .Rn /n k D
kf j jAsp;q .Rn /k:
(6.45)
j D1
Then Lv .0; T /; b; Xn is normed by (6.42) with Xn in place of X D Asp;q .Rn /. There is an obvious counterpart of (6.43). This is the specification of (5.9)–(5.12) to global spaces. There are obvious counterparts with AVsp;q .Rn / in place of Asp;q .Rn /. Now Corollary 5.16 can be transferred from smooth wavelets to Haar wavelets specifying hybrid spaces LrAsp;q .Rn / by global spaces Ln=p Asp;q .Rn / D Asp;q .Rn /. The Banach space E.RnC1 as in (5.77). It is the collection of all C / has the same meaning D Rn .0; 1/ such that functions u.x; t/ D u1 .x; t/; : : : ; un .x; t/ in RnC1 C ku jE.RnC1 C /k Z tZ h j 2 n=2 t ju .x; t/j C t D sup x2Rn ;t >0; j D1;:::;n
0
p jxyj t
juj .y; /j2 dy d
i1=2
(6.46)
is finite. As before we deal with Navier-Stokes equations in the version of (5.1), (5.2). There one finds also related explanations. Recall that Qk;m D 2k m C 2k .0; 1/n, k 2 N0 , m 2 Zn . Theorem 6.8. Let n 2. Let 1 < p < 1, 1 q 1 and s > n=p. Let 2 < v 1; 0 < g 1
and a D 1
1 v
~g with 0 < ~ < 1:
(6.47)
Let C > 0. Then there are numbers T > 0 and k 2 N0 such that @t u u C P div .u ˝ u/ D 0 u.; 0/ D u0 .Rn /n with has for any u0 2 As1Cg p;q ku0 jAs1Cg .Rn /n k p;q
C
in Rn .0; 1/, in Rn ;
(6.48) (6.49)
Z u0 .x/ dx D 0;
and
for all m 2 Zn ;
Qk;m
(6.50) a unique solution
s n u 2 E.RnC1 C / \ L2v .0; T /; a=2; Ap;q .R /n :
(6.51)
.Rn /n and 1=2 ~ < 1 .1=2 < ~ < 1 in case of If, in addition, u0 2 AVs1Cg p;q v D 1/ then n Vs u 2E.RnC1 C / \ L2v .0; T /; a=2; Ap;q .R /n (6.52) \ C Œ0; T /; AVs1Cg .Rn /n \ C 1 Rn .0; T / n : p;q
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6 Navier-Stokes equations in global spaces
Proof. Compared with Corollary 5.16 one has to show that one can replace the smooth wavelets used there by Haar wavelets. With D s 1 C g one has (6.40). As explained in (6.41) the second conditions in (6.50) make sense for all admitted , hence > pn 1. We choose numbers w and % with w p, n 1 < w < 1;
1 < % <
n1 w
and
n > %: p
(6.53)
Then one has for the related inhomogeneous spaces the well-known embeddings n
n
C%
1
w w .Rn / ,! Bw;2 .Rn /; Ap;q .Rn / ,! Bw;2
(6.54)
[T83, Theorem 2.7.1, p. 129]. Furthermore, n n 1 1 1< 1< C%< (6.55) w w w w (here n 2 is needed). By Proposition 6.1 the B-spaces in (6.54) can be expanded by Haar wavelets. If one inserts the second condition in (6.50) (scalar case) in (6.16) with (6.10) then one has j;G m .u0 / D 0 for j < k. Hence X X X j n=2 j j;G HG;m u0 D m .u0 / 2 j k G2G m2Zn
D
X X X
(6.56)
j n=2 j j;G hG;m m .u0 / 2
j k G2G m2Zn
where we used the notation (6.6), (6.7) and (6.8), (6.11). The inital data u0 can be j g without starting terms and by expanded by the inhomogeneous Haar system fHG;m the homogeneous Haar system fhjG;m g without infrared tail (terms with j < 0). By Proposition 6.1, (6.14) and (6.54) one obtains for some c > 0, n
n
1
C%
w w ku0 jBw;2 .Rn /n k 2k.1C%/ ku0 jBw;2 .Rn /n k c C 2k.1C%/ ;
(6.57)
where C has the same meaning as in (6.50). By Remark 6.2 the functions ' 2 D.Rn / n n w C% w 1 .Rn / and Bw;2 .Rn / spanned by with (6.21) are dense in the subspaces of Bw;2 n C% (6.56). Furthermore one has (6.24) for the related homogeneous spaces BP w .Rn / w;2
n
w 1 .Rn /. But then we are in the same situation as in Section 3.4.5 where we and BP w;2 proved Theorem 2.33. In particular, one can interpret (6.56) as expansions in terms of n n j p C% w 1 .Rn / and BP w;2 .Rn /. Then it follows the homogeneous Haar system fhG;m g in BP w;2 from (6.26) with q D 2, (6.28) and (6.57) that n
w 1 .Rn /n k c 0 C 2k.1C%/ : ku0 jBMO1 .Rn /n k cku0 jBP w;2
(6.58)
Recall that C has the same meaning as in (6.50). Furthermore 1 C % > 0 (whereas c 0 is independent of C and k 2 N0 ). If k is large then one has (5.76). The rest is now the same as in the proofs of Proposition 5.14 and Corollary 5.16.
6.1 Haar wavelets and Reynolds numbers
153
Remark 6.9. One can replace the cubes Qk;m D 2k m C 2k .0; 1/n in (6.50) by the cubes z C d 2k m C d 2k .0; 1/n for fixed z 2 Rn and d > 0. This can be seen if one modifies the Haar functions in (6.8), (6.11), Proposition 6.1 and the above proof appropriately. Otherwise we followed largely [Tri14a]. There arise several questions. In contrast to Chapter 5 we deal in Chapter 6 exclusively with global spaces Asp;q .Rn /. One may ask whether Theorem 6.8 can be extended to the more general hybrid spaces. As far as expansions by Haar functions are concerned one can rely on Theorem 3.41 and, in case of Morrey spaces, on Theorem 2.33 and its proof in Section 3.4.5 in terms of homogeneous Haar systems. In addition one would need suitable counterparts of the distinguished embeddings in Proposition 6.3. There are s; some sharp embedding assertions for the spaces Bp;q .Rn / which can be reformulated s .Rn /, D p1 C nr . One according to Theorem 3.38 in terms of hybrid spaces LrBp;q may consult the recent paper [YHSY14] and the literature mentioned there. But this does not cover so far an adequate counterpart of Proposition 6.3. Remark 6.10. So far we have always assumed that the underlying spaces Asp;q .Rn / in Theorem 6.8 and Corollary 5.4, but also their hybrid generalizations LrAsp;q .Rn /, are multiplication algebras. Then u 7! u2 is a (nonlinear) map within these spaces. This is convenient, surely the first choice, but not necessary if one checks our approach in [T13], now transferred to hybrid spaces, including Asp;q .Rn /. One can try to rely on more sophisticated multiplication properties. Restricted to the above needs, hence u 7! u2 , we recall some related assertions. Let s > 0, 0 < p2 < p1 < 1, 0 < q 1, 1 s s 1 2 1 0< D < 1: and D (6.59) r p1 n r p2 n Then Bps 1 ;q .Rn / Bps 1 ;q .Rn / 7! Bps 2 ;q .Rn / if, and only if, Furthermore
Fps1 ;q .Rn / Fps1 ;q .Rn / 7! Fps2 ;q .Rn /:
0 < q r:
(6.60) (6.61)
This is a special case of [ET96, Theorem 2.4.3, p. 52/53], proved by paramultiplication with a reference to [SiT95]. Based on this observation, Corollary 5.4 has been extended in [Baa14] to a larger class of underlying spaces Asp;q .Rn /. One can expect that this applies also to Theorem 6.8 (but we have not yet done so). As far as the initial data u0 2 Ap;q .Rn /n with D s 1 C g are concerned one has again the restriction (6.40), hence the supercritical case according to Section 5.5.1. But there are some modifications and improvements as far as the other parameters in (5.18) = (6.47) are concerned. Remark 6.11. The Navier-Stokes equations describe the motion of an incompressible fluid in Rn where n D 2 or n D 3. The case of a planar fluid, hence n D 2, is well understood. The standard reference here is [Lad69]. One may also consult [GGS10, Section 2.2, pp. 42–47]. The recent literature deals quite often exclusively
154
6 Navier-Stokes equations in global spaces
with n D 3, whereas the extension to n > 3 is considered as a technical matter. The case n D 1 in (1.1)–(1.3) does not make much sense because of u0 D div u D 0. Furthermore (1.7), (1.8) reduces to the linear heat equation in R .0; T / because of P D 0. The situation might be different with the model case (1.11), (1.12). But usually there was no reason in our considerations to exclude n D 1 explicitly. However the situation in Theorem 6.8 is different. One has to assume n 2. If n D 1 then the critical spaces 1
1
p 1 p 1 Bp;q .R/ and BP p;q .R/;
1 < p; q < 1;
(6.62)
cannot be characterized by Haar wavelets. More generally, neither 1
1
1
p p Bp;q .Rn / nor Bp;q .Rn /;
1 < p; q < 1;
n 2 N;
(6.63)
can be expanded by Haar wavelets. This follows from the necessary and sufficient conditions (3.173) ensuring that the characteristic function of a half-space is a pointwise multiplier in Asp;q .Rn /. But we do not know whether it is of interest to have a closer look at the counterpart of Theorem 6.8 with n D 1 for, say, the nonlinear heat equation (1.11), (1.12) with T D 1. Of course, there is no physical relevance. It seems to be a little bit like the mockery of the inhabitants of Flatland when looking at the people of Lineland in [Abb78].
