Sobolev Spaces on Domains (Teubner-Texte zur Mathematik; 137)

Il/II Wo(n),

where c3, c4 > 0 are independent of f . This follows, with c3, c4 depending only on n, p and 1, from Holder's and Jenssen's inequalities for finite sums. If p = 2, then W21(S2) is a Hilbert space with the inner product

(f,9)ww(n) = f (f9+ E DwfDwg) dx n

IQI=1

and Ilf 11(') W,l n) is a Hilbert norm, i.e., Ilf IIwt(n) _ (f,

Let us consider the weak gradient of order I

ow f - (( 8xi, Then

= L`. n

I

oa f12

I(

i =1

n

f

axi,

a xis ... axis

) w ) i,....,y=1 l!

) I2 = w

a' ! 101=1

and norm (1.22) is equivalent to

Ilfllw;(n) _ (f (IfIP + IawfI) dx) `. n

We also note that for even I Vf E C000(f))

f

IV1fI2dx=fI1if12dx,

n

n

where A is the Laplacian. Hence, for such f, IIfII(2)

=

(f (If n

I2+Io{f12)dx)12.

'

f12 ID I

1.3. SOBOLEV SPACES (BASIC PROPERTIES)

31

We shall also need the following variant of Sobolev spaces.

Definition 6 Let 12 C R" be an open set, l E N, 1 < p < oo. The function f belongs to the semi-normed Sobolev space w,(f2) if f E L, 0c(12), if it has weak derivatives D: f on fl for all a E l satisfying I a I= I and IID*fIILP(n) < oo.

IIfIIwi,(o)

(1.23)

1a,=(

The space wp(fZ) is also a complete space (the proof is similar to the proof of Theorem 3). Thus w,(1l) is a semi-Banach space, because the condition If llwP(n) = 0 is equivalent to the following one: on each connected component of an open set ) f is equivalent to a polynomial of degree less than or equal to l - 1 (in general different polynomials for different components).

Remark 9 Let Sl C R" be a bounded domain and B be a ball such that R C 12, 1 E N, 1 < p < oo. We denote by LP' L (0) the Banach space, which is

the set w,(l), equipped with the norm IIf1IL;(n) = IllIIL,(B) + IllIIw,(n)

(It is a norm, because if If IIL;(n) = 0, then from If Ilwt(n) = 0 it follows that f is equivalent to a polynomial of degree less than or equal to I - 1, and from

Ill IIL,(B) = 0 it follows that f - 0 on Q.) For different balls with closure in I? these norms are equivalent. (This will follow from Section 4.4). One can replace IllIIL,(B) by Of IIL,(B) and the corresponding norms will again be equivalent. Note that by definition L,(SZ) = WP(1l) 13.

Remark 10 Clearly Wp(Q) C w,(fl). In general WW (fl) 96 w,(12), but locally they coincide, i.e., for each open set G with compact closure in SZ Wp(Sl)Io = wp(SZ)IG. This will follow from the estimates in Section 4.4. In that section the conditions on fl also will also be given ensuring that WA()) = wp(fl). Remark 11 The semi-norm II' IIW;(an)(in contrast to the norm II Ilwa(Rn)) posesses the following homogenity property: Vf E w,(R") and Ve > 0

IIf(ex)IIw;(R) = e'-""PIIf(x)IIw;(s.) Is Here and in the sequel for function spaces ZI (fl), Z2(fl) the notation ZI (fl) Z'(fl), ZI(fl) C Z2(fl) means equality, inclusion respectively, of these spaces considered only as sets of functions (see also Section 4.1).

CHAPTER 1. PRELIMINARIES

32

Moreover, V f E WD (R") If (Ex)IIWw(R") ' e'"/PIIf (x)IILp(R")

ase->0+and e)-n/PIll(x)jjw,(R") jIf(ex)jjW;(R") ^

ase-++oo. The number 1- n/p, which is called the differential dimension of the spaces Wp(S2) and wp(Q), plays an important role in the formulation of the properties of thebe spaces (see Chapters 4,5) 19. It will also appear in the next statement.

Example 8 Let n, I E N, µ, v E R, 1 < p < oc. Denote by 1%,e the set of all nonnegative even integers. Then jxjµilogIxIj" E Wo(B(0,1/2)) if, and only if, 1xIµIlogjxj I" E w,(B(0,1/2)) and if, and only if, the following conditions on > the parameters are satisfied. If 1 < p < oo, then in the case µ V No,,

l - n/p, v ERor µ=l-n/p, v<-1/p and in thecase µENo,,:v=0or

µ>1-n/p, vERorµ=l-n/p, v
µ§ENo,,:µ>1, vEllfor IA =l, v < 0 and in the case p E No,, : v = 0 or

µ>1, vERorp=1, v<1.Inparticular, for1
1) jxI" E Wp(B(0,1/2)) if, and only if, either p 0 No,e and µ > I - n/p, or µ E 1No,e;

2) IloglxIj' E Wy(B(0,1/2)) where l = n/p if, and only if, v < 1 - 1/p. 14 Let Z(R") be a semi-normed space of functions defined on R". One may define the differential dimension of the space Z(R") as a real number p posessing the following property: V f E Zo(R") there exist ee, c5, co > 0 such that VE > co cae°IIf(x)IIZ(R") <- IIf(E2)IIZ(R") <- cse'IIf(Z)IIZ(R")

If the semi-norm II

is homogenuous, i.e., for some v E R Vf E Z(R") and VE > 0 then the differential dimension of Z(RI) is equal to Y. The

differential dimension of Ln(R") is equal to -n/p, the differential dimensions of both W, (R") and wy(R") are equal to 1- n/p (which follows from the above relations). This notion may be usefull when obtaining the conditions on the parameters necessary for validity of the inequality II/IIZ,(R") <- cr IIfIIZ,(R-),

where cy > 0 does not depend on f. From this inequality it follows that the differential dimension of Z1(R") is less than or equal to the differential dimension of Z2(R" ). If, in addition, both of the semi-norms II IIZ,(R") and II IIz,(R") are homogenuous, then their differential dimensions must coincide.

1.3. SOBOLEV SPACES (BASIC PROPERTIES)

33

Idea of the proof. Apply Example 4. Let M = No for n = 1 and M = 1%,e for n > 1. Prove by induction that Va E N, a 0, and Vx E R", x 54 0, lal

Da(IxlµlloglxlI") = Ixlu-I0I E Pk,a k=o

( \

l

xI/ jlogjxjj

v-k'

where Pk,a are polynomials of degree less than or equal to Jai, P0,a A 0 and

o=0forpVMorpEM, Jai

A;a=1forpEM, Jai >p, v540 (the

case in which p E M, jai > p, v = 0 is trivial: Da(jxjµjlogjxjj") = 0). Deduce

that VxER,x00, jDa(IxjµjlogjxllV)I

Ixlµ-Iall loglxlj O,

c8

where c8 > 0 does not depend on x. Moreover, if n > 2, then for some t E R",

where I{I=1,e>0andb'xEK-{xER":x00, I11-eI<E} jDa(jxjµjlogjxjjv)j ? c9

jxja-1Q1jlogjxjI"-O,

where cg > 0 does not depend on x. Finally, use that for some clo, c,, > 0 I/2

f g(axl)dx = cio f 9(p)p"-idp, B(0,1/2)

f

1/2

g(IxD)dx = cii

B(0,1/2)nK

0

f g(p)p"-idp. 0

Example 9 Let 1 < p < oo.

Under the suppositions of Example 8 jxjµ(logjxj)" E Wo(eB(0, 2)) if, and only if, p < -n/p, v E R or p = -n/p, v < -1/p. On the other hand, (xjµ(logjxj)" E wp(CB(0, 2)) and if, and only if, in

thecase p0No,e:p<1-n/p, vERorp=l-n/p, v<-1/p and in the

case.ENo,e:v=0orp
Let F f denote the Fourier transform of the function f : for f E L1(R" )

and V R" (21r)-"' f e-;='{ f (x)dx;

(1.24)

R"

for f E L2(R")

Ff = lim F(fxk),

(1.25)

where Xk is the characteristic function of a ball B(0, k) and the limit is taken in L2(R"). It exists for each f E L2(R") and JIFf jIL2(R^) = IIf IIL2(R") (Parseval's equality).

(1.26)

CHAPTER 1. PRELIMINARIES

34

Lemma 8 For all I E N and f E WW (R") (1.27)

IlowflDL,(R.) = II and

IlfllW;(R') =11(1 +

(1.28)

IIL,(R').

Idea of the proof. For f E L1(R") n WZ(R") starting with Definition 4 prove

that F(Dwf)({) = (il;)°(Ff)(1;) on R". To obtain (1.27) and (1.28) apply (1.26) and the identity 1'2-1

Ial=(

('1)

... (fin)

=

ICI21.

Inl=t

Lemma 9 Let c2 C R" be an open set, M > 0 and suppose that Vx, y E S2

If(x)-f(y)1<-MIx-y I.

(1.29)

Then f E w;o(S2), the gradient (vf)(x) exists for almost every x E S2 and

I v f(x) 1< M a.e.

on 12.

(1.30)

If, in addition, S2 is a convex set, then the condition (1.29) is equivalent to the following: f E C(c) n wl 00(S2) and (1.30) holds.

Idea of the proof. Let j E { I_-, n}, x = (x0), x;), x(>> _ (xl, ..., x;_1 i x;+l, ...,xn), c0) = Prxj=oc C R"-1 and dx0) E 0) 5201(x(')) = Pro,Sinlz(,) C R, where lz(,) is a straight line parallel to the axis Ox; and passing through the point (x0),0). Deduce from (1.29) that for almost every x; E 120)(x0)) there exists L, (x) = s,j (x0>, x;) and ( f.L (x) ( < M. Integrating by parts (which is possible because dx0> E c0) the function f (x0), ) is locally absolutely

continuous on 520)(x0))) show that the ordinary derivative a (existing thus almost everywhere on 0) is a weak derivative (E )w on a If 12 is convex, then to obtain the converse result use Lemma 4 and (1.7) to prove that Vx, y E S2 and 0 < 6 < dist ([x, y], 8f2) the following inequalities for the mollifier A6 with a nonnegative kernel are satisfied is

I(Asf)(x) - (Aof)(y)I -< II V A6fIIC(Iz,vuIx - yI 15 When writing II v 911c(c) we mean that I I V 911c(c) = I I I v 91 IIC(c) =

J-1

(II v 911L.,(o) is understood in a similar way).

L99 1 2 ) 1 1 2 11

f

(c)

1.3. SOBOLEV SPACES (BASIC PROPERTIES) = II A6 vw f Ilc([x, ]) Ix - y1:5 II

35

V. f II L-([x,y)6) I x - yl

Y15 MIX - yl

< II V. f IIL (n)IX - y1 = II v f

(note also that for f E C(Sl) fl wl(Q) the gradient of exists a.e. on Q and v f = vw f on Sl ). Now it is enough to pass, applying (1.5), to the limit as

6->0+. Corollary 3 If SZ C R" is a convex open set, then g E w.(Sl) if, and only if, it is equivalent to a function f satisfying (1.29) with some M > 0. (Given a function g, the function f is defined uniquely.) Moreover, denote by M' the minimal possible value of M in (1.29). Then M' and, hence, V9 M' < II9IIw- (n) < n M*.

Idea of the proof. The first statement is just a reformulation of Lemma 9 for the case of convex open sets. The second one follows from the definitions of II9IIwi (n) and vwg.

Lemma 10 (Minkowski's inequality for Sobolev spaces) Let S1 C W" be an open set and A C Wn a measurable set, l E N, 1 < p < oc. Moreover, suppose that f is a function measurable on Sl x A and that f y) E WP(I) for almost every y E A. Then 11f f(x,y)dyll wo (n) S A

f

pn)dy

(1.31)

Ilf(x,y)llw,(

A

(the norm Ilf (x, y) II wo(n) is calculated with respect to x).

Idea of the proof. Use Lemma 3 and Minkowski's inequality for Lp(Sl). Proof. Let the right-hand side of (1.31) be finite, then by Holder's inequality for each compact K C 11

f ( f if (x, y)ldx)dy < oo A

K

and 1(1 IDwf(x, y)Idx)dy < oo A

K

Va E N where Ial = I. Hence by Fubini's theorem the function f, being measurable on K x A, belongs to Ll(K x A). Now the inequality (1.31) follows from Lemma 3 and Minkowski's inequality for Lp(SQ): 11f f(x, y)dy11 Wp(n) = 11f f (x, y)dy11Lp(n)

+ E 11Dw f f (x, y)dyll Lv(n)

CHAPTER 1. PRELIMINARIES

36

f IIf(x,y)IIL,(n)dy+E f II(Dwf)(x,y)IIL,(n)dy= f IIf(x,y)Ilwp(n)dy. I°I=1 A

A

A

Lemma 11 (Multiplication by Co -functions) Let S2 C R" be an open set, I E N, I < p < oo. Then V E CO' (Sl) there exists c,p > 0 such that V f E W,(0) (1.32)

114 16"(f)) <_

Idea of the proof. Use Lemma 6, Leibnitz' formula and the Ly-estimates of the derivatives of lower order. Proof. Let a E R satisfy Ial = 1. By Lemma 6 d$ E N where 1,3 1<- 1 there exist Du, f , therefore on S1 Leibnitz' formula 16 holds:

()Do-sDf. o
(1.33)

Q

Let Q, C Q. j = 1...., s. be open cubes with faces parallel to the coordinate b

planes such that supp W C U Q,. Then, applying twice the inequality in footj=1

note 11. we get

IID°W)IIL,(n) <- 2'maxIID'c Iic(supp,,)

I1D1 fI1 L,(suppsv) I0I
x

< 2'

x

E IID°0IL,(Q,)) I7I5' i=1

I0I51 i=1

<- M IIIPIICI(n) 11f 16"n,

where Al depends only on 1,11 and supp W. (See also Lemma 15 of Chapter

4.)0 Lemma 12 Let 11 C R" be an open set, l E 1V, 1 < p < oo. Then the E Cu (s1) andVf E w,(S1) Wf E w,(S2). Idea of the proof. Since locally w,(S1) and Wp(S1) coincide (see Remark 9) and W is compactly supported in fl, it is enough to apply Lemma 11. The estimate

(1.32) does not hold if Wy(11) is replaced by wp(fl). (Take any nontrivial polynomial of degree less than or equal to 1 - I as f to verify this.) n

16 Here (p) _ , o

°i

a! = a1!...a"!; note that E (p) = j'[ E s=1 31=o o«<°

!

21°

.

1.3. SOBOLEV SPACES (BASIC PROPERTIES)

37

Lemma 13 Let I E N, 1 < p < 00, rt E Co (R") be a function of "cap-shaped" type such that g = 1 on B(0,1) and Vs E N, Vx E R" 77,(x) = i (8) . Then Vf E Wp(R")

rl,f -4 f in WW(W)

(1.34)

ass-4oo. Idea of the proof. Use the definition of the norm in li n(R") and Leibnitz' formula.

Proof. First of all Vg E Lp(R") where 1 < p < oc 1101, -1)gIIL,(Rn) <- IIgIIL,(
0

as s -i oc. From (1.33) it follows that Va E No where I R = I IIDw(nef - f)IIL,(R")

511('i -1)DwfIIL,(Rn)+ IIrlmDwf

- D,°u,f II

(a,)

a! E IIDmf II/.,(a.. ). OG,9
where M does not depend on f and s. By footnote 11 Dwf E Lp(R"), consequently we have (1.34).

Remark 12 For p = oo Lemma 13 does not hold, because, for instance, for f = 1 on R' Vs E N II r), f - f II 1. However, rl, f -* f a.e. in R" and II rl,f IIw;,(Rn) - If IIw' (Rn) as s - oo, which sometimes is enough for applications.

It is well-known that if S1 C R" is a measurable set and 1 < p < oo, then each function f E Lp(11) is continuous with respect to translation (__ continuous in the mean), i.e.,

lo II fo(x + h) - f(x) IIL,n= 0.

(1.35)

The analogous result is valid for Sobolev spaces. We recall that for an open set S2 C R" the space (W1',)o(S2) is the set of all functions f E Wp(Q) compactly

supported in n.

CHAPTER I. PRELIMINARIES

38

Lemma 14 (Continuity with respect to translation for Sobolev spaces) Let 9 c R" be an open set, l E Ill, 1 < p < oo. Then V f E W, (fl) lim II f (x + h) - f (x)

0,

(1.36)

where h E R", f2ih) _ {x E Q: x + h E S1}, and Vf E (W,)o(S2) lim II fo(x + h) - f (x) Ilwa(n)= 0.

(1.37)

Idea of the proof. Use the definition of the norm in Wp(R") and (1.35). 0 Proof. (1.36) follows from (1.35) because II f (x + h) - f (x) IIwn(n{,,})

=11 f (x + h) - f (x) IILp(na.}} + Ell (Dwf)(x + h) - (Dwf)(x) IILp(n{n}) IQI=1

<_ II fo(x + h) - f (x) IILp(n) + Ell (Dwf)o(x + h) - (Dwf)(x) IILP(n) 101=l

If f E (Wp)o(S2), then Va E N satisfying Ial = I we have (Da f )o = DO-(fo) on R", which easily follows from Definition 2, and thus fo E Wp(R"). Therefore II fo(x + h) - f (x) ilw;(n)<-II fo(x + h) - fo(x) IIw;(Rn)

and (1.38) follows from (1.37). 0

Remark 13 In contrast to the situation in L,(1)-spaces the relation (1.37) is not valid for all functions in W, )). For example, if n = 1, fa = (0, 1), f = 1, then on (0, 1) we have fo(x + h) - f (x) = -X(}-h,h)(x) 0 Wp(0,1) for every h E (0, 1). Moreover, Lemma 14 does not hold for p = oo. For example, if n = 1, 1 = 1, ) = (-1,1), f (x) = lxl, then II f (x + h) - f (x) for every h E (0,1).

f,,(x + h) - f,,(x)

Chapter 2 Approximation by infinitely differentiable functions 2.1

Approximation by C01-functions on Rn

Let Aa be a mollifier with the kernel w defined in Section I.I. We start by studying the properties of Aa in the case of Sobolev spaces.

Lemmal LetlEN. Thendf EWP(R") for t
Moreover, for 1 < p < 00

Aaf -1 f in Wy(r)

(2.1)

as b -> 0+. For p = oo (2.1) is valid `df E C'(R"). If f E W;,(R"), then in general Aaf -.+ f in W;,(R"), but in the case of nonnegative kernels of mollification

Aaf -+ f in W.-'(W), IIAafIIw.(vn) - IIfIIW (R-)

(2.2)

as ' E - 0+. ' By footnote 11 of Chapter 1 it follows that A4 f - f in Wp (R" ), where m = 0,..., l if

1
CHAPTER 2. APPROXIMATION BY C°°-FUNCTIONS

40

Idea of the proof. Apply (1.6), (1.8), (1.9), (1.10) and (1.20). 0 Proof. Using the above properties we find that for 1 < p:5 00 II A6f IIW,(R")=11 Ajf IIL,(R") + E II A6Dwf IIL,(R") lot=1

S c(II f IIL,(R") + E Ii Dwf IIL,(R")) = c II f IIWP(R") Ia1=1

If1
as 6-*0+. If p = oc, then the same argument works Vf E (R"). It follows from (1.8), because w(6, 0 as 6 -* 0+ for these f. If f E W;°(R"), then by (1.8) IIA6f - f II W

(R") = IIA6f - f IIL (R")

+E II AoDwf - Dwf IQI=1-1


W(&,

-f

6 IIf IIw;,(R").

Similarly for IaI = I - 1 W(6, Dwf )L0 (Rn) <- 61IfIIwL(R")-

0 for IaI = I - 1 as 6 -+ 0 +. It also follows that w(6, A- W) -- 0, since by footnote 11 of Chapter 1 Consequently, w(6, DW

IIlIIw;,(R") << MIIfIIwi (R"),

where M is indepent of f. The second statement of (2.2) follows from (1.10) with p = oo and (1.20).

Finally by Remark 2 below it follows that for f E W.(R") in general A6 -« f in W;°(R"). 0

2.1. APPROXIMATION BY Co -FUNCTIONS ON RN

41

Remark 1 If 11 is a proper open subset of R", 1 < p < oo and f E WP(cl), then we can prove only that Ve > 0

A6f -> f in Wp(0f)

(2.3)

as 6 -* 0+. We next aim to construct more sophisticated mollifiers, which will allow us to prove the analogous assertion for l itself.

Lemma 2 Let I E N, 1 < p < oc. Then Co (R") is dense z in Wp(R") Idea of the proof. Let f E W,(R") and rI s E N, have the same meaning as in Lemma 9 of Chapter 1. Set spa = A 1(rja f ). Then V, E Co (R") and Va

f in

WP(R") as s -1 oo. 0 Proof. By (2.1), (2.2) and (1.34)

II A;(rlaf) - f IIW;(R')<-II Al f - f IIWo(R") + II A1(nsf) - At f IIWW(R-)


as 6-+0+. 0 Remark 2 For p = oo Lemma 2 is not valid. The counter-example is simple: f = 1 on R". Moreover, C°°(R") also is not dense in W,,,,(R"). In order to prove this fact, for example, for n = 1 and I = 1, it is enough to consider the function f (x) = IxIrI(x), where t1 is the same function as in Lemma 13 of Chapter 1. Then dip E C°°(R) II f -'P II WJ(R) >- II f a

-

'

IIL00(-1,1)= II sgnx

IIL.(-1,1) > 2

However, by Lemmas 1- 2 it follows that Co (R") is dense in W0O (R") in a weaker sense, namely,'df E W;° (R") functions W. E C' (W), s E N, exist such

that

ova - f in woo i(R"), II'PsIIW;,(R") -' IIfIIw (R") as8-100. 2 Thus *y(R") = W, (R"), where T pl(R") is the closure of Co (R") in WW (R").

CHAPTER 2. APPROXIMATION BY Coo-FUNCTIONS

42

2.2

Nonlinear mollifiers with variable step

We start by presenting four variants of smooth partitions of unity, which will be constructed by mollifying discontinuous ones.

Lemma 3 Let K C R" be a compact set, s E N, Slk (: R", k = 1, ..., s, be open sets and 9

K C U Stk.

(2.4)

k=1

Then functions 1Jik E Co (Slk), k = 1, ..., s, exist such that 0 < ok < 1 and a

t tpk = 1

K.

on

(2.5)

k=1

Idea of the proof. Without loss of generality we may assume that the 1k are bounded. There exists b > 0 such that K C G =_ 6 (S1k)a. Set Gk = (1k)6 \ k=1 k-1

a

U (Slm)6 and consider the discontinuous partition of unity: E co,, = Xc on k=1

m=1

R". Mollifying it establishes the equality t A¢Xck = AIXc on R", which ] 2 k=1

implies (2.5), where 'pk = Al Xck. (Here A6 is a mollifier with a nonnegative kernel.) 0 Lemma 4 Let SZ C R" be an open set and S1k C R", k E N, be bounded open sets such that 00

C S1k+1,

k E N,

U Slk = 0.

(2.6)

k=1

Then functions 1Pk E Co (Sl), k E,/'N, exist such that Gk C 3upp Wk C Gk-1 U Gk U Gk+l

,

(2.7)

where Gk=Slk\Ilk- I (fork=0 we set ilk= 0),0
E k=1

tpk = 1

on

S1.

(2.8)

2.2. NONLINEAR MOLLIFIERS WITH VARIABLE STEP

43

Idea of the proof. Starting again with the discontinuous partition of unity

0

E k=1

Xck = 1 on ft, choose ek = 4 dist (Gk, i9(Gk_l u Gk U Gk+1))

= i min{dist (f2k-1, Oak ), dist (f)k, O1k+l))

(ifQ

(2.9)

R", then ek-a0as (k-+ oo) and set (nk)vt_, , ok - ( Ae._, Xc, on on ft \ (000k, l Aek Xck

(2.10)

where A6 is a mollifier with a nonnegative kernel w.

So the characteristic function Xck is mollified with the step pk-1 "in the direction of the set Gk_1 " and with the step pk "in the direction of the set Gk+1". Let Gk = Gk u Gk U Gk , where

G+k = k \

Gk = (1k-1)ek-1\ ck-1, Gk =

Then tik = 1 on Gk, supp t+I'k C CT- 1 U Gk u G1k+1, therefore, 7p,,, = 0 on Gk where m 96 k - 1, k, k + 1. Moreover, on Gk U Gk+1 00

E PPm = lOk + 1Gk+1 = Aek (Xck + XCk+,) = 1. M=1

Lemma 5 Let SZ C R' be an open set, n 0 Rn, Gl = {x E 12 : dirt (x, Of) > 2-2} and f o r k E N, k > 1, let Gk = {x E SZ : 2-k-1 < dist (x, O1) < 2-k }

(fork < 0 Gk = 0) Then functions lPk E C°°(Sl), k E Z, exist (fork < 0 we set ik = 0) such that "0 < t/ik < 1, 82-k-1 < dist (x, OI) < 12-k }

Gk c supp y k c {x E fl :

C Gk-1 U Gk U Gk+1, 00

1:

ko-00

00

Pk =

10, = 1 on 0

(2.11)

k=1

and Va E N there exists cQ > 0 such that Vx E R" and Vk E Z ID°+'k(x)I .5

k

(2.12)

CHAPTER 2. APPROXIMATION BY C°°-FUNCTIONS

44

Idea of the proof. The same as in Lemma 4. Now the Ilk are defined via the k

Gk: Qk = U

m=-oo

G," and ok = 2-k-3. Estimate (2.12) follows from the equality

10i

D°V,k = 0k (D°W ek) * XGk on Q \ (Ilk_1)ek and the analogous equality on (Dk)ek-i . 0

Remark 3 Sometimes it is more convenient to suppose that the functions &k in Lemmas 4 and 5 are defined on R" and supp'k C Il. (We shall use the same notation ikk E Co (Il) in this case also). Then equality (2.8) can be written in the following form: 00 E Ok = Xn (the same refers to equality (2.11)). k=1

Remark 4 There may exist an integer ko = k0(Il) > 1 such that Gk = 0 for k < ko (in this case we assume that '1k - 0) and (2.11) takes the form 00

00

E V)k=EOk=1 on k=-oo

Q.

(2.13)

k=ko

For Il = R" we shall apply the following analogue of Lemma 5.

Lemma 6 For nonpositive k E Z let

Go={xEW':Ixj <1}, Gk={xER":2-k-' 0) such that the properties (2.7) and Vc E No (2.12) are satisfied, 0 < tPk < 1 and 00

0

k=-oo

k=-oo

E4 =E Vk = 1

R".

(2.14)

(supp Qek C (Gk-1 U Gk U Gk+1)ek

(2.15)

on

Idea of the proof. The same as in Lemma 5. 0

Remark 5 Note that in Lemmas 4 - 6

Moreover, in the case of Lemma 4 for any arbitrarily small ryk > 0, k E N, one can construct functions 'kk, k E N, satisfying the requirements of Lemma 4

such that suppikk C (Gk)7k,

Ok = 1 on (Gk),.

(2.16)

2.2. NONLINEAR MOLLIFIERS WITH VARIABLE STEP

45

To do this it is enough to replace pk defined by (2.9) by pk = min{ dist (Gk, 8(Gk_l U Gk U Gk+1)),'Yk}. a

In the case of Lemmas 5 and 6 for any fixed y > 0 one can construct functions yk, satisfying the requirements of those lemmas, such that suPPiPk C (Gk)ry2-k,

lPk = 1 on (Gk),2-k.

(2.17)

Remark 6 From (2.15) it follows, in particular, that the multiplicity of the covering {supp iik} in Lemmas 4-6 is equal to 2, i.e., Vx E Q there are at most 2 sets supp tbk containing x and there exists x E Il such that there are exactly 2 sets supp t/ik containing x. (From (2.7) it follows only that the multiplicity of this covering does not exceed 3.) Of course 2 is the minimal possible value (if supp ,bk J Gk and the multiplicity of covering is equal to 1, then iik = XGk). Moreover, from (2.15) it follows that for 6 E (0,'-g] the multiplicity of the covering {(supp

Vlk)62_k

} is also equal to 2.

In Chapter 6 we shall need a variant of Lemma 5 for 0 = {x E R" : x,, > W(x1,...,x"-1)}, where W is a function of class Lip 1 on R"-1, - that variant will be formulated there.

Let Q C R" be an open set and Qk C S2, k E N, be bounded open sets, possessing the properties (2.6), Gk = ilk \ S1k_1. Suppose that pk is defined by (2.9) and {k}kEN is the partition of unity in Lemma 4 defined by (2.10).

Definition 1 Let b = {bk}kEN, where

0
(2.18)

and f E L'-(fl). Then VX E 11 00

(B7f)(x)_E(A6r(lkf))(x)_: J k=1

Ok(x-bkz)f(x-6kz)w(z)dz, (2.19)

k=1 B(0,1)

where w is a kernel of mollification defined by (1.1).

Remark 7 The functions ' k f E L1(S2), therefore, A6R (z/ik f) E C°° (R" ). We note that we assume that 1/ik(y) f (y) = 0 for all y 0 supp Ok even if y 0 St and f (y) is not defined (for this reason in contrast to (1.2) in (2.19)1Iik(x-dkz) f (x-

CHAPTER 2. APPROXIMATION BY C°°-FUNCTIONS

46

bkz) is written instead of (1bk)o(x-bkz)fo(x-8kz)). By (1.4), (2.18) and (2.15) it follows that supp A6,: (i f) C (supp +Gk)6k C (Gk-1 U Gk U Gk+1)6k,

(2.20)

therefore,

A6k(kf) E Co (1l) (2.21) and the sum in (2.19) is finite. For, let `dx E f2 a number s = s(x) be chosen such that x E G,. Then fork # s -1, s, s + 1 we have x E supp A6k (ipk f) and 3 B(X)+l iGk(x - akz)f(x - 6kz)w(z)dz.

(B1f)(x) = E

(2.22)

J k=a(x)-1B(o,1)

For the same reason dm E N m+1

BBf = E A6k(Okf)

on G..

(2.23)

k=m-1

Moreover, by Remark 5, for any given ryk E (0, pk], k E N, a partition of unity {1 }kEN can be chosen in such a way that for all sufficiently small 6k, k E N, we have V f E Lt°C(I) and dm E N (2.24)

B6f = A6,,,f on (Gm)7,,,.

Lemma 7 Let 1 C R" be an open set and f E L10c(fl). Then for each d = {dk}kEN satisfying (2.18) B7f E C(11) and Va E NB 00

D"(BB f) = E D°(A6k (iyk f ))

on 0-

(2.25)

k=1

Idea of the proof. Apply (2.21) and (2.23).0 Proof. From (2.21) and (2.23) it follows that `dm E N and Va E Plo m+1

00

D"(B8f) = E D"(A6k(Vkf)) _ ED"(A6k(,Pkf)) k=m-1

k=1

on Gm. Hence Bgf E C°°(S2) and (2.25) holds on 12. 0 3 Moreover, from (2.20) it follows that Vx E (I in the sum (2.16) no more than 2 summands

are not equal to 0.

2.3. APPROXIMATION BY Coo-FUNCTIONS ON OPEN SETS

47

When applying the mollifiers Ba f we shall choose bk satisfying (2.18) depending on f. For this reason we call them nonlinear mollifiers with variable

step (though, of course, BI is a linear operator for fixed b). The variableness of the step follows from (2.19): t/x E Si the mollification is carried out

with the steps b,_1, b 6,+1 depending on x (and these steps tend to 0 as x approaches the boundary 8c). We can also say that By is, in some sense, a mollifier with piecewise constant step, because by (2.23) the same constant steps bm_1i bm, bm+1 are used for the whole "strip" G,,,. Moreover, by (2.24) only one step bm is used for the whole "substrip" (Gm).r,,,. Remark 8 In a number of cases it is more suitable to apply the mollifiers CC, which are similar to the mollifiers Bg and are defined by the equality: Vx E Il 00

ck(x) f f(x - Skz) w(z) dz.