6.1.4 Reynolds numbers We modify (1.7), (1.8) (with T D 1) by @t u
1 u C P div .u ˝ u/ D 0 Re u.; 0/ D u0
in Rn .0; 1/,
(6.64)
in Rn ;
(6.65)
as the preferred version of @t u C .u; r/u
1 u C rP D 0 Re div u D 0 u.; 0/ D u0
in Rn .0; 1/,
(6.66)
in Rn .0; 1/; in Rn :
(6.67) (6.68)
All notations have the same meaning as in Chapter 1 and Section 5.1. Again we do not assume div u0 D 0 in (6.65). Here Re > 0 is the Reynolds number. The case n D 3 is of physical relevance and of utmost interest. The Reynolds number Re is a dimensionless physical quantity which governs the flow u in (6.66)–(6.68) in a decisive way. Following [BoF13] one calls the flow u laminar if Re > 0 is small and turbulent if Re is large. Discussions about the role of Re may be found in [Lad69, Con06]. In particular if Re ! 1 then one expects turbulences, possible loss of uniqueness and regularity of related solutions u, blow-up effects and breakdowns. We quote here a few sentences from [Con06, p. 1] where one finds with an explicit reference to Reynolds original paper [Rey84]:
155
6.1 Haar wavelets and Reynolds numbers
The Reynolds equations are still a riddle. They are based on the NavierStokes equations, which are still a mystery. The Navier-Stokes equations are a viscous regularization of the Euler equations, which are still an enigma. Turbulence is a riddle wrapped in a mystery inside of an enigma. According to [Con06] the last sentence adapts Sir W. Churchill’s opinion, I cannot forecast to you the action of Russia. It is a riddle wrapped in a mystery inside an enigma, (Radio Broadcast, Oct. 1939). In [Con06] one finds also a discussion, supported by outstanding references, about the mystery in the Navier-Stokes equations: the infinite time behavior at finite but larger and larger Reynolds numbers. If one replaces (6.48), (6.49) in Theorem 6.8 by (6.64), (6.65) then the above quotations suggest that one needs k ! 1 in (6.50) for the cubes Qk;m D 2k m C 2k .0; 1/n, k 2 N, m 2 Zn , if Re ! 1 to ensure solutions global in time. It is just this point which we wish to discuss in a quantitative way. We are only interested in this peculiar effect (and not in the other parts of Theorem 6.8) and which can be said in our context of Haar expansions. For this purpose it is sufficient to assume n
C%
w u0 2 Bw;2 .Rn /n ;
n 1 < w < 1;
1 < % <
n1 ; w
(6.69)
n 2. By wn C % wn D % > 1 these spaces are supercritical according to (6.40) with a reference to Section 5.5.1. Otherwise we have the same situation as n n w C% w 1 .Rn / and Bw;2 .Rn / admit expansions by in (6.54), (6.55), in particular both Bw;2 Haar wavelets according to Proposition 6.1. Let again Qk;m D 2k m C 2k .0; 1/n, k 2 N, m 2 Zn , whereas E.RnC1 C / has the same meaning as in Theorem 6.8, normed by (6.46). Corollary 6.12. Let n 2, n1<w <1
and
1<%<
n1 : w
(6.70)
Let C > 0, Re 1 and k 2 N with Re 2k ;
C .Re/% 2.%C1/k :
(6.71)
Then there is a number k0 2 N0 which is independent of C , Re and k such that @t u
1 u C P div .u ˝ u/ D 0 Re u.; 0/ D u0
in Rn .0; 1/,
(6.72)
in Rn ;
(6.73)
156
6 Navier-Stokes equations in global spaces n
C%
w has for any u0 2 Bw;2 .Rn /n with Z n w C% .Rn /n k C and ku0 jBw;2
u0 .x/ dx D 0
for all m 2 Zn ; (6.74)
QkCk0 ;m
a unique solution u 2 E.RnC1 C /. Proof. We reduce Corollary 6.12 to Theorem 6.8 by scaling. For this purpose we assume that uRe D u is a solution of (6.72), (6.73) with the initial data u0 as in x t ; Re . The Leray projector P according to (5.4) based (6.74). Let uRe .x; t/ D uRe Re on (5.5) is homogeneous of degree zero. Then one obtains @t uRe uRe C P div .uRe ˝ uRe / D 0 uRe .; 0/ D
u0 Re
in Rn .0; 1/,
(6.75)
in R :
(6.76)
n
We expand u0 as in (6.56) now with k C k0 in place of k. For sake of simplicity one may assume Re D 2l with l 2 N0 , l k. One has X X X u0 j n=2 j u0 j;G HG;m D /2 m . Re Re j k0 Ckl G2G m2Zn (6.77) X X X u0 j n=2 j D j;G hG;m : /2 m . Re n j k0 Ckl G2G m2Z
j
This is again an expansion by the inhomogeneous Haar system fHG;m g in (6.8) without starting terms which can be interpreted as an expansion by the homogeneous Haar system fhjG;m g in (6.11) without infrared tail. In particular one obtains by the n
w C% homogeneity (6.24) for the homogeneous spaces BP w;2 .Rn /,
n n uRe .; 0/ jB w C% .Rn /n .Re/% u0 jB w C% .Rn /n :
w;2
w;2
(6.78)
From (6.57), (6.58) based on (6.71) and (6.74) follows kuRe .; 0/ jBMO1 .Rn /n k c C .Re/% 2k.%C1/ 2k0 .%C1/ ı;
(6.79)
if k0 for given ı > 0 is chosen sufficiently large. One obtains by the same arguments as in the proof of Theorem 6.8 that (6.75), (6.76) has a unique solution in E.RnC1 C /. Re Re-transformation from uRe to u proves the corollary. Remark 6.13. As indicated at the beginning of Remark 6.9 one can replace the cubes QkCk0 ;m in (6.74) by the cubes zCd 2.kCk0 / mCd 2.kCk0 / .0; 1/n for fixed z 2 Rn and d > 0. This makes also clear that the assumption Re D 2l , l 2 N0 , in the above proof is immaterial. Furthermore similarly as in Remark 5.19 it is not necessary that the integrals in (6.50) and (6.74) are zero. It is sufficient that their absolute values are small. We return to this point in greater detail in Section 6.1.5 below.
6.1 Haar wavelets and Reynolds numbers
157
Remark 6.14. The outcome of the above corollary is the expected one. The uniform behaviour in tiny cubes of side-length 2k according to (6.74) prevents the solution u from becoming turbulent (no longer globally in E.RnC1 C /) even if Re is large. This is in good agreement with expectations of researchers working in this field, especially in the case of physical relevance n D 3 and also with the quotations at the beginning of this Section 6.1.4. If C D 1 in (6.71) then one has for optimally chosen k 2 N, ( % .Re/ %C1 if 1 < % < 12 ; k 2 (6.80) ; Re if 12 % < n1 w 12 . The exponent in the first line in (6.80) where the second line is empty if n1 w is monotone in % and larger than 1. If % # 1 then 2k ! 1. This improves monotonically in the interval 1 < % 1=2. There is no further improvement if % > 1=2. But this additional observation may depend on our approach.