(Cif)(x) = 1:001N(x)(Aakf)(x) = k=1

k=1

(2.26)

B(0,1)

For instance, in contrast to the mollifiers B6, for f = 1 on Sl and arbitrary bk E (0, pk] we have Cg = 1 on Q. On the other hand, for CCf the equalities analogous to (2.22), (2.23) and (2.24) are valid and Cg f E C°°(c). Furthermore, if the kernel of mollification w is real-valued and even, then the operator CT is the adjoint of Bg in L2(ci), because V f, g E L2 (n) c00

00

(BBf, 9) = L,(Aak ('+l)kf ), 9) _ I:(Okf, Ask9) = k=1

k=1

"

l

f, F, 1GkAak9 f = (f, Ca9) k=1

/

(note that for these kernels w the operator Aa is self-adjoint in L2(fl) - see Section 1.2).

2.3

Approximation by C°-functions on open sets

In this section our main aim is to prove the following statement. Theorem 1 Let Cl C R" be an open set, I E N. Then CO' (0) f1 W1(Sl) is dense in W,(Sl) where 1 < p < oo and C°°(Sl) f1 Cis dense in C°(11). '(Cl)

CHAPTER 2. APPROXIMATION BY C°°-FUNCTIONS

48

Moreover, Vf E W, ,(Q) if 1:5 p < oo and Vf E C((SZ) if p = oo there exist functions pk E CO°(Sl) fl WP(fl) such that 4

Wk -4 f in WP '(0), m = 0,1, ...,1. The set CI(I) fl WW(Q) is not dense in W;°(S)). However. df E W.(Sl) there exist functions Wk E COO (0) fl W;O(Sl), k E N, such that

Wk -> f in WW(Sl), m = 0,1, ...,1- 1,

IIcvkHlw, (a) -+ IIf IIwd,(nl

ask-4 oo. Later, in Section 2.6, we shall see that bf E W,(Sl) (or &(Q)) functions gyp, E COO (11) fl WD(S1), gyp, E C°°(11) fl ?71 (11) respectively, exist, which depend

linearly on f, do not depend on p and are such that for 1 < p < oo

G, -Y f in Wy(1l),

(2.27)

in C'(c) respectively. Moreover, the functions W. may possess additional useful properties.

We shall deduce the statement of this theorem, in the case in which 1 < p < oc, from a much more general result, which holds for a wide class of semi-normed linear spaces Z(Sl) of functions defined on SZ with semi-norms II' 11z(n) such that Co (Sl) C Z(fl) C Li°°(Sl). Let Zo(fl) denote the subspace of Z(Sl) that consists of all functions f E Z(fl), which are compactly supported in

SZ. Moreover, let Z(SZ) denote the space of all functions f E Lr(12), which are such that VW E C01(fl) we have cof E Z(Sl). From these definitions it follows, in particular, that Co (Sl) C Zo(fl) C (L()o(SI) and

C°°(Sl) C Z(°`(Sl) C L10°(Sl).

"Under additional assumptions on fl (see Theorem 6 of Chapter 4), IIfIIw, (n) 5 M II/IIw;(n), m = 1,..,1- 1, where M is independent of f, and this statement follows from the density of COO (0) nW.(fl) in W,(fl). However, for arbitrary opens sets it is not so (see Examples 8 - 9 of Chapter 4), and this statement needs a separate proof. We also note that

it is possible that f ¢ W.'(11). In that case also ,k V Wa (fl) but f - wk E W, '(fl) and

f -Ws -0in WD (Sl) as k-4oo.

2.3. APPROXIMATION BY Coo-FUNCTIONS ON OPEN SETS

49

Remark 9 For any I E No we have (C1)'a(c2) = (C,)'°°(c2) = C1(cl). Moreover, for 1 < p:5 oo the following equivalent definition of the space (Wp)'-(Q) can be given: (WP)" (c2) = (f E Li°`(c1): for each open set G compactly embedded into 1l f E Wp(G)}.

Theorem 2 Let Q C R" be an open set and suppose that the semi-Banach space Z(n) satisfies the following conditions:

1) Co (cl) C Z(c2) C Li°L(1), 2) (Minkowski's inequality) if A C 1R function measurable on c2 x A, then

II f f (x, y)dyl z(n) 5 A

f

is a measurable set and f is a

IIf (x, y)Ilz(n)dy,

A

3) if p E C0 00(Q) and f E Z(cl), then W f E Z(S2),

4) all functions f E Zo(S2) are continuous with respect to translation, i.e.,

u oIIfo(x +

h) - f(x)IIz(n) = 0.

(2.28)

Then C°°(Q) is dense in Z"(cu) (and, hence, C°°(Q) f1 Z(Q) is dense in Z(11)), i.e., V f E Z11(c2) functions cp, E C°°(c2) fl Z'-(c2), 8 E N, exist such that

cp, - f in Z(c2)

(2.29)

ass -4 oo.

Idea of the proof. Apply Minkowski's inequality to the right-hand side of the equality (Baf)(x) - f(x) =00E

f

(fk(x - bkz) - fk(x)) w(z) dz,

(2.30)

k-1B(o,1)

where fk = I k f and the mollifier BI is constructed with the help of a nonnegative kernel of mollification, and prove the inequality It Baf - f Itz(n)5 F, w(bk, fk)Z(n) k=1

(2.31)

CHAPTER 2. APPROXIMATION BY C°°-FUNCTIONS

50

Here

w(b, f)z(n) = sup Ilfo(x + h) - f (x)llz(n) Ihj<6

is the modulus of continuity of the function f in Z(Q). (Compare with (1.8).) Using condition 4) choose dk in such a way that w(bk, fk) < e 2-k. Then II Baf - f 11z(n) < e. 0 Proof. 1. From 4) it follows that V f E Zo(SZ) there exists y = y(f) > 0

such that Vh E R" satisfying IhI < 'y the function fo( + h) - f E Z(SZ). Let us suppose, in addition, that y < dist (suppf, 6Q), then suppfo( + h) C (suppf )Ihi C Sl and fo( + h) E Zo(SZ). For, first of all fo( + h) E Z'-(Q). Consider, furthermore, a function of "cap-shaped" type 'n E C0 00(Q) such that n = 1 on (see Section 1.1), then by definition of Z'°°(1) rlfo(' + h) E Zo(SZ).

Let A(h) = Il fo(x+h) -f (x)IIz(n) for h E B(0, ry). Condition 4) means that the function A is continuous at the point 0. Moreover, A E C(B(0, ry)). Indeed,

let u E B(0, ry). Then, by the continuity of the semi-norm, in order to prove

that A(h) -+ A(u) as h i u it is enough to prove that fo(x + h) - f (x) fo(x + u) - f (x) or fo(x + h) - fo(x + u) = go(x + h - u) - g(x) -r 0 ash -

u

where g(x) = fo(x + u). And this is valid because g E Zo(SZ). 2. Let us consider the mollifiers B1, which are constructed with the help

"

of any nonnegative kernel. Since 00 E )k = 1 on SZ and f wdx = 1 we have k=1

NnEi1

B(0,1)

00

(Baf)(x) - f (x) = E,((A6k (Okf )) - Wk(x)f (x)) k=1

00

_ F I

(fk(x - 6kz) - fk(x))w(z)dz =

k=1 B(0,1)

Fk(x),

k=1

where fk = 'kf and Fk(x) = f (fk(x - akz) - fk(x)) w(z)dz. B(0,1)

By 3) and (2.15) we have that fk, Ft E Zo(S1) and supp fk, supp Fk C Gk_1 U Gk U Gk+1. Applying Minkowski's inequality for infinite sums (which holds besause of the completeness of the space Z(Il)) we have 00

IIFkllz(n)

II Baf - f Ilz(n) <_ k=1

2.3. APPROXIMATION BY C°°-FUNCTIONS ON OPEN SETS

51

Now suppose that, in addition to (2.18), (2.32)

bk < 2 y(fk),

then the function Ilfk(x - bkz) - fk(x)II z(n) is continuous on B(0,1) and

f 11f,&

- bkz) - fk(x)II z(n) w(z) dz < oc.

B(0,1)

Moreover, the function [fk(x - bkz) - fk(x)]w(z) is measurable on SZ x B(0,1) (see footnote 9 of Chapter 1). Therefore, we can apply Minkowski's inequality in condition 2) and establish that IIFkIIz(n) <

f

II fk(x - 5kZ) - fk(x)II Z(n) w(z) dz

B(0,1)

< sup 11f& + h) - fk(x)IIz(n) = w(dk, fk)Z(n). Jhl!5hk

Thus, (2.31) follows. 3. Now `de > 0 by 4) choose 5k such that, in addition to (2.18) and (2.32), w(bk, fk)Z(n) < e 2

k

.

(2.33)

(we note that (fk)o = fk). With this choice ofd = a(e, f) (depending on a and f) we have Bgf E C°°(SZ) and II Baf - f 11z(n) < e.

(2.34)

Thus, Theorem 2 is proved (in (2.29) one can take cp, = Bra f , where S.

_

a (;, f)). o Remark 10 The functions gyp, in the given proof are constructed in such a way

that they depend on f, in general, nonlinearly. Moreover, they may depend, of course, on the space Z(f ). For example, in the case of Z(Q) = Wp(f2) they may depend on n,1, p, SZ and f . Idea of the proof of Theorem 1. The density of C°°(12) f1 WP(SZ) in W,(SZ)

where 1 < p < oo and of C- (Q) f1 C'(SZ) in C1(1l) follows directly by applying Theorem 2 to Z(S2) = WW(S1), GV'(S1) respectively. In order to prove the second

CHAPTER 2. APPROXIMATION BY COO-FUNCTIONS

52

statement of the theorem take Z(1) = WP(sl), where WP(n) = n Wn (Sl), m=0

hence I I f Il wp(n) = E II f II wp (n)

Note that by Corollary 14 of Chapter 4

=0

(WP)°Oc(Q) D WP(sl). The case of the spaces C'(Sl) is similar.

In the case of the spaces W;,(1) take open sets Ilk C Sl, k E N, which are such that mess OSlk = 0. Consider any ryk, E (0, pk), k, s E N, such that lim ryk, = 0. Choose a partition of unity {0k}kEN in such a way that 8-400

for any sufficiently small Sk,

Ba,f = A6,,,.f on (Gm)7.,,., where b, _ {6k,}kEN. This is possible by Remark 7. Moreover, assume that the kernel of mollification w is nonegative.

Proof of Theorem 1. By Lemmas 10, 11 and 14 of Chapter 1 the conditions

of Theorem 2 are satisfied for Z(Sl) = WP(Q) where 1 < p < oo and for

Z(Sl) = C (a). Hence the first two statements of Theorem 1 follow from Theorem 2.

However, if p = oo, then condition 4) of Theorem 2 is not satisfied, and Theorem 2 is not applicable. In this case we need a more sophisticated argument. Let f E W.1(11) and m < I - 1. Then for any mollifier Ba, which is constructed with the help of a nonnegative kernel, and m = 0, 1, ...,1 - 1 we have

IIBaf -

f

G E '(6k, fk)w,TM,(n) :5 E J(bk,''kfo)W,m(R") k=1

k=1

By Lemma 11 of Chapter 1 Okfo E W;O(R") and as in the proof of Lemma 1, applying, in addition, footnote 11 of Chapter 1, we establish that w(4,0kf0)W;(R^) :5 M1 is independent of f and k. Consequently, b's > 0 there exist akl) > 0, k E N, such that Wk E (0, akl)) we have w(bk, 1kfo)w;(R") < e 2-k, m = 0, ...,1 - 1, and hence IIBaf - f 11w,,_.(n)

< s, m = 0, ...,1- 1.

Furthermore, for Va E No satisfying j al = I by (2.25), Lemma 4 of Chapter 1 and Leibnitz' formula we have 00

E (*)EAj,,(D--0,0kD.#f), 0
0

k=1

2.3. APPROXIMATION BY C00-FUNCTIONS ON OPEN SETS n

where

,=n

0!(&"!

,

00

If 3 96 a, then E

°''

i=1

53

0 on Q and

k=1

by (1.8) 00

00

(D°-9,bkDwf)-D°-,e1,kDB,f)II

1

lIEA6.(D°-flakDwf)

flX k=1

k=1

Le(n)

`(A6,

00

00 D",f)-D°-6,iIjk Dwf

II A6. (D°-R?k

w(bk,

IILm(f1)

k=1

D°-l3tPk Dwf)Le,(n)

k=1 00

< > w(bk,

D°-dV

k Dwfo)LW(R')

k=1

Since by Lemma 11 of Chapter 1 D`011k Dw f0 E W.'-"l (R") as in the proof of Lemma 1 we establish that w(bk, D°-p k L'w0 f0)Loo(R°) < M2bkIID°-1 Vk DwfolI

W:181(R") ,

where M2 is independent of f and k. Consequently, there exist ak2) E (0, ak1)), k E N, such that dbk E (0, ak2)) we have w(bk, D°-",Pk DwfO)Loo(R") < c 2-k-"(1 + E 1)-1 101=1

and, hence,

II E 00 A6.(D°-O*k Dwf)IIL,o(n) < e 2-"(1 + E 1)-1. k=1

I°I=1

If Q = a, then since 10k, k E N, and the kernel of mollification is nonnegative we have 00

00

IIEIA6k(PkD,°of)IIIL-(n)

II

k=1

k=1

00

00

S


k=1 00

< III + E(A6k'Ok - *k)IIL,.,(n)IIDwf IIL (n) k=1

CHAPTER 2. APPROXIMATION BY C00-FUNCTIONS

54

< (1+EIIA6k7Gk-')kIIL (n))IIDwfIIL..(n) k=100

< (1 +

(J (bk+ Y k)

II Dwf II

k=1

Since ?k E C000(0), as in the proof of Lemma 1 we have w(6k,Ok)L (n))

W(6k,Ok)L (R"))

6k

and it follows as above that there exist ak3) E (0, ak2)), k E N, such that Wk E 3

(0+ ak )

00

IIA6k(VkDwf)IIL (n) < (1+e)IIDwfIIL.,(n) k=1

(This inequality also holds for cr = 0.) Thus, if bk E (0, ak3)), then Il Baf 11w, (n) < e + (1 + E)IIfIIw;,(n)

(We have applied the equality E (;) = 2".) 0<0
In particular, if 6k, E (0, ak3)), k, 8 E N, then

IIBi.fllw;,(n) <e+(1+e)IIfllw;,(n) On the other hand by construction of the mollifier b, and by Lemma 4 of Chapter 1 IIBa,fllw, (n) >_ IIB8,f llw;,((Gk).,k,) = IIA6k,fIIL..((Gk),k,) +

IIA6k.fllw;,((C4)ryk.)

II 4. DwfIIL.,((Gk),k, ).

By relation (1.9) for p = oo there exist a(4) E (0, ak3)), k E N, such that for 6k, E (0,

ak4))

IIBJ,fIIW'(n)

IIfIIW;,((Gk),,)

2

and, hence,

IIBa.fIlwa,(n) ? sup IIfIIW.((Gk),k,) - z = IIf IIW., (kU kEN

(G,,),,,.)

-z

2.3. APPROXIMATION BY Coo-FUNCTIONS ON OPEN SETS

55

Since meas (U 00 09%) = 0 we have k=1

Ilfllw;,(n) =

IIfIIWW,(n\U k

8O)

IIfIIWW(U

(Gk)., )

Consequently, there exists s E N such that Ul(Gk),,) >- IllIIW;,(n)

z'

k

Thus, W > 0, there exist s E N and 5ks E (0,ak4)) such that IlBd,f - fIIw,1(n) <e and

IIfUIW,(n) -s < IIBa.fllw.,(n) and the statement of Theorem 1 in the case p = oo follows.

Corollary 1 Let 11 C R" be an open set, l E No

Then C°°(S2) is dense in

(WP)'0°(&1) where 1 < p < oo and in C'(S2).

Idea of the proof. Apply Theorem 2 to Z'«(S2) _ (WW)'°°(SZ) and Z(SZ) _ C'(Sl).

Remark 11 If p = oo, then C°°(SI) fl W/,(SZ) is not dense in W;°(1l) (see Remark 2).

Remark 12 The crucial condition in Theorem 2 is condition 4). It can be proved that under some additional unrestrictive assumptions on Z(Sl) the density of C°°(Il) in Z'°°(n) (or the density of C°°(a)flZ(SZ) in Z(Sl)) is equivalent to condition 4).

Remark 13 Theorem 2 is applicable to a very wide class of spaces Z(fl), which are studied in the theory of function spaces. We give only one example. Consider positive functions ao, as E C(fl) (a E 1%, lal = 1) and the weighted Sobolev space Wp.{°o)(S2) characterized by the finiteness of the norm IlaofIIL,(n)+

lla.DwfIIL,(n)

By Theorem 2 it follows that C°°(fl) fl WP(°Ql(S2) is dense in this space for 1 < p < oo without any additional assumptions on weights ao and a.. Such generality is possible due to the fact that the continuity with respect to translation needs to be proved only for functions in this weighted Sobolev space, which are compactly supported in 0.

CHAPTER 2. APPROXIMATION BY Coo-FUNCTIONS

56

Now we give one more example of an application of Theorem 2, in which the spaces Z'°`(SI) (and not only Z(SZ)) are used.

Example 1 Let SI C R" be an open set, then dµ E C(SI) and de > 0 there exist µE E C°°(Q) such that Vx E SI we have µ(x) < p,(x) < µ(x) + e.

To prove this it is enough to set µe = Bg(,u + Z) with 3 = b(z, µ + a) in the proof of Theorem 2 for Z(S)) = C(SI) (hence, Zb°`(SI) = C(SI)) and apply inequality (2.34).

2.4

Approximation with boundary values

preservation

of

In Theorem I it is proved that for each open set SI C R" and Vf E W'(SI) (1 < p < oo) functions W. E C°°(i) n WW(SI), s E N, exist such that (2.27) holds. In this section we show that it is possible to choose the approximating functions cp, in such a way that, in addition, they and their derivatives of order a E No satisfying jal < 1 have in some sense the same "boundary values" as the approximated function f and its corresponding weak derivatives. The problem of existence and description of boundary values will be discussed in Chapter 5. Here we note only that for a general open set SI C R" it may happen that the boundary values do not exist and even for "good" 0 boundary values of weak derivatives of order a satisfying lad = 1, in general, do not exist. For this reason in this section we speak about coincidence of boundary values without studying the problem of their existence - we treat the coincidence as the same behaviour, in some sense, of the functions f and cp, (and their derivatives) when approaching the boundary of SI. Theorem 3 Let SZ C R" be an open set, I E N, 1 < p < oo. Then V p E C(O) anddf E WP(fl) functions W, E C°°(0)f1Wp(1I), s E N, exist such that, besides (2.27), Va E N satisfying Jal < 1 II (Dwf - D°cpd)pI1L,ini -> 0

(2.35)

ass -* oo. For p = oo this assertion is valid V f E C'(SI). Corollary 2 Let SI C R" be an open set, fI 34 R" and I E N. Then V f E U1 (fl) functions cp, E COD(A) n s E N, exist, which depend linearly on f and

2.4. PRESERVATION OF BOUNDARY VALUES

57

are such that, besides (2.27) where p = oo, s

D*v.Ion = D°f Ian,

(al < 1.

Idea of the proof. Choose any positive p E C(12) such that

(2.36)

lim

y-rx,yEf

µ(y) = oo

forall xE812. Proof. One may set, for example, p(x) = dist (x, 812)-1. For a continuous function II - 11C(n) = II - IIL,(n), therefore, from (2.35) it follows that for some

M>OVsENandVyE0

ID°w9(y) - D°f (y)I S M(.u(y))-1.

Passing to the limit as y -+ x E all, y E S2 we arrive at (2.36).

Corollary 3 Let 12 C R" be an open set, 1 E N, 812 E C1 and 1 < p < oc. Then V f E W(1l) functions cp, E C°°(Q) n WW(Q), s E N, exist such that, besides (2.27), D°wal an = Dwf Ian,

j al < 1 - 1.

(2.37)

Idea of the proof. Take again µ(x) = dist (x, 812)-'. By Chapter 5 it is enough to consider the case, in which 12 = R+ = {x E R" : x" > 01 and µ(x) = x;1. In this case the statement follows by Lemma 13 of Chapter 5.

Remark 14 The function p in Theorem 3 can have arbitrarily fast growth when approaching M. Let, for instance, u(x) = g(Q(x)), where e(x) = dist (x, 80) and g E C((0, oo)) is any positive, nonincreasing function. Then

forl
II Dwf - D°w,ll L,(n\n,) <_ (9(6))-' II (D0 f - D°w,)9(e)II L,(n) <- M(9(6))'-'

with some M > 0, which does not depend on s and 6. It implies that for a fixed f E Wp(12) where 1 < p < oo one can find a sequence of approximating 5 We recall that Vf E G'(12), Vx E OQ and Va E IT satisfying jal < I there exists lim I#

Z'Ven

D° f (y) and, thus, the functions D° f , which are defined on n can be ex-

tended to ?I as continuous functions. It is assumed that D° f den are just restrictions to Oil of these extensions. The same refers to the functions gyp, E C°0(11), because by (2.27) where p = oo we have gyp, E &(fl). From Theorem 8 below it follows, in particular, that w, can be chosen in such a way that they depend linearly on f. Here by D,°, f Ian and D°w, Ion the traces of the functions Dl.f and D°,p, on 80 are denoted (in the sense of Chapter 5, they exist if lal < 1- 1). See also Theorem 9 below.

CHAPTER 2. APPROXIMATION BY Coo-FUNCTIONS

58

functions V which is such that, besides (2.27), Va E l satisfying Ial < 1 the norm IIDwf - D' a$II c,(n\n,) tends to 0 arbitrarily fast as 6 -+ +0. Thus, condition (2.35) with arbitrary choice of p means not only coincidence of boundary values, but, moreover, arbitrarily close prescribed behaviour of the functions f and cp, and their derivatives of order a satisfying Ial < 1 when approaching the boundary O 1. For unbounded fZ we have the same situation with the behaviour at infinity. Choosing positive µ E C(Q) growing fast enough at infinity, we can construct the functions cp, E C°°(1) f1 W,(SZ) such that IIDwf - D°WaII j,(n\B(o,*)) where

j al < 1 tends to 0 arbitrarily fast as r -+ +oo, i.e., D,f and D°cp, have arbitrarily close prescribed behaviour at infinity.

As in Section 2.4 we derive Theorem 3 from a similar result, which holds for general function spaces Z(SZ).

Theorem 4 In addition to the assumptions of Theorem 3, let the following condition be satisfied: 5) bf E C0 00(Q) there exists cy > 0 such that df E Z0(SZ) Ikof Ilz(n) <- c,II f IIz(n)

Then Vp E C°°(SZ) and `If E Z(Q) functions gyp, E C00(11) f1 Zl°°(SZ), a E N, exist such that

W. - f in Z(Q)

(2.38)

11(f - WW.)PIIz(n) -+0

(2.39)

and

as s -+ coo.

Idea of the proof. Starting with the equality that differs from (2.30) by the factor p show, applying 5), that 00

II (BIf - f)µllz(n) <- E Ckw(6k, fk)z(n),

(2.40)

k.1

where the ck > 0 are independent of 6k. El Proof. In addition to the proof of Theorem 2, we must estimate the expression 00

(Baf - f )µ = E uFk. k=1

2.4. PRESERVATION OF BOUNDARY VIAL LIES

59

Recall that Fk E Zo(SZ) and supp Fk C Gk = Gk_, U Gk U Gk+1. Let us denote by 77k E Co (SZ) a function of "cap-shaped" type, which is equal to 1 on Gk (see Section 1.1), then 1Fk = µ71kFk. By condition 5) where cp = µ77k there exists ck > 0 depending only on µ77x (and, thus, independent of bk), such that IIIFkIIZ(n) < CkIIFkIIz(n) < CkW6,t(fk)Z(ft)

and (2.40) follows (without loss of generality we can assume that ck > 1).

Choosing Ve > 0 positive numbers 5k in such a way that in this case w6,t(fk)z(n) < e2_kckl (instead of (2.33)), we establish, besides (2.34), the inequality II (B3f - f)µllz(n) < c.

Remark 15 From the above proof it follows that b'µ1i...,µm E C' (Q) (m E N) and Vf E Zb0C(Sl) functions cp, E C°°(Q) f1 Zf0C(c), s E N, exist such that

II(f - e)WIIz(f:) -> 0,

i = 1,...,in.

as s -+ oo. (Theorem 4 corresponds to m = 2, it, = 1, µ2 = µ.)

Idea of the prof of Theorem 3. Apply Theorem 4 and Remark 15 to the space Z(Q) = WW(c) and to a set of the weight functions (D'µ,)I,I<,, where µ, E C°°(c) and IµI < µl on Q. Proof of the Theorem 3. The existence of the function µ, follows by Example 1. By Remark 15 bf E (W,)(0C(S2) 7 functions gyp, E C°°(c), s E N, exist such that ip, -+ f in WI (fl) and dry E N satisfying Iyl < 1. 11(f - w$)D'pIII `wP(n) + 0

as s -* oo. Hence, Va E N satisfying Ial < I IIDW((f - co$)D'µl)II L,(n) -+ 0.

Applying "inverted" Leibnitz' formula 8, we have

(Dwf - D°v,)µl =

(-1)191

(c) DW((f -,pa)D---'{ll)

0
We recall that this space was also considered in the proof of Theorem 1. s For n = 1 and ordinary derivatives it has the form

f(k)9 = E (-1)m (m) (f9(k m))(m) M=0

\/

and is easily proved by induction or by Leibnitz' formula.

(2.41)

CHAPTER 2. APPROXIMATION BY Coo-FUNCTIONS

60

and from (2.34) it follows that Vf E (Wp)1oc(Q) and Va E 1

satisfying Ial < l

L,(11) <_ II(Dwf - D°Ve)121IIL,(n) -+ 0

II

as s -+ oo. Finally, as in the proof of Theorem 1, it is enough to note that W,(c) C (W,)1oc(cl).

Linear mollifiers with variable step

2.5

We start by studying the mollifiers A6 having kernels with some vanishing moments. The main property of the mollifier A6 is that Yf E LP(I) where

1
(2.42)

IIA6f - IIIL,(n) = 0(1)

as 6 -3 0+. As for the rate of convergence of A6f to f, in general, it can be arbitrarily slow. However, under additional assumptions on f one can have more rapid convergence.

Lemma 8 Let S2 C Rn be an open set, 1 0 and G C S2 be a pleasurable set such that G6 C SZ. Then Vf E wp(Q) (2.43)

IIA6f - f II L,(c) <- c1bIIf where

cl = Max i=l,...,n

(2.44)

Idea of the proof. For f E C°O(Sl) apply Taylor's formula and Minkowski's inequality. For f E wp(f) approximate f by Aryf and pass to the limit as

ry-r0+.

Proof. For f E C°°(12)

IIA6f - f II L,(c) = II

f

(f (x - 6z) - f (x)) w(z) dz 114(G)

9(0,1)

=

II

(

f f

B(0,1)

of (x - t6z)(-6z;) w(z)) dt

/

o

dz IIL,(c)

afIz;w(z)I dt <6 f J(II_(x_toz)il ax; L (0) 1 1

9(0,1)

0

n

dz

2.5. LINEAR MOLLIFIERS WITH VARIABLE STEP

of

<5 i=1,...,n max f Iziw(z)Idz

8xi

i=1

B(0,1)

61

= C151IfIIw;(G6). L, (GS)

Now let f E wP(Sl) and first suppose that p = dist (G6, 8Sl) > 0. Then for 0 < y < P we have that Af E C°°(f) on G6 C Sly and A,(A6 f) = A6(A.1 f ) on G (see Section 1.1). Consequently, II A,(A6f - f)IIL,(G) = II A6(A,f) - A,f II L,(G)

5

axjf 11 L,(C6)-C15jl1 Ary(8xj)wII Lv(G6) by Lemma 4. Passing to the limit as y - 0+ (see (1.10)) we obtain (2.43). If dist (G6, 811) = 0, we choose measurable sets Gk, k E N, such that c15IIAf11w;(G6)=C15EII

Gk C Sl, Gk C Gk+1 and kJ Gk = G. Inequality (2.43) is already proved for k=1

Gk replacing G. Passing to the limit as k -+ oo we obtain (2.43) in this case also.

Estimate (2.43) is sharp as the following examples show.

Example 2 Let for some j E { 1, .., n} f zjw(z) dz 3k 0 and 0 < meas Sl < oo. R

Then IIA6xj - xjIIL,(G) = c26, c2 =

fz,w(z)dz I(meas G)o > 0.

I

R

Remark 16 This example shows also that for some kernels of mollification cl is the best possible constant in inequality (2.43). Let us choose j = 1, ..., n, Ilziw(z)IIL,(R) Moreover, let G be a bounded such that Ilzjw(z)IIL,(R) = iMax

measurable set such that 0 < mess G = mess G < oo. Then sup 6>O:G6cn

5-'

sup

>- 6i 6

'IIA6xj -xjIIL,(G)IIxjII;P(G6)

=If zjw(z) dz 16lli+m I R

II A6f - f IIL,(G) II f II W'(G6)

IUII.'Imoo

mess G mess

G6' PL

=

zjw(x) dz I

R

Thus, if, in addition to (1.1), w(z) < 0 if zj < 0 and w(z) > 0 if zj > 0, then cl is the best possible constant in inequality (2.43).

CHAPTER 2. APPROXIMATION BY C°°-FUNCTIONS

62

Example 3 Let n = 1, G = St = R, p = 2 and f E W2 (R). Then by the properties of the Fourier transform

IIA6f - fIIL,(R) = (2ir)26II6-'((Fw)(6f) - (Fw)(0))(Ff)(f)IIL,(R). We have

6-1((Fw)(bf) - (Fw)(0)) -* (Fw)'(0)t

ash -0+ and sup 6>0

6-'I (Fw) (6f) - (Fw)(0)I

m ax I (Fw)'(z)I

Therefore, by the dominated convergence theorem

II6-'((Fw)(6f) - (Fw)(0))(Ff)(f)IIL2(R)

- I(Fw)'(0)I IIe(Ff)(f)IIL,(R) = I Jzw(z)dz I IIf'IIL,(R). R

Hence, if f zw(z) dz 0 0 and f E W2 (1R) is not equivalent to zero, then for R

some c3 > 0 (independent of 6) and II A6f -f IIL,(R) >- c36 for sufficiently small J.

Let us make now a stronger assumption: f E WP(f) where I > 1. In this case, however, in general we cannot get an estimate better than IIA6f - f II L,(c) = O(5)

(which is the same as for I = 1), if for some j E {1, ..., n} f z,w(z) dz # 0, R

as Examples 2 - 3 show. Thus, in order to obtain improvement of the rate of convergence of A6f to f for the functions f E Wp(f2) where 1 > 1, some moments of the kernel of mollification need to be equal to zero.

Lemma9 LetnCW bean open set, 10andGC1 be a measurable set such that G6 C 11. Moreover, assume that the kernel of the mollifier A6 satisfies, besides (1.1), the following condition:

f

z°w(z) dz = 0, a E ?V, 0 < Ial < 1-1,

(2.45)

B(o,1)

where z° = zi'

. z.*-. Then V f E wip(11) II A6f - f II L,(o) <- c46'IIf Ilw,(6),

(2.46)

2.5. LINEAR MOLLIFIERS WITH VARIABLE STEP where

c4 = max IaI=1

r

z°

J R^

a!

w(z) dz <

63

(2.47)

IIwIIL,(a^).