6.1.5 Oscillation, persistency, lattice structure of initial data The proofs of Theorem 6.8 and Corollary 6.12 rely on the interplay between inhomogeneous Haar systems without starting terms and homogeneous Haar systems without infrared tails as indicated in (6.56). The same has been done in connection with Corollary 5.16 based on smooth inhomogeneous and homogeneous wavelet systems with (5.55) in place of the second condition in (6.50). As already mentioned in Remark 5.17 the assumption that the initial data u0 2 X.Rn /n are infrared-damped according to Definition 5.10 and its Haar counterparts in Theorem 6.8, Corollary 6.12 may be considered as wavelet versions of oscillating (small) initial data as discussed in [Mey01, Mey01a, Mey06] and [Can04, Chapter 4]. In [Mey01a, pp. 71, 72, 84] and in [Mey06, pp. 121–123] initial data u0 2 X.Rn /n are called oscillating if in addition ku0 jAPp;q .Rn /n k is small in some critical homogeneous spaces APp;q .Rn /n with pn D 1 and < 0, especially in case of related Besov spaces BP p;q .Rn / and BMO1 .Rn /. By (5.93), (5.94) and (6.58), (6.79) the initial data in Corollary 5.16, Theorem 6.8 and Corollary 6.12 are oscillating in this terminology. If the initial data u0 2 X.Rn /n are in addition small in BMO1 .Rn /, maybe as a consequence of assumed oscillation resulting in (6.58), then one has for some spaces nC1 X.Rn/ that the global solution u 2 E.RC / according to (6.46) belongs also to n L1 .0; 1/; X.R /n . This remarkable property is called persistency. We continue our discussion in Remark 5.18. There one finds distinguished spaces X.Rn / having this property and also related references. This will not be repeated here. Assuming that Theorem 6.8 can be extended to g D 0 (which is desirable but not covered by our approach) then one may ask whether (6.51) and (6.52) can be complemented by n n s n w 1 u 2 E.RnC1 (6.81) C / \ C Œ0; 1/; Bw;2 .R /n \ L2v .0; 1/; a=2; Ap;q .R /n n
1
w .Rn / is a critical space. As far as the where n < w < 1 and w p. Here Bw;2 second space in (6.81) is concerned one may consult [Mey06, Theorem 6.1, p. 128],
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6 Navier-Stokes equations in global spaces
the reference to [Can95] and [Can04, Theorem 3, pp. 195–196]. The last space in (6.81) asks for some persistency properties similar to the above-mentioned references and as in Remark 5.18. The above-suggested assumption that the initial data should oscillate is rather qualitative whereas the requested lattice structure of u0 in (6.50) might be too rigid. The question arises whether this lattice structure of the initial data u0 is not only an artifact of our approach but a natural property of initial data to ensure global solutions of Navier-Stokes equations. In any case a more moderate qualitative version of the second conditions in (6.50) and in (5.55) is desirable. But this might be possible. To ensure (6.58) we used (6.56), (6.57) based on (6.14). If one admits terms with j k in (6.56) then one can again rely on Proposition 6.1, the related coefficients in (6.10) and the adapted sequence spaces in (6.14). We do not discuss these possibilities systematically, but we describe an example. Let ˇZ ˇ ˇ ˇ 2j n ˇ u0 .x/ dx ˇ cm 2.kj /.1C%/ ; j 2 N; j k C 1; m 2 Zn ; Qj;m
(6.82) ˇZ ˇ ˇ
and
Q0;m
ˇ ˇ 0 k.1C%/ u0 .x/ dx ˇ cm 2 ;
X
such that
jcm jw < 1;
m2Zn
X
m 2 Zn ;
0 jcm j < 1;
(6.83) (6.84)
m2Zn
with w and % as in (6.53)–(6.55) instead of the second conditions in (6.50). Then j;G one has by (6.10), based on (6.8), similar estimates for the coefficients m .u0 / with j k. One obtains for the corresponding terms in (6.14) with j 2 N applied to n
1
w .Rn /, 0 < q < 1, bw;q
2
k.1C%/
X m2Zn
jcm j
w
k 1=w X
2j q 2j.1C%/q
1=q
c 2k.1C%/
(6.85)
j D1
where we used (6.53). But then one is in the same situation as in the proof of Theorem 6.8 with q D 2. However to step from (6.57) to (6.58) under these modified conditions for the initial data u0 one must be sure that the starting terms for the inhomogeneous Haar system (6.8), hence the characteristic functions of the cubes mC.0; 1/n, m 2 Zn , fit in the scheme (which is obvious for the other terms belonging also to the homogeneous Haar system (6.11)). But this is covered by Corollary 6.7, where one needs again n 2 and the second condition in (6.84). Afterwards one can generalize Theorem 6.8 to these initial data. Similarly one can replace (5.55) by j corresponding conditions for the coefficients u0 ; ‰G;m in (5.56). One may extend the notion of infrared-damped to conditions of type (6.82)–(6.84) for the initial data u0 . There is again a (moderate) lattice structure.
6.2 Initial data in spaces with dominating mixed smoothness
159
One may ask again whether these two conditions, to be infrared-damped and having a moderate lattice structure, are a natural price to pay in order that the solution of the Navier-Stokes equations for arbitrarily large initial data belonging to a supercritical space can be extended globally in time (at least in E.RnC1 C /). The approach in this Section 6.1 relies on the interplay between homogeneous and inhomogeneous isotropic spaces APsp;q .Rn / and Asp;q .Rn /. These spaces have many symmetries. In particular they are rotationally invariant. But one does not really need this property. One may ask for spaces having less symmetries which are equally good or maybe even better adapted to Navier-Stokes equations. After the Cartesian coordinates are fixed it seems to be indispensable that related spaces are invariant against permutations of coordinates, but maybe not more. This suggests dealing with Navier-Stokes equations in the context of spaces with dominating mixed smoothness, subject of Section 6.2.