Condition (2.45) is necessary in order that inequality (2.46) be valid for all f E WW(G) with some c4 > 0 independent off and J. Idea of the proof. By condition (2.45) Yf E COO(1)

(Asf)(x) - f (x) = f (f (x - bz) - f (x)) w(z) dz B(o,1)

f

(Daa)(z)

P x) - E

(-bz)a w(z) dz.

IaI«

B(o,1)

Now multidimensional Taylor's formula (see section 3.3), Minkowski's inequal-

ity and direct estimates (close to those which were applied in the proof of Lemma 8) imply (2.46). If f E w,(S2), then pass to the limit in the same manner as in the proof of Lemma 8.

As for necessity of condition (2.45) for bounded G it is enough to take in (2.46) successively f (x) = xj,j = 1,...,n, f (x) = xjxk,j, k = 1, ..., n, ..., f (x) = , ji-i = 1, ..., n. If G is unbounded, then one needs to multiply the above functions by a "cap-shaped" function 77 E Co (R"), which is equal to 1 on a ball B such that mess B f1 G > 0. O In the sequel we shall apply the following generalization of inequality (2.46).

Lemma 10 Let S2 C R" be an open set, 1 < p:5 oo, l E N, b > 0 and G C fl be a measurable set such that G6 C S2. Assume that the kernel of the mollifier As satisfies, besides (1.1), condition (2.45). Then V f E w,(S)) and Va E N IID°(Asf) - DafUL..,(o) <_

C56'-I°1IIfIIwp(G

),

Ial < 1,

(2.48)

and II Da(Aaf)II L,(C) 5 C66'-'Q1 11f1014(01),

lal ? 1,

(2.49)

where c5, c6 > 0 do not depend on f, 5, G and p. (For instance, one can set ca = II(IIL,(n") and ce = max Is1=IaI-1

CHAPTER 2. APPROXIMATION BY COO-FUNCTIONS

64

Idea of the proof. Inequality (2.48) follows by Lemma 4 of Chapter 1 and inequality (2.46) applied to Dwf E w1 I°I(Q). Estimate (2.49) does not use condition (2.45). It is enough to apply Young's inequality to the equality (see (1.21))

D°(A6f) = 6IRI-I7I(D°-Iw)6 * Dwf ,

where ry E too is such that 0 < ry < a and (rye = 1. 0

Let S2 be an open set and let the "strips" Gk be defined as in Lemma 5 if i1 0 R° and as in Lemma 6 if 11 = )R". Moreover, let {ii*}kEZ be partitions of unity constructed in those lemmas.

Definition 2 Let 0< 6<

l E N and f E Ll-(f2) . Then dx E 0

(E6f)(x) _ (E6,lf)(x) _ k=-oo

0

r

= E 'pk(x) J f (x - a2-IkIz) w(z) dz, k=-oo

(2.50)

B(0,1)

where w is a kernel satisfying, besides (1.1), condition (2.45) 9.

Remark 17 For boundedfZ the operator E6 is a particular case of the operator C1 by Remark 7. As in Section 2.2 in (2.50) in the last term we write f and not fo, assuming that iik(x)g(x) = 0 if 'k(x) = 0 even if g(x) is not defined. (This can happen if dist (x, Of)) < 62-I1I). Since 0 < 6 < a we have supp kkA62-Iklf C (Gk-, U Gk U Gk+1)62-IkI

(2.51)

1ikAd2-IkI f E C°°(fl).

(2.52)

and

(If SZ is bounded, then 0jtA62-IkI f E Co (fl).) As in the case of the operators BI and Cg the sum in (2.50) is finite. If Vx E fl the number s(x) is chosen in

such a way that x E G then 8(x)+1

(E6f) (x) =

t,b (x) f f (x - 62-1klz) w(z) dz. k=8(2)-1

B(0,1)

9 If I = 1, then there is no additional condition on the kernel w.

(2.53)

2.5. LINEAR MOLLIFIERS WITH VARIABLE STEP

65

Moreover, Vm E Z m+1

E6f = E 1kA62-iki f

on

(2.54)

Gm.

k=m-1

We call the E6 a linear mollifier with variable step. The quantity E6(x) is an average of ordinary mollifications with the steps 52-I3(s)I-1, 52-I4052-I3(=)I+1 which (in the case 1196 R") tend to 0 as x approaches the boundary M. Again

we can say that the E6 is a mollifier with a piecewise constant step since the steps of mollification, which are used for the "strip" Gm, namely 52-I-I, 52-1"'1+', do not depend on x E Gm. Moreover, by Remark 5 for any fixed -y > 0 we can choose a partition of unity {z'k}kEz in such a way that, in addition to (2.54), Vm E Z 52-ImI-1

E6f = A62-1-If on

(Gm)ry2-m.

Remark 18 Changing in (2.50) the variables x - 52-1kIz = y we find

(E6f)(x) = JK(xyio)f(y)dy, n

where

K(x,y,5) _ E Vk(x)(82-jkj) "w k= -00

(E)

Comparing these formulae with formula (1.2) we see that, similarly to the mollifiers A6i the mollifiers E6 are linear integral operators, however, with a more sophisticated kernel K(x, y, 6) replacing 5-"w (Z-811)

-

The mollifier E6 inherits the main properties of the mollifier A6i but there are some distinctions.

Lemma 11 Let S2 C R" be an open set and f E L1 `(S1). Then V6 E (0, 8] E6f EC0O(51) andVaE1g 00

D°(E6f) = E D°(OkA62-ikif) on 11.

(2.55)

k=-ao

Remark 19 In contrast to the mollifier E6 we could state existence and infinite differentiability of A6 f for f E Li00(11), in general, only on 526.

CHAPTER 2. APPROXIMATION BY Coo-FUNCTIONS

66

Idea of the proof. The same as for Lemma 7.

Lemma 12 Let 11 C R" be an open set and f E L,°`(11). Then E8 f -+ f

a.e. on it

(2.56)

ash->0+. Idea of the proof Apply (2.54) and the corresponding property of the mollifier A6.

Lemma 13 Let 11 C R' be an open set and 1 < p:5 oo. Then db E (0, 8] and f E Lp(il) (2.57)

IIEaf IIL,(n) <_ 2c7 IIf IIL,(n) ,

where c7 = IIwIIL,(R-)

In order to prove this lemma we need the following two properties of Lpspaces where 1 < p < oo. 1) If 11 C R' is a measurable set and Vx E 1 a finite or a denumerable sum

E ak(x) of functions ak measurable on Il contains no more than x nonzero k

summands, in other words, if the multiplicity of the covering {supp ak} does not exceed x, then 1

I I E ak II L,(n) <_ xl = E I Iak IIi,(n) k

(2.58)

k

(This is a corollary of Holder's inequality.) 2) If it = U Ilk is either a finite or a denumerable union of measurable sets k

ilk and the multiplicity of the covering {ilk} does not exceed x, then for each function f measurable on i1 IIfII,,(nk)

<_ xaIIfIIL,(n)

(2.59)

k

In particular, if p = 1 and f = 1, then we have E mess Slk < x mess il. k

(2.60)

2.5. LINEAR MOLLIFIERS WITH VARIABLE STEP

67

For p = oo these inequalities take the following form II E akllL (n) <_ csup IIakIIL-(n), k

k

IIL (n)

SUP Ill k

Idea of the proof of Lemma 13. Apply (2.58), (1.7) and (2.59). Proof of Lemma 13. By (2.7) and (2.15) V6 E (0, el 00

00

U suPP Ok = U (suPP Ok)62 k=-oo

k=-oo

and the multiplicities of the coverings {SUpp10k}kEZ and are equal to 2 (see Remark 6). Therefore, by (2.58) and (2.59)

{(SUppok)62-IkI}kEZ

00

IlE6fIIL,(n) = II

VbkA62-I.IfIIL,(n) k=-oo

/

<21 p

00

( ao IIA62I'll fll'L(SUpp 0,) =-00

00

< 2'-a

IIfIIP

l

y

2c711fIIL,(n)

=-oo

Now for an open set S2 C R" and Vx E fl we set p(x) = dist (x, 8Sl) if SZ96 R" and 10 p= (1+Ixl)-1} if S2=IRS.

Lemma 14 Let 1l C Rn be an open set and 0 < 6 <

Then

1) for 1 < p < oo and `d f E Lp(S2)

E6f -+ f in

LP(S2)

(2.61)

2) for p = oo relation (2.61) holds V f E G (SZ), io It is also possible to consider Q(x) = min{diet (x, tifl), (1 + IxI)-' }. However, in that case one must use a partition of unity constructed on the base of altered p and verify that estimate (2.12) still holds.

CHAPTER 2. APPROXIMATION BY COO-FUNCTIONS

68

3) for 1
Ewp(S2)

(2.62)

II Eaf - f IIL,(n) 5 ca. 011f

4) for 1 < p < oo, ! E N and df E w,'(52) II (Ed - f)e 11IL,(n) <- c901fhIW',(n) where c8 and c9 do not depend on f , 6,

(2.63)

,

n and p.

Idea of the proof. To prove the statements 1) and 2) establish, by applying the proof of Lemma 13, that the series (2.50) converges in LP()) uniformly with respect to 6 E (0, e] and use the corresponding properties of the mollifiers Aa. To prove (2.62) and (2.63) follow the proof of Lemma 13, applying inequality

(2.48) with a = 0 instead of (1.7). In the case of inequality (2.62) apply, in addition, the fact that there exist B1, B2 > 0 such that B12-k

5 o(x) < B22-k

dkEZif(

on supp tpk

(2.64)

R" andVk<0ifS2=R°. 0

Proof. We start with the proof of inequality (2.62). If f E ww(11), then using, in addition, Minkowski's inequality for sums we find 00

IIEaf - fIIL,(n) = II E +Gk(Aa2-Iklf - f)IIL,(n) k=-ao 00

< 2' P

IIA62-Iklf - fIILP(80PPtlk) k=-0o i

E

< 2' Pcs61 (J-0

1bk)'2-Ikl )

a0

00

5 2' Pcs6l k=-0o

z-lkl'PIIfIIWy((.PP

P\ s

E

IIDwhhL,((wPP16k)a=-Ikl)

1011=1

1 P

<

21-ac,,a1

E CC IIDwfIIPLP((.PP*k)62-Ikl)J 101=1

=-oo

69

2.5. LINEAR MOLLIFIERS WITH VARIABLE STEP < 2c5a'

IID`f1IL,(n) =

c86`Ilfllw;(a),

Ial=1 {(supp1Pk)a2-"" }kEZ is equal to 2 (see because the multiplicity of the covering Remark 6). In the case of inequality (2.63) we find with the help of (2.64) that

II (Eaf - f )P 'IILP(n) = II = 'iPk(Aa2-ikif - f)IILo(n) k=-oo

_ 2' PBl'

Ia

0o

21k1'pIIAa2-Iklf k=-oo

<

2'_0Bl'c56'

0

11i11°

(k=-000

The rest is the same as above (c9 = 2Bi c5). 0 Lemma 15 Let Il C Rn be an open set, 1 E N and 0 < b <

Then for each

polynomial pl-1 of degree less than or equal to 1- 1 (Eapr-1)(x) = Pr-1(x), x E S2.

Idea of the proof. Apply multidimensional Taylor's formula (see Section 3.3) to pt-1(x - 62-IkIZ) in (2.50) and use (2.45), (1.1) and (2.11) or (2.14). 0

Remark 20 In Lemmas 13 - 14 the property (2.45) of the kernel of mollification was not applied. It was applied in Lemma 15, but this lemma will not be used in the sequel. The main and the only reason for introducing this property is connected with the estimates of norms of commutators [Dw, Ea] f , which will be given in Lemma 20 below. In its turn these estimates are based on Lemmas 9-10, in which the mollifiers Aa with kernels satisfying the property (2.45) were studied. Let us denote the commutator of the weak differentiation of first order and the mollifier Ea in the following way:

(esf). (Ej) = [(a )m, EE] This operator is defined on (W1)1O1(0)

(

)w EE

- Ea

(k)w.

CHAPTER 2. APPROXIMATION BY C°°-FUNCTIONS

70

Furthermore, for I E N, l > 2, we define the operators b.0 (E6), where a E I ` and jal =1, with the domain (Wi)'O°(12): b.-(E6) =

((aI)w'

...

(E6).

Lemma 16 Let 0 C R" be an open set, 1 E ICI, 0 < b < s and f E (Wi)i0c(Sl) Then Va E N' satisfying dal < I 00

(D°(E6)) f = E (D°,0k)A62-Ik1f k=-oo

Idea of the proof. Induction. For Ian = 1 by (2.55)

l\ax,) w (Ea))f = (((

(A62-ikif+V)kaxj(A62-ikif))

00

k=-oo 00

00

-kA62-iki (e ii) k=-oo

on S2, because by (2.7) and (1.19) ikk

371

_ W

k=-oo

821

A62-ikif

(Ab2_j,,jf) = tPkAd2-iki (e )W on 11. E3

Remark 21 For the mollifiers A6 we have (Dw(A6)) f =_ 0 but, only on 126 (see Section 1.5), while for the mollifiers E6 in general (Dw(E6)) f $ 0 even on 526, but on the whole of S2 the quantity (Dw*(E6))f is in some sense small (because 00

E D°9fik = 0 on S2) and, as we shall see below, it tends to 0 in Lp(S2) fast k=-oo

enough under appropriate assumptions on f .

Remark 22 On the base of Lemma 16 we define for Va E N satisfying a 96 0 the operator E6°) with the domain L"(1) directly by the equality

Ea°if = E (D°*k)A62_Iki f.

(2.65)

k=-00

Lemma 17 Let SZ C W' be an open set and 0 < b <

Then Va E Pla

satisfying a 96 0 and V f E Li«(12) Es(*) f -s 0

a.e. on S2.

(2.66)

2.5. LINEAR MOLLIFIERS WITH VARIABLE STEP

71

m+1

Idea of the proof. Since Vm E Z we have E(°) f =

F,

Da1PkA62-IkI f and

k=m-1 m+1

E Da0k = 0 on Gm, the relation (2.66) follows from (1.5). 0

k=m-1

Lemma 18 Let fZ C JW be an open set, 1 < p:5 oo and 0 < 6 < e. Then 1) V f E w,(Q) and Va E N satisfying 0< IaI < 1 (2.67)

IIEea)f IIL°(n) <_ C1 blllf

2) Vf E w,(11) and Va E N satisfying jai > 0 (2.68)

II(E6(a)f)P'a1-1IIL°(n) < clIa1llfllwp(!1),

where c10, c11 > 0 do not depend on f, 6, 0 and p.

Idea of the proof. Starting for a 36 0 from the equality

EE°)f =

= D0,0k(A62-141 f - f)

(2.69)

k=-oo

follow the proof of Lemma 13, apply estimate (2.12) of Lemmas 5 and 6 and inequality (2.48), in which a = 0, G = supp tpk and 6 is replaced by 62-A. In the case of inequality (2.68) apply, in addition, (2.63). O Proof. For IaI < 1 from (2.69) it follows that °

IIE6a)fllL°(:z) < 21

°

IIDabk(A62-Iklf -

f)IIL°(n))

k=-oo

< 21 sca

f IIL°(supp*h)

F,

k=-oo

< 21 'c0c5d1 (E 00

21k1(1al-1)PIIf

=-00

21

.

r

IIp

y

k

)6r1k1) 1

00

Ca4g61 E IIf lip

=-

+°((sappd.)69-1.1 )

°I

CHAPTER 2. APPROXIMATION BY C°°-FUNCTIONS

72

<

2c°c56'lifllwy(f) = c1061IIfIIwy(n)-

We have taken into account that the multiplicity of the covering (suPPOk)62ik'}kez

{

is equal to 2 (see Remark 6). In the case of inequality (2.68) a

II (E6 f)Q

IIL,(n) < 2

1

a

P n E IIQI oftD °''k(A62-ikif - f)IIL,(n)

(J-000 21-1

(max {Bj 1, B2}) II°I-A ca

I

21kh1P IIA62-ikif

(J-000

- f 11 v( i ,,,

The rest is the same as above (c11 = 2c°cs (max {B1',

B2})11°I-111

O

Lemma 19 Let I C R" be an open set, l E N and f E (Wi)1oc (S2). Then V E N satisfying 0< f ai < I

D°(Esf) _

(;)E-(Df)

(2.70)

and

(a)E6(--0)(D1Pf). 0

[Dw, E6]f =

(2.71)

0:50:50,000

Idea of the proof. Apply (2.55), Leibnitz' formula, Lemma 4 of Chapter 1 and the definition of the operator E6") (see Remark 22). 0

In the sequel we shall estimate D°(E6 f) and D°(E6 f) - D.*f with the help of (2.71) and the following obvious identities: D°(E6f) = [Dc, EB) f + E6(Dwf)

(2.72)

D°(E6f) - D. ,f = [D,°n, E6] f + E6(D"f) - Dwf-

(2.73)

and

Lemma 20 Let 11 C R?' be an open set, 1 < p < oo and 0 < 6 <

Then

df E wy(SZ) :

1) Va E 1q satisfying 0 < iai < I II [D.*, E6)f IIL,(n) S c1261-1°I+1IIf IL4(n),

(2.74)

2.6. THE BEST POSSIBLE APPROXIMATION

73

2) Va E N satisfying IaI > 0 II (D ,

)PIaI-`IIL,(n)

C1381-IaI+1

Ifllwy(R)

(2.75)

+

where c12, C13 > 0 do not depend on f , 8, SZ and p.

Idea of the proof. Starting from equality (2.71) apply inequalities (2.67), respectively (2.68), with 1- 1,61 replacing 1, a - p replacing a and DO ,,f replacing

f. Take into consideration that plat-1 =

2.6

19la-P1-(1-IaI) and IQI < IaI

- 1.

The best possible approximation with preservation of boundary values

We start by studying some properties of the mollifiers E6 in Sobolev spaces.

Theorem 5 Let Q C R" be an open set, I E N, 0 < 8 < e and 1 < p < oc. Then V f E wn(S2) II E6f IL,,(n) < c14IIf (Iw;(s1)

(2.76)

II E6 f Ilwy(n) :5 c14IIf IW,1(n) ,

(2.77)

anddf E Wa(S2) where c14 > 0 does not depend on f , 8, 11 and p.

Idea of the proof. Apply (2.71) and Lemmas 13 and 20. Proof. By (2.72), (2.74) and (2.57) IIE6f 11w;,(n)

_

II DwE6f IIL,(n) IaI=1

E II[Dw, E6] f IIL,(n) + II EoDW f IIL,(n) IaI=1

(c12IIf Ilwo(n) + 2C7II Dw f IIL,(n)) = c14IIf 11,4(n)

<_

IaI=1

Inequality (2.77) follows from (2.57) and (2.76).

Theorem 6 Let) C R" be an open set, 1 E N and 0 < 8 < 1) If 1:5 p:5 oo, then df E Wp(11) and Va E N satisfying IaI 5 1 Da(E6f) -+ D.Of a.e. on 11

(2.78)

CHAPTER 2. APPROXIMATION BY Coo-FUNCTIONS

74

asb-r0+. 2) If 1 < p < oo, then V f E WD(D)

Eaf -> f in Wo (S2), m = 0, ...,1,

(2.79)

ash->0+. 3) If p = oo, then Vf E W;°((2)

Ea f -> f in WW (S2), m = 0,..., l -1,

(2.80)

as 6 -+ 0+ (if f E C1((t) then (2.79) holds). Idea of the proof. Relation (2.78) follows from equalities (2.72), (2.71) and Lemmas 14 and 20; relations (2.79) and (2.80) follow from (2.72) and Lemmas 14 and 20. Proof. Let us prove (2.79). From (2.72), (2.74), (2.62) and, in the case m = 1, (2.61) it follows that IIEaf - Ill w; (n) = II Eaf - f IIL,(n) +

II D°Eaf - Dwf IIL,(n) IaI=m

<- IIEaf - f IIL,(n) +

(II[D.*, Ea]f IIL,(n) + IIEa(Dwf) - Dwf IIL,(n)) -' 0 IaI=m

asb-90+. The same argument works to prove (2.80). Since in this case m < 1, it is enough to apply only inequalities (2.74) and (2.62).

Theorem 7 Let 11 C R" be an open set, I E N, 1 < p < oo, 0 < b <

aEK.

and

1) If IaI < 1, then Vf E wlp(SZ) Il (Da(Eaf) - Dwf)d"1_1IIL,(n) -< c16b1-Ia1llf Ilwp(sl) ,

(2.81)

where cls > 0 does not depend on f, 6, 11 and p. 2) If Ial > 1, then V f E w,(1l) II (Do(Eaf

))p101-'IIL,(n) <- c166' -IaI ilf Ilwp(n) ,

(2.82)

where c16 > 0 does not depend on f , 6, Sa and p.

3) There exists an open set (1 such that for any e > 0 inequality (2.82) with 0101-1-E replacing OI01-1 does not hold.

2.6. THE BEST POSSIBLE APPROXIMATION

75

Idea of the proof. Inequality (2.81) follows from equality (2.73) and the inequalities (2.74) and (2.62) with D. *f replacing f and I - IaI replacing 1. Inequality (2.82) follows from equality (2.72), inequality (2.75) and the inequality II(Ea(Dwf))Plal-'IIL,(n)

(2.83)

5 0170'-1°l1IfIJt4(n)

for IaI > 1, where c17 > 0 does not depend on f, b, SZ and p. In order to prove (2.83) apply the proof of inequality (2.63). The third statement will be considered in the proof of the Theorem 8 below. 0 Proof. It is enough to prove (2.83). Applying Lemma 4 of Chapter 1 and the inequalities (2.63), (2.58), (2.49) and (2.59) we establish that 00 II(E6(Dwf))P101-1IILP(n)

=

Pl°1-11kDaA62-Iklf

II

ll

k=-oo W

LP(n)

P

< 21-=B2°"-1

B2°I-tc6at-IaI

2

k=-oo

Iif IIPw ((eup P

P k

)6t-Ikl )

< 2B2°I-'Ca'-I°IIIfIIw4(n) _ C176`-°1I1J Iw,(n)

(For details see the proof of Lemma 14.) O

Theorem 8 I. Let SZ C R' be an open set, l E N and 1 < p < oo. Then df E Wp(SZ) functions cp, E C0(S2) f1 Wp(I), s E N, exist, which depend linearly on f and satisfy the following properties:

1) for1
a.e. on 0,

IaI < 1,

ass-+00,

2) fort
in Wp (f ), m = 0, ...,1,

(2.84)

in W.(0), m = 0,..., 1 - 1,

(2.85)

ass -aoo, 3) for p = 00

w, -+ f

CHAPTER 2. APPROXIMATION BY COO-FUNCTIONS

76

as s -+ 00 (if f E &(S1), then relation (2.84) also holds),

4) for1
IaI < 1,

(2.86)

IaI > 1,

(2.87)

ass-100,

5) for1
where c°,, are independent of f, Sl and p.

II. There exists an open set SI C R°, for which, given e > 0 and m > 1, a function f E Wp(SI) exists such that , whatever are the functions gyp, E COO (11) fl

Ytip(SI), s E N, satisfying property 4), for some V E IN satisfying IaI = m II

'°l -'-EII L,(n) = 00.

(2.88)

Idea of the proof. The first part of Theorem 8 is an obvious corollary of Theorems 6 and 7: it is enough to take cp, = Ei f The second part will be proved in Remark 14 of Chapter 5.

Remark 23 The second part of Theorem 4 is about the sharpness of condition (2.87). We note that since in (2.87) p(y)IOH --10 as y approaches the boundary BSI, the derivatives D°v,(y) where IaI > I can tend to infinity as y approaches BSI. By the second part of Theorem 8 for some SI C R" and f E WW (Q) for any appropriate choice of gyp, some of the derivatives D°cp,(y) where IaI = m > l do tend to infinity as y approaches a certain point x E on. Indeed, for bounded 11 from (2.88) it follows that for some Va E NB satisfying IaI = m, for some x E BSI and for some yk E Sl such that yk --- x as k -+ oo lim

(D°Ws)(yk)P(yk)1Q1-'_, = 00,

(2.89)

i.e., (D°w,)(yk) tends to infinity faster, than d-101-(yk). (We note that the higher order of a derivative is the faster is its growth to infinity.)

Remark 24 This reveals validity of the following general fact: if one wants to have "good" approximation by COO-functions, in the sense that the boundary values are preserved, then there must be some "penalty" for this higher quality. This "penalty" is the growth of the derivatives of higher order of the approximating functions when approaching the boundary. The "minimal penalty" is given by inequality (2.87).

2.6. THE BEST POSSIBLE APPROXIMATION

77

Remark 25 By Theorems 6 and 7 the functions cp, = E1 f satisfy the statements of the first part of Theorem 8. Thus, by the statement of the second part of this theorem the mollifier E6 is the best possible approximation operator, preserving boundary values, in the sense that the derivatives of higher orders of E6f have the minimal possible growth on approaching 852. Now we formulate the following corollary of Theorem 8 for open sets with sufficiently smooth boundary, in which the preservation of boundary values takes a more explicit form.

Theorem 9 Let I E N,1 < p < oo and let 12 C R" be an open set with a C'-boundary (see definition in Section 4.3). I. For each f E Wp(12) functions W. E C°°(12), s E N, exist, which depend linearly on f and are such that 1) gyp, - f in Wp(12) as s -> oo,

"

2) D°welen = Dwf I. 3)

IID°w,d°I-`IIL,(n) <

lal < 1- 1, cc,

j al > 1.

II. Given e > 0 and m > 1, a function f E WP(12) exists such that, whatever are the functions W. E C°°(12), s E N, satisfying 1) and 2), for some Va E I' satisfying Ial = m IIL,(n) = cc.

(2.90)

Idea of the proof. As in the proof of Corollary 3, by Lemma 13 of Chapter 5, propety 2) follows from relation (2.86). 0 The most direct application of Theorem 7, for the case in which p = oo, is a construction of the so-called regularized distance. We note that for an open set S2 C R", SZ # R1, the ordinary distance p(x) = dist (x, 812), x E 12, satisfies Lipschitz condition with constant equal to 1:

I p(x) - p(y)I 5 Ix - yI,

x, y E S2.

(2.91)

(This is a consequence of the triangle inequality.) Hence, by Lemma 8 of Chapter 1 (2.92) p E wolo(12), Ivpl < 1 a.e. on 12. The simplest examples show (for instance, p(x) = 1- Ixi) for 12 = (-1, 1) C R) that in general the function p does not possess any higher degree of smoothness than follows from (2.91) and (2.92).

CHAPTER 2. APPROXIMATION BY C°°-FUNCTIONS

78

Theorem 10 Let Sl C R" be an open set, 11 36 R. Then Vl E (0,1) a function ea E C°°(11) (a regularized distance) exists, which is such that (1 - 6)e(x) < ea(x) < e(x),

x E 0,

(2.93)

I Pa(x) - e6(y) 15 Ix - yI,

x, y E Sl,

(2.94)

on fl

I oea(x)I < 1

(2.95)

and We E N satisfying IaI > 2 and Vx E Q I (D°ea)(x)I 5

cool-lalg(x)1-IaI

,

(2.96)

where cc, depends only on a.

Idea of the proof. In order to construct the regularized distance it is natural to regularize, i.e., to mollify, the ordinary distance. Of course, one needs to apply mollifiers with variable step. Set ea = aE"g and choose appropriate a, b > 0. Here EM is a mollifier defined by (2.50) where l = 1 and the kernel of mollification w is nonnegative.

Proof. First let Da = E6e. Since p E w;o(Q), from (2.81) and footnote 4 on the page 12 it follows that sup Ioa(x) - e(x)I e(x)-1 < c156 'En

or bxESl (1 - c156)0(x) 5 Do(x) 5 (1 + ctsb)e(x),

where c15 > 0 depends only on n. Moreover, from (2.82) it follows that Va E N satisfying a 96 0 sup

IDaoa(x)le(x)IaI-1 <

c16b1-IaI

Zen

or Vx E Cl

IDao6(x)I 5

c1861-Iale(x)'-IaI

where c16 > 0 depends only on n and a. Furthermore, by definition of Ea and by (2.11) or (2.14)

Aa(x) -,Na(y) = E (0k(x)(Aaz-i,ie)(x) -Ok(y)(Aaa-i+ie)(y)) k--o°

2.6. THE BEST POSSIBLE APPROXIMATION

79

00

k(x)((A62-Iklp)(X) - (A62-IklQ)(0) k=-oo 00

+ E ('ibk(x) - Vk(y))((A62-IkIQ)(Y) - 9(y)) k=-0o

Hence, V)k(x)I(A62-Iklp)(x) - (A62-I410)(y)I

Io6(x) - o6(Y)I G k=-oo

+ E I)k(x) -'k(y)I f I A(y kES(x,y)

62-Iklz)

- p(y)I w(z) dz.

B(0,1)

Here by (2.53) S(x, y) = {s(x) -1, s(x), s(x) + 1, s(y) -1, s(y), s(y) + 11. From (2.12) it follows that n

I'+hk(x) -'+Gk(y)I = E(x9 - y3) f 8xk (x + t(y - x)) dt G c182kIx - yI , =1

0

where c18 = ( E c2)1/2 with ca from (2.12) depends only on n. Now, applying (1.13), (2.11)1-or1=1(2.14), and (2.91) we have

IO6(x) - O6(y)1 :5 Ix - yI

1 + C38 E 2k(62-IkI) kES(x,y)

f

Izi w(z) dz

B(0,1)

G (1 + 6c186)Ix - yI.

Finally, it is enough to set p6 = aEb6p, where, for instance, a = (1 +

z)-' and

b = min{cig , (6c18)0 Remark 26 The regularized distance can be applied to the construction of linear mollifiers with variable step. It is quite natural to replace the constant step 6 in the definition of the mollifiers A6 by the variable step bp(x), i.e., to consider the mollifiers (Hof)(x) = (A6Q(x)f)(x) =

f B(0,1)

f (x - 6p(x)z) w(z) dz

CHAPTER 2. APPROXIMATION BY C°°-FUNCTIONS

80

for 0 < b < 1. (In this case B(x, bp(x)) C 11 for each x E ) and, therefore, the function f is defined at the point x - bp(x).) If p E CO°(11), it can be proved that H6 f E C°°(S1) for f E L Oc(S1) and that Ha f -+ f a.e. on fl. This is so, for instance, for 0 = R" \R'", 1 < m < n, in which case p(x) = (xm+I+...+xn)1/2 However, as it was pointed out above "usually" gEC°°(SZ). This drawback can be removed by replacing the ordinary distance p by the regularized distance p' = p4 with some fixed 0 < So < 1 (say, 6o = 1). We set

(H6f)(x) _ (A6 =)f)(x) = J f (x - be(x)z) w(z) dz. B(0,1)

Then Vf E L'°`(Sl) we have H6 f E C°°(Q) and lib f -p f a.e. on 11. As for results related to the properties of the derivatives D°H6 f , in this case estimate (2.96) is essential. Some statements of Theorems 8-9 can be proved for the operator H6 as well. The main difficulty, which arises on this way is the necessity to work with the superposition f (x - 5p60(x)z). For this reason the mollifiers E6 with piecewise constant step are more convenient, because in their construction superpositions are replaced by locally finite sums of products. Another advantage of the mollifiers with piecewise constant step is that it is possible to choose steps depending on f . This is sometimes is convenient inspite

of the fact that the mollifiers become nonlinear. (See the proofs of Theorems 1- 4 of this chapter and Theorems 5 - 7 of Chapter 5.)