6.2 Initial data in spaces with dominating mixed smoothness 6.2.1 Introduction We asked at the end of the last Section 6.1.5 whether homogeneous and inhomogeneous isotropic spaces APsp;q .Rn / and Asp;q .Rn / are natural or optimal in the context of Navier-Stokes equations. These spaces have some distinguished properties for which one has little use in the framework of Navier-Stokes equations. We mention as a typical example that all these spaces are rotationally invariant. On the other hand it seems to be quite natural that underlying spaces are invariant against permutations of coordinates (after the Cartesian coordinates are fixed in which the Navier-Stokes problem is stated with initial data having similar properties). Adopting this point of view one comes to spaces with dominating mixed smoothness as a distinguished choice. There are some striking examples where seemingly isotropic problems can be treated better in spaces with dominating mixedRsmoothness than in isotropic spaces. One may think about numerical integration of Q f .x/ dx in cubes Q or in Rn , related sampling numbers, tractability and discrepancy (the unavoidable deviation from uniformity of the distribution of points, say, in cubes, coming from number theory). Later on we say what is meant by sampling numbers. Then we give also some references. It is the main aim of this Section 6.2 to describe a link between sampling in spaces with dominating mixed smoothness and solutions of Navier-Stokes equations, global in time. First we describe our method keeping temporarily the technicalities as simple as possible. Generalizations, greater details and references are shifted to later sections. Let Sp1 W .Rn /, 1 < p < 1, be the classical Sobolev space with dominating mixed
160
6 Navier-Stokes equations in global spaces
smoothness of first order, collecting all f 2 S 0 .Rn / such that X kD ˇ f jLp .Rn /k kf jSp1 W .Rn /k D
(6.86)
ˇ 2Nn 0; ˇj 2f0;1g
is finite. Here D ˇ has the usual meaning, hence Dˇ D
@jˇ j @x1ˇ1 @xnˇn
;
jˇj D
n X
ˇj ;
ˇ j 2 N0 :
(6.87)
j D1
In particular any ˇj in (6.86) is either 0 or 1. This explains also why constructions of this type are called spaces with dominating mixed smoothness, in contrast to, for example, the classical Sobolev spaces Wpk .Rn / normed by (3.18). Let w ˛ .x/ D
n Y
.1 C xj2 /˛=2 ;
˛ 2 R;
x D .x1 ; : : : ; xn / 2 Rn :
(6.88)
j D1
Then Sp1 W .Rn ; ˛/, 1 < p < 1, collects all f 2 S 0 .Rn / such that w ˛ f 2 Sp1 W .Rn /, obviously normed by kf jSp1 W .Rn ; ˛/k D kw ˛ f jSp1 W .Rn /k:
(6.89)
Let again C.Rn / be the collection of all complex-valued bounded continuous functions in Rn , normed by kf jC.Rn /k D sup jf .x/j (6.90) x2Rn
and let
C0 .Rn / D ff 2 C.Rn / W jf .x/j ! 0 if jxj ! 1g:
(6.91)
Sp1 W .Rn ; ˛/
If ˛ > 0 then is continuously, even compactly, embedded in the space n C0 .R /. Let n 2, 2 p < 1, 1 u p and ˛ > 2 p1 . Then all embeddings id W
Sp1 W .Rn ; ˛/ ,! Lu .Rn /
(6.92)
are compact. We dealt in [T12] with sampling numbers of these embeddings. (What is meant by sampling numbers will be explained in Section 6.2.3 below.) For this N purpose we constructed for any 2 N 2 N distinguished sets ˛;p of N sampling points n o ˚ ˛;p N N k1 kn n n ˛;p D N;j 2 m ; : : : ; 2 m ; m 2 Z ; k 2 N (6.93) 1 n 0 j D1 such that 1
kf jLu .Rn /k c N 1 .log N /.n1/.2 p / kf jSp1 W .Rn ; ˛/k
(6.94)
6.2 Initial data in spaces with dominating mixed smoothness
161
N for f 2 Sp1 W .Rn ; ˛/ with f j˛;p D 0. Since Sp1 W .Rn ; ˛/ is continuously embed˚ ˛;p N N D f .N;j / j D1 makes sense. It is the main ded in C.Rn / the restriction f j˛;p aim of this Section 6.2 to employ this assertion in the context of the Navier-Stokes equations in the version of
@t u u C P div .u ˝ u/ D 0 u.; 0/ D u0
in Rn .0; 1/, in Rn :
(6.95) (6.96)
Here we use the same notation as in Section 5.1 where one finds the necessary explanations. One may also consult Section 6.1.3. Similarly as there we are interested in conditions uniquely determined solutions for the initial data u0 in (6.96) ensuring n u.x; t/ D u1 .x; t/; : : : ; un .x; t/ in RnC1 D R .0; 1/ in E.RnC1 C C / as before, normed by ku jE.RnC1 C /k Z tZ h D sup t juj .x; t/j2 C t n=2 x2Rn ;t >0;
0
j D1;:::;n
p jxyj t
juj .y; /j2 dy d
i1=2 :
(6.97)
Otherwise we use the same notation as in Section 6.1.3. In particular, if X.Rn/ is a quasi-Banach space then X.Rn /n collects all f D .f 1 ; : : : ; f n / with f j 2 X.Rn / quasi-normed by n X kf jX.Rn/n k D kf j jX.Rn /k: (6.98) j D1
Let
˚ N N / D f 2 Sp1 W .Rn ; ˛/ W f j˛;p D0 Sp1 W .Rn ; ˛I ˛;p
(6.99)
N has the same meaning as above. This is a finite co-dimensional subspace where ˛;p 1 of Sp W .Rn ; ˛/. It is the main aim of Section 6.2 to prove assertions of the following prototype.
Theorem 6.15. Let 2 n < p < 1 and ˛ > 2 a number N 2 N such that @t u u C P div .u ˝ u/ D 0 u.; 0/ D u0
1 . p
Let C > 0. Then there is
in Rn .0; 1/, in Rn ;
(6.100) (6.101)
has for any N /n u0 2 Sp1 W .Rn ; ˛I ˛;p
a unique solution in E.RnC1 C /.
with
ku0 jSp1 W .Rn ; ˛/n k C
(6.102)
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6 Navier-Stokes equations in global spaces
This is a special but nevertheless typical case we are dealing with below. There we weaken the assumptions about ˛ and extend these assertions from Sobolev spaces r Sp1 W .Rn ; ˛/ to more general spaces Sp;q A.Rn ; ˛/ of dominating mixed smoothness. In some sense Theorem 6.15 is the counterpart of Theorem 6.8. Essentially we are doing the same as there: We use (6.94) to prove a counterpart of (6.58) and hence of (5.76) which ensures that (6.100)–(6.102) has a unique solution in E.RnC1 C /. But in contrast to Theorem 6.8 we do not discuss further properties of the solutions similarly as in (6.52) (which are not yet available). Instead of (6.58), (5.76) one can use (6.94) to ensure that (6.103) ku0 jLn .Rn /n k ı for any given ı > 0 if N in (6.102) is chosen sufficiently large. Recall that Ln .Rn /, n 2, is a critical space for Navier-Stokes equations, (5.41). If ı > 0 in (6.103) is small then (6.100), (6.101) has a solution, global in time. This celebrated result goes back to [Kato84]. Uniqueness in suitable spaces has been proved much later in [FLT97, FLT00]. Nowadays this cornerstone of Navier-Stokes theory as well as several modifications and generalizations are treated in detail in corresponding books and surveys. We refer the reader in particular to [Mey06, Sections 5–7] and the related parts in [BCD11, Can95, Can04, Lem02]. It should be mentioned that these assertions are usually formulated in terms of the original Navier-Stokes equations (1.1)–(1.3) with div u0 D 0 and quite often restricted to n D 3 (the case of physical relevance). But [Lem02] covers all dimensions 2 n 2 N. As we have noted the conditions in Theorem 6.15 ensure not only (5.76) but also (6.103). This applies also to the more general assertions below. But we stick in what follows to the indicated reduction to BMO1 .Rn /. On the one hand we wish to be coherent with Section 6.1 where one has no such possibilities, at least not in general. On the other hand, further modifications and in particular combinations with some assertions in Section 6.1 (expansions by Haar functions) may result in estimates of type (5.76), but not of type (6.103). We return to this point in Section 6.2.6 below. In this Section 6.2 we follow largely [Tri14b].
6.2.2 Definitions and basic properties The Sobolev spaces Sp1 W .Rn / and Sp1 W .Rn ; ˛/, normed by (6.86), (6.89), are distinguished but special cases of spaces with dominating mixed smoothness. First we recall some definitions. Let '0 2 S.R/ with '0 .t/ D 1 if jtj 1
and '0 .v/ D 0 if jvj 3=2;
(6.104)
and let 'l .t/ D '0 .2l t/ '0 .2lC1 t/;
t 2 R;
l 2 N;
(6.105)
6.2 Initial data in spaces with dominating mixed smoothness
163
be the one-dimensional resolution of unity according to (3.9)–(3.11). Let 2 n 2 N and 'k .x/ D
n Y
k D .k1 ; : : : ; kn / 2 Nn0 ;
'kj .xj /;
x D .x1 ; : : : ; xn / 2 Rn :
j D1
(6.106) Since
X
'k .x/ D 1
for x 2 Rn
(6.107)
k2Nn 0
P the system ' D f'k gk2Nn0 forms a resolution of unity. Let again jkj D njD1 kj if k 2 Nn0 . Let A 2 fB; F g, 0 < p; q 1 (p < 1 for F -spaces) and r 2 R. Then r Sp;q A.Rn / collects all f 2 S 0 .Rn / such that 8 1=q P ˆ rjkjq b/_ jLp .Rn /kq < ; B-spaces; .' f n 2 k k2N r A.Rn /k' D P 0 kf jSp;q 1=q ˇ ˇ ˆ b/_ ./ˇq : 2rjkjq ˇ.'k f jLp .Rn / ; F -spaces; k2Nn 0 (6.108) is finite (usual modification if q D 1). These spaces are independent of ' (equivalent quasi-norms). This is the counterpart of Definition 3.1. The Fourier-analytical theory of these spaces and further (historical) references may be found in [ST87, ScS04, Schm07, Vyb06, T10]. Recall that 0 F .Rn / D Lp .Rn /; Sp;2
1 Sp;2 F .Rn / D Sp1 W .Rn /;
1 < p < 1;
(6.109)
where Sp1 W .Rn / can be (equivalently) normed by (6.86). Furthermore, r r r B.Rn / ,! Sp;q F .Rn / ,! Sp;max.p;q/ B.Rn / Sp;min.p;q/
and
r Sp;q B.Rn / ,! C.Rn /;
r > 1=p;
(6.110) (6.111)
for all admitted parameters. This may be found in the above-mentioned literature. We refer the reader in particular to [ST87, pp. 89, 132]. Let w ˛ with ˛ 2 R be as in r r A.Rn ; ˛/ collects all f 2 S 0 .Rn / such that w ˛ f 2 Sp;q A.Rn /; (6.88). Then Sp;q quasi-normed by r r kf jSp;q A.Rn ; ˛/k D kw ˛ f jSp;q A.Rn /k;
(6.112)
[T12, pp. 9, 98]. It follows from (6.109) that the Sobolev spaces Sp1 W .Rn ; ˛/ as introduced in Section 6.2.1 are special cases. If is a bounded domain in Rn then r r A. / is the restriction of Sp;q A.Rn / to , quasi-normed by Sp;q r r A. /k D inf kg jSp;q A.Rn /k kf jSp;q
(6.113)
r A.Rn / with gj D f 2 D 0 . /. This is where the infimum is taken over all g 2 Sp;q the counterpart of Definition 3.16 where one finds further explanations.
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6 Navier-Stokes equations in global spaces
6.2.3 Sampling N We wish to provide a better understanding of where the distinguished sets ˛;p in (6.93) and the resulting estimates of type (6.94) are coming from. First we recall what is meant by linear sampling numbers adapted to our specific situation. Let G1 .Rn / and G2 .Rn / be two quasi-Banach spaces such that
S.Rn / ,! G1 .Rn / ,! G2 .Rn / ,! S 0 .Rn /: We assume that is compact and that
(6.114)
id W
G1 .Rn / ,! G2 .Rn /
(6.115)
id W
G1 .Rn / ,! C.Rn /
(6.116)
is continuous. Here C.Rn / is again the space of all complex-valued continuous bounded functions in Rn , normed by (6.90). The embedding (6.116) must be interpreted in the usual way: In each equivalence class of G1 .Rn / (consisting of regular distributions) there is a (uniquely determined) continuous function belonging to C.Rn / to which (6.116) applies. Let N 2 N. Then the N -th linear sampling number of the compact embedding id in (6.115) is given by lin id W G1 .Rn / ,! G2 .Rn / gN h ˚ i (6.117) D inf sup kf SN f jG2 .Rn /k W kf jG1 .Rn /k 1 n where the infimum is taken over all N -tuples fx j gN j D1 R and all linear maps
SN f D
N X
f .x j / hj ;
hj 2 G2 .Rn /;
f 2 G1 .Rn /:
(6.118)
j D1
The (nonlinear) sampling numbers gN .id/ are defined similarly where now arbitrary (nonlinear) maps SN f D ˆN f .x 1 /; : : : ; f .x N / 2 G2 .Rn /; f 2 G1 .Rn /; (6.119) in place of (6.118) are admitted. But this will not be needed here. The abstract background of sampling, information-based complexity and their relations to numerical integration and discrepancy may be found in [NoW08, Chapter 4] and [NoW10]. Let be a bounded domain in Rn . Then we introduced the spaces Asp;q . / and r Sp;q A. / in Definition 3.16 and (6.113) by restriction. Then one can ask again for related sampling numbers where one has to replace Rn in (6.115)–(6.118) by . r Sampling for Asp;q . / and Sp;q A.Q/ with Q D .0; 1/n and the relations to numerical integration and discrepancy have been studied in [T10]. There one finds also the related literature. We refer the reader in particular to the survey [Vyb07]. The step from bounded domains and cubes to Rn requires that some weights are coming in to ensure that embeddings of type (6.115) are compact. This has been done
165
6.2 Initial data in spaces with dominating mixed smoothness
r in [T12] for some weighted spaces Sp;q B.Rn ; ˛/ and Sp1 W .Rn ; ˛/ as introduced in Sections 6.2.1, 6.2.2. We formulate corresponding assertions which cover (6.94) as a special case.
Proposition 6.16. Let 2 n 2 N. Let ˛ > 0, u 1;
0
1 1 1 < C˛ p u p
1 1
and
(6.120)
0 < q 1. Then id W
r B.Rn ; ˛/ ,! Lu .Rn / Sp;q
is compact. (i) If, in addition, p D q and ˛ > r c1 , c2 such that for 2 N 2 N,
1 p
C
1 u
(6.121)
then there are two positive numbers
1
1
lin c1 N r .log N /.n1/.1 p / gN .id/ c2 N r .log N /.n1/.rC1 p / :
(6.122)
(ii) If, in addition, ˛ < r p1 C u1 then there are numbers c > 0 and for any ", 0 < " < 1, numbers c" > 0 such that for 2 N 2 N, 1
1
1
1
˛
1
lin c1 N ˛ p C u .log N /.n1/˛ gN .id/ c" N ˛ p C u .log N /.n1/.2˛C r .1 u /C"/ : (6.123)
Remark 6.17. This follows from [T12, Theorems 4.5, 4.20, pp. 82, 98]. The restriction r > 1=p ensures the embedding (6.111) whereas the restriction r < 1 C p1 comes from our use of Faber bases. (Whereas r > 1=p is indispensable, the assumption r < 1 C p1 is immaterial when it comes to Navier-Stokes equations.) The N according to (6.93) play a crucial role in Theorem distinguished sets of points ˛;p 6.15 and in the more general assertions below. They can be constructed explicitly looking for optimal approximations in (6.117), (6.118) ensuring the right-hand sides of (6.122), (6.123). We outline the procedure, but refer the reader for details to [T12], r which in turn is based on [T10]. One expands f 2 Sp;q B.Rn ; ˛/ in terms of Faber bases fvk;m g, X k;m vk;m C C C : (6.124) f D n k2Nn 0 ;m2Z
We do not need vk;m explicitly but we remark that @1 @n vk;m .x/ D ck
n Y
hM 2kl xl ml ;
k 2 Nn0 ;
m 2 Zn ;
(6.125)
lD1
where hM is the one-dimensional Haar function (6.5) and ck are suitable normalizing factors. Hence vk;m is the (tensor) product of one-dimensional hat functions.
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6 Navier-Stokes equations in global spaces
This must be complemented by some starting terms indicated in (6.124) by C C C. Furthermore X k;m D am0 f 2k m C 12 2k m0 ; k 2 Nn0 ; m 2 Zn ; (6.126) m0 2Zn ;jm0 ja
for some constants a > 0 and am0 2 R. Here 2k m D 2k1 m1 ; : : : ; 2kn mn . Hence k;m evaluates f at 2k m and some neighbouring (tensor)-lattice points. It is the basis to determine optimal operators SN in (6.118) resulting in the right-hand N sides of (6.122), (6.123). This leads under the indicated specifications to sets ˛;p;u of the same type as in (6.93). The calculations in [T12] are somewhat involved, but nevertheless constructive. It is an interplay between the weight w ˛ in (6.88), ˛ > 0, and lattice points 2k m D 2k1 m1 ; : : : ; 2kn mn ; k D .k1 ; : : : ; kn / 2 Nn0 ; m D .m1 ; : : : ; mn / 2 Zn ; (6.127) in the context of the hyperbolic cross. Let n Y ˚ n Vl D x 2 R W .1 C xj2 /˛=2 < l ;
Wl D VlC1 n Vl ;
2 l 2 N; (6.128)
j D1
and W1 D V2 . Let for L 2 N, M1 : : : ML ;
Ml 2 N;
(6.129)
be a decreasing sequence of natural numbers and l D
n X
˚ k 2 1 m1 ; : : : ; 2kn mn 2 Wl ;
kj Ml ; m 2 Zn
(6.130)
j D1
S be related lattice points within Wl . Then D L lD1 l is the set of sampling points in (6.118) we are looking for. If ˛, p, u and N are given then one identifies SN in (6.118) with partial sums of (6.124), (6.126) with optimally chosen L 2 N, Ml in (6.129) and l in (6.130). In this way one can prove the right-hand sides of (6.122), S N (6.123) where the underlying sets of points ˛;p;u DD L lD1 l may also depend on u. Corollary 6.18. Let 2 n 2 N. Let ˛ > 0, 1 < p < 1, u 1 and Then id W Sp1 W .Rn ; ˛/ ,! Lu .Rn / is compact. (i) If, in addition, p 2 and ˛ > 1 c1 , c2 such that for 2 N 2 N, c1 N 1 .log N /
n1 2
1 p
C
1 u
1 p
1 u
<
1 p
C ˛.