Example 4 For each open set 0 C R" a function f E C°O(R") exists such that it is positive on S1 and equal to 0 on R" \ 0. The function f can be constructed

in the following way: f (x) = exp(-ee(=)) with some fixed b E (0,1). The property (D° f)(x) = lim (D° f)(y) = 0 for x E 80 follows from (2.96). y-4z,yEft

This function f possesses, in addition, the following property, which sometimes

is of importance: Vy > 1 and da E N there exists c19 = cis(ry, a) > 0 such

that dxER" I (D°f)(x)11 < clef (x) This also follows from (2.96).

Another application of a regularized distance for extensions will be given in Remark 17 of Chapter 6.

Chapter 3 Sobolev's integral representation The one-dimensional case

3.1

Let -oo
Jwdx=1

w E L1(a,b),

(3.1)

a

and suppose that the function f is absolutely continuous on [a, b]. Then the derivative f exits almost everywhere on [a, b], f E L1 (a, b) and Vx, y E [a, b] we

have f (x) = f (y) + f f'(u)du. Multiplying this equality by w(y) and integrating

with respect to y from a to b we get

f b

f (x)

f (y)w(y) dy+

fbx

(J f'(u) du)w(y) dy

a

a

y

Interchanging the order of integration we obtain x

x

b

z

b

yv

f (f f(u) du)w(y) dy = f (f f'(u) du)w(y) dy - f ( f f'(u) du)w(y) dy a

a

V

z

x

V

u

b

z

b

b

-

= f (f w(y) dy) f'(u) du f ( f w(y) dy)f'(u) du = f A(x, y)f'(y) dy, a

a

s

u

a

CHAPTER 3. SOBOLEV'S INTEGRAL REPRESENTATION

82

where V

fw(u)du, a
a

- fw(u)du, a<x
Hence Vx E (a, b) b

6

f (x) = f f (y) w(y) dy + f A(x, y)f'(y) dy. a

(3.3)

a

This formula may be regarded as the simplest case of Sobolev's integral representation. We note that A is bounded: Vx, y E [a, b)

(3.4)

jA(x, y)j < II)IIL.(a,b)

and if, in addition to (3.1) w > 0, then 1 IA(x, y)j< A(b, b) = 1.

dx, y E [a, b]

(3.5)

Let us consider two limiting cases of (3.3). The first one corresponds to w = const, hence, Vx E (a, b) we have w(x) = (b - a)-1. Then Vx E [a, b]

f

x

6

f (x) = b

a

a

b

f (y) dy + f b- a f'(y) dy - f b- a f'(y) dy. a

(3.6)

x

To obtain another limiting case we take w = +X(6_,6)), where X(.,p) denotes the characteristic function of an interval (a, p), m E N and m > 2(b - a)-1. Letting mn -> oo we find: dx E [a, b]

f b

f(x) =

f(a)

f(b)

2

+1

sgn(x -

y)f'(y) dy.

(3.7)

0

Of course both of formulas (3.6) and (3.7) can be deduced directly by integration by parts or the Newton-Leibnitz formula. 1 Ifs is symmetric with respect to the point $4t, then Vy E [a, b] we have IA(

°-, y)j < 1.

3.1. THE ONE-DIMENSIONAL CASE

83

Obviously, from (3.6) it follows that b

If(x)I

b

<_bla f IfIdy+ f If'IdY a

a

for all x E [a, b]. 2

If f E (Wi)1-(a, b), then f is equivalent to a function, which is locally absolutely continuous on (a, b) (its ordinary derivative, which exists almost everywhere on (a, b), is a weak derivative f,', off - see Section 1.2). Consequently, (3.3), (3.6) and (3.8) hold for almost every x E (a, b) if f' is replaced by f,,.

Let now -oo < a < b < oo, xo E (a, b), I E N and suppose that the derivative f(1-1) exists and is locally absolutely continuous on (a, b). Then the derivative f (1) exists almost everywhere on (a, b), f (1) E L", (a, b) and by Taylor's formula with the remainder written in an integral form t/x, x0 E (a, b) 1-1

f (x) = E

f ikk!

(x - x0)k +

(1 1

k_0

_

f lkk x0

1)!

f (x - u)'-1 f i'l(u) du io

(x - x°)k + ((1- 1)!!

k=0

J(i - t)1-1 f <
Theorem 1 Let 1 E N, -oo < a < a <,3< b < oo and w E LI(R), suppw C [a,Q],

f wdx = 1.

(3.11)

R

Moreover, suppose that the derivative f (1-1) exists and is locally absolutely continuous on (a, b). Y By the limiting procedure inequality (3.8) can be extended to functions f, which are of bounded variation on [a, b]: Vx E [a, b] 6

If(x)I <-

b

l

f Ifl dy+ a&f a a

One can easily prove it directly: it is enough to integrate the inequality I f (x)I < I f (y)I + If (z) - f (y)I <- If (y)I + ar f with respect to y from a to b.

CHAPTER 3. SOBOLEV'S INTEGRAL REPRESENTATION

84

Then Vx E (a, b)

f fik'(y)(x -

f(x) X=O

f b

b

y)kw(y) dy + (j

11)1

(x - y)'-'A(x, y)fi`i(y) dy

a

a

(3.12) 0

bs

11),

f f(k'(y)(x - y)kw(y) dy + (l

t

k=0

(x - y)'-'A(x, y)f(')(y) dy, ( 3.13)

for xE (a, 3); a= = as

where aZ = x, bs = a for xE (a, a]; as = a, b; b= = x for x E [Q, b).

Idea of the proof. Multiply (3.10) with xo = y by w(y), integrate with respect to y from a to b and interchange the order of integration (as above). Proof. The integrated remainder in (3.10) takes the form in (3.12) after interchanging the order of integration: b

f (f (x - u)'-'f(')(u) du)w(y) dy = f w(y) (f (x - u)'-' f I>(u) du) dy a

a

y

y

b

f

(

- u)' f (') (u) du) dy =

s

-

i

(x -

u)'-' ( f

a b

b

b

- u)(f w(y) dy) f (u) du

w(y) dy) f (') (u) du =

f (x -

y)1-1A(x,

y)f (')(y) dy

u a s Finally, since supp w C [a, p], it follows that A(x, y) = 0 if y E (a, as)U(b=, b)

and, hence, (3.13) holds.

Remark 1 If in Theorem 1 a > -oo and f (1-1) exists on [a, b) and is absolutely continuous on [a, 61) for each b1 E (a, b), then equality (3.12) - (3.13) holds for x = a and a = a as well. To verify this one needs to pass to the limit as x -4 a+ and a -+ a+, noticing that in this case f() E Li (a, bi) for each bi E (a, b). The analogous statement holds for the right endpoint of the interval (a, b). If, in particular, -oo < a < b < oo, f (1-1) exists and is absolutely continuous on [a, b], then equality (3.12) - (3.13) holds Vx E [a, b] and for any interval (a,fl) C (a, b).

3.1. THE ONE-DIMENSIONAL CASE

85

Remark 2 Suppose that -oo < a < b < oo, P-') exists on [a, b] and is absolutely continuous on [a, b]. Then the right-hand side of (3.12) is actually defined for any x E R, if to assume that A(x, y) is defined by (3.2) for any x E R and for any y E [a, b]. Since in this case for any x < a and for any b

f w(u) du, the right-hand side of (3.12) for

y E [a, b] we have A(x, y)

a

x < a is the polynomial pa of order less than or equal to l - 1 such that = f(') (a), k = 0, 1, ..., I - 1. Respectively, for x > b the right-hand side is the polynomial pb of order less than or equal to I - 1, which is such that pbk) (b) = f (L) (b), k = 0,1, ..., 1 - 1. Thus the function pakT(a)

1_1

F(x) _

b

b

k1 f flki(y)(x - y)kw(y) dy + (l 11)1 f (x - y)'-'A(x, y)fi'i(y) dy a

is an extension of the function f with preservation of differential properties, since F<'-1) is locally absolutely continuous on R. See also Section 6.1, where the one-dimensional extensions are studied in more detail. Corollary 1 Suppose that l > 1, condition (3.11) is replaced by wE

supp w c [a,#],

C(1-2) (R),

Jwdx = 1

(3.14)

R

and the derivative w(1-2) is absolutely continuous on [a, b]. Then for the same f as in Theorem 1 Vx E (a, b) p

f(x)= f( a

; 1-1

(-1) k[(x-y)kw(y)](")f(y)dy k,

b.

+(l 11)1 J(x -

y)1-1A(x,

y)f(1)(y) dy

(3.15)

as

Idea of the proof. Integrate by parts. 0 From (3.14) it follows, in particular, that w(a) = ... = w(t-2)(a) = w(3) = ... = w(i-2)(Q) = 0.

(3.16)

CHAPTER 3. SOBOLEV'S INTEGRAL REPRESENTATION

86

Corollary 2 Suppose that 1, m E N, m < 1. Then for the same f and w as in Corollary 1 Vx E (a, b) 9

1m-1 _

f(-)(X)=f( E ( a

+m

k!

((x -

y)kw(y)1(k+1fl))

/

k=0

f (y) dy

bs

f (x - y)'-m-lA(x, y)f(L)(y) dy.

+(1 - m - 1)! ,

(3.17)

a,

Idea of the proof. Apply (3.15), with I - m replacing 1, to f (m) and integrate by parts in the first summand taking into account (3.16). Remark 3 The first summand in (3.15) may be written in the following form:

a.(x-y)°wl°i(y))f(y)dy' a-= LS_! ) a

k=s

s-0

k

(3.18)

It is enough to apply Leibnitz' formula and change the order of summation in order to see this. By the similar argument the first summand of (3.17) may be written in the following form: 9 1_1

f (E Qs,m(x - y) mw(s)(y))f (y) dy, a s-m

(3.19)

where vs,m

= (-1)' (s - m)!

1

s+k

°-1

(

).

k

(3.20)

From (3.18) and (3.19) it is clearly seen that the first summand in (3.17) is a derivative of order m of the first summand of (3.15) and thus (3.17) can be directly obtained from (3.15) by differentiation. (In order to differentiate the second summand one needs to split the integral into two parts - see the proof of Theorem 1.)

Corollary 3 Let -oo < a < b < oo, l E N, m E N0, m < 1. Moreover, suppose that the derivative j(1_1) is absolutely continuous on (a, b]. Then 11x E (a, b] b

flm)(x)

dy + f

ix-ylt-m-lIflll(y)Idy)

3.1. THE ONE-DIMENSIONAL CASE

87 B

< cl (b - a)1-m-1((fi -

f 6

a)-t Jii dy +

If ("I dy)

(3.21)

a

at

and, consequently, b

b

If(m)(x)I <_ ct((b-a)-m-' f IfI dy + f Ix-ylt-m-'If(o(y)I dy) a

a

6

6

< c1 ((b - a)-m-' f If I dy + (b - a)t-m-' f If (1)1 dy) a

and 3

f

a

0

If(-)(X)I <- C2(

(3.22)

b

IfI dy +

°

fix - yI`-m-'If(')(y)I dy) a

Q

b


(3.23)

a

where cl > 0 depends only on 1, while c2, c3 > 0 depend on 1 and, in addition,

depend on Q - a and b - a.

Idea of the proof. In (3.17) take w(x) = rµ(

), where x0 =

,r=2

and u E C0 '10(R) is a fixed nonnegative function, for which supp µ C (-1, 11 and

f µ dx = 1. In order to estimate the first summand in (3.17) apply (3.19) and R

the estimate iw(') (x)I < M r-'-' for m < s < I - 1, where M depends only on 1. To estimate the second summand in (3.17) apply (3.5). 0 3 From (3.23) it follows, by Holder's inequality, that for I < p:5 00 Iif(m)IIL,(a,b) <- M1 (IIIIIL,(a.b) + Ilft`)IIL,(a,b)),

where Ml = c3(b - a), and, after additional integration, that

II

Bx; IIL,(Q)

Mz (IIfIIL.(Q) + II8 i IIL,(Q))'

where Q C lit" is any cube, whose faces are parallel to the coordinate planes, f E &(Q) and M3 > 0 is independent of f. These inequalities were used in the proof of Lemmas 5-6 of Chapter 1.

CHAPTER 3. SOBOLEV'S INTEGRAL REPRESENTATION

88

Remark 4 We note a simple particular case of the integral representation (3.17): if w is absolutely continuous on [a, b], w(a) = w(b) = 0 and fwdx = 1, a

then for each f such that f' is absolutely continuous on [a, b], for all x E [a, b] b

b

(3.24)

f'(x) = - f w'(y)f (y) dy + f A(x, y)f"(y) dy. a

a

It follows that

f

6

If'(x)I <- IIw IIL (a,b)

b

IfI dy + IIA(x,')IIL.(a,b)

a

Choosing w in such a way that w(x)

=

is minimal we find

_ a+b

(b-a

4 (b

111,9dy.

a

- a)2 \

-

2

2

Ix

)

and, hence, b

If'(x)I S

(b

6

4a)2 Jutdy +

1-2 (minx- a,b- x)2) f If"I dy. (3.25) a

a

In particular b

If'(a)I,If'(b)I <-

6

(b 4a)2 a

a

and b

If'(a2 b)I

`- (b 4a)2

b

flfldy+if If"Idy.

a

a

From (3.25) it follows that Vx E [a, b] b

b

If'(x)I 5 4 ( (b la)2 f III dy+ f If"I dy).

(3.26)

89

3.1. THE ONE-DIMENSIONAL CASE

This is a particular case of (3.22) with the minimal possible constant c, = 4. The latter follows from setting f (y) = y - aZb. The same test-function shows that the constant multiplying fa If I dy in (3.25), (3.26) also cannot be diminished even if the constant multiplying fQ If"I dy is enlarged. We note that the constant muliplying fQ I f"I dy in (3.25) also cannot be

diminished. 4 This can be proved in the following way. For a < x < b and d > 0 consider the function 5 ga,=(y) = (x - y + b)+, y E [a, b], if a < x <2+6 and ga,x (y) = (y - x + b)+, y E (a, b], if a2 < x < b. In (3.24) take f = A2 where Aa is a mollifier, and pass to the limit as b -+ 0 +. Finally, as in the case of the integral representation (3.3), we consider a limiting case of (3.24). We write wm for w, where m E N, m > b?a, wm(x) _

m(x - a)(b - a - L)-' for a < x < a +

wm(x) _ (b -a - m)-` for

a+m <x
fI(x) =

f(b)-f(a) b-a

s

b

y-a b+f b-af (y)dy- f b - -yaf(y)dy

(3.27)

x

a

Here x E [a, b] and f is such that f' exists and is absolutely continuous on [a, b]. Again, as in the case of representations (3.6) and (3.7), (3.27) can be deduced directly.

Corollary 4 Let I E N, m E Nlo,1>2 and m<1-1. 1. If -oo < a < b < oo and the derivative f(1-0 is absolutely continuous on [a, b], then Vx E [a, b] and VC E (0, ci (b - a)s-'"'1], b

b

If lml (x) I <- c4K(e) f If I dy + e f a

if (')I dy,

(3.28)

a

where c4 > 0 depends only on t and

K(e) = e-

.

(3.29)

b

4 In contrast to the constant multiplying f If I dy it can be diminished if to enlarge approa b

priately the constant multiplying fIfI dy - see Corollary 4.

a

5Here and in the sequel a = a fora>0anda+=0fora<0.

CHAPTER 3. SOBOLEV'S INTEGRAL REPRESENTATION

90

2. If I = [a, oo) where -oo < a < oo, I = (-oo, b) where -oo < b < oo or I = (-oo, oo) and the derivative f (1-') is absolutely continuous on each closed interval in I, then Vx E I and `de E (0, oo)

If(m)(x)I <- c5 K(e) fiii dy+e f If ()I dy, r

(3.30)

r

where c5 > 0 depends on I only.

Idea of the proof. In the first case for x E [a, b] apply (3.22) replacing [a, b] by any closed interval [a1, b1] C [a, b] containing x, whose length is equal to b,

where 0 < b < b - a, and set clot-'"-' = e. The second case follows from the first one.

There is an alternative way of proving (3.28). Given a function f, it is enough to apply (3.22) to the functions fs,s, where 0 < 6 < b -a and x E [a, b],

which are defined for y E [a,b] by fb,z(y) = f(x+b(')), change the variables putting x + 6(y a) = z and set c16I-1-1 = e.

Corollary 5 If -oo < a < b < oo, I E N and fV) is absolutely continuous on [a, b], then there exists a polynomial p1_1(x, f) of degree less than or equal to I - I such that for each m E No, m < l and tlx E [a, b]

If(m)(x) -pi(_)(x,f)I S

(b-a)1-'"-'

b

(1-m-1). JIfIdx.

(3.31)

Idea of the proof. It is enough to take Taylor's polynomial T...1 (x, f) as P1-1(x,f): 1-1

PI-1(x, f) = Ti-1(x, f) _

(k}

>k=0f Is

xo (x

- xo)k

with an arbitrary xo E [a, b], apply to f(m) Taylor's formula with I-m replacing I (with the same xo) and take into account that

(

Ti-m-i (x, f (m)) = Tm) (x, f).

It is also possible to take 1_1

Pl-1(x, f) = S,-1(x, f)

_ k-(1

1

b

kl f f(*) (y) (x - y)kw(y) dy,

(3.32)

3.1. THE ONE-DIMENSIONAL CASE

91

where w satisfies (3.11) and is nonnegative. (Notice that this is the first summand in (3.12).) One must take into account that in this case also Si_m_1(x, f(m)) = S('i)(x, f). Both of the choices lead to (3.31) (in the second case according to (3.5)).

Theorem 2 Let I E N, -oo < a < a <,0 < b < oo, w satisfy condition (3.11) and f E (W11)b0c(a, b). Then for almost every x E (a, b) 1-1

b:

R

I f fwki(y)(x - y)kw(y) dy + f (x) _ Z- k! k=0

11)! (I

J

(x - y)`-'A(x, y)fu,'(y) dy,

a.

(3.33)

where a= and b= are defined in Theorem 1.

Idea of the proof. Set a(&) = max{a+b, -11, b(b) = min{b-b, 1) for sufficiently small S > 0, write (3.13) for A7 f E C°°(a(b), b(6)), where 0 < 7 < b, and pass to the limit as y -y 0+. Proof. Since for sufficiently small 6 > 0 (a,,31 C (a(b), b(b)) and (a(b))= _ a1, (b(S))x = by for each x E (a(S), b(b)), we have Vx E (a(b), b(b)) 1-1

(A,f)(x) _ k=O

J(1&f)(k)(y)(x

- y)kw(y) dy

a

b.

e

(1- 1)!

f (x - y)'

A(x, y)(A,f)(')(y) dy.

By Lemma 5 of Chapter 1 f , ( k ) exists on (a, b) where k = 1, ..., l - 1, and by Lemma 4 of Chapter 1 (A, f )iki = on (a(S), b(S)) where k = 1, ...,1. Consequently, Vx E (a(S), b(b)) a

a

if (A,f(y)(x - y)kw(y) dy - f f(k) (y)(x - y)kw(y) dyI a

a

d

5 f I A,(fwkl)(y) a

f B

-

fwk)(y)I I (x

- y)kw(y)I dy 5 M1

fwkiI dy --> 0

a

as 7 -+ 0+, where k = 1, ...,1- 1 and M1 is independent of y and x0.

CHAPTER 3. SOBOLEV'S INTEGRAL REPRESENTATIO.\'

92

Analogously, in view of (3.4), Vx E (a(b), b(b)) e.

b.

lI (x - y)'-'A(x, y)(A,f)(`)(y) dy - J(x - y)'-'A(x, y)f.()(y) dyl a.

b(6)

<MMf IA7(fw!))-f,iThdy->0 a(6)

as ry -a 0+, where M2 is independent of ry and x.

Finally, by (1.5) A, f -+ f almost everywhere on (a(b), b(b)). Thus (3.33) is valid almost everywhere on (a(b), b(b)) and, hence, on (a, b) since U (a(6), b(6)) = (a,b). 0 6>0

Remark 5 By Theorem 2 it follows that if in Corollaries 1-2 f E (W1)'a(a, b) and in Corollaries 3-5 f E WW(a, b), then equalities (3.15), (3.17) and inequalities (3.21) - (3.23), (3.28), (3.30) and (3.31) hold almost everywhere on (a, b), the ordinary derivatives f (1) and f (m) by the weak derivatives f,(`)

if

3.2

Star-shaped sets and sets satisfying the cone condition

A domain Sl C V is called star-shaped with respect to the point y E fl if `dx E fl the closed interval [x, y] c fl. A domain fl c R" is called star-shaped with respect to a point if for some y E Sl it is star-shaped with respect to the point y. A domain fl C Rn is called star-shaped with respect to the ball 6 B C Sl if Vy E B and dx E Sl we have [x, y] C fl. A domain SZ C R" is called star-shaped with respect to a ball if for some ball B C S] it is star-shaped with

respect to the ball B. If 0 < d < diam B < diam Sl < D, we say that Cl is star-shaped with respect to a ball with the parameters d, D. We call the set V. = V:,B = U (x, y) pEB

Recall that by "ball" we always mean "open bail".

93

3.2. STAR-SHAPED SETS AND THE CONE CONDITION

a conic body with the vertex x constructed on the ball B (if x E B, then V, = B). A domain St star-shaped with respect to a ball B can be equivalently defined in the following way: Vx E Q the conic body V= C S2. Let us consider now the cone n-1

I

K=K(r,h)={xER":0<x,)1
(3.34)

i=1

We say also that an open set S2 C R" satisfies the cone condition with the parameters r > 0 and h > 0 if Vx E 0 there exists 7 a cone Kx C 0 with the point x as vertex congruent to the cone K. Moreover, an open set Sl C R" satisfies the cone condition if for some r > 0 and h > 0 it satisfies the cone condition with the parameters r and h. Example 1 The one-dimensional case is trivial. Each domain SZ = (a, b) C R is star-shaped with respect to a ball (__ interval). An open set ft = 6 (ak, bk), k=1

where s E N or s = oo and (ak, bk) fl (am, bm) = 0 for k # m, satisfies the cone condition if, and only if, inf (bk - ak) > 0.

Example 2 A star (with arbitrary number of end-points) in R2 is star-shaped with respect to its center and with respect to sufficiently small balls (= circles) centered at its center. It also satisfies the cone condition.

Example 3 A convex domain 11 C R" is star-shaped with respect to each point y E fl and each ball B C C. A domain fZ is convex if, and only if, it is star-shaped with respect to each point y E 0. Example 4 The domain ci C R2 inside the curve described by the equation lxi If + lxs l' = 1 where 0 < y < 1 (the astroid for y = 2/3) is star-shaped with respect to the origin, but it is not star-shaped with respect to any ball B c Cl. It does not also satisfy the cone condition.

Example 5 The union of domains, which are star-shaped with respect to a given ball, is star-shaped with respect to that ball. The union (even of a finite number) of domains star-shaped with respect to different balls in general is not star-shaped with respect to a ball. In contrast to it the union of a finite number of open sets satisfying the cone condition satisfies the cone condition. Moreover, the union of an arbitrary number of open sets satisfying the cone condition with the same parameters r and h satisfies the cone condition. 7"Vx E !2" can be replaced by "Vx E Il" or by "Vx E Oft" and this does not affect the definition.

CHAPTER 3. SOBOLEV'S INTEGRAL REPRESENTATION

94

Example 6 The domain 0 = {z E Rn : IxI7 < x < 1,

Jxj < 1}, where for ry > 1 is star-shaped with respect to a ball and satisfies

Y = (xl, ...,

the cone condition. For 0 < y < 1 it is not star-shaped with respect to a ball. Furthermore, it cannot be represented as a union of a finite number of domains, which are star-shaped with respect to a ball, and does not satisfy the cone condition.

Example 7 The domain 0 = {x E Rn : -1 < xn < 171-1, 171 < 1} satisfies the cone condition for each 'y > 0. It is not star-shaped with respect to a ball, but can be represented as a union of a finite number of domains, which are star-shaped with respect to a ball.

Example 8 The domain SZ = ((XI, x2) E R2 : either - 2 < x1 < 1 and - 2 < x2 < 2, or 1 < x1 < 2 and - 2 < x2 < 1} is star-shaped with respect to the ball B(0,1). For 0 < b < %F2 - 1 the domain fl D B(0,1), but it is not star-shaped with respect to the ball B(0,1). (It is star-shaped with respect to some smaller ball.)

Lemma 1 An open set S2 C R" satisfies the cone condition if, and only if, there exist s E N, cones Kk, k = 1, ._a, with the origin as vertex, which are mutually congruent and open sets SZk, k = 1, ..., s, such that ,

U

k=1

2)b'xEIk the cone' x+KkCQ. Idea of the proof. Sufficiency is clear. To prove necessity choose a finite number

of congruent cones Kk, k = 1, ..., s, with the origin as a vertex, whose openings are sufficiently small and which cover a neighbourhood of the origin, and consider the sets of all x E 1 for which x + Kt C Q. 0 Proof. Necessity. Let fl satisfy the cone condition with the parameters r, h > 0. We consider the cone K(rl, h1) defined by (3.34), where h1 < h and r1 < r is such that the opening of the cone K(rl, h1) is half that of the cone K(r, h). Furthermore, we choose the cones Kk, k = 1, ..., s, with the origin as a vertex, which are congruent to K(rl, h1) and are such that B(0, h1) C 6 Kk. Hence, k=1

Vx E Sl the cone Ks of the cone condition contains x + Kk for some k. Denote by Gk the set of all x E St, for which K. contains x + Kk. Finally, there exists b= > 0 such that Vy E B(x, 6) we have y + Kk C fl. Consequently, the open 8 Here the sign + denotes a vector sum. The cone z + Kk is a translation of the cone Kk and its vertex is x.

3.2. STAR-SHAPED SETS AND THE CONE CONDITION

95

sets S2k = IJ B(x, a=), k = 1, ..., s, satisfy conditions 1) and 2). zEG4

Let a domain SZ C R" be star-shaped with respect to the point x0. For t; E S"'', where S"-' is the unit sphere in R", set w(t;) = sup{p > 0 : xo+pt; E S2}. Then

52={XER":x=xo+pt where DES"-'. 0
Moreover, set R1 =

to the angle -y between the vectors 0 t and O q, where 0 is the origin.

Lemma 2 Let a bounded domain SZ C W be star-shaped with respect to the point xo E Q. Then it is star-shaped with respect to a ball centered at .ro if. and only if, the function V satisfies the Lipschitz condition on S" -'. i.e.. for some

M>0and9Vt,rlES"-1 I

v(77)1:5 M d(C, n).

Idea of the proof. Sufficiency. Consider the conic surface C(C) with the point f = xo + v(C)t; as vertex, which is tangent to the ball B(xo, r). Suppose that

0< p < oo} intersects C(t;) at two points a and e. Denote d = xo + Since f, d E 852 it follows that f V bd and d V V'. Therefore, d E [a, e]. Necessity. For fixed t; E S"'' consider two closed rotational surfaces L+ and L_ defined by the equations p = F* (q), where F* (q) = p(t;) ± M d(t;, rl). Then the boundary 852 lies between L+ and L_. Let the (n - 2)-dimensional sphere

E be an intersection of L_ and the surface of the ball B(xo, R1). Consider two conic surfaces, which both pass through E and whose verticies are x0, f respectively. Let 6 denote the angle at the vertex of the conic surface D.,, (We assume that M > 0 and p(t;) > R1, since the cases, in then 6 = 'P(()-R. M set r(l;) = R1. which M = 0 or R1, are trivial.) If 6 > do = arccos 9 Since

If - ql <- d(f, n) = 2 s in z kk - nl <s

2

IE - nI

this condition is equivalent to: for some M1 > 0 and V(, q E S" WE) - W(17)1:5 M1 It -

I.

96

CHAPTER 3. SOBOLEV'S INTEGRAL REPRESENTATION

Otherwise, let r(t;) be such that the ball B(xo, r(f )) is tangent to Dj. Then the conic body with the point f as vertex constucted on the ball B(xo, r(t;)) lies in Q. 0 Proof. Sufficiency. Denote c = x0 + 4Q,7. Since d E [a, c] or d E [c, eJ we have iv(f) - w(77)1:5 max {ICI, Ic-tI}.

Since I a-tI < II 'o we establish that Iw(f) - V(77)1:5 Ic I = __(a > - w(f) 2w(()sin2 sin B-7

'0(07C-0-2) ein(9-ry

- w(f)

-ry

sin

d(f,+])Con

Vt;, 7 E 51-1 such that -y < Q I

'(q) I = R2 '

(a-7)

d(f, n).

Hence, given e > 0, there exists (e) > 0 such that V , r! E S"-1: -y < b(e) we have Iw(f) - W(17)1:5 (R2 cot,0+e)d(t;,7)

= (RZ

Rz

- r2 +e)d(la,r7).

Now let t; and' be arbitrary points in S"-1, f 96 ri. We choose on the circle, centered at x0 and passing through f and 17, the points to = f 1 -< ... -< f,,,-j -< t;,,, = 17 such that all the angles between the vectors O6_1 and 01, i = 1, m, are less than 6(e). Then m

Iw(f) - w('7)I <- F,

W&-1) - wWI

i=1

1e One can see that In I = I& I(oot (9 +'y) + tan J) while IccI = IbI(cot (,6 -1') - tan J), where 10. The inequality (aa$I < Icc-tj follows from the inequality

cot(p+-y)+tan2
2 tan 2 < 2

g,

. This inequality is equvalent to

sin 27

2 sin try

sin (p - ') sin (p +) ry

cos 2,y - c os 2,0'

to cos 2ry - cos 2,6 < 4 cost I cosy and to - cos 20 < 2 coe ti + 1, which is obvious since 0< y< 12 .

3.2. STAR-SHAPED SETS AND THE CONE CONDITION

<(Rr-2

RZ-r2+e)(RZ

97

RZ-r2+E)d(,rl)

Passing to the limit as e -* 0+ we find that the Lipshitz condition is satisfied with

Rz-r2.

M=R2

r

(3.35)

Necessity. If 0 < 6 < 6g. then

= p(t:)R1 sin6 (cp({) - Ri cos 6)2 + (RI sin 6)2 cp(C)R1 sin 6

=

cp({)R1 sin6

(tp(0 - RI)2 + >

w(t)R1

2

M2 + yo(C)Ri

62M2 + 4cp( )R1 sine

R2

2

a

M + Rl

sin26

=

r0

One can verify that for any point g E L_ \ B(xo, RI), g 5A f , the interval (g, f ) lies "" in 52. Therefore, the conic body Vf with the point f as vertex constructed on the ball B(xo, ro) lies in St. Hence, SZ is star-shaped with respect to the ball

B(xo,ro) Remark 6 The constant M given by (3.35) is the minimal possible, because, for example, for any conic body V., defined by (3.34) we have jw(f) tt-

sup d{,,7ES^-`,f#9

d(S, 17)

R2

r

- r2.

If a domain f C R", which is star-shaped with respect to the ball B(xo, r), is unbounded, then set S' = {t E Sn-1 : ip(t) < oo}. u Consider the curve L obtained by intersecting L_ \ B(xo, RI) by the two-dimensional plane passing through g and the ray going from zo through f. Let this ray be the axis Ox of a Cartesian system of coordinates in this plane. Suppose that y = '(x) is a Cartesian equation of the curve L. We recall that its polar equation is g = W(f) - MIyI and note that Iyj < J. The part of the curve l_, for which 0 < y < 6, is convex and the part of L, for which -6 < y < 0, is concave since, for example, for 0 < y < 6 2M2 + (IG(f) - My)2 ((W(f)

- My)sinjp+Mcosrp)3

Hence, for any g E t_,g 0 f, the interval (g, f) lies in fl.