(6.131)
then there are two positive numbers 1
lin gN .id/ c2 N 1 .log N /.n1/.2 p / :
(6.132)
6.2 Initial data in spaces with dominating mixed smoothness
167
(ii) If, in addition, ˛ < 1 p1 C u1 then there are numbers c > 0 and for any ", 0 < " < 1, numbers c" > 0 such that for 2 N 2 N, 1
1
1
1
1
lin c N ˛ p C u .log N /˛.n1/ gN .id/ c" N ˛ p C u .log N /˛.n1/.3 u /C" : (6.133)
Remark 6.19. This follows from [T12, Corollaries 4.9, 4.21, pp. 90, 99]. But (6.133) and the right-hand side of (6.132) are also covered by Proposition 6.16, (6.109) and (6.110), (6.111) with r D 1. Remark 6.20. Later on we will rely on Proposition 6.16 and Corollary 6.18 with u D p and u D 1 under the assumption 2 n < p < 1 and ˛ > 1 p1 . Then one N N N D ˛;p;p [ ˛;p;1 in (6.93), Theorem 6.15 and the more general can choose ˛;p N assertions below (consisting now of at most 2N points). Here ˛;p;u has the same meaning as at the end of Remark 6.17.
6.2.4 Sobolev spaces After the above preparations one can prove Theorem 6.15 in a more general version using Corollary 6.18 instead of (6.94). For given 1 < p < 1, ˛ > 0 and u D p the distinguished sets n o ˚ ˛;p N N k1 kn n n D N;j 2 m ; : : : ; 2 m ; m 2 Z ; k 2 N ˛;p ; (6.134) 1 n 0 j D1 N 2 N, of the related hyperbolic cross, have always the meaning as explained in Remark 6.17, resulting in the right-hand sides of (6.122), (6.123) and (6.132), (6.133). If one needs Proposition 6.16 or Corollary 6.18 both for u D p and u D 1 (as in TheN orems 6.21 and 6.23, but not in Theorem 6.27 below) then ˛;p must be understood as indicated in Remark 6.20. Let again ˚ N N / D f 2 Sp1 W .Rn ; ˛/ W f j˛;p D0 (6.135) Sp1 W .Rn ; ˛I ˛;p with Sp1 W .Rn ; ˛/ as in (6.86)–(6.89) and (6.109), (6.110). The Navier-Stokes equations (6.95), (6.96) have the same meaning as explained in detail in Section 5.1. As indicated there and also after (5.77) we do not need the initial data u0 to be divergencefree, div u0 D 0. We use the notation (6.98). Let again E.RnC1 C / be the above Banach space normed by (6.97). We extend now the formulation of Theorem 6.15 from ˛ > 2 p1 to ˛ > 1 p1 . Theorem 6.21. Let 2 n < p < 1 and ˛ > 1 a number N 2 N such that @t u u C P div .u ˝ u/ D 0 u.; 0/ D u0
1 . p
Let C > 0. Then there is
in Rn .0; 1/, in Rn ;
(6.136) (6.137)
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6 Navier-Stokes equations in global spaces
has for any N u0 2 Sp1 W .Rn ; ˛I ˛;p /n
with
ku0 jSp1 W .Rn ; ˛/n k C
(6.138)
a unique solution in E.RnC1 C /. Proof. It is sufficient to deal with the scalar case of (6.138). Let ˛ > 2 p1 . From (6.132), based on (6.117), the above explanations and (6.138) it follows that 1
ku0 jLp .Rn /k C ku0 jL1 .Rn /k c C N 1 .log N /.n1/.2 p /
(6.139)
where c > 0 is independent of C and N , 2 N 2 N. If 1 p1 < ˛ < 2 p1 then one has to use (6.133) (˛ D 2 p1 can be incorporated by the monotonicity of Sp1 W .Rn ; ˛/ with respect to ˛). In what follows we assume ˛ > 2 p1 , the necessary modifications in the other cases are obvious. According to Proposition 6.1 0 .Rn / by the inhomogeneous and Remark 6.2 one can expand u0 2 Lp .Rn / ,! Bp;p j Haar basis fHG;m g, u0 D
X
m .u0 / m C
X
m .u0 / m C
D
C
1 X X X
(6.140)
j n=2 j;G hG;m m 2 j
j D0 G2G m2Zn
m2Zn
u00
j j n=2 j;G HG;m m 2
j D0 G2G m2Zn
m2Zn
D
1 X X X
u10 :
Here m is the characteristic function of the cube Qm D m C .0; 1/n, m 2 Zn , Z Z m .u0 / D u0 .x/ dx D u0 .x/ m .x/ dx; m 2 Zn ; (6.141) Rn
Qm
whereas u10 (the second sum) can be likewise expanded by the inhomogeneous Haar j j system fHG;m g without starting terms or by the homogeneous Haar system fhG;m g in (6.11) without infrared tail (terms with j < 0). Otherwise one is in the same situation as in (6.56)–(6.58). One has by the elementary embedding for inhomogen
1
p 0 .Rn / ,! Bp;2 .Rn / (recall 2 n < p < 1), ku10 jLp .Rn /k neous spaces Bp;p c ku0 jLp .Rn /k (as a consequence of Haar expansions) and (6.139), n
1
p ku10 jBP p;2 .Rn /k c C N 1 .log N /.n1/.2 p / ; 1
2 N 2 N:
(6.142)
Using (6.139) and (6.141) one obtains 1
ku00 jL1 .Rn /k ku0 jL1 .Rn /k c C N 1 .log N /.n1/.2 p / ;
2 N 2 N: (6.143)
6.2 Initial data in spaces with dominating mixed smoothness
169
From Proposition 6.3 with q D 2 and Corollary 6.7 follows 1
ku0 jBMO1 .Rn /k c C N 1 .log N /.n1/.2 p / ;
2 N 2 N:
(6.144)
The rest is now the same as at the end of the proof of Theorem 6.8 with the references given there in particular to (5.76), (5.77). As mentioned above one can replace in the above argument ˛ > 2 p1 and the right-hand side of (6.132) by ˛ > 1 p1 and the right-hand side of (6.133). Remark 6.22. We discussed in Section 6.1.5 that the rigid second condition in (6.50) can be replaced by more moderate assumptions with (6.82)–(6.85) as an example. One may ask for a similar possibility in connection with the assumption u0 .x j / D 0 N if x j 2 ˛;p in the above theorem. This can be done. We outline a corresponding procedure. We switch again to the scalar case and expand u0 2 Sp1 W .Rn ; ˛/ by the Faber system fvk;m g as indicated in (6.124), (6.126) adapted to our situation, u0 D
X
k;m vk;m C
n k2Nn 0 ;m2Z ; k N 2 m2˛;p
X
k;m vk;m C CC
n k2Nn 0 ;m2Z ; k N 2 m62˛;p
(6.145)
D u0;N C uN 0 ; where 2k m D .2k1 m1 ; : : : ; 2kn mn / and CCC indicates again some unimportant N starting terms. We outlined in Remark 6.17 how to construct the finite set ˛;p . The N term u0 coincides essentially with u0 in (6.138) and one has (6.144) with uN 0 in place of u0 . As for u0;N one can show that X
ku0;N jBMO1 .Rn /k c
X
jk;m j c 0
N 2k m2˛;p
ju0 .2k m/j (6.146)
e
0 2k m2 N ˛;p
0 N where e N ˛;p is the modification of ˛;p according to (6.126). For this purpose one needs vk;m 2 BMO1 .Rn /. But this can be obtained by the same arguments as in Proposition 6.5 and (6.28). We refer the reader for details about expansions in terms of Faber bases to [T10, T12]. In any case if the right-hand side of (6.146) is small then one can argue as in the above proof. This shows that Theorem 6.21 remains valid ˛;p if one replaces the assumption u0 .N;j / D 0 by 0
N X ˇ ˛;p ˇ ˇ ˇu0 N;j
small;
j D1
where N 0 N indicates that some neighbouring points are coming in.