< 0.

CHAPTER 3. SOBOLEV'S INTEGRAL REPRESENTATION

98

Corollary 6 Let an unbounded domain 11 C R" be star-shaped with respect to the ball B(xo, r). Then S' is an open set (in Sn-1) and the function p satisfies the Lipschitz condition locally 12 on S'.

Idea of the proof. Note that if cp(l;) = oo fort E S"`1, then the whole semiinfinite cylinder, whose axis is the ray R(t) and whose bottom is the hyperball {x E B(xo, r) : X-ol 1 O}, is contained in Q. Deduce from this that S' is open and apply Lemma 1.

Example 9 For the domain S2 C R2, that is obtained from the unit circle B(0,1) by throwing out the segment {x1 = 0, 1 < x2 < 1) and which is starshaped with respect to origin, but is not star-shaped with respect to a ball, the function V is not even continuous.

Example 10 For the domain 11 = {xl,x2) E R2 : Ixlx2I < 1), which is starshaped with respect to the origin, but is not star-shaped with respect to a ball, the function c' is locally Lipschitz on the set S' = S' \ ((0,±1), (±l,0)}. Lemma 3 If a bounded domain S2 C R" is star-shaped with respect to a ball, then it satisfies the cone condition. Idea of the proof. Let 11 be star-shaped with respect to the ball B(xo, r). and r. (It follows Then S2 satisfies the cone condition with the parameters because the cone KZ with the point x as vertex and with axis that of the conic body L, which is congruent to the cone K (Rs , r), is contained in Q). Now we give characterization of the open sets, which satisfy the cone condition with the help of bounded domains star-shaped with respect to a ball.

Lemma 4 1. A bounded open set 0 C Rn satisfies the cone condition if, and only if, there exist s E N and bounded domains 11k, which are star-shaped with a

respect to the balls Bk C Bk C Slk, k = I,-, s, such that S2 = U Stk. k=1

2. An unbounded open set S2 C Rn satisfies the cone condition if, and only if, there exist bounded domains S2k, k E N, which are star-shaped with respect to the balls Bk C Rk- C SZk, k E N, and are such that

0

1) S2 = U f2k, k=I

11 I.e., Vt E S' there exist M(f) > 0 and v(f) > 0 such that Vq E S', for which If -ql < v(t) we have IW(f) -W(i)I 5 M(E)d(f,+)

3.2. STAR-SHAPED SETS AND THE CONE CONDITION

99

2) 0 < inf diam Bk < sup diam Ilk < 00 kEN

kEN

and 3) the multiplicity of the covering x({S2k}k 1) is finite. Idea of the proof. Sufficiency. By Lemma 3 ) satisfies the cone condition with the parameters c 26q' and c6, where 13

c6 = inf diam Bk,

C7 = sup diam S2k,

k=1,s

k=TTs

s E N for bounded 12 and s = oo for unbounded Q.

Consider for x E 52, in addition to the cone Kx, the conic body K. with the point x as a vertex, which is constructed on the ball B(y(x), r1) inscribed into the cone Kx (here r1 = rh/(r + r2 + h2)) and Necessijr.

the conic body K= with the point z(x) = x + exas a vertex, where ex = 2 min{rl,dist (x, 8Q)}, which is constructed on the same ball B(y(x), r1).

Then S2 = U K. Choose xk E R', k E N, such that R" = U B(xk, 11) and xEfl

kEN

the multiplicity of the covering 14 x({B(xk,

Wk = IlnB (1k,2

,

Z)}kEN)

Gk =

<- 2". Set

U

Ks .

lEfl: y(x)Ewk 00

Then S2 = U Gk. Renumber those of Gk which are nonempty and denote k=1

them by 521, 522i .... 0

Proof. Necessity. Suppose that Gk # 0 and t; E Gk, then there exists x E Q

such that y(x) E wk and t; E K. Let us consider the conic body Kt with the point { as a vertex, which is constructed on the ball B (xk, ). Since

z Hence, y(x) E B (xk, z) we have B(y(x), rl) B (xk, z) and KK C KZ C 11. the set Gk is star-shaped with respect to the ball B (xk, 2 ). Furthermore, IC - xkl < Iz(x) - xki = Iz(x) -.TI + Ix - y(x)I +- Iy(x) - xkl < h because Ix - y(x) I = h - rl, therefore Gk C B(xk, h) and diam Gk < 2h.

13 Here and in the sequel k = f, where s E N means k E {1,...,s} and k = 1 oo means

kEN. 14 This is possible because the minimal multiplicity of the covering of R" by balls of the same radius does not exceed 24.

CHAPTER 3. SOBOLEV'S INTEGRAL REPRESENTATION

100

Let us consider those of the sets Gk which are nonempty. If 0 is bounded,

then there is a finite number of these sets - denote this number by s. If SZ is unbounded, then there is a countable number of these sets (s = oo). Renumbering them and denoting by Sgt, Q2, ..., we have S1 = U Gk = U Stk. k=1

k=1

Thus, for S2k, k = 1, s, the properties 1) and 2) are satisfied. Finally, 1s 1 X({S2k}k=1) < x({B(xk, h)}k=1) < 2" (1 +

n

Ti

.O

n < Remark 7 In the above proof c6 = r1 and c7:5 2h. Furthermore, dim lam Bk 4 (1 + r) , k = T ,-s. It is also not difficult to verify that x ({Slk}k=1) < cs 6" (1 + h)n.

3.3

Multidimensional Taylor's formula

Theorem 3 Let I C W be a domain star-shaped with respect to the point xo E 11, l E hl and f E C' (0). Then Vx E 12

(D°f)(xo) ( x - xo)°

f (x) = IoI<1 1

(x - -L o ) Q

+l

Jo

a! 101=1

C!

(1 - t)'" (D" f)(xo + t(x - xo)) dt

(3.36)

(here in addition to multi-notation used earlier we mean that xo + t(x - xo) _ (x01 + t(xi - x01), ..., xOn + t(xn - xOn)))15 We use the inequality

t

n

(1+ QI

where 0 < Q1 < Q2 < oo. Since for x E R" the number x(x) of the balls B(zk, o) 3 x is equal to the number of the points zk E B(x, Qs), by inequality (2.60) we have

x(x) mesa B(O, Q)) = E mess B(zk. QI) k: B(zh, e)3z

< x((B(zk, Qi))k.1) mesa

U k: B(a,.p2)3z

and the desired inequality follows.

B(.., ,p,) 5 x({B(zk, Ql)k=1) measB(x, of + Q,),

3.3. MULTIDIMENSIONAL TAYLOR'S FORMULA

101

Idea of the proof. Consider for fixed x and x0 the function V of one variable defined for 0:5 t < 1 by w(t) = f (xo+t(x-xo)) and apply the one-dimensional Taylor's formula (3.10) with the remainder in integral form: i_i sv(1) _ k=0

(k)

k(0) + (1

11)! J(i -

dt.

0

If I = 1, then (3.36) takes the form: V f E C'(S2) and Vx E S2

f (x) = f (xo) + j=1

(x, - x0t) f

(afaxe ) (xo + t(x - xo)) dt. °

The analogue of this formula for the functions f, which have all the weak derivatives (e )w of the first order on S2 cannot have the same form, even for almost every x E S2 because it contains the value f (xo) at a fixed point x0. (For, suppose that this formula is valid for some such function f. Then it will not be valid for any function g, which coincides with f on S2 excluding the point x0.) For this reason we write the above formula in a different way. Suppose, that S2 C R" is an arbitrary open set and h E R", then it follows that V f E C' (S2) and Vx E S2IhI

f(x+h)= f(x)+Eh3 f° 1(af )(x+th)dt. i=1 ax' Lemma 5 Let S2 C R;" be an open set and h E R". If f E L10c(11) and for each j = 1, ..., n the weak derivative (LL )w exists on S2, then for almost every x E 1111,1

f(x+h)=f(x)+1:hj f 1(8x)w(x+th)dt i=1 Ihl

=f(x)+Ihl f 0

1

(f)w(x+ST)r=f(x)+ f 0

) w and V w f are the weak derivative in the direction of the weak gradient respectively.

where t; = Th and (

CHAPTER 3. SOBOLEV'S INTEGRAL REPRESENTATION

102

Idea of the proof. Apply mollification and pass to the limit. 0 Proof. Since A6f E C°°(54) for each 0 < d < y, by Lemma 4 of Section 1.2 we have: `dx E Qlhl+, n

f

h) = (A6f)(x) + E j=1

'(A,(f)w)(x+th)dt. oxi

We claim that

(f)

f'(A6(.-i_) ) (x + th) dt -4 pf 8x j w

OX) w

(x + th) dt

in L,°C(52,) as 5 -* 0 + . Indeed, for each compact K C St, by Minkowski's inequality

f ,(4

-lw)(x+th)dt-

l

f'(8x

f'11 (A6(8xj)w)(x+th) -

(aLxjf

)w(x+th)dtIIL'(K)

L,(K)dt

< IA6(C72)w-( aLx!f by (1.9)as 5-a0+. Consequently, there exists a sequence 6k > 0 such that bk -+ 0+ as k -4 oo and

L

(A6k(a )w)(x+th)dt- L_)x+th)dt

almost everywhere on Sty. Moreover, by (1.5) (A,, f)(x + h) -+ f (x + h) and (A6k f)(x) -> f (x) almost everywhere on Sty. Thus, passing to the limit as k - oo, we obtain the desired equality for almost every x E Sty and, since y > 0 was arbitrary, for almost every x E St . 11

We note that one can prove similarly that if f E L10C(1) and `da E N satisfying jai = I there exist weak derivatives D° Of on St, then for almost every x E StjhI i

h°

(Dw°f)(x) h° + I

f (x + h) _ I°I<1

al

I

jai=1

(1 - t)1-1(DW f)(x + th) dt. 0

3.3. MULTIDIMENSIONAL TAYLOR'S FORMULA

103

Corollary 7 Let 0 C R' be an open set, 1 < p < oo, h E R" and f e wp(S2). Then

IIf(x+h) -f(x)IIL,(n,,,l) 5 II IowfIIIL,(n)lhl <_ IIfIIwy(n)Ihl.

Idea of the proof. Apply Lemma 5 and Minkowsli's inequality. Proof. By Lemma 5

IIf (x + h) - f (x)IIL,(nlhi) = I f ((owf)(x + th)

. h) dtIIL,(nlhl)

0

f

I

I

lI(owf)(x + th) . hlIL,(n,h,) dt < IhI

f

II Iowf I(x + th)Iln,h,) dt

0

0

= IhI f II Iowf I Ilnih,+th) dt < II owf I IIL,(n) IhI <_ If Ilw,(n) IhI 0

since c1h1 + th C St and Iowf I <'E K-20.1. Next consider for I E N and h E Rn the difference of order I of the function

f with step h:

(Ahf)(x) = (-l)

1-k (k') f (X + kh).

k=0

Corollary 8 Let c C Rn be an open set, I E N, 1 < p < oo, h E R" and f E Wy(S2). Then IlohfIIL,(n(ihi) 5 2'I1fIIL,(n),

IlohfllL,(nghl) <- n'-'Ihl'Ilfllwo(n)

Idea of the proof. The first inequality follows by Minkowski's inequality. To prove the second one apply Corollary 7 and take into account that Q', < n1-1 if

a E N satisfies lal = I. Proof. By induction we get IIAhfIIL,(n,lh,) =

I(a(axj,

n

`-IhI j1=1

ll

Iloh(oh'f)IILP((n(j-I)Ihl)Ihl)

\

f) / wIIL,(n(t-j)Ihl)

hI('

...r L.r ll (ox,, Of ... axj, )wllLa(n)

= IhI' E IIDwf IIL,(n) <- n'-' IhI' llf Ilwa(n) Iol=1

CHAPTER 3. SOBOLEV'S INTEGRAL REPRESENTATION

104

3.4

Sobolev's integral representation

Theorem 4 Let S2 C R" be a domain star-shaped with respect to the ball B = B(xo, r) such that B C S2,

fdx=1 ,

suppw CB,

w E LI(R"),

(3.37)

R.

I E N and f E C'(/S'2). Then for every x E S2

f (x) =

III

I0I<1

J (D°f)(Y)(x - Y)°w(Y) dy + B

J

D° f)I(y) w. (x. y) dY,

I°I=1 J(3.38)

where for x,yER",x0y, Ial (x-Y)°w(x,Y)

w°(x,Y) _ a! Ix - Y1101 and

(3.39)

00

r w(x>Y)= f w(x+eiY-xl)f-'do

(3.40)

IS-VI

(for x = y E Sl we define w°(x, x) = w(x, x) = 0).

Remark 8 The first summand in right-hand side of (3.38) is a polynomial of degree less than or equal to I - 1 while the second one (the remainder) has the form of an integral of potential type. Both summands in right-hand side of (3.38) consist of integrals containing the function f and its derivatives and does not involve the values of the function f and its derivative at particular points - thus, in contrast to Taylor's formula,

this is an integral representation of the function f via the function f and its derivatives up to the order 1. Let 1=-v1 and k E N. Consider the derivative of the function f in the direction of C of order k: ( 26 )(y) _ E (k) (D° f)(y){°. Then one may write I°I=k

(3.38) in the following way 1-1

f(x) = f (E I

B k

x k!y lk

() (y)) w(y) dy

3.4. SOBOLEV'S INTEGRAL REPRESENTATION

+

?f) (?/w(x, y)

1

(l - 1)!

f ('9V

)

Ix - yI11-1

105

dy

V.

In particular,

f(x) = f f(y)w(y)dy+ f((Vf)(y) - (x - y)) B

w(x,y) dy. Ix - yI"

v,

Remark 9 Let us denote for x i4 y by MZ,y the ray, which goes from the point x through the point y, and by Lx,y the "subray" of M=,, which goes from

the point y. As the variable p in (3.40) changes from Ix - yI to infinity, the of the function w runs along the ray L,,,,y. We note argument z = x + p that p = I z - xI and that (3.40) may be written with the help of a line integral: w(x, y) =

J

w(z)Iz - xI"-t dL.

The ray Lx,y intersects the ball B if, and only if, y E K. For this reason VxER" suppy wa(x,,y) = suppy w(x, y) C K.

(3.41)

(if w is positive on B, then there is equality in (3.41)). Furthermore, Vx E R" and Vy E K. suppe w (x

+ p y - x) C B fl Lz,y = [d1, d2]. Iy - xI

Here d2 = d2(x, y) is the length of the segment of the ray M=,y contained in K= while d1 = dl (x, y) = max{ Ix - yl, d1 }, where d1 = dt (x, y) is the length of the segment of the same ray contained in K = \ B. 16

Therefore, actually, the integral in (3.40) is equal to 0 if y ¢ K= and is an integral over the finite segment [d1, d2] if y E K. We note that d2 < D,

d2 - d1 < d,

(3.42)

where D = diam 1 and d = diam B. 16 If Ix - zoI = h and sp is the angle between the vectors xx and 2-1, then 0 < tamp < and d1 and d2 are the minimal root, the maximal respectively, of the quadratic equation d2 - 2dh cos,o + h2 - r2 =0.

106

CHAPTER 3. SOBOLEV'S INTEGRAL REPRESENTATION

Remark 10 If Sl is bounded, then IIw(x,y)IIC(R"xR") <- II"IIL (R")dz n

I <-

IIwIIL-(R")D"-1d.

Moreover, Va E 1% satisfying IaI = l (3.43)

IIw°(x,y)IIC(R"xR") <_

We have taken into account that

I (x - y)° Ix - YI+

I_

-

Ixl - Y11 Y, Ix - yI J

(Ixn - ynl \ on < 1

` Ix - yI J

and that a! > n for IaI = 1. Hence, if w is bounded, then for bounded 11 the functions w and w,, are bounded on R" x R". If fl is unbounded, then these functions are bounded on K x R" for each compact K. If w E COO(R" ), then the functions w(x, y) and w°(x, y) have continuous derivatives of all orders Vx, y E R' such that x y. 17 Remark 11 In the one-dimensional case for Sl = (a, b) and B = (x0-r,x0+r) (a, d3), where -oo < a < a < $ < b < oo, we have V. = (ax, b=), where a= and b= are defined in Theorem 1. Moreover, L1,, = (-oo, y) if x < y and LZ,y = (y, oo) if x > y. Furthermore, Y

f w(u) du,

w(x, y) = f w(z) dL =

b f w(u) du,

a < y < x, x < y:5 b,

Y

17 At the points (x, x), where z f I3 they are discontinuous. For n > 1 it follows from the fact that for each y E K. \ B lying in some ray going from the point x (for all these y the vectors have the same value, say, v = (v1, .... v")) the function w(x, y) has the same value y(x, v) = fcow(x + Qv)Q"'1 dQ. Hence, the limit of w(x,y) as y tends to z 0

along this ray is also equal to y(x,v). Respectively for the function w°(x,y) this limit is equal to 4(-v1)°1 ... The discontinuity follows from the fact that these limits depend on v. For, if the ray defined by the vector v does not intersect the ball B, then y(x, v) = 0. On the other hand, there exists v such that 7(x, v) = 0, otherwise

f w(z) dz = f f w(x + Q

R.

v)Q"-1 do) dS = 0, where S is the unit sphere in lit", which

S(000

contradicts (3.37). For n = 1 the discontinuity follows from the formulas for w and w° given in Remark 11.

3.4. SOBOLEV'S INTEGRAL REPRESENTATION

107

and 1 (x - y)'w(x, (sgn (x - y))'-' A(x, y). y) _ (l - 1)! (l - 1)! Ix - yl'

W1 (X, y) =

Thus, (3.38) takes the form (3.13).

Idea of the proof of Theorem 4. Multiply Taylor's formula (3.36) with xo = y and y E B, by w(y) and integrate with respect to y over Rn. (We assume that for y 34,0 w(y)g(y) = 0 even if g(y) is not defined at the point y.) The lefthand side of (3.36) does not change and the first summand in the right-hand side coincides with the first summand in the right-hand side of (3.38). As for the second summand it takes the form of the second summand in (3.38) after appropriate changes of variables. 0 Proof. After multiplying (3.36) with xo = y by w(y) and integrating with

respect to y over R" the second summand of the right-hand side of (3.36) takes the form 1

I E -i f w(y) (f (1 - t)'-'(D°f)(y+t.(x - y))dt)dy I°1=1

R.

0

1 J°.

li f (1 - t)'-' ( f (D°f)(y + t(x - y)) w(y) dy) dt = l

=l 1°I=1

0

.

101=1

fit"

Changing variables y + t(x - y) = z and taking into account that (x - y)° _ dy = tl' ", we establish that (1 i

J. = f (D°f)(z)(x - z)° R.

x) -tt \ /

(f w I 0

1

(1

dt)n41) dz.

Replacing _ by p, we have J

f (D°f)(z)(Ix

- zz-x

-zii ( f w(x+q'-xl)e,,_ldp)dz, Iz-aI

which by (3.41) gives (3.38). 13

108

CHAPTER 3. SOBOLEV'S INTEGRAL REPRESENTATION

Remark 12 One can replace the ball B in the assumptions of Theorem 4 by some other open set G. depending in general on x E 11 such 18 that G= C SZ and replace the function w by some function w such that

w.E L1(R"),

suppwx C G=,

f

w. (y) dy = 1.

Rn

In this case the same argument as above leads to the integral representation (3.38) in which B, w and the conic body V. are replaced by GwS1 the conic body

(J (x, y) respectively. We shall use this fact in Corollaries 10 -12 yEG.

below.

Remark 13 Set w(r) = r"w(xo - rx). Then w(,.) is a kernel of mollification in the sense of Section 1.1 (in general, a non-smooth one since we have only that w(r) E L1(R")). The polynomial SS_1(x,xo), the first summand in Sobolev's integral representation (3.38), is Taylor's polynomial averaged over the ball B =- B(xo, r) in the following sense: Sl-1(x, xo) = (A,TI-1(x, ))(xo).

Here T1_ 1(x, y) is Taylor's polynomial of the function f with respect to the point y, A8 is the mollifier with the kernel w(r) and the mollification is carried out with respect to the variable y. For,

(ArPt-1(x, -)) (xo) = f

w(,.)

B(o,1) 1

I°I<1

(D°f)(xo - rz)(x - xo + rz)°r"w(xo - rz) dz B(o,1)

1

I°I<1

f (D°f)(y)(x-y)°w(y)dy=(S1-lf)(x,xo) B(xo,r)

This allows us to characterize Sobolev's integral representation as a "mollified (averaged) Taylor's formula with the remainder written in the form of an integral of potential type" of briefly "averaged Taylor's formula". 18 It is also possible to suppose only that G: C fl replacing the assumption f E C,(fl) by f E C(f1) in the case in which a.-n8fl # 0. See a detailed Remark 1 for the one-dimensional case.

3.4. SOBOLEV'S INTEGRAL REPRESENTATION

109

Theorem 5 Let Q C R" be a domain star-shaped with respect to the ball B = B(xo, r) such that ,9 C Q, WE L,,.(R" ).

fWdx = 1,

supp w C B,

(3.44)

R.

I E N and f E (W )t°`(52). Then for almost every x E Q

f (x) = E °I
D.0 )(

(D' f)(J)(x - iJ)*w(3/) dy +

a!

I°I=tt

B

twa(x,Y) dy.

1

(3.45)

Idea of the proof. For 0 < 5 < dist (B, on), which is such that - a < IxoI - r < Ixol + r < set 19 X161 = {x E S2 : Yi c 526 fl B (0, 6) } c Q6, write (3.38) for .4y f E C°O(Q161) where 0 < y < b and pass to the limit as y -+ 0+. 0 Proof. For each 6 and y such that 0 < y < 6 and Vx E 52161 we have

c, 1 f.__..

..

._

,

.

, f (DoA,f)(y) x-yl n t w°(x,y)dy I°I=t i ,

.

I°I
1°I<1

jf

1

/ (A, (D.* f)) (y)

J

(A7(Dwf))(y)(l - y)°w(y) dy +

g

Ix - yl"_t

wa(x, y) dy.

101=1 v

We have applied Lemma 4 of Chapter 1 and the fact that the weak derivatives DW f where Ial < l exist (Lemma 6 of Chapter 1). By (1.5)

A, f -+ f

a.e. on

5216)

(3.46)

as -y -+ 0+. We shall prove that R.,-, (x) =

f(A(Df))(y)(x - y)°w(y) dy B

'-

f

(Dwf)(y)(x - y)-w(y) dy = R°(x) on 52161

(3.47)

B

is The necessity of introducing of these more complicated sets than fl6 arises in connection with Example 8.

CHAPTER 3. SOBOLEV'S INTEGRAL REPRESENTATION

110

and

Sa,7(x) = J

(A7(Dwf))(y)wa(x,y)dy

Ix - yl-

K.

f)(Y)wa(x,y)dy=Sa(x) - JfIx(D'- yin-1

-) (3.48)

in L1(Sl[al).

K,

The relation (3.47) follows from the estimate

IR.,7(x) - R.(x)I < Ml

f

I (A7(Dwf))(y) - (D.* f)(y)I dy,

B

where M1 = II(x - y)allc(n,a,xB)II w(y)IIL (R") < oo and the property (1.9). Set M2 = Ilwa(x,V)IIL (n,d,xR") Then by Remark 10 M2 < oo. Applying the inclusions V. C 116 = St6 fl B (0, s) for x E nlq and 0[6[ C Q6 we obtain II Sa,,,(x) - Sa(x)IILi(n,a,) < M2

I (A1(Dw I))(y) - tD.0f)(V)I

= M2 I I A7(Dwf) - Dwf l n;

(f

dx IxyI"-i

dy IIL,(ne)

dy

n;

< M2 II IZI` "IIL,(na-na)IIA_l(Dwf) - D.f IIL,(na)

and 20 (1.9) implies (3.48).

From (3.48) it follows that, for some ryk > 0, k E N, such that 'Yk -+ 0 as

k ->oo,wehave

S.,,, (x) -1 Sa(x) a.e. on ft[,q.

(3.49)

Now passing to the limit as k -> oo in equality for Al f with 'Yk replacing ry, by (3.46), (3.47) and (3.49) we obtain that (3.45) holds almost everywhere on S2[k[. Since U 11[o[ = f2 we establish that (3.45) is valid almost everywhere on

Q.0

6>0

20 In the last inequality in the expression 116 - nj* the sign - denotes vector subtraction

of sets inR",i.e.,A-B=(zER":z=x-y where xEA,VEB). Clearly, flj - n6' C B(o, 8-1) - B(o,6-') =

B(o,2a-1).

3.5. COROLLARIES

3.5

111

Corollaries

Corollary 9 In Theorems 4, 5 let conditions (3.37), (3.44) respectively, be replaced by

fdx=1.

supp w C B,

w E Co (S2),

(3.50)

Rn

Then Vf E CI(Q) for every x E 11 and Vf E (Wi)'-(fl) for almost every x E S2

f (x) = f (E B

(_ aDy a

Ial<1

+E

[(x - y)aw(Y)1)f (y) dy

("'of '(Y) wa(x, y) dy

Ix - yl-

lal=1 V:

(3.51)

with D° If replacing DI f in the case in which f E (Wi)!°`(f2)

Idea of the proof. For f E C'(Sl) - integration by parts in the first summand of the right-hand side of (3.38). For f E (W1)b0c(Sl) - the same limiting procedure as in the proof of Theorem 5, starting from (3.51) with A, f replacing f.

Corollary 10 In addition to the assumptions of Corollary 9 let Q E l and 0 < 1,61 < 1. Then V f E C'(S1) for every x E ) and V f E (WW)!°`(SZ) for almost every x E Sl (-1)1.1+191Dv+'[(x-y)aw(y)])f(y)dy

(D1'f)(x)_f(E B

+

a!

Ia1<1-191

f

(D°f)(y) Ix - y1n-i+191 wQ-9(x> y) dy

(3.52)

Ial=1,a>9 V.

with DO f replacing DO f and D. *f replacing D* f if f E (Wi)'0C(11).

Idea of the proof. For f E C'(S)) write equality (3.38) with Dyf E CHVI(1l) replacing f and with l - 1,61 replacing 1, integrate additionally by parts in the first summand of the right-hand side, change the multi-index of summation a

to -y -,6 (then E = E ) and write a instead of ry. For f E (Wl)'0c(1)) IaI=1-I0I

171=1,'Y>>0

apply the limiting procedure from the proof of Theorem 4.

CHAPTER 3. SOBOLEV'S INTEGRAL REPRESENTATION

112

Remark 14 For n = 1 equality (3.52) takes the form (3.17). As in the onedimesional case the first summand in (3.51) and (3.52) can be rewritten in the form

f(

a,"#(x -

y)°-1(D°w(y))f

(y) dy,

(3.53)

where o°,s depends only on a and p.

Corollary 11 Let I E N and SZ C R" be a bounded domain star-shaped with respect to the ball B with the parameters d, D, i. e., d < diam B < diam SZ < D < oo. Then there exists c9 > 0, depending only on n, 1, d and D, such that Vf E C1(5Z) for every x E St and df E (W)"(0) for almost every x E 0 (D°f)(y

1(DIf)(x)1 <- cs( f IfIdy+: f I°I=1 v,

B

1dy(3.54)

Ix - yl

where # E Non and 0 < 1,31 < 1, with DW f replacing DI f and Dm f replacing D* f in the case in which f E (W,1)10c(c ). Moreover, I (Dsf) (x) l 5 clo (

(W D)

J

If I dy

e (D)n-1

E f I(D°f)(y)I Ix

+d

yln-1+Ie1

-

dy )

(3.55)

where c10 > 0 depends only on n and 1.

Idea of the proof. Let p be a fixed kernel of the mollifier defined by (1.1). In equality (3.52) take w(x) = (s)np(2 xdxo) and apply (3.53) and (3.43) (see also Corollary 3). Proof. First of all Vx E f and dy E B

I (x - y)°-a(D°w)(y)15 <

()n+°Ix

- y1l°I-191I(D°µ)(2(x d xo))

M1IID°,uIIc(R')(d)10'D-10Id-n

<

M2(a)1-1D-I$Id-n,

where M1 and M2 depend only on n and I (since µ is fixed). Hence

( Ia1<1,0>d a°,6(x -

y)°-B(D°w)(y))

15 Ms(a

)1-1D-101d-",

(3.56)

3.5. COROLLARIES

113

where M3 depends only on n and 1. Secondly from (3.43) it follows that Vx E St and y E V.

D n-1

iw°-0(x,y)I < M4()

(3.57) ,

where M4 depends only on n and 1. So (3.53), (3.56) and (3.57) imply (3.55) and hence (3.54).

Remark 15 For n = 1 inequalities (3.54), (3.55) take the form of the first inequality (3.23), the form of (3.21) respectively. Moreover, if St = B and diam B = d, then (3.55) implies that I(D0f)(x)I <_

M.(d-IoI-n Jutdy B

f

+

I(D°f)(y)I dy),

Ix - yIn-1+101

(3.58)

1°1=1 B

where Mb depends only on n and 1, which is a multidimensional analogue of the first inequality (3.22). If, in addition, 1- 1,61 - n > 0, then dy+d!-101-n1

I(D0f)(x)I: M5 (d-101-n J If I

J I(DQf)(y)I dy),

1°I=1 B

B

which is a multidimensional analogue of the second inequality (3.22).

Remark 16 If I -IQI - n > 0 (in the one-dimensional case this condition is always satisfied), then there is no singularity in the integrals of the second summand in the righ-hand side of (3.54). In this case (3.54) implies the inequality

Wf)(x)I <- cll (f IfI dy+ E JI(JYf)(Y)ldv) B

r

V

JI(D°f)(v)Idv),

< cll (J If I dy + B

(3.59)

Ia1=! n

where c11 > 0 depends only on n, 1, d and D. Applying the same procedure as in the second proof of Corollary 4 one can

obtain the related inequality with a small parameter: if 1 - IQI - n > 0, then

I(D'f)(x)I 5 c12K(e) Jiii dy + e B

ft (Df)(y)I dy,

(3.60)

I°I=i B

where now 0 < e < MS d1-191-n,

K(c) = E and c12 > 0 depends only on n and 1.

7F

(3.61)

CHAPTER 3. SOBOLEV'S INTEGRAL REPRESENTATION

114

Corollary 12 Let 1 E N and f? C Rn be a bounded domain star-shaped with respect to a ball with the parameters d, D. Suppose that f E C' (0) (or f E (W;)b0c(fz) ). Then there exists a polynomial pl_1(x, f) of degree less than or equal to l - 1 such that V,3 E 1

satisfying 101 < 1 and Vx E 11

I (Daf)(x) - (D5Pi-1)(x, f)I < c13

(Dd)n-1

x(D-

IQI=1

a f)-y)IBI

J

dy,

(3.62)

n

where c13 > 0 depends only on n and 1. (If f E (Wf)t0C(fZ), then (3.62) with DO f and D° f replacing DO f and DI f holds for almost every x E f'). Idea of the proof. Set

1aiIQlDDI(x-y)aw(y)))f(y)dy

P1-1(x,f)=S1-1(x,f) f (E( Ial
B

(this is the first summand in (3.51)), where B C fl is such that n is star-shaped with respect to B and w is the same as in the proof of Corollary 11. Note that Sl-lal-1(x, D1 f) = (D1 S1-1)(x, f)

(3.63)

and apply (3.51), (3.52) and (3.56). 0

Corollary 13 Let SZ be a bounded convex domain. For x, y E fZ (x t y) denote by d(x, y) the length of the segment of the ray, which goes from the point x through the point y, contained in fZ . Then V f E U'(11) and for all x E fZ and Vf E (W11)t°C(0) for almost all x E Cl 1 f(x) = measf2 l 1 -t f (Daf)(y)(x-y)ady

n

IoW

1, +1 n lol=l a.