(6.147)
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6 Navier-Stokes equations in global spaces
6.2.5 Besov spaces One can replace the Sobolev spaces Sp1 W .Rn ; ˛/ in Theorem 6.21 by the Besov r B.Rn ; ˛/ with dominating mixed smoothness. Then one has to use Propospaces Sp;q sition 6.16 instead of Corollary 6.18. Otherwise we are in the same position as at the r beginning of Section 6.2.4. Let Sp;q B.Rn ; ˛/ be the weighted Besov spaces with dominating mixed smoothness as introduced in Section 6.2.2. We have, in particular, N (6.111) and Proposition 6.16. Let again ˛;p be the distinguished sets of points as recalled at the beginning of Section 6.2.4 resulting in the right-hand sides of (6.122), (6.123). One may consult again the explanations given in the Remarks 6.17 and 6.20. Instead of (6.135) we introduce now ˚ r N r N Sp;q B.Rn ; ˛I ˛;p / D f 2 Sp;q B.Rn ; ˛/ W f j˛;p D0 ;
(6.148)
where r > 1=p and ˛ > 0. Otherwise we use the same notation as in Theorem 6.21. Theorem 6.23. Let 2 n < p 1, 0 < q 1, r > 1=p and ˛ > 1 C > 0. Then there is a number N 2 N such that @t u u C P div .u ˝ u/ D 0 u.; 0/ D u0
in Rn .0; 1/, in Rn ;
1 . p
Let
(6.149) (6.150)
has for any r N B.Rn ; ˛I ˛;p /n u0 2 Sp;q
with
r ku0 jSp;q B.Rn ; ˛/n k C
(6.151)
a unique solution u 2 E.RnC1 C /. Proof. One can copy the proof of Theorem 6.21 now based on Proposition 6.16 inr B.Rn / are monotone in r stead of Corollary 6.18. The (inhomogeneous) spaces Sp;q such that the previous restriction r < 1 C p1 is not needed as far as the right-hand sides of (6.122) and (6.123) are concerned. This applies also to p D q in part (i) of Proposition 6.16. (The additional restrictions are needed for the left-hand sides of (6.122), (6.123)). On the other hand one has to rely on the counterpart of (6.139), hence u D p and u D 1 in Proposition 6.16. This explains the restriction ˛ > 1 p1 (as in Theorem 6.21). Remark 6.24. By the same arguments as in Remark 6.22 one can replace the as˛;p sumption u0 .N;j / D 0 in the above theorem by 0
N X ˇ ˛;p ˇ ˇ ˇu0 N;j j D1
small:
(6.152)
6.2 Initial data in spaces with dominating mixed smoothness
171
6.2.6 Complements We discuss a few questions arising in connection with the main assertions of Chapter 6. Remark 6.25. In Section 5.5.1 we discussed some aspects of Navier-Stokes theory and collected a few relevant books, surveys and recent papers. As far as underlying function spaces are concerned these books and surveys rely on isotropic (homogeneous and inhomogeneous) spaces APsp;q .Rn / and Asp;q .Rn /. Preference is given to s homogeneous Besov spaces BP p;q .Rn /. But in recent times there seems to be a growing interest in stepping from isotropic to some types of non-isotropic spaces (called sometimes anisotropic, which contradicts occasionally the historical and traditional meaning of this notation that may be found, for example, in [ST87], going back to the Russian literature, [Nik77, BIN75]). There might be different reasons to deal with non-isotropic spaces. If the initial data in R3 are supposed to have a distinguished behaviour in one fixed direction then it is quite natural to ask for non-isotropic spaces (of Besov-type) which are better adapted to the given problem. One may consult in this context the recent paper [CPZ14] and the references within. But this does not apply to our approach. Nearer to us is the recent paper [BaG13]. The spaces introduced there have dominating mixed smoothness in the context of Fourier-analytical decomposition in R3 . There it is said explicitly that this deviation from isotropy is an important new aspect in the analysis of Navier-Stokes equations. We share the opinion although the topics treated above and in [BaG13] have little in common. We discussed this point also at the beginning of Section 6.2.1. Problem 6.26. In Theorems 6.21, 6.23 we described conditions ensuring that the Navier-Stokes equations (6.136), (6.137) have a unique solution u in E.RnC1 C / if u0 belongs to some spaces with dominating mixed smoothness. In contrast to Corollary 5.16 (hybrid spaces) and Theorem 6.8 (global spaces) we do not know whether u has further desirable properties. In particular it is not clear whether u is a strong solution, local or global in time, similar to (5.92), (6.52) or to the question (6.81). This r requires further properties of Sp;q A.Rn / and a direct approach to heat and NavierStokes equations in the framework of these spaces. A counterpart of Theorem 3.14 is r of interest. In other words, based on (6.110), (6.111) one may ask whether Sp;q A.Rn / with r > 1=p is a multiplication algebra, hence r r r kf1 f2 jSp;q A.Rn /k c kf1 jSp;q A.Rn /k kf2 jSp;q A.Rn /k
(6.153)
or, better, r r kf1 f2 jSp;q A.Rn /k c kf1 jSp;q A.Rn /k kf2 jL1 .Rn /k r A.Rn /k C c kf1 jL1 .Rn /k kf2 jSp;q
(6.154)
r if f1 ; f2 2 Sp;q A.Rn /. We ask a corresponding question in Problem 3.63 for the hyr s brid spaces L Ap;q .Rn /. As explained in Remark 5.18 and Problem 3.63 the interest in (6.154) comes from the persistency property of Navier-Stokes equations. There one
172
6 Navier-Stokes equations in global spaces
finds also some related references in particular to [Can04, Theorem 10, pp. 209–211] in connection with (6.154). Furthermore one needs a counterpart of Theorem 3.52, hence r r Rk W Sp;q A.Rn / ,! Sp;q A.Rn /; 1 < p < 1; (6.155) where Rk is the Riesz transform in (3.267)=(5.5). This can be extended to PW
1 < p < 1;
r r A.Rn /n ,! Sp;q A.Rn /n ; Sp;q
(6.156)
where P is again the Leray projector according to (5.4). We reduced the NavierStokes theory in Chapter 5 in hybrid spaces to corresponding assertions in Chapter 4 for heat equations (based in turn on [T13]). It seems to be reasonable to do the same r in case of the spaces Sp;q A.Rn / with dominating mixed smoothness. Then one may ask for mapping properties of the Gauss-Weierstrass semi-group Wt as introduced in Section 4.1. Properties as in Theorem 4.1 are of interest with Wt W
r r Sp;q A.Rn / ,! Sp;q A.Rn /;
1 p; q 1
(6.157)
(p < 1 for F -spaces) uniformly in t, 0 < t < 1, as a minimal request. Some of these properties can be transferred from Lp .Rn / to Sp1 W .Rn /, 1 < p < 1, normed by (6.86). This applies to (6.155)–(6.157), but it is not so clear in general. This may pave the way to deal with heat and Navier-Stokes equations in the distinguished spaces Sp1 W .Rn /, 1 < p < 1, as a substitute for Lp .Rn /, n < p < 1, in the supercritical case. Spaces with dominating mixed derivatives have been studied in recent years with great intensity. Related references may be found in Section 6.2.2. But this does not apply to the above-listed questions (at least we have no references). This is a little bit surprising, especially in connection with (6.153), (6.154). To deal with these problems seems to be of interest for its own sake, but also for possible applications to heat and Navier-Stokes equations. The methods in Sections 6.1 and 6.2 are different. But one can combine them to improve the restriction ˛ > 1 p1 for the weight w ˛ in (6.88), (6.112) by the more natural assumption ˛ > 0. We needed ˛ > 1 p1 to ensure (6.144), based on (6.143) and (6.140), (6.141). But this is rather crude and can be improved if one applies the arguments from Section 6.1.5. We formulate the outcome. Let again Qm D m C .0; 1/n, m 2 Zn . Let o n ˚ ˛;p N N k1 kn n n ; (6.158) D N;j 2 m ; : : : ; 2 m ; m 2 Z ; k 2 N ˛;p 1 n 0 j D1 N 2 N, be the distinguished subsets of the related hyperbolic cross in Rn as explained in Remark 6.17 where now only the case u D p, ˛ > 0, in Proposition 6.16, Corollary 6.18 will be needed (the modifications indicated in Remark 6.20 are no longer necessary). For brevity we put N ˇ ˇ ˇ X ˇ ˇu0 j N ˇ D ˇu0 . ˛;p /ˇ ˛;p N;j j D1
(6.159)
6.2 Initial data in spaces with dominating mixed smoothness
173
0 N where now ˛;p must be understood as the slight modification e N ˛;p according to r B.Rn ; ˛/ have the same meaning (6.146), (6.147). The spaces Sp1 W .Rn ; ˛/ and Sp;q as in Sections 6.2.1, 6.2.2.