_

a

J(Df)(v) Ix - yin (d(x, y)n - I x - yI n) dy

(3.64)

n

and hence

If(x)I <

measfZ

at f d(x,y)nI(Daf)(y)I dy IaI
n n

+

n

W! j I0I=1

Ix ! fn d(x

In-1 I(Daf)(y)I

dy.

(3.65)

3.5. COROLLARIES

115

In particular,

Ifx)I <

1

DI°I

E

measS2

a!

Ial<_n

f RD°f)(y)I dy,

(3.66)

n

where D = diam S2. (If f E (W,)'Oc(f ), then D" f must be replaced by DW 'f.)

Idea of the proof. Suppose that in Theorems 4 - 5 supp w C S2 instead of suppw C B. Then in (3.38) we have Vn = S2 instead of VZ V1,B - see Remark 12. Take w(x) = x E 1, then (3.64) follows from (3.38) - (3.40). (measS2)-1 for

Corollary 14 Let S2 C R" be an open set, l E N. Then V f E Co(S2) for every x E 1 and df E (WW)0(11) for almost every x E S2

E a!

f (x) = o

IaI=,

f

and hence

If(x)I S

_

a

Ix - yIn

(D°f)(y) dy

I I(D° f)(y)l d a!J Ix - yln-i y 1

1

On 161=

(3.67)

(3.68)

n

with D. Of replacing DI f for f E (W1)0(Q), where vn is the n-dimensional measure of the unit ball in Rn and Un is the surface area of the unit sphere in Rn (an = nvn) 21 Idea of the proof. Since supp f is compact in l one can assume, without

loss of generality, that the function f is defined on R" and f = 0 outside 11. Replace in Theorems 4 - 5, keeping in mind Remark 12, B by Gx = B(x, r2) \ B(x, r1), where r1 < r2 are such that supp f C B(x, r1) and w by w.(y) = (measG=)-1, y E G1. Then V1,G = B(x,r2). Moreover, from (3.40) it follows that Vy E supp f r2

w(x, y) =

nl

n)

vn(r2 - r1)

f Pn-1 dp = 1 . an

rl

Since f = 0 on G. equality (3.67) follows from (3.28). Furthermore, inequality (3.68) follows from (3.67) because a! > ? for IaI = 1. 21 We recall that vn =

a+1

where r(u) = f 0

x°-1e_= dx,

u > 0, is the gamma-function.

CHAPTER 3. SOBOLEV'S INTEGRAL REPRESENTATION

116

Remark 17 For n = 1, 1 = 1 and ) = (a, b) equality (3.65) reduces to the obvious equality: V f E Co (a, b) and dx E (a, b) b

f (x) = 2 f sgn (x - y) f '(y) dy

(3.69)

(see (3.7)). Starting from this equality it is possible to give another proof of equality (3.67). Remark 18 Consider the following particular case of (3.67) an

J

ix - y1"

x.

1

(Of)(y) . (x -1l)

1

f(x) =

dy - an

lyln * Vf.

(3.70)

We note that

Ixi2 = ax e (in Ix I), n = 2,

n>3,

Ixln

and for p E (Wi)'-(Rn), -0 E Co(Rn) we have Consequently, V f E C0 2(R) )

f (x) = - (n -12)an

lxi2-n

* i f, n2:3

(3.72)

(the Newton potential).

Corollary 15 Let SZ C Rn be an open set satisfying the cone condition with the parameters r > 0 and h > 0 and K. be the cone of that condition. Suppose that Vx E fl W. E Co (R"),

suppw: C IT.,

Jiz(v)dY = 1,

(3.73)

Rn

I E N, 0 E No and lal < 1. Then V f E C1(Q) for every x E 1 and V f E (W11)(f2) for almost every x E 12

[(x-y)°w=(y)])f(y)dy

(D"f)(x) =J K.

io<<-I91

117

3.5. COROLLARIES

r

+

(D°f)(y) .w°-F,:(x, y) dy yln-1+191

(3.74)

J Ix I°I=1, °>9 K.

(with DO f and D. *f replacing DO f and D° f for f E (Wi)1oc(Sl)), where n +r

f wylx+ely - xl)y

lal(x -

y - x

//

00' a! Ix - yIn

n-1

(3.75)

dp.

I=-Y1

Remark 19 In contrast to other integral representations the first summand in (3.74) is no more a polynomial.

Idea of the proof. Apply Remark 12 with G. = Kt and wz replacing w. Note that V:,K= = K. and d2 < h2 + r2 (d2 is defined in Remark 8). Corollary 16 Let Sn C Rn be an open set satisfying the cone condition with the parameters r > 0 and h > 0, l E N, 0 E No and 101 < 1. Then there exists c14 > 0, depending only on n, 1, r and h, such that V f E C' (Q) for every x E S2 and Vf E (Wl)1ac(0) for almost every x E SZ (D8f)(x)I <_ C14

(f If I dy + F n

f

D* I

I°I=1:z

Dye -1191

dy)

(3.76)

(with D. Of and D. *f replacing D, f , D' f respectively, for f E (Wl)1Oe(11)) Moreover,

I(Daf)(x)I <- c15 (I

h

)-191rn f

K,

hy-'

Ifl dy+ (r

f

I

)(Y)I dy), I(D°in

1+191

101=1 K:

(3.77)

where r1 = min{r, h} and c15 > 0 depends only on n and 1. Remark 20 Compared with (3.54) inequality (3.76) is valid for a wider class of open sets satisfying the cone condition. On the other hand, (3.54) is a sharper version of (3.76) (for in the right-hand side of (3.54) f If I dx replaces f if I dx) B a for a narrower class of domains star-shaped with respect to the ball B.

Idea of the proof. Let K = K(r, h) if h > r and K = K(h, h) if h < r, and let B(xo, r2) be the ball inscribed into the cone K. Here r2 = rh( r + h +r)-1 >-

CHAPTER 3. SOBOLEV'S INTEGRAL REPRESENTATION

118

r(l+f)-' if h > r and r2 = h(1+f)-' if h < r. Hence B(xo, rl(l+f)-') C K. Moreover, let w be a fixed function defined by (3.50). Suppose that in (3.74) wx is defined by: Wx(y) = (i+ 22)-1(y-£y)), where £_ = A1(xo) and

A. is a linear transformation such that Kx = A.(K). Following the proof of estimates (3.56) and (3.57), establish that b'x E Sl and Vy E K.

(-1 )'IQH Ifll D,+R[(x - y)awx(y)]I < Ml (h

r1)'-ih-101r1

(3.78)

Ial
h n-'

Iwa-#,.(x, y)I <- M2 (rl)

(3.79)

,

where Ml and M2 depend only on n and 1. Estimates (3.77) and hence (3.76) follow from (3.74), (3.78) and (3.79). 0 Remark 21 From (3.77) it also follows that in (3.76) f If I dx can be replaced n

by If IIL,(n) for any p E [1, oo].

Chapter 4 Embedding theorems The main aim of this chapter is to prove various inequalities related to those of the form II1 fIIL,(n) < lIfIIW ini, where a E K, lal < l (D,°, f a f) and cl > 0 does not depend on f. These inequalities may be also presented in an equivalent form as the socalled embedding theorems.

4.1

.

Embeddings and inequalities

We start with the consideration of the notion of a continuous embedding as it relates to the general theory of function spaces, which are, in the framework of this book, normed or semi-normed vector spaces. First let Z1 and Z2 be normed vector spaces. We say that Zl is embedded in Z2 if Z1 C Z2.

(4.1)

The identity operator I, considered as an operator acting from Z1 in Z2: Vf E Z1

If = f, I :

Zl -+ Z2,

(4.2)

which is possible because of (4.1), will be called the embedding operator corresponding to the embedding (4.1).

Definition 1 Let Z1 and Z2 be normed vector spaces. We say that Z1 is continuously embedded in Z2 and write Z1

C:;

Z2

(4.3)

CHAPTER 4. EMBEDDING THEOREMS

120

if, in addition to (4.1), the corresponding embedding operator is continuous, i.e., there exists c2 > 0 such that V f E Z1 illiIZ, <_ c llfliz,

(4.4)

In the cases we are interested in relations (4.1) and (4.3) are equivalent,

which follows from the next statement.

Lemma 1 Let Z1 and Z2 be Banach spaces such that Z1 C Z2. Suppose that the corresponding embedding operator is closed, i.e., for any fk E Z1 where

kEN,g,EZ1andg2EZ2 lion fk = 91 in Z1,

k-+oo

Urn fk = 92 klao

in Z2 = 91 = 92.

(4.5)

Then (4.4) is satisfied and, hence, Z1 Ci Z2.

Idea of the proof. Since the embedding operator I : Z1 -a Z2 is defined on the whole Z1 and is closed, by the Banach closed graph theorem the operator I is bounded.

Remark 1 Let us introduce for Banach spaces Z1 and Z2 such that Z1 C Z2 one more norm on Z1, namely, df E Z1 U 11z" = ill liz, + Iif lIz,.

It is a norm on Z1 considered as the intersection of Z1 and Z2. Condition (4.5) is equivalent to the following: Z1 is complete with respect to the norm II' liz Indeed, if { fk}kEN is a Cauchy sequence with respect to II liz,,, then

lim IN - fmiiz, = k,m-400 urn Ilfk - fmiiz, = 0.

k,m-oo

Consequently, by the completeness of Z1 and Z2 elements g1 E Z1 and 92 E Z2

exist such that limk,o fk = 91 in Z, and limk-wo fk = g2 in Z2. By (4.5) 91 = 92 and, hence, slim li fk - g1 li z = 0. Conversely, let the above relations be satisfied. Then { fk}kEN is a Cauchy sequence with respect to the norm II By the completeness of Z12 an element g E Zl exists such that

' liz

klirnoo(Iifk - 9112, + 11A - 9ii2,) = 0. From the uniqueness of limits in Z1 and Z2 it follows that gl = 92(= g). We note also that the closedness of the embedding operator I is a necessary and sufficient condition for the equivalence of (4.1) and (4.3). The sufficiency is proved in Lemma 1, while the necessity is obvious, since the boundness of I implies its closedness.

4.1. EMBEDDINGS AND INEQUALITIES

121

Now let Z, and Z2 be semi-normed vector spaces and

Bk= If E Zk: IIf IIz. =0}, k = 1,2. Definition 2 Let Z, and Z2 be semi-nonmed vector spaces, which are subsets of a linear space Z. We say that Z1 is continuously embedded in Z2 and write Z, C=y Z2

(4.6)

if' Z, C Z2 + 81,

and the corresponding embedding operator I : Z1 -+ Z2, defined by If = g, is bounded. 2

Remark 2 This means that V f E Z, there exists g E Z2 such that g is equivalent to f in Z1, i.e., g - f E 81, and there exists c3 > 0 such that Vf E Z, II9IIz, <- c31IfiIz,.

(4.7)

Remark 3 If 8, C 82 (in particular, if Z1 and Z2 are normed vector spaces), then Z2 + 01 = Z2, 1I9I1z, = Ill IIz, and Definition 2 has the same form as Definition 1.

Remark 4 Assume that the semi-normed vector spaces Z, and Z2 possess the following property:

the semi-norm IIf IIz, makes sense for each f E Z, with IIf IIz, < oo or IIf IIz, = oo and V f E Z, there exists g E Z2 such that

inf IIf - hllz, = II9IIz2-

hEel

In this case Definition 2 is equivalent to: there exists c3 > 0 such that Vf E Z1

inf IIf - hllz, 5 csllf Ilzi.

heo,

' The sign + denotes the vector sum of sets. 2In this case, in general, the embedding operator is not unique. However, one may easily

verify that for different embedding operators, say I, and 12, Vf E Z, we have IIIiflIz, _ 1112flIs,-

CHAPTER 4. EMBEDDING THEOREMS

122

Lemma 2 Let Z1 and Z2 be semi-Banach spaces such that Z1 C Z2. Suppose that for any fk E Z1, k E N, g1 E Z1 and g2 E Z2 lim fk = 91 in Z1i lim fk = g2 in Z2 k-+oo

k-+oo

91 - 92 E 01.

(4.10)

Then (4.9) is satisfied.

Idea of the proof. Apply the Banach closed graph theorem to the factor spaces Zl = Z1/81 and Z2 = Z_2/81 and 3 the embedding operator I : Z1 -> Z2. Proof. We recall that Z1 is a Banach space and b'/ E Z1 If 112, = 11f 11z"

(4.11)

where f is an arbitrary element in f. (If fl, f2 E f, then 11f, 11 = 11f211.) As for Z2 it is, in general, a semi-Banach space and

IIJ II2, =h0i inf If - hllz2

(4.12)

where f E j. (The right-hand side does not depend on the choice of f E f.) From Z1 C Z2 it follows that Z1 C Z2 and by (4.10) the corresponding embedding operator I is closed. For, let 1k E ZZ1, k E N, 91 E Z1, 92 E Z2 and

k

Ik=91

in Z1, k Ifk=kl Ik=g2 in Z2.

Suppose that fk E fk, 91 E 91 and g2 E 92 Then fk - 91 E It - 91, fk - 92 E fk - 92 and

limllft -91IIz, = 0,

lim (hnf I1fk-92-hIlz,) = 0.

Therefore, Vk E N there exists hk E 81 such that 1lim Ilfk-92-hkllz, = 0. Thus fk - hk -+ 92 in Z2 ask -+ oo. Moreover, since Ilft - hk - 91IIz, = lift - 9111z we also have that fk - hk --1 91 in Z, as k -3' oo. By (4.10) 91 - g2 E 81 and, hence, g1 = 92 Now by the Banach closed graph theorem the operator I is bounded: for some c4 > 0 we have b'/ E Zl 111112,=IIIJII2,:5 04II!II2,.

Consequently, by (4.10) and (4.12) the desired inequality (4.9) follows. 3 The spaces Z'1 and Z2 consist of nonintersecting classes f C Z1, f C Z2 respectively, such that fl, f2 E 1 . = fl - f2 E 01, fl - fs E B2 respectively.

4.1. EMBEDDINGS AND INEQUALITIES

123

Corollary 1 If, in addition to the assumptions of Lemma 2, (4.8) is satisfied, then Z, C Z2 is equivalent to Z, C:; Z2. Idea of the proof. Apply Remark 4.

Corollary 2 In addition to the assumptions of Lemma 2, let 01 C 02.

(4.13)

Then there exists c5 > 0 such that V f E Z1 IIfIIz2 <- Cs IIfIIz,

(4.14)

Idea of the proof. Since 01 C 02, we have IIhIIz2 = 0 for each h E 01. Furthermore, Vf E Z, we also have IIf - hIIz2 = IIf 1122 and (4.9) coincides with (4.14).

Corollary 3 Let Z be a semi-normed vector space, equipped with two seminorms II III and II . 112 and complete with respect to both of them. Moreover,

suppose that for any fk E Z, k E N, g1, g2 E Z HM MIIfk-91111=0,

kllfk-92112=0 =* g1- 92E91n82.

(4.15)

Then the semi-norms II 'III and II.112 are equivalent 4 if, and only if, 01 = 82.

(4.17)

Suppose, in particular, that Z is a normed vector space, equipped with two norms 11.11, and II.112 and complete with respect to both of them. If for any

fkEZ, kEN, 9,,92EZ

kIIfk-91111=0, k IIl-92112=0

(4.18)

then the norms 11 'III and II 112 are equivalent. 4Le., there exist ce, or > 0 such that V f E Z CO IIf112 <- IIfI11 <_ CT IIf112.

(4.16)

CHAPTER 4. EMBEDDING THEOREMS

124

Idea of the proof. Necessity of (4.17) follows directly from (4.16). To prove sufficiency apply Corollary 1 to the semi-Banach spaces Z, and Z2, which are the same set Z, equipped with the semi-norms II - IIl and II.112 . Now we pass to the case of function spaces. Let SZ C Ri" be an open set. Moreover, suppose that Z(ft) is a semi-nonmed vector space of functions defined on Q. Denote

ez(n) = If E Z(Q) : IIf1Iz(n) = 0} and

9(0) = If : f (x) = 0 for almost every x E 0}. All function spaces Z(0), which are considered in this book, possess the following property:

Z(n) Ci L' -(Q),

(4.19)

i.e., Z(S2) C L'-(O) and for each compact K C 0 there exists ca(K) > 0 such that Vf E Z(S2) hEin ezf

(4.20)

IIf - hIIL,(x) <- c8(K) IIf 112(n).

For many of the function spaces considered 8z(a) = t)(ft).

(4.21)

If this property is satisfied, then (4.20) takes the form (4.22)

IIf IIL,(x) <- c8(K) IIf 112(n).

Remark 5 If two semi-normed vector spaces Zl(f1) and Z2(SZ) satisfy (4.19) and 0z,(n),9z,(n) C 0(ft), then f o r a n y f k E Z I (SZ) n Z2(S2), k E N, gl E Z1(()) and g2 E Z2(ft)

lim fk = gi in Zl (ft), k m fk = 92 in Z2(1)

9i - 92 on S2,

i.e., 91 - 92 E 9(S2).

Lemma 3 Let Sl C R" be an open set and let Zl(f2) and Z2 (fl) be semi-Banach function spaces satisfying (4.19) and (4.21). If Zi(p) C Z2(1Z) then there exists cs > 0 such that 'If E Z1(f2) IIfIIzs(n) < cs II/IIz,(n)

(4.23) (4.24)

and, hence, (4.23) is equivalent to ZA(0) C4 Z2(11).

(4.25)

4.1. EMBEDDINGS AND INEQUALITIES

125

Idea of the proof. Apply Corollary 2. Proof. If fk E Z1(12), k E N, gl E ZI(f ), 92 E Z2(S2), and

urn fk = gi in Zi(I), k fk = 92 in Z2(Q),

(4.26)

then by (4.22)

lim fk = gl in L"(9),

k-+oo

lim fk = 92 in L1°`(SI) kloo

(4.27)

and gi - g2 E 9(e) = 9z,(n). Hence, (4.24) follows from (4.13) From (4.21) it follows that Z2(S2) + Bz,(n) = Z2(St). Moreover, for each g E Z2(11), for which g- f E Oz,(n) we have II9IIz2(n) = IIf1Iz2(n) Consequently,

(4.23) coincides with (4.6), (4.24) coincides with (4.7) and, hence, (4.23) is equivalent to (4.25).

Corollary 4 Let 9 C R" be an open set and Z(i2) a semi-normed vector space equipped with two semi-norms and complete with respect to both of them.

Moreover, suppose that conditions (4.19) and (4.21) are satisfied. Then these semi-norms are equivalent. Idea of the proof. Apply Lemma 3 to the semi-normed vector spaces Z1(il) and Z2(0), which are the same set Z(1l), equipped with the given semi-norms. Finally, we collect together all the statements about equivalence of inequalities, embeddings and continuous embeddings in the case of Sobolev spaces.

Theorem l Let l E N, m E No, m <1,1 < p, q < oo and let Il C R" be an open set. 1. The continuous embedding WW ((1) Ci WQ (0),

(4.28)

Ill II Wr(o) < CIO llf llww(n) ,

(4.29)

i.e., the inequality where cto > 0 does not depend on f, is equivalent to the embedding Wp(11) C WQ (11).

(4.30)

2. The continuous embedding W,(11)C:; Cb (c),

(4.31)

CHAPTER 4. EMBEDDING THEOREMS

126

i.e., the statement: Vf E WP(SZ) there exists a function g E Cb (S1) such that

g - f on SZ and Ilgllcm(n) <_ cil Of Ilwp(n),

(4.32)

where c11 > 0 does not depend on f, is equivalent to inequality (4.29) and embedding (4.30), where q = oo, and in (4.29) c1o1q = cil 3. If inequality (4.29) holds for all f E WW(I) fl C°°(fl), then it holds for all f E W,(SZ).

Idea of the proof. Apply Definition 2 and Lemmas 2 and 3. To prove the equivalence of (4.32) and (4.29) when q = oo apply Theorem 1 of Chapter 2 and the completeness of the spaces under consideration. If m > 0 apply also the closedness of the differentiation operator D° where Iai = m in Cb(Q). To prove the third statement of the theorem again apply Theorem 1 of Chapter 2, the completeness of W, ,'(Q) and, for m > 0, the closedness of the weak differentiation operator DW where Jai = m in Lq(Sl). 0 Proof. 1. Clearly, (4.30) follows from (4.28). As for the converse, it is a direct corollary of Lemma 3, because 8wp(n) = ow-(A) = 0(g). 2.

Furthermore, (4.31) implies (4.29) with q = oo, which, by the first

statement of the theorem, is equivalent to (4.30) with q = oo. Let us prove that (4.29) with q = oo implies (4.32) where c11 = c1019=00.

First suppose that m = 0 and 1 < p < oo. Then V f E W,1(0) there exist fk E C°°(Q) n WP(SZ), k E N, such that fk -4 f in Wp(SZ) - see Theorem 1 in Chapter 2. By (4.29) with q = oo, fit E Cb(SZ), k E N, and'dk, s E N

Ilfk - fallC(n) = Ilfk -

5 C11 Ilfk - fsIIW (n)

(see footnote 4 on page 12). Hence, { fk}kEN is a Cauchy sequence in Cb(SZ). Since Cb(fl) is complete, there exists g E Cb(11) such that fk -+ g in Cb(SZ) as k -> oo. Since both WP(1) and Cb(SZ) are continuosly embedded into L10Q(0), it follows that g - f on SZ - see Remark 5.

If m = 0 and p = oo, then 'If E W00(SZ) there exist fk E C00(11) n W,10(SZ), k E N, such that fk -+ f in WO0(f) for r = 1, ...,1 - 1 and IIfkIIw., (n) - Ilf Ilw;,(n) as k -> oo (see Theorem I in Chapter 2). Since llfk - fallC(n) = Ilfk - fell L (n), {fk}kEN is again a Cauchy sequence in C6(SZ).

The rest is the same as for the case in which 1 < p < oo.

If m > 0 and 1 < p < oo, then the same argument as above shows that there exist h E Cb(S2) and h° E Cb(fl) where Ial = m such that fk -+ h in Cb(SZ)

and D° fk -+ h° in Cb(SZ). Since the differentiation operator D° is closed in Cb(11), it follows that h° = D°h. Hence h e Cr(fZ) and fk -+g in Cr(fZ).

4.2. THE ONE-DIMENSIONAL CASE

127

Finally, for all m > 0 and 1 < p < oo, we take f = fk in (4.29) where q = oo and passing to the limit as k -+ oo we get (4.32) since IIgIICm(n) =

kliM -+00

liM oc Ilfkllw(n)

IIfkIICm(11) = k

< CIO lim IIfkIIwn1(s1) = C10IIf IIw;(n) k-4oo

3. The proof of the third statement of the theorem is analogous.

4.2

The one-dimensional case

We start with inequalities for intermediate derivatives.

Theorem 2 Let -oo < a < b < oo, 1E N and 1 < p < oo. 1. For each function f E W,(a, b) and m = 1, ..., l - 1 (4.33)

IIfw'")IIL9(a,b) <- C12IIJ W,(a,b),

where c12 > 0 depends only on I and b - a.

2. If -oo
Ilf,(`'IIL,(ab)),

(4.34)

where c13 > 0 depends only on 1, b - a, fi - a and f is such that the right-hand side is finite.

Remark 6 If b - a = oo and B - a < oo, then inequality (4.34) does not hold. This follows by setting f (x) = xk where m < k < 1. Idea of the proof. Apply inequality (3.21) and Remark 5 of Chapter 3. Proof. Let -oo < a < b < oo. From (3.21), by Holder's inequality and Remark 5 of Chapter 3, it follows that

Ml (b -

a)'_m

(0 - a)-'IIfIIL,(a,O) + (b - a)i' Ilfw')IIL,(a,b)), (4.35)

where M1 depends only on n. This inequality implies (4.34).

Now let b - a = oo. Say, for example, -oo < a < oo and b = oo. If 1 < p < coo, then by (4.33) ao

=(E k=1

1 Ilf-(_)IIL,(a+k-1,a+k))

CHAPTER 4. EMBEDDING THEOREMS

128

M2 ( E(IIf IIL,(a+k-1,a+k) +

Ilfwl) IIL,(a+k-1,a+k)))

k=1

= M2(IIfIIL,(a,-) + Ilfw`)IIL,(a,.))° M2 (IIfIIL,(a,aa) + Ilfw`)IIL,(a.ao)) Here M2 is the constant c12 in (4.33) for the case in which b - a = 1. If p = oo, then Ilfwm'IILoo(a,ao) = Sup kEN

I1fwm)IIL.(a+k-1,a+k)

M2 SUP(IIf

llfw`)IIL,o(a+k-1,.+k))

kEN

< M2 (IllIILW(a.00) +

Corollary 5 Let 1 < p < oo. The norm 1

F, Iifw)IIL,(a,b)

(4.36)

M=0

is equivalent to If IIw;(a,b) for any interval (a, b) C It. The norm IIf IIL,(a,0) +

Ilf,

lIL,(ab)

(4.37)

is equivalent to Ill II W (a,b) if -oo < a < a <,8:5 b < oo.

Idea of the proof. Apply inequality (4.33) and inequality (4.34) with m = 0.

Corollary 6 Let -oo < a < b < oo, t E N and 1 < p < oo. Then w4(a, b) = WW(a, b).

(4.38)

( If b - a = oo, then 1 E w,(a, b) \ W, (a, b). Thus, the embedding Wp(a, b) C wp(a, b) is strict.) Idea of the proof. If f E wp(a, b), then f E L11-(a, b) , and Corollary 5 implies

(4.38).0 Remark 7 Equality (4.38) is an equality of sets of functions. Since B,,,D(a,b) Bwo(a,b), the semi-norms II - IIw;(a,b) and II - IIw;(a,b) are not equivalent. (See Corollary 3 of Section 4.1.)

4.2. THE ONE-DIMENSIONAL CASE

129

Corollary 7 Let 1, m E N, m < l and 1 < p < oo. 1. If -oo < a < b < oo, then the validity of inequality (4.33) with some c12 > 0 independent off is equivalent to the validity of Ilfwm)IILP(a,b)

C14

((b - a)-mIIfIIL,(a,b) + (b -

156 >= C156-"7-

(4.39)

a)'-mlIfw')IIL,,(a,b)/1,

(4.40)

IIfIIL,(a,b) + IF IIff')IIL,(a,b)

for 5 0 < e < c14(b - a)'-' and (4.41)

IIfwm)IIL,,(a,b) <- C16IIfIIL,,(a,b) IIfIIW,(a,b)

with some c14, c15 > 0 independent of f, a and b and some c16 > 0 independent of f. 2. If b - a = oo, then inequality (4.33) is equivalent to ii;(m)1LP(a,b) <_ C12

e

' m IIfIILP(a,b) +e IIfW')IILP(a,b)

(4.42)

l / 1 T -IIIfIILP(a,b) Ilf,;`'IIL,ca,br

(4.43)

for0<e<00, II c(m)IIL,ca,b) <_ C12

t (1

b

b llm

r

J

If-)IPd2 <'12(( [(

a

//t)1-T11-P

) (1

f (IfIP+lf.(')IP)dx

(4.44)

a

if1
6

fIf(m)IPdx
l(!

b

m'

T(1

t /1

a

1-'T]

'

e-'-

b

f IfIPdx+e f Ifw')IPdx a

a

(4.45)

for0<e 0 depends on eo as well. It also follows that Ve > 0 there exists C(e) such that Ilfwm)IIL,(a,b) < C(e) IIfIIL,(a,b) +f Ilf, IIL,(a,b)

Note that one cannot replace here C(e) by a and a by C(e). (If it were so, then taking f (z) = xm and passing to the limit as a - 0+ would give a contradiction.) From (4.40) it also follows that Ilfwm)IIL.(a,b) < M((b - a) m + 1) II/II W,(a,b),

where M > 0 is independent of f,a and b.

CHAPTER 4. EMBEDDING THEOREMS

130

Idea of the proof. 1. Changing variables reduce the case of an arbitrary interval

(a, b) to the case of the interval (0, 1) and deduce (4.39) from (4.33). For

0
p

IIf

IIf II L,(a,o) =

(4.46)

IILp(ak,ak+,))

k=1

and the inequality s < 61 < b to deduce (4.40) from (4.39). Minimize the right-hand side of (4.40) with respect to e in order to prove (4.41). Finally, note that inequality (4.39) follows from (4.41) and the inequality x°y1-Q < a°(1 - a)'-* (x + y),

(4.47)

where x,y > 0, 0 < a < 1. 2. Apply (4.33) to f (a + 5(x - a)) if b = oo, to f (b - b(b - x)) if a = -oo or to f (bx) if a = -oo, b = oo where b > 0, and deduce (4.42). Minimize the right-hand side of (4.42) to get (4.43). Note that (4.33) follows from (4.43) and (4.47). Raise (4.43) to the power p and apply (4.47) to establish (4.44). Deduce, by applying dilations once more, (4.45) from (4.44). Minimizing the right-hand side of (4.45), verify that (4.45) implies (4.43). 0 Proof. 1. Setting y = e-a we get Ilfwm)IIL,(a,b) _ (b - a)',-mII(f (a + y(b -

a) p-m(IIf(a + y(b - a))IILp(o,1) + 11(f (a + y(b - a)))(') jILp(o,1))

= M1((b - a)-'11f IIL,(a,b) + (b -

where M1 is the constant c12 in (4.33) for the case in which (a, b) = (0, 1).

Moreover, for 0 < a < b - a and 1 < p < oo from (4.46), (4.39) and Minkowski's inequality it follows that N Ilfmm)IILp(a,b)

,

t mIIJ IILp(aiAk+1) + o1 mIIfwj)IIL,(ak,ak+t))Pj p

k=1

C14 (al

m (E IIf IILp(ak,ak+i)) k=1

'If p= oo this means that IIIIIL,.(o,b) =

al-m (E II fw1) IIPLP(ak,ak+i)) p ) k=1

kmaxN

UfHL..(ok,ak+,)

4.2. THE ONE-DIMENSIONAL CASE

<

C14(2'6-m IIIIILp(a,b)

131

+ b'-' 11f(')IILp(a,b))-

Setting e = c1461-' we establish (4.40). The minimum of the right-hand side of (4.40) is equal to

C15 T((';) (1- -1)1-

)-IIIf IILp(ab) Ilfwl)IILp(a,b)

and is achieved for e = e1, where e1= (1

C15 U114(0) IVw`'IIL,(a,b))1

If el < (b - a)l-', then, setting e = el in (4.40) we get (4.41). Now let el > (b - a)"-'". This is equivalent to Iifw')IILp(a,b) <-

Cl5 (b -

,,

a)-'IIfIILp(a,b).

Since

IllIILp(a,b) 5 IIfIILpa,b) IIlIIw

(a,b)'

_T Ilfw''IILp(aa) <_ Ilfw`'IILp a,b)

T IIfIIWy(a,b)'

inequality (4.39) with e = c14(b - a)I-' implies that Iifw')IILp(a,b) 5 M2((b -

a)-m

+ 1)

IIfIILp(,b)

IIfIIW;(a,b)'

where M2 depends only on 1, and (4.41) follows. In its turn (4.41) and (4.47) imply (4.33). 2. Let b - a = oo, say a = -oo, b = oo. Given a function f E WP'(-00' oo) and b > 0, by (4.33) we have Ilfw''IIL,(-aa,aa) - b-m+;II(f(bX)),am)IILp

< C12 b-'+= IIf = C12

(bx)IILp(-00,00) +

II(f(bx))w')IILp(-oo,oo))

(b 'IIfIILp(-00,00) +8' ' Ilfw')IILp(-.,.)

Setting c1261-m = e, we get (4.42).

The rest of the proof is as in step 1. 0 Corollary 8 Let 1, m E N, m < 1. Then Ilfwm)IILa(-ao,oo)

IIfIIL2(-

,oo)

and the constant 1 in this inequality is sharp.

Il1w')IIL

(-00,00)

(4.48)

CHAPTER 4. EMBEDDING THEOREMS

132

Idea of the proof. Prove (4.44) when p = 2, a = -oo, b = oo by using Fourier transforms and Parseval's inequality, and apply Corollary 7. Proof. By Parseval's equality and inequality (4.47) 00

Ilfwm)IIL3 (-00,00) - IIF(fwm))IIL2 (-00,00) = J -00 00

< ('!')m (1 - m)1-T f(i -00

T (IIFf11L2(-00,00) +

_ (';)T(1 - m)1

_ (m)'(1 - it)1 T

IIF(f.())IIL2

2(-o,0))

2

(IIfIIL2

2(-00,00) +

I1fw!)IIL2(-00,00)).

if, and only if, ICI = (1 m) - Co we set

Since Z2in = ('")T(1 -

f = f, where ff(x) = (F-1(x(Eo-,Fo+E)))(x) = A stti zz a-'{0y. Passing to the limit as e -+ 0+ we obtain that (m)9 (1 - m)1-T is a sharp constant.

Remark 8 Let 1, m E N, m < l and 1 < p < oo. The value of the sharp constant cm,,,P in the inequality m,l,p IIfwIIL,(_oo,oo) Ilfwt)IILp(-00,00)

IlAm)II4(-00,00)

(4.49)

is also known in the cases p = oo and p = 1: KK-m cm,1,1 = Cm,l,oo =

K-m , t

where for j E N 4

K2,i-1 =

r

00

i=O

4X00`

1

(2i + 1)2j

K21

_±

(-1)'

` =o (2a + 1)2j+1

The following inequality holds 1 = Cm,1,2 < Cm4,P < C ij,l = Cd,00 <

Thus, df E Wp(-oo, oo) for each 1 < p < 00 Ilfwm)IILp(-00,00) 5

2 IIf

(-oo,oo)'

(4.50)

133

4.2. THE ONE-DIMENSIONAL CASE

Remark 9 We note a simple particular case of (4.49): (4.51)

IIfwIILoe(-oo,oo) :5 1/2 If

where f is a sharp constant. One can easily prove this inequality by applying the integral representation (3.27) with a = x + e, b = x - e, e > 0. It follows that for each function f whose derivative f' is locally absolutely continuous If'(x)I <-

11AL,,(-oa,o) + 2

E

We get the desired inequality by minimizing with respect to e > 0 and applying Definition 4 of Chapter 1. In the sequel we shall need a more general inequality: for 1 < p:5 oc 1(1-1)

11+1 IIfwIIL,(-oo,oo)

(4.52)

To prove it we apply Holder's inequality to the integral representation (3.24) and get If'(x)I <- IIw IIL,,(a,b)IIfIILp(a,b) + IIA(x,')IIL,,(a,b)IIfwIILp(a,b)

almost everywhere on (a, b). Choosing w in such a way that IIw'IIL,,(a,b) is min-

imal, we establish that' +1)(b-a)p(6 I2x -(a+b)I°

W(X)

+pb 1

a

and

IIw IIL,,(a,b) = 2 (p + 1)0. Moreover, b

b,y)I

IA(a

2

f w(u) du < (1 + y - )',

for a < y < °ab

b f w(u) du < (1 + p b-a

for Sib-
Y

b

b

7Euler's equation for the extremal problem f lw'(x)IP' dx - min, f w(x) dx = 1 where a

1 < p < oo has the form (Iw'(x)I" sgnw'(x))' = A. So, m'(x) = Jalx + A2IP 'sgn (AIx + A2) Here A, A1, A2 are some constants.

CHAPTER 4. EMBEDDING THEOREMS

134

and

Taking a=x+s,b=x-e, we get If'(x)I lIL,(_,,,oo) °IIf almost everywhere on (-oo, oo). By minimizing with respect to e > 0 and applying Definition 4 of Chapter 1 we have

IIf

AP IllIILp(-00,00) IIf wIILp(±oo,00)'

where 8

2e< <2.

p'+1\p'+1Ja)- " <

AP-

Remark 10 By (4.52) and (4.50) it follows that VI E N, I > 2,

Ilf<_

(1-00,00)

< 2 1 2 IIf IIL,(-oo,oo) Ilfwr)IILp 00,00))

-

27r IIf

11Lp(_oo,00)

Il

'1'

Ilf(`)III('+;)

La(-00,00)

w

( a)

lfw+)IILa(±.+.)

(4.53)

IIL; 00,00)'

Remark 11 Let I, M E N, m < I and 1 < p:5 oo. Then `df E Wp(0, oo) IIf (')IIL°(O,ao) :5

2

& IllIILp(00)Ilf

2IIL(0,-).

(4.54)

This can be proved with the help of the extension operator T2, constructed in Section 6.1 of Chapter 6 (see Remark 1): IIT2f IILa(-oo,w) lI(T2f)((a')IILa(-OO,oo)

IIfwm)IILp(0,C0) C II(T2f)wm)IILp(-.1.) <-

2

8This inequality is equivalent to 2p-1 < Qv 'T)P(1 - a)P, where v = clear since for p > 1

f

/

P-1<2<2et<ei11

l .az)P(1-2p/a.

which is

4.2. THE ONE-DIMENSIONAL CASE

2

IIfIILp O,oo)

135

IIfw"11ILp(O,oo)'

It can be proved that, in contrast to the case of the whole line, the best constant in the case of the half-line for fixed m tends to oo as I -+ oo. To verify this one x)' for 0 < x < Y l-!, and may consider the function f defined by f (x)

f(x)=0forx> '1!. Corollary 9 Let -oo < a < b < oo, I, m E N, m < I and 1 < p:5 oo . If a sequence { fk}kEN is bounded in WP (a, b) and converges in Lp(a, b) to a function f, then it converges to f in Wpm (a, b).

Idea of the proof. Apply inequalities with a parameter (4.40) and (4.42) or multiplicative inequalities (4.41) and (4.43). 0 Proof. Let IIfkIIWP(a,b) < K for each k E N and fk -+ f in Lp(a,b) as k -+ oc. 1. By footnote 5 it follows that Ve > 0 there exists C(c) such that

5 C(c)

IllIILp(a,b) + e

for each f E W, (a, b). Consequently, Vk, s E N II(fk)y,'") - (fs);p"`)IILp(a,b) <- C(e) Ilfk - fall Lp(a,b) + 2eK.

Given 6 > 0 we choose a in such a way that 2eK < 6. Since fk is a Cauchy sequence in Lp(a, b), there exists N E N such that C(e) Ilfk - fsIILp(a,b) < s if k,s > N. Hence, bk,s > N we have II(fk)W'") - (fs)w'")IILp(a,b) < 6, i.e., the sequence (fk)(W'") is Cauchy in Lp(a, b). Because of the completeness of Lp(a, b) there exists g E Lp(a, b) such that (fk)(W") -+ gas k -+ oo in Lp(a, b). Since the weak differentiation operator is closed (see Section 1.2), g is a weak derivative

of order m off on (a, b). Consequently, fk -4 f in Wpm (a, b) as k -+ oo. 2. By (4.41) and (4.43) it follows that dk, s E N II(fk)y,'") - (fs)w'' llL; (a,b) <- M Ilfk - fall cp',b) Ilfk - Jsllwp(a,b)

< M(2K)'` llfk - fsllip(a,b) , where M depends only on 1. Consequently, we can again state that (fk)W('") is a Cauchy sequence in Lp(a, b). The rest is the same as in step 1. 0

Theorems Let -oo < a < b < 00,1 E N, m E N, m < I and l Then the embedding W, (a, b) c Wp (a, b)

p:5

CHAPTER 4. EMBEDDING THEOREMS

136

is compact, i.e., the embedding operator I : W,(a, b) -+ Wp (a, b) is compact. 9

Idea of the proof. Let S be an arbitrary bounded set in W,(a, b). In the case m = 0 consider any bounded extension operator T : WW (a, b) -i Wp(-oo, oo). (See Section 6.1.) For d > 0 set f j = Aa(T f ), where A6 is a mollifier with

a nonnegative kernel. Prove that there exists All > 0 such that Vf E S and `db>0 if - faiIL,(a,b) <- M15.

(4.55)

Moreover, prove that there exists M2(5) > 0 such that b' f E S and Vx, y E [a, b]

I fa(x)15 M2(6), Ifa(x) - f6(y)15 M2(5) Ix - yI.

(4.56)

Finally, apply the criterion of compactness in terms of c-nets and Arzela's theorem. 10 In the case m > 0 apply Corollary 9. Proof. By (1.8) we have IIf - f3II L,(a,b) 5 II Ab (Tf) - T f I I L,(-oo,aa)

< sup lnl<s

I I (T f) (x + h) - (T f) (x) II L,(-aa..).

By Corollary 7 of Chapter 3 and inequality (4.33)

II(Tf)(x+h) - (Tf)(x)IIL,(-,,oa) <- IhI II(Tf)',IIL,(-.,.) < M3 IhI IITfIIw,,(-oo,.) <- M4 IhI IIfIIw,,(a,b),

where 1W3 and Af4 are independent of f. Since S is bounded in WW (a, b), say 11f IIW;(a,b) < K for each f E S, inequality (4.55) follows. Furthermore, by Holder's inequality Vx E [a, b] x+6

If6(x)I 5

f w(_-)

I(Tf)(y)I dy 5 ds (26)' IITfIIL,(R)

X-4

< M6(6) IIf IIW (a,b) S K M6(b) 9 This means that each set bounded in W,(a, b) is compact in W, (a, b) (e precompact), i.e., each of its infinite subsets contains a sequence convergent in Wy (a, b). 10 Let it C R" be a compact. A set S C C(f)) is compact in C(fl) (= precompact) if, and only if, S is bounded and equicontinuous , i.e., Ve > 0 36 > 0 such that Vf E S and Vx, y e fl satisfying Ix - yj < 6 the inequality if (x) - f (y)l < e holds.

4.2. THE ONE-DIMENSIONAL CASE

137

and the first inequality (4.56) follows. Here M5 = max Iw(z)I and M6(b) is independent of f. Moreover, Vx, y E [a, b]

If6(x) -f6(y)I =

b

I f ( (x

8

) -w(y d u)) (Tf)(u)du

R

< 6

f

Iw(xb!)-6u)

(x-6,x+6)U(y-6,y+6)

I(Tf)(u)Idu

1

<M'I662 -yI

KM8(6)Ix-yi

and the second inequality (4.56) follows. Here M7 = maxlw'(z)I and M8(6) is independent of f .

Given e > 0 by (4.55) there exists 6 > 0 such that if - f6IIL,(a,b) < z for each f E S. Then an z-net for the set S6 = { f6 : f E S} will be an a-net for S. If we now establish the compactness of S6 in L,,(a, b) it will imply the existence of a finite i-net for S6. This means that we may construct finite c-nets for S for an arbitrary e > 0, which implies that S is compact in Lp(a, b).

Finally, it is enough to note that from (4.56) it follows that the set S6 is bounded and equicontinuous in C[a, b]. Hence, by Arzela's theorem S6 is compact in C[a, b] and consequently in Lp[a, b] since convergence in C[a, b] implies convergence in Lp[a, b]. 2.

Let m > 0. By step 1 each infinite subset of S contains a sequence

{fk}kEN convergent to a function f in Lp(a, b). By Corollary 9 f E Wp (a, b)

and fk -+ f ask -*oo inWp (a,b).0 Example 1 If b - a = oo, then Theorem 3 does not hold. Let, for example, (a, b) = (0, oo). Suppose, that cp E Co (-oo, oo) is such that supp cp C [0, 1] and W ; 0. Then the set S = {,p(x - k)}kEN is bounded in Wp(0, oo) since II P(x - k)IIw;(o,w) = IIwIIwo(o,0o) However, it is not compact in W(0, oo) because for each k, m E N, k

m

IIw(x - k) - p(x - m)Ii w; (o,oo) >- IIw(x - k) - W(x - m)IIL,(o,co) = 20.

(Consequently, any sequence in S, i.e., {w(x - k,)}$EN, is not convergent in wpm (0,

00))

CHAPTER 4. EMBEDDING THEOREMS

138

Next we pass to the embedding theorems in the simplest case of Sobolev spaces Wp (a, b). In this case it is possible to evaluate sharp constants in many of the relevant inequalities.

Theorem 4 Let -oo < a < b < oo and 1 < p < oo. Then each function f E Wp (a, b) is equivalent to a function h E C(a, b). Moreover,

1) if-oo
(4.57)

and, consequently,

II!IIL-(..b) where C17 = max{(b 2)

C17IIfIIW, (a,b),

(4.58)

a)-D, (1/+1) '(b -

if-oo
IIf (x) -

b 1 a

f f (y) dyIIL (a,b) < (p' +

1)-'

(b - a) ° IIf,IIL,(a,b);

(4.59)

a

3) if -oo < a < b = oo, then lim h(x) = 0 and IIfIIL (a,oo) <- (p')-'IIfIIwo(a,aa);

4) if (a, b) = (-oo,oo), then

Ill

IILm(-oo,oo)

(4.60)

Zlimah(x) = 0 and

< 2 v (Y')

Of II Wo(-oo,oo).

(4.61)

All the constants in the inequalities (4.57), (4.59) - (4.61) are sharp. The constant c17 in inequality (4.58) is sharp if b - a < (p' + 1) '.

Remark 12 For p = 1 inequality (4.57) takes the form

Ill

(b - a)-'IIf IIL,(a,b) + IIf,,IIL,(a,b).

(4.62)

We also note that for p = 1 inequalities (4.60) and (4.61) are equivalent to IIfIIL (a,.) <- IIfwIIL,(a,oo)

(4.63)

IIlIIL ,(-oo,oo) <- z IIfwIIL,(-aa.aa)

(4.64)

and

respectively.

4.2. THE ONE-DIMENSIONAL CASE

139

Idea of the proof. Apply Definition 4 and Remark 6 of Chapter I. In order to prove inequality (4.59) apply the integral representation (3.6). Deduce (4.57) and (4.58) from (4.59). Inequality (4.63) follows from (4.62) and implies inequality (4.64). Apply (4.63) and (4.64) to If Ip and deduce (4.60) and, respectively, (4.61). Set f a 1 to prove sharpness of c17 in (4.58) and of the constant multiplying IIfIIL,(a,b) in (4.57). Set f (x) = (x - a)"' to prove sharpness of the constant in (4.59). Set f (x) = f, (x) = 1 + e(x - a)"' and pass to the limit as e -a 0+ to prove sharpness of the constant multiplying IIfti,IIL,(a,b) in (4.57). Set f (x) = e-,,(z-a) in (4.60), f (x) = e-µI=I in (4.61) respectively, and choose an appropriate µ to prove sharpness of the constants in those inequalities. 0 Proof. 1. Let f E Wp (a, b). By Definition 4 of Chapter 1 there is a function h equivalent to f on (a, b), which is locally absolutely continuous on (a, b). Moreover, if a > -oo or b < oo, then the limits lim h(x) and lim h(x) exist. x_+b-

x_+a+

If a = -oo or b = oo, then as will be proved in steps 3-4 lim h(x) = 0, x_+-00

lim h(x) = 0 respectively. Hence h is bounded and uniformly continuous on

2_+00

(a, b), i.e., h E V(a, b) for all -oo < a < b < oo. 2. By applying Holder's inequality to (3.6) we obtain that for almost every

xE(a,b) If (X)

- b a f f(y)dyI 0

< (b - a)-' (f (y - a)"' dy + n

(b - y)'' dy)'

_ (p' + 1)-a (b - a)-'[(x - a)"'+' + (b Since max [(x - a)"'+' + (b - x)"'+'] a<x
IIf,,IIL,(a,b)

-

(4.65)

(b - a)'+a we have established (4.59).

Inequality (4.57) and, hence, (4.58) follow since b

b

If (x)I5Ibla f f(y)dyl+If(x)-blaf f(y)dyl a

a b

< (b - a) = IIf IIL,(a,b) + I1(x) -

b

1

a

f f (y) dyl. a

3. By letting b -+ +oo in (4.62) we obtain (4.63). Moreover, Vx E (a, oo) Ih(x)I <- IIhItC(x,oo) = IIfIIL (x,.) <- ZOO If,. (y)Idy x

CHAPTER 4. EMBEDDING THEOREMS

140

and it follows that lim h(x) = 0. x-+oo 4. If f E W1(-oo, +oo), then

lh(x)I <- f Ifaldy -00

and lim h(x) = 0 as well. Adding the last inequality and the previous one, W-00

we get Ih(x)I

IIfwIIL,(-oo,oo)

and (4.64) follows since

IIhIIc(-oo,oo)

5. If p > 1, then by (4.63) and Holder's inequality II(IfIP)wlli,(a,oo) =PPII

II IJ

IfIP-1fwlli,(a,oo)

s Pp

(4.66)

Il/IILp(a,oo)IIfwIIL,(a,oo)*

We establish (4.60) by applying inequality (4.47). Inequality (4.61) is proved in a similar way. 6. Setting f = 1 we obtain that the constant (b-a)-P multiplying IIf IIL,(a,b) in (4.57) and the constant c1. in (4.58), if b - a < (p' + 1)', cannot be dimin-

ished. If f (x) = (x - a)P', then one can easily verify that there is equality in (4.59).

Now let us consider the inequality IllIILo(a,b) 5 (b - a) o IIf IIL,(a,b) + AIIf,IIL,(a,b) (p'+1)-P(b-

a) ', which means that the constant multiplyWe prove that A > ing IIf,IlL,(a,b) in (4.57) is sharp. Indeed, set f (x) = f,(x) = 1+e(x-a)"', where

e > 0, then and II fc 1I

IIfAIIL (a,b) =

l+e(b-a)P',II(fe)'IIL,(a,b) =ep'(p'+1)-P(b-a)"'-

(b-a) P (1+e (p'+1) -'(b- a)P' +o(e)) as a -+ 0+. Consequently,

A > a.-o+ lim llfsl$L (a,b) - (b - a) PIIfaIIL,(a,b)

+ 1)"a (b - a)

II(fe)'IIL,(a,b)

Finally, for f (x) = e-µ( 2-a) inequality (4.60) with 1 < p < oo is equivalent to the inequality 1 < (p')-gyp-o (,u P + pt ). For p = P11 the quantity p o + µ is minimal and this inequality becomes an equality. Hence, for f (x) = e-

4.2. THE ONE-DIMENSIONAL CASE

141

there is equality in (4.60). Analogously for f (x) =

e-Pu

there is equality in

(4.61).

If p = 1, then equality in (4.60) and (4.61) holds if, and only if, f is equivalent to 0. This follows from inequalities (4.63), (4.64) respectively. However,

the constants 1 in (4.60) and z in (4.61) are sharp, which easily follows by setting f (x) = e-a(x-a), f (x) = e-lIx[ respectively, and passing to the limit as

µ -4 +00. 0 Remark 13 We note that for a function It, which is equivalent to f on (a. b) and which is absolutely continuous on [a, b], inequality (4.63) may be rewritten as

max Ih(x)I < Var h.

a<x
[a,oo)

The maximum exists since h(x) -r 0 as x -- +oo. The inequality is clear since for each function h of bounded variation Vx E [a, oo) we have

Ih(x)I = limy Ih(x) - h(y)I < Var It. Y

[a.00)

It is also clear that for f E Wp (a, oo) equality is achieved in (4.63) if, and only if, f is equivalent to a nonnegative and nonincreasing function or a nonpositive and nondecreasing one on [a, oo). Similarly for f E Wp (-oc, oo) equality is achieved in (4.64) if, and only if, f is equivalent to a function, which is nonnegative, nondecreasing on (-oo, xo) and nonincreasing on [xo, oo) for some xo or nonpositive, nonincreasing on (-oo, xo) and nondecreasing on [xo, oo).

Remark 14 Analysis of the cases, in which there is equality in Holder's inequality, 11 suggests the choice of test-functions, which allows one to state the

sharpness of the constants. In the case of inequality (4.59) we take x = b in (4.65). If (f')" = Ml(x - a)9' on (a, b) for some M1 > 0, and, in particular, f (x) = (x - a) e, then there is equality in inequality (4.65) and, consequently, in (4.59). Let f > 0 and f < 0 on (a, oo) in the case of inequality (4.60). Then by Remark 13 there is equality in the first inequality (4.66). Furthermore, if n_1)p' (- f')P = M2(f on (a, oo), M2 > 0, then there is equality in the second ii Let f and g be measurable on (a, b) and 1 < p < oo. The equality IIf9IIL,(a,b) = IIfIIL,(a,b)II9IIL,,(a,b)

holds if, and only if, Alf I° = B191° almost everywhere on (a, b) for some nonnegative A and B.

CHAPTER 4. EMBEDDING THEOREMS

142

inequality (4.66). All solutions f E Lp(a, oo) of this equation have the from f (x) = e-"(x-a) for some a > 0. A more sophisticated argument of similar type explains the choice of testfunctions f (x) = 1 + E(y - a)" in the case of inequality (4.57). Corollary 10 (inequalities with a small parameter multiplying the norm of a derivative) Let -oo < a < b < oo,1 < p < oo. 1) If -oo < a < b < coo, then Ve E (0, Eo), where Eo = ((p' + 1) If IIL (a,b) 5 W+ 1) °E

2) If -oo < a < b = oo, then

IIfIIL°(n,b)

(b - a)) ', (4.67)

+EIIfwIIL°(a,b)

VE E (0, oo)

(4.68)

If (a, b)

_

(-oo, oo), then Ve E (0, oo) (p')-'(2E)-

(4.69)

IIfIIL°(-aa,aa)+6IIfwIIL°(-aa,aa)

The constants in inequalities (4.68) and (4.69) are sharp.

Idea of the proof. Apply the proofs of inequalities (4.40) and (4.42). Verify exp(-`-dr that there is equality in (4.68) for f (x) = -6) and in (4.69) for

f (x) = exp (-

Qa

r II) . See also Remarks 15 -16 and 18 below. C3

Corollary 11 (multiplicative inequalities) Let -oo < a < b:5 oo, 1


1) If -oo
where cls = (b - a) a 2) If -oo < a < b = oo, then

IIf

(4.70)

1)-' < (b- a)-'p + 2.

II t (a,oo) 11fw11 i°(a,aa)

(4.71)

3) If (a, b) = (-oo, oo), then 5 (Z)° IIfII (-,a,,,) IIfwIIL°(a,b)

The constants in inequalities (4.71) and (4.72) are sharp.

(4.72)

4.2. THE ONE-DIMENSIONAL CASE

143

Idea of the Proof. Inequalities (4.71) and (4.72) have already been established in the proof of Theorem 4. If f (x) = exp (-(x - a)) or f (x) = exp (-IxI), then inequalities (4.71) and (4.72) become equalities. Apply the proof of inequality (4.41) to prove (4.70). See also Remarks 15-16 and 18 below. 0

Remark 15 We note that for 1 < p < oo the additive inequality (4.60), the inequality with a parameter (4.61) and the multiplicative inequality (4.71) are equivalent. Indeed, (4.68) was derived from (4.60) with the help of dilations and (4.60) was derived from (4.71) with the help of inequality (4.47). Finally (4.68) implies (4.71) by minimizing the right-hand side of (4.68) with respect to a parameter. These inequalities are also equivalent to the following ones: 00

IIfIILW(a.aa) < (P- 1)

7(f (IfIP+ Ifv,I°)dx)=

(4.73)

a

and, Ve > 0,

I fI dx+e I a

IfwIpdx)P.

(4.74)

a

For, (4.73) follows from inequality (4.71) raised to the power p and (4.47), (4.74) follows from (4.73) with the help of dilations and (4.71) follows from (4.74) by minimizing its right-hand side. For the same reasons inequalities (4.61), (4.69), (4.72) and the inequalities 00

i 2- (p-1)--' ( f (IfIPdx+If'IP)

(4.75)

-00

and

00

F-IT

1

+e f If1IPdx)°

(4.76)

-CO

with an arbitrary e > 0, are equivalent as well. Equalities in (4.73) - (4.74), (4.75) - (4.76) respectively, hold for f (x) = exp (-µ(x-a)), f (x) = exp (-pIxI) respectively, with appropriate choice of A. For example, in the case of inequality

(4.75) p = (p - 1)=. Moreover, the listed inequalities for the halfiine and for the whole line are also equivalent. This follows from the equivalence of (4.73) and (4.75). For, if

CHAPTER 4. EMBEDDING THEOREMS

144

(4.73) holds, then, replacing x by 2a - x, we have also that

f a

IIfIILoo(-oo,a) < (p- 1)7 (

(Iflp+ If' I°) )dx)o.

-00

Consequently, 00

(y

z (IIfIIL (-oo,a) + IIfIIL (a,0a)) _

2

f (Iflp+

Ifwlp)

_00

and (4.75) follows. Conversely, if (4.75) holds and f E Wp (a, oo) we apply

(4.75) to the even extension F of f: F(x) = f(x), if x > a, and F(x) _ f (2a - x), if x < a. Then 00 P

IIfIILo (0,00) = IIFIIL-(-00,00) <- 2(p- 1);- (f (IFI°+IF' IP)dx) I -00 00

=(p-1)T (f (Ifly+IfwI")dx)° a

Remark 16 For p = 2 all the inequalities discussed in Remark 15 may be deduced by taking Fourier transforms since by Parseval's equality II F-'Ff II L,,(-,,..) =

IIf

27r

IIFfIIL,(-00,00) =

7217 (f

r2-,,

00

II(1

-00

+t2)j(Ff)(t) IIL,(-00,00)

00

(1+e2)-'4)}'(f -00

00

_

II f eu{(Ff)( ) d

+t2)-#(1

-00

f

2

00

f

00

(IFfI2+IFffI2)dS)3

-00

Thus we obtain (4.75) for p = 2.

=

-00

(If l2+Ifwl2)dx)'

4.2. THE ONE-DIMENSIONAL CASE

145

There is an alternative way of using Fourier transforms, which leads to two other inequalities 12 00

1

If ±f ,I2dx)5, -00

which hold for each 13 f E W2 (-oo, oo). The constant 7 is sharp. Indeed, by the properties of Fourier transforms and by the Cauchy-Schwartz inequality we have 1

(11

Of IILm(-oo,ao) =IIF-1

27

IIF-1(1

^t F(f

,

± ft) ) IILm(-oo,00)

t g) * (f f

00

(

27r II

1

F'

(1

-00

f

(x - y) (f (y) f fW(U)) d1/

IIF-1(1

fg)IIL,(-oo,ao)

27r

27r II

II

2(-oo,oo)

Of ± fwllLz(-oo,ao)

1±t

IIL,(-oo,oo)

and the desired inequality follows. If f (x) = e-1=t, then all three inequalities under consideration become equalities. The second approach is applicable to the case 1 < p:5 oo as well and leads to the inequalities a

00

/ ,,

If ±fwlpdx)v -00

for each f E WP (-oo, oo) since, for example,

IIF(-1) (1 I 9)

IILd(-oo,oo)

=

27rIIexIIL,,(o,oo) =

If 1 < p < oo, f (x) = e-= for x > 0 and f (x) = e9 T for x < 0, then these inequalities become equalities. 12 If we square and add them, we obtain the previous inequality. Is We note that this inequality does not hold for each function f, which is such that the right-hand side is finite. (It does not hold, say, for f (x) = e*=.)

CHAPTER 4. EMBEDDING THEOREMS

146

R.emark 17 We note two corollaries of (4.65) under the supposition that f is absolutely continuous on [a, b]:

If(a)I S (b - a)° IIfIIL,(e,b)+(p +1)-o(b-a)tIIf'IIL,(a,b)

(4.77)

(for p = 1 (4.77) coincides with (3.8)) and

(a If

2 b) I

(b- a) 'IIfIIL,(a,b)+2

(p'+1)-o(b-a) IIf'IIL,(a,b).

(4.78)

Both constants in (4.77) are the same as in (4.57) and are sharp. This is proved by using the same test-functions as in the case of inequality (4.57). In inequality (4.78) both constants are also sharp. The test-function f =_ 1 shows that the constant multiplying If IIL,(a,b) is sharp. Moreover, the test-

functions f = fE, where e > 0 and fE is defined by ff(x) = 1 + e(x - a)", if a < x < a2, and fE(x) = 1 +e(b- a)"', if azb < x < b, show by passing to the limit as a -4 0+ that the constant multiplying IIf'IIL,(a.b) is sharp.

Corollary 12 Let I E N,1 < p < oo and -oo < a < b < oo. Then each function f E W,,(a, b) is equivalent to a function h E

-b) and

Ct-1(a,

IIg(m) IIC(a,b) <- C19 IIf II WW(a,b), m = 0, ..., 1 - 1,

(4.79)

where c19 > 0 is independent of f, i.e., W, (a, b) C; V'4 (a, b).

If a = -oo, then lim h(m)(x) = 0, if b = oo, then lim h(m)(x) = 0, where

m=0,...,1-1.

x-r-oo

x-ioo

Idea of the proof. Apply Remark 6 of Section 1.3 and Theorems 4 and 2.

Theorem 5 Let lEN,mENo,m<1,1
holds if, and only if, b - a < oo, or b - a = oo and p < q.

(4.80)

Moreover,

this embedding is compact if, and only if, b - a < oo and the equalities m = I - 1, p = 1 and q = oo are not satisfied simultaneously. Remark 18 As in the simplest case discussed in Corollary 7, from the inequality, accompanying embedding (4.80), Ilfy,m)IILa(a,b) < M IIf IIWW(a,b),

(4.81)

4.2. THE ONE-DIMENSIONAL CASE

147

where M > 0 is independent of f, it follows that, for b - a < oo, Ilf,m)IIL,(a,b) 5 c20 (b -a) 9

0 ((b - a)--Ilf IIL,(a,b) + (b (4.82)

where c20 > 0 is independent of f, a and b.

If q > p, then, excluding the case in which m = l - 1, p = 1 and q = oo, it also follows that IIL,(a,b) <- C21 E

I1f IIL,(a,b) + EIIf,S,) IIL,(a,b) , a)1_m_ o+Q and

where 0 < s < c22(b b, and

c21, c22 > 0 are independent of f, a and

(I-1)+,

C2311fII1, (Q

11fwm)IIL,(a,b) <-

(4.83)

V)

ykm-l 11fIIWp(a,b)+°',

(4.84)

where c23 > 0 is independent of f . Moreover, inequalities (4.81) - (4.84) are equivalent.

The proof is similar to the proof of Corollary 7. One should notice, in addition, that since q > p, by Jensen's inequality V

II911L,(ak ak+,)) k=1

< ( II9IIL,(ak,ak+.))

a

= II9IILp(a.b).

k=1

If b - a = oo, then inequality (4.83) holds Ve > 0 and in inequality (4.84) IIfIIW,(a,b) can be replaced by Ilfw')IIL,(a,b)

Idea of the proof. To prove (4.81) apply Corollary 12 and Holder's inequality if

b - a < oo and the inequality IIfIIL,(a,b) s IIfIIL,(a,b)IIf11L1 (a,b),

whereifb-a=ooandp
f(x) _

(4.85)

kEZ:(k,k+1)C(a,b)

where W E Ca (R), W 0 0 and supp V C [0,1], to verify that (4.81) does not hold.