Theorem 6.27. (i) Let ˛ > 0, 2 n < p < 1. Let C > 0. Then there are numbers ı > 0 and N 2 N such that @t u u C P div .u ˝ u/ D 0 u.; 0/ D u0
in Rn .0; 1/, in Rn ;
(6.160) (6.161)
has for any u0 2 Sp1 W .Rn ; ˛/n and
with
X ˇˇ Z ˇ m2Zn
Qm
ku0 jSp1 W .Rn ; ˛/n k C
ˇ ˇ ˇ ˇ N ˇ u0 .x/ dx ˇ C ˇu0 j˛;p ı
(6.162)
(6.163)
a unique solution u in E.RnC1 C /. (ii) Let ˛ > 0, 2 n < p 1, 0 < q 1, r > 1=p. Let C > 0. Then there are numbers ı > 0 and N 2 N such that (6.160), (6.161) has for any r B.Rn ; ˛/n u0 2 Sp;q
with
r ku jSp;q B.Rn ; ˛/n k C
(6.164)
and (6.163) a unique solution u in E.RnC1 C /. Proof. We rely again on the decomposition (6.140). The terms in u10 referring to the second sum can be treated as there, modified according to Remark 6.22 resulting in (6.147). This is covered by P (6.163) if ı > 0 is small. As for the first sum we rely now 0 on (6.83) with k D 0 and m2Zn jcm j small. Then one can apply Corollary 6.7 in the same way as there. Remark 6.28. Compared with Theorem 6.8, modified according to Section 6.1.5 as indicated, on the one hand, and Theorems 6.21, 6.23, modified by (6.147) on the other hand, one has now a mixture of conditions. The assumption that the first sum in (6.163) is small may be considered as a mild version of uniform oscillation. It is the substitute of the L1 -term in (6.139). For the corresponding Lp -term it is sufficient to assume ˛ > 0, hence a moderate decay at infinity for the weight w ˛ in (6.88). The second term on the left-hand side of (6.163) requires again the control of u0 at finitely many points (near the origin in terms of the hyperbolic cross).
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Symbols
Symbols Sets C, 7 G 0 , G j , G , 39, 40, 52, 57, 144 N , 160, 167 ˛;p N, N0 , 7 Nn0 , 46 N D N, 59 N1 D N0 [ f1g, 59 Pj , 71 PJ;M , PJ;M , PP J;M , 40, 63, 83 Qm , 143 Qj;m , 8, 49 R, Rn , 7 RnC1 C , 112 P 52, 57, 83 ‰, ‰, Z, Zn , 7
Spaces s .Rn /, 53 ap;q s ap;q .Rn /, 49 bmo.Rn /, 76, 105, 106 bmo1 .Rn /, bmos .Rn /, 78, 105 s .Rn /, 53, 145 bp;q b sp;q .Rn /, 49 s s .Q/, fp;q .Q/, 72 bp;q c0 , c0 .A/, 25, 30 cmo.Rn /, 138 s .Rn /, 53 fp;q f sp;q .Rn /, 49 Asp;q .Rn /, 47 APsp;q .Rn /, 82 Asp;q .Rn /n , 150 Asp;q . /, 56
s; n n As; p;q .R /, Ap;q .R /, 69 e Asp;q . /, 56 Asp;q . /, e
BMO.Rn /, 107 BMO1 .Rn /, 106 s .Rn /, 15, 47 Bp;q s s .Rn /, FPp;q .Rn /, 103 BP p;q s; s; Bp;q .Rn /, Bp;q .Rn /, 69 C.Rn /, C0 .Rn /, 160 C s .Rn /, 15, 48 C01 .Rn / D D.Rn /, 7 C01 . / D D. /, 55
C k .Rn /, 100 C Œ0; T /; X.Rn/ , 120 D 0 .Rn /, D 0 . /, 7, 55 E.RnC1 C /, 139, 151 s Fp;q .Rn /, 47 s; s; .Rn /, Fp;q .Rn /, 69 Fp;q Hps .Rn /, 48 H % Lp .Rn /, 9 H% Lp .Rn /, 9 Lp .Rn /, Lp .M /, 7, 46 n Lloc p .R /, 7 n Lp .R ; w /, 8
Lrp .Rn /, LV rp .Rn /, 7
Lrp .Rn /, LV rp .Rn /, 7 Lp .log L/a .Q/, 14 Lv .0; T /; b; X , 114 Lv .0; T /; b; Xn , 150, 151 L1 .0; T /; X , 115 L1 .0; T /; Xn , 127 s .Rn /, 64 Lr ap;q s .Rn /, 63 Lr bp;q s .Rn /, 63 Lrfp;q
183
184
Symbols
s Lrbp;q .Rn /, 64 r s L fp;q .Rn /, 64
Lr b sp;q .Rn /, 66 Lr f sp;q .Rn /, 66 s .Rn /, 84 Lr aP p;q s s Lr bPp;q .Rn /, Lr fPp;q .Rn /, 83, 84 Lr hsp .Rn /, Lr hsp .Rn /, 40 Lr hP s .Rn /, 40 p
L Asp;q .Rn /, LV r Asp;q .Rn /, 58, 62 LrAsp;q .Rn /, 60, 62 s s Lr BP p;q .Rn /, Lr FPp;q .Rn /, 88, 89 r
Lr Wpm .Rn /, 98 S.Rn /, S 0 .Rn /, 7, 46 r A.Rn /, 163 Sp;q r Sp;q A.Rn ; ˛/, 163 r A. /, 163 Sp;q r N Sp;q B.Rn ; ˛I ˛;p /, 170
Sp1 W .Rn /, 160 Sp1 W .Rn ; ˛/, 160 N Spr W .Rn ; ˛I ˛;p /, 161
Functions, functionals Ap .Rn /, 19 D ˛ , D ˇ , 21, 160 @t , @j , 1, 112 Df , 2 1h f , lh f , 48 div u, 1 div .u ˝ u/, 1 hM .y/, hF .y/, 41, 144 j .x/, hjG;m .x/, 41, 71, 144 HG;m r, 1 .u; r/u, 1 w .x/, 16 w ˛ .x/, 160 x ˇ , 46 'k , 163 k'kK , 116 F , M , 52, 57 ‰m , 83 j ‰m , 57 j ‰G;m , 52, 83 j;m , 49
‰ VJ;u , 57
Wpk .Rn /, 47
Operators b ' , F ', 15 ' _ , F 1 ', 15, 46 I , 48 P, 1 Rk , 1 Wt , 102, 113
Numbers, relations , equivalence, vii p , p;q , 50, 58 Re, 154 lin , 164 gN
Index
185
Index anisotropic, 171 atom, 50
operator, Calder´on-Zygmund, 31 oscillation, 157
basis, Haar, 41, 74, 109 basis, Faber, 165
persistency, 142, 157
Churchill, Sir W., 155
resolution of unity, 15, 47, 163 Reynolds number, 154
diffeomorphism, 100 distributional, 113 Duhamel formula, 113 equivalence, vii extension, 32 Fatou property, 62 function, scaling, 52 heat equation, inhomogeneous, 114 heat equation, nonlinear, 117 homogeneity, 33, 103, 146 hyperbolic cross, 166 infrared damped, 133, 135
sampling number, 164 semi-group, Gauss-Weierstrass, 102, 113 solution, strong, 120 space, Besov, 15, 48 space, critical, 132 space, dominating mixed smoothness, 160 space, global, 46, 47 space, H¨older-Zygmund, 15, 48 space, Morrey, 7, 13 space, Morrey, dual, 9 space, Morrey, global, 8 space, Morrey, local, 8 space, Sobolev, 48 space, Sobolev, classical, 48 space, subcritical, 133 space, supercritical, 132 space, Zygmund, 14
k-infrared damped, 135 laminar flow, 154 Leray projector, 1, 125 Muckenhoupt class, 19 multiplication algebra, 54, 95 Navier-Stokes equations, 1, 125, 154
theorem, Littlewood-Paley, 43, 47 thermic characterization, 102 transform, Fourier, 15 turbulent flow, 154 wavelet, 52 wavelet, Meyer, 81