To prove the compactness apply Theorem 3 and inequality (4.83) or (4.84). If b - a < oo, m = ! - 1, p = 1 and q = oo, consider the sequence

fk(x)=kl'iq(a26+k(x-a2

))'

(4.86)

CHAPTER 4. EMBEDDING THEOREMS

148

where k E N, q E Co (-oo, oo), supp q C (a, b) and (°-2) = 1. Finally, if b - a = oo, apply Example 1. Proof. The proof of the statements concerning embedding (4.80) being clear, we pass to the proof of the statements concerning the compactness. 1. Let b - a < oo and fk, k E N, be a sequence bounded in W,(a, b). Then by Theorem 3 there exists a subsequence ft., s E N, and a function f E L,(a, b) such that fk. -* f in LD(a, b). If I - m - v + o > 0, then from (4.83) or (4.84) it follows that fk, -i f in WW (a, b).

2. Ifb-a
then for the functions ft defined by (4.86) we have IIfkIIW;(a,b) = k-' II 7 L,(a,b) +

II?7(')IIL,(a,b) - IInIIW;(a,b)

and k m f(k'-')(x) = h(x), where h(0) = 1 and h(x) = 0 for x 96 0. Consequently, the sequence fit, k E N, is bounded in Wi (a, b), but none of its subsequences fk., s E N, converges in Laa(a, b). Otherwise, for some subsequence fk., lim 11A. - fkeIICIa,b) =h-m 11A. - fk,IIL (.,b) = 0. Hence, fk. a,o-wo

ao

00

convergers uniformly on [a, 6) to h, which contradicts the discontinuity of the function h. 3.

If b - a = oo, then Example 1 shows that embedding (4.80) is not

compact for any admissible values of the parameters.

4.3

Open sets with quasi-resolvable, quasicontinuous, smooth and Lipschitz boundaries

We say that a domain ) C R" is a bounded elementary domain with a resolved boundary with the parameters d, D, satisfying 0 < d:5 D < oo, if

11={xER":an<xn<tp(x), 2EW},

(4.87)

where14 diam H < D, 2 = (xi, ..., xn_i), W = {2 E R"-1 : ai < xi < b;, i = 1, ...n - 1}, - oo < aj < b, < oo, and

an+d<
(4.88)

1a Since fl is a domain, hence measurable, by Fubini's theorem the function v is measurable d2. w

on W and mess fl = f

4.3. CLASSES OF OPEN SETS

149

If, in addition, w E C(W) or w E C'(W) for some l E N and IID°wllc(w) 5 M if 1 < lal < I where 0 < M < oo or w satisfies the Lipschitz conditon

I w(i) - w(y)I 5 M la - yl, x, y E W,

(4.89)

then we say that Sl is a bounded elementary domain with a continuous boundary

with the parameters d, D, with a C'-boundary with the parameters d, D, M, or with a Lipschitz boundary with the parameters d, D, M respectively. Moreover, we say that an open set Sl C R" has a resolved boundary with

the parameters d, 0 < d < coo, D, 0 < D < oo and x E N if there exist open parallelepipeds Vj, j = !-,s, where s E N for bounded Sl and s = oo for unbounded Sl such that 1) (Vj)d n Sl i4 0 and diamV, < D, 2) Sl C U (V )d, j=1

3) the multiplicity of the covering {Vj}'=1 does not exceed x, 4) there exist maps )1j, j = 1, s, which are compositions of rotations, reflections and translations and are such that

Aj(Vj)={xER": aij<xi
Aj(f2 n vj) = {x E R: a"j < x" < cpj(-*), 2 E Wj},

where 2 _ (x1, ..., x"-0, Wj = {i E R"-1 and

aij < xi < bij, i = 1, ..., n - 11,

a,,,+d
(4.90)

(4.91)

0. If Vj C ft, then wj(:i) _- b"j. (The left inequality (4.91) is

a"j > 2 d.) satisfied autimatically since by 1) W e note that Aj(SlnVj) and, if Vjnofl 34 0, also ) ((`N)nVj) are bounded elementary domains with a resolved /boundary with the parameters d, D, where

A (x) = (41(x), ..., Am-1W , -,\j,.W )Since by 1) by -ai,j > 2d, i = I,-, n-1, by 4) it follows that meas (Stnvj) > d", j = 1, s. So by 3), for unbounded 0, mess Sl = oo, because by (2.60) 0 E meas (Stn vj) < x mess Q. =1

If an open set Sl C R" has a resolved boundary with the parameters d, D, x and, in addition, for some I E N all functions cpj E C'(Wj) and IID°'pjilc(wt) <_

M if 1 < Ial < I where 0 < M < oo and is independent of j or all functions wj satisfy the Lipschitz condition

Iwj(x)-wj(v)I :5 M12-9I, 1,9 EWj,

(4.92)

CHAPTER 4. EMBEDDING THEOREMS

150

where M is independent of x, 9 and j, then we say that Q has a C'-boundary (briefly O E C1) with the parameters d, D, x, M, or a Lipschitz boundary (briefly 8S2 E Lipl) with the parameters d, D, x, M respectively. If all functions wj are continuous on W we say that 11 has a continuous boundary with the parameters d, D, x. Furthermore, an open set S1 C R" has a quasi-resolved (quasi-continuous) a

boundary with the parameters d, D, x if 11 = U Ilk, where s E N or s = oo, k=1

and 51k, k = 1-.s , are open sets, which have a resolved (continuous) boundary with the parameters d, D, x, and the multiplicity of the covering {Qk}k=1 does not exceed x. (We note that if 0 is bounded, then s E N.) Finally, we say that an open set S2 C R" has a resolved (quasi-resolved, con-

tinuous, quasi-continuous) boundary if for some d, D, x, satisfying 0 < d <

D < oo and x E N, it has a resolved (quasi-resolved, continuous, quasicontinuous) boundary with the parameters d, D, x). Respectively an open set Sl C W has a C'- (Lipschitz) boundary if for some d, D, x, M, satisfying 0 < d < D < oo, x E N and 0 < M < oo, it has a Cl- (Lipschitz) boundary with the parameters d, D, x, M. Example 2 Suppose that S1 = 1(X1, X2) E R2 : -1 < x2 < 1 if -1 < x1 <

0, -1 < x2 < xi if 0 < x1 < 1} where 0 < y < 1. Then 11 is a bounded elementary domain with a resolved boundary, which is not a quasi-continuous boundary.

Example 3 Let 11 = {(x1i x2) E R2 : 0 < x1 < 1, xj < x2 < 2 xi } where 0 < y < oo, -y 34 1. Then 811 is not a quasi-resolved boundary while IN satisfies the cone condition.

Example 4 For the elementary domain 11 defined by (4.87) the Lipschitz condition (4.89) means geometrically that Vx E 811 the cones

Ki = {y E R" : yn < W(2)-MIa-yI}, Ky = {y E R" : V(2)+MI2-91 < yn} are such that

Kz n W C 1, K= n W C `S1,

(4.93)

where W=Ix ER":2EW,a"<xn
and y E Q. Similarly, K; n W C °N. Suppose that (4.93) is satisfied. Since (y,,p(y)) V f1 the inclusion K; n W C 0 implies that w(9) > W(2) - MI! - 91.

4.3. CLASSES OF OPEN SETS

151

(For, if p(y') < gyp(.t) - Ml - 91, then (9, sp(y)) E 0.) Similarly, Kz n W C` Q implies that cp(y) < ap(t) + MI± - 91 and (4.89) follows.

We note also that the tangent of the angle at the common vertex of both cones K= and K= is equal to may.

Example 5 Let 52 = {(x1,x2) E R2 : x2 < cp(xl)}, where tp(x1) _ Ix, 17 if x1 < 0, cp(x1) = xi if x1 > 0 and 'y > 0. Then the function cp satisfies a Lipschitz condition on R if, and only if, y < 1, while 0 has a Lipschitz boundary in the sense of the above definition for each y > 0.

Example 6 Let y > 0. Both the domain 521 = { x E R" : 1xI" < x,, < 1, Ixl < 1 } and the domain 522 = { x E R" : -1 < x,, < 1x1 < 1 } have a Lipschitz boundary if, and only if, y > 1. (Compare with Examples 6 and 7 of Chapter 3.)

Lemma 4 If an open set 52 C R" has a Lipschitz boundary with the parameters d, D, x and M, then both 52 and `S2 satisfy the cone condition with the parameters r, h depending only on d, Al and n. Idea of the proof. Let z E Vj n 852 and x = ad(z). Consider the cones K= and K= defined in Example 4, where 1P is replaced by tp,. Proof. By Example 4 we have K= n A, (V;) c.\;(V; n 52),

K= n a; (V,) c A, (V; n' 52).

By 1) bid - aid > 2d and (4.90) implies that there exist r, h > 0 depending only on d, M and n such that ) (V, n c) and \j(Vj n` S2) satisfy the cone condition with the parameters r and h. (The cone condition is satisfied for the largest cone with vertex the origin, which is contained in the intersection of the cone K(d, defined by (3.34) and the infinite rectangular block x1, ..., x"_1 > 0). SinceM) aj is a composition of rotations, reflections and translations, the sets Vj n 52, Vj n` S2 and, hence, the sets 52 and `S2 also satisfy the cone condition with the parameters r and h.

Example 7 Let 52 = Q1 U Q2, where Qt and Q2 are open cubes such that the intersection iU1 n Z72 consists of just one point. Then both i and `S2 satisfy the cone condition, but the boundary of S2 is not Lipschitz. (It is not even resolvable.)

Lemma 5 A bounded domain i C R" star-shaped with respect to the ball B C Cl has a Lipschitz boundary with the parameters depending only on diam B, diam 52 and n.

CHAPTER 4. EMBEDDING THEOREMS

152

Idea of the proof. Apply the proof of Lemma 2 of Chapter 3 and Example 1. 0 Proof. Let SI be star-shaped with respect to the ball B(xo, r) and z E BSI. We consider the conic body V. = U (y, z) and the supplementary infinite cone yEB(xo,r)

V= _

U

(y, z) , where (y, z)" _ {z + Q(z - y) : 0< @< oo} is an open ray

yE B(xo,r)

that goes from the point z in the direction of the vector yd. Then V2 CSI and by the proof of Lemma 2 of Chapter 3 V2 C` N. Without loss of generality we assume that the vector x is parallel to the axis Ox, hence, z = (xo, zn), where to = (xo,1, ...) xo,n-1) and z,+ > xo,,,, and consider the parallelepiped U2 = {y E R" : xo,n < y,, < 2zn - xo,n, y E U,}, where U= = {y E Rn-1 : jy; - xo,;j < Z, i = 1, ..., n - 1}. Then Vp E U, the ray that goes from the point (y, xo,n) in the direction of the vector x intersects

the boundary 8) at a single15 point, which we denote by y = (y, p(p)). In particular, p(I) = zn. Since the tangent of the angle at the common vertex of Vs and V2 is greater where R2 = max Ixo - yl, it follows (see Example 1) that than or equal to VEOn

y E U=. We note that if 9 E U,, then the conic body V, contains the cone Ky with the point y as a vertex, whose axis is parallel to Ox,, and which is congruent to the cone defined by (3.34) with the parameters xo,n. Moreover, V. contains the supplementary infinite cone Ky. The tangent of the angle at the common vertex of these cones is equal to r 2 iP OT x0.n

2(p(s)-xo,n+ ? Ir-flU

4R,

x-

U.4

Consequently (see Example 1), 1'P (.t) - V(9)

Moreover, since V, C a and Vs C` SI, we have x0,, + z < V(2) < 2zn - xo,n z 2 E Us . We note also that B(z, E) C Us C B(z, (Rl + (n - 1)2(x)2){).

(4.94)

Finally, we consider a minimal covering of R" by open balls of radius (Its multiplicity is less than or equal to 2".) Denote by B1, ..., B, a collection of those of them, which covers the 6-neigbourhood of the boundary BSI. Each of e.

'5 Suppose that q E 8fl, rr # y and Q = P. If nn > yn, then y E V,r C fl. If nn < yn, then y E Vs C° 3f. In both cases we arrive at a contadiction since y E M.

153

4.3. CLASSES OF OPEN SETS

these balls is contained in a ball of the radius z centered at a point of 852. Since Vz E 8Sl we have U= D B(z, a), we can choose Us ..., Us,, where zk E 8i2 in such a way that UEk D Bk. Consequently, the parallelepipeds UZ, , ..., U, cover the B-neigbourhoods of 852. From (4.94) it follows that the multiplicity of this (2)2)

covering does not exceed x = 2' (1 + r 15 of Chapter 3.)

+ (n - 1)2

in )

.

(See footnote

Thus, 0 has a Lipschitz boundary with the parameters d = e, D =

diam il, M = 4 f

< 4D and x.

Lemma 6 1. A bounded open set it C R" satisfies the cone condition if, and only if, there exist s E N and elementary bounded domains 12k, k = 1, ..., s, with

Lipschitz boundaries with the same parameters such that S2 = U Stk. k=1

2. An unbounded open set 52 C R" satisfies the cone condition if, and only if, there exist elementary bounded domains ilk, k E N, with Lipschitz boundaries

with the same parameters such that 00

1) iz = U Qk, k=1

and 2) the multiplicity of the covering x({ilk}k 1) is finite.

Idea of the proof. To prove the necessity combine Lemma 4 of Chapter 3 and Lemma 5. Note that if the boundaries of the elementary domains ilk, k = !-,s are Lipschitz with the parameters dk, Dk and Mk then they are Lipschitz with the parameters d = inf dk, D = sup Dk, M = sup Mk as well if d > 0, D < 00 k=1s

k=1,-s

k=Ti

and M < oo. To prove the sufficiency apply Lemma 4 and Example 5 of Chapter

3.0 Remark 19 If in Lemma 6 ilk are elementary bounded domains with Lipschitz boundaries with the same parameters d, D, M, then fl satisfies the cone condition with the parameters r, h depending only on d and M. Remark 20 If we introduce the notion of an open set with a quasi-Lipschitz boundary in the same manner as in the case of a quasi-continuous boundary, then by Lemma 6 this notion coincides with the notion of an open set satisfying the cone condition. If we define an open set satisfying the quasi-cone condition,

then this notion again coincides with the notion of an open set satisfying the cone condition.

CHAPTER 4. EMBEDDING THEOREMS

154

Lemma 4 of Chapter 3 and Lemma 6 allow us to reduce the proofs of embedding theorems for open sets satisfying the cone condition to the case of bounded domains star-shaped with respect to a ball or to the case of elementary bounded domains having Lipschitz boundaries. To do this we need the following lemmas about addition of inequalities for the norms of functions.

Lemma 7 Let mo E N, 1 < p1, ...pm, q < oo and let f? = 6 Slk, where Slk C k=1

max pm ands E N or s = co

R" are measurable sets, s E N for q <

m=1,...,mo

otherwise. Moreover, ifs = oo and q < oo, suppose that the multiplicity of the covering x e x({Qk }k=1) is finite. Furthermore, let fm, m = 1, ..., mo, and g be functions measurable on Q. Suppose that for some am > 0, m = 1, ..., mo, for each k mo

II9IIL,(nk) <_ E am

(4.95)

llfmIILp,,,(nk)

m=1

Then

mo

(4.96)

IIgIIL,(n) < Aa M=1

when A = s if q <

max pm and A = x otherwise.

m=1,...,mo

Idea of the proof. If pi = ... = pm = q = 1 add inequalities (4.95) and apply inequality (2.59). In the general case apply Minkowski's or Holder's inequalities for sums (for q > pm, for q < pm respectively) and inequality (2.59). 0 Proof. Let q < oo. 16 By (4.95) and Minkowski's inequality it follows, that a

1

IIgIIL,(n) < ( II9II9Lq(nk)) k=1

-`

fmllLp(f1k))Q) a

S

m=1 k=1

mp

a

V

a

k=1 m=1

=

Qm

m=1

( IIfmIILp,,,(nk)) k=1

le The case q = oo is trivial and the statement holds for f2 = U f4, where I is an arbitrary if f

set of indices:

Me 11911L- (n) = SUP

F, QmhIfmHIL,.,(n). M=1

4.3. CLASSES OF OPEN SETS

155

If q > p,,,, denote by Xk the characteristic function of Stk. Since E Xk(x) < x, k=1

by Minkowski's inequality we have s

i

a

IIfmIILvm(flk)) X k=1

k=1

_ ((u(f k=1

Ifm(x)I°mXk(x) dx)

) q) Pm

(J ( I fm(x) I Q7Ck(x) 1 ' dx) n k=1

_ (f Ifm(x)IP- (:L Xk(x)) " dx) "- < x4IIfmIILvm(n). k=1

0

If q < p,,, < oc, then by Holder's inequality with the exponent (2.59)

(

sa am (E IIfmIILvm(nk))

k=1

> 1 and

oM

k=1 <s

xD

IIfIIL,,,,(n) <_ 3aIIfIIL,,(n)

and inequality (4.96) follows. The case in which some pm = oo is treated in a similar way with suprema

replacing sums. 0

Corollary 13 Let I E N, Q E 1

satisfy 101 < l,1 < po, p, q < oo, f = U S2k, k=1

where11kCW' are open sets,sENifq<poorq<pandsENors=coif q > po, p. Moreover, ifs = oo and q < oo, suppose that the multiplicity of the covering x = is finite. Suppose that f E Lpo(Il) f1 w,(11), c25, c26 > 0 and II Dwf II Lq(nk)

s C25IIIIIL,o(nk) + ccIIf Iiwi(nk),

k = 7s-

(4.97)

Then IIDwOfIILq(n) <_ Aa (c

IfIIL,a(n)+02611f11wp(n)).

Idea of the proof. Direct application of Lemma 7. O

(4.98)

CHAPTER 4. EMBEDDING THEOREMS

156

Lemma8 Let lEN,m ENo,m
k=1

Wp(11k) CZ; Wq (f1k), k = 1, ..., s.

(4.99)

Then

W, (1l) C.; WQ (Sl).

(4.100)

Moreover, if embeddings (4.99) are compact, then embedding (4.100) is also compact.

Idea of the proof. Apply Theorem 1 and Lemma 7 to prove embedding (4.100). To prove its compactness consider a sequence of functions bounded in Wn(Sl) and, applying successively the compactness of embeddings (4.99), get a subsequence convergent in W91401).

Proof. 1. By Theorem 1 (4.99) is equivalent to the inequality Ill IIW (n4) <- Mk IlfIIW (n,k),

where k = 1, ..., s and Mk are independent of f. By Lemma 7 it follows that 1If11w, (n) < Mo kmax.Mk IIfIIW;(n),

where Mo depends only on n, m, p, q, and (4.100) follows. 2.

Let M > 0 and II f fl w (n) < M for each i E N. Then , in particular,

11f II wP(n1) < M. Consequently, there exist a function gl E W. 'n(111) and a subsequence f,w -+ 91 in W9 n(111) as j - oo. Furthermore, lI IIw,(n,) 5 M and, hence, there exist g2 E W9 (S12) and a subsequence f,(,) of f;(1) such j , that f1(2) -i g2 in WQ (S)2). Moreover, f(2) -+ g1 in W."'(111). Repeating this procedure s - 2 times, we get functions gk E W9 (Slk), k = 1, ..., s and a subsequence f,, such that fj -a gk in WQ (Slk) as j -4 oo. We note that gk is equivalent to go on Slk f1 Sl,. Hence, there exists a function g, defined on Sl, such that g - gk on Slk, k = 1, ..., s. By the properties of weak derivatives (see Section 1.2) g E W, 'n(11) and

IIA - 9IIw; (n) < E 11fii - 9kIIW (nr) -+ 0 k=1

as jioo.

4.3. CLASSES OF OPEN SETS

157

Lemma 9 Let I E Id, m E No, m < 1 and 1 < p, q < oo. Suppose that for each bounded elementary domain G C Rn with a resolved (continuous) boundary there exists c26 > 0 such that for each Q E R

satisfying IQI = m and

Vf E WW(G) (4.101)

IID!fIIL,(G) :5 C26IIf1IWD(c)

Then for each bounded open set

C Rn having a quasi-resolved

11

(respectively, quasi-continuous) boundary there exists c27 > 0 such that (4.102)

II D,'f IIL,(n) <- C27 Ill IIw;(n)

satisfying 1#1 = m and V f E Wp(0).

for each 13 E I

If p:5 q and c26 depends only on n, 1, m, p, q and the parameters d and D of a bounded elementary domain with a resolved (continuous) boundary, then for each unbounded open set Sl C Rn having a quasi-resolved (quasi-continuous) boundary there exists C27 > 0 such that inequality (4.102) holds.

Idea of the proof. Apply (4.101), where G, f are replaced by \,(n fl V,), f, _ f (A,) respectively, and the parallelepipeds V, and the maps a, are as in the definition of a resolved (continuous) boundary. Change the variables, setting y= and obtain (4.101) where G = SZ f1 V j. Apply Corollary 13 twice to prove (4.102) succesively for open sets 0 with a resolved (continuous) and quasi-resolved (quasi-continuous) boundary. Proof. First suppose that St has a resolved boundary. We notice that A,(x) =

A71x - A,1b,, where b, E Rn, A, _

A,x + b,, (bV))n

ik=1

,

A-1 =

Ia k)I, Ibb)I < 1 and Idet A)I = Idet A; 1I = 1. Consequently, we have

(D!f)(x)I = I Dw(f;(at ')(x)))I =

=I

ik=1

(aa;,)w... (a ,,)w(f,(A(_1)(x)))I

I

n k,,...,km=1

In!

< I7 y! I (Dwfj)

(,\(7') (x))

I (Dwfj)

I < n'" 17

Setting y = Af-1)(x) we establish that IIDwf IIL,(nnvj) <_ n-

II D.7fj II L,(a;(nnv,))-

I,I=m

1)(x))

CHAPTER 4. EMBEDDING THEOREMS

158

Similarly, for a E Nt) satisfying j al = 1, IID,fjIIL,(1nvj).

IIDwfjIIL,(A,(invtn 5 n` Iti1=1

Hence, inequality (4.101) implies that IIDwfIIL,(S2nv,)
If s2 is bounded, then the number of parallelepipeds Vj is finite, say, s. Hence, by Corollary 13, IIDwfIIL,(n) < nm+l sv max c26(Aj(1l n Vj)) IIf IIw;(a)

Let SZ be unbounded, then the set of parallelepipeds Vj is denumerable. Suppose that n, 1, m, p, q are fixed (p< q). Then in (4.101) c26(G) = c (d, D). Hence Vj E N IIDwfIIL,(nnvj)


By Corollary 13 it follows that IIDwf IIL,(n) <_ nm+l x+ c ;6(d, D) IIf Ilwo(n) = c;(d, D, x)II f Ilwo(n).

Thus, (4.102) is proved for an SZ with a resolved boundary. If sZ has a quasiresolved boundary one needs to apply Corollary 13 once more, in a similar way. The case of 11 having a quasi-continuous boundary is similar. O Lemma 10 Let I E N, m E 1%, m < 1 and 1 < p, q < oo. Suppose that for each bounded domain !Q C R" star-shaped with respect to a ball there exists c26 > 0 such that Vf E WI,(G) inequality (4.101) holds. Then for each open set 11 C W' satisfying the cone condition there exists C27 > 0 such that Vf E WW(ft) inequality (4.102) holds.

If p < q and c26 depends only on n, 1, m, p, q and the parameters d and D of a domain star-shaped with respect to a ball, then for each unbounded open set s2 C R" satisfying the cone condition there exists c2? > 0 such that inequality (4.102) holds. Idea of the proof. Apply Lemma 4 and, if SZ is unbounded, Remark 7 of Chapter

3 and Corollary 13. 0

4.3. CLASSES OF OPEN SETS

159

Proof. Let 1 satisfy the cone condition with the parameters r, h. By Lemma 4

and Remark 7 of Chapter 3, St = U 1k, where s E N for bounded 1, s = 00 k=1

for unbounded Sl, and S1k are bounded domains star-shaped with respect to the balls Bk C Wk C Stk. Moreover, 0 < M1 < diam Bk < diam Slk < M2 < oo and x({Slk}k=1) < M3 < oo, where MI, M2 and M3 depend only on n, r and h. If Sl is bounded, then IIDwfIILq(nk) <_ G26(Qk) IIfllW;(nk),

k = 1,...,s.

(4.103)

Hence, by Corollary 13, IID°fIIL,,(n)1<_ 89 kmax C26(Qk) IIlIIW"(n)

(4.104)

Suppose that Sl is unbounded. Denote by A(d, D) the set of all domains, whose diameters do not exceed D and which are star-shaped with respect to balls whose diameters are greater than or equal to d and set C& (d, D) = sup c26(G). Clearly A(d, D) C A(d1, D1) if 0 < d1 < d < D < D1 < oo. GE A(d,D)

Then bk E N II Dwf IILq(nk) < c26(dk, Dk) IIf II W (nk) <_ c26(Ml, M2) Ill IIw;(nk)

and, by Corollary 13, IIDWf IIL,(n) <_ M3 c26(Ml, M2) 11f Ilw;(n)

Lemma 11 Let I E N, m E No, m < l and 1 < p, q < oo. Suppose that for each bounded elementary domain G C R' with a Lipschitz boundary there exists c26 > 0 such that for each 0 E No satisfying 1/31 = m and V f E Wp(G) inequality (4.101) holds. Then for each bounded open set St C R satisfying the cone condition there exists c27 > 0 such that for each 03 E I satisfying 101 = m and V f E Wp(sl) inequality (4.102) holds.

If p:5 q and c26 depends only on n, 1, m, p, q and the parameters d, D and M of a bounded elementary domain with a Lipschitz boundary, then for each unbounded open set St C R" satisfying the cone condition there exists c27 > 0 such that inequality (4.102) holds. Idea of the proof. Apply Lemma 6, Remark 19 and the proof of Lemma 9.

CHAPTER 4. EMBEDDING THEOREMS

160

Proof. Let Q satisfy the cone condition with the parameters r, h. By Lemma a

6 and Remark 19, St = U IZk, where s E N for bounded Q and s = oo k=1

for unbounded St. Here Qt are bounded elementary domains with Lipschitz boundaries with the same parameters d, D, M depending only on n, r and h. Moreover, x({SZk}k=1) < Mg, where M3 also depends only on n, r and h. If SZ is bounded , then as in the proof of Lemma 10 we have inequalities (4.103) and

(4.104). Let fl be unbounded. Suppose that n, 1, m, p, q are fixed (p < q). Then in (4.101) c26(G) = c28(d, D, M). Hence, Vk E N IIDWf IILq(nk) < c26(d, D, M) If II W;(nk)

and, by Corollary 13,

I

IID.'f IILq(nk) 5 M c28(d, D, M) IIf Ilwgn) = cze(r, h)11f11W;(n)

4.4

Estimates for intermediate derivatives

Theorem 6 Let I E N, 8 E N satisfy 1(31 < l and let 1:5 p:5 oo. 1. If 11 C R" is an open set having a quasi-resolved boundary, then Vf E WW (n)

IIDwfIIL,(n) 5 cas Ilfllwl(n),

(4.105)

where c28 > 0 is independent of f. 2. If SZ C R" is a bounded domain having a quasi-resolved boundary and the ball B C Sl, then df E wP(S2) IIDmfIIL,(n) <_ 2a (II/IIL,(B)+IlfIlwa(n)),

(4.106)

where c29 > 0 is independent of f. 3. If 11 C R" is a bounded open set having a quasi-continuous boundary, then be > 0 there exists cgo(e) > 0 such that df E WW(S1) (n) <_ cso(e) Ill 11n> +e llfllw;(n)

(4.107)

Idea of the proof. Apply successively the one-dimensional Theorem 2 to prove

(4.105) and (4.106) for an elementary bounded domain Q with a resolved boundary. In the general case apply Lemma 9 and the proof of Lemma 7. Deduce inequality (4.107) from Theorem 8 and Lemma 13 below.

4.4. ESTIMATES FOR INTERMEDIATE DERIVATIVES

161

Proof. 1. Suppose that S2 is a bounded elementary domain (4.87) with the parameters d, D. By inequality (4.35) it follows that VQ E N satisfying IQI < 1 and V2 E W

MI \ ll(De, f)( x,

'

s

a

) + II \ 8xn w

\Dw.f

(

x'

) II

J'

0 < 6 < 2 and MI depends only on 1, p, b and D. (We recall that +p(2) - an < D.) where Q = (.8 1,

By the theorem on the measurability of integrals depending on a parameter1 ' both sides of this inequality are functions measurable on W. Therefore, taking Lp-norms with respect to ± over W and applying Minkowski's inequality for sums and integrals, we have IID ,f IIL,(n) <- MI (11 11 (Dwf) (x+ xn) II L,,,n

-6,w,+ d+6) II L,s(W )

"I Dwf +1I\

\a2n /

< MI (II

IIfIIwp(n)).

Let a = (Q, an), where an = an + Z, 8 E W5, and Q = ((i, Q= (QI, ..., Qn-a). We consider the cube Q(a, 6) _ {x E R" Ixj - ail < 6, j = :

1, ..., n} and set U = {x = (xi, ..., xn_z) E R"-I : aj < xj < bj, j = 1, ..., n- 2). Applying the same procedure as above, we have II II(Dwf)(x,xn)IIL,.i(w)IIL,(a + -a a++ +a)

S MI(II it +Q Ilflu g,' (W)

IIL,,=n(?n-6,on+s))

Substituting from this inequalty into the previous one and applying Holder's inequality, we get Il D.f II L,(n)

Let E C R"' and F C R" be measurable sets. 17 We mean the following statement. Suppose that the function f is measurable on E x F and for almost all y e F the function f Y) is integrable on E. Then the function f f (x, ) dx is measurable on F. E

CHAPTER 4. EMBEDDING THEOREMS

162

,,+6)

M2(1111

+IlfIIwi(n)),

where M2 depends only on 1, p, b and D.

Repeating the procedure n - 2 times, we establish that IIDwfIIL,(n) <- M3(IIf0IL1(Q(a,d))+IIfIIi4(n)),

(4.108)

where M3 depends only on n, 1, p, 6 and D. 2. Taking b = 4 and applying Holder's inequality, we establish that inequality (4.105) holds for all Q E N satisfying 101 < 1 for each bounded elementary

domain with a resolved boundary and c2s depends only on n, 1, p, d and D. Hence, by Lemma 9, the first statement of Theorem 6 follows. 3. Suppose that B - B(xo, r) C fl, where 12 is an elementary domain considered in step 1. Without loss of generality we assume that xo,n > an and set or = (x0,1, ..., xo,n_1 i an), 6 = min{ , f). Then Q(xo, 6) C B(xo, r) and the parallelepiped G = {x E Rn : 1 xj - xoi 1 < 6, an - 6 < xn < xo,n + b} C S2. Applying inequality (4.108) in the case, in which ,0 = 0, p = 1 and SZ, Q(o, 6) are replaced by G, Q(xo, 6) respectively, we get IllIIL,(c) -< M4 (11f11L,(Qt=o,o)) + Ilf ll.,(c))

5 M5(IIIIIL,(B)+IIf114(c)),

where M4 and M5 are independent of f. Hence, from (4.108) it follows that (4.105) holds for each Q E No' satisfying 1131 < 1.

4. From step 3 and the proofs of Lemmas 7 and 9 it follows that for each bounded domain 1 having a resolved boundary

(t s

IIDwflIL,(n) 5 Ms

11f11 L,(8;) + I1f111,(n)),

(4.109)

j=1

where s E N and Bj are arbitary balls in n fl Vj. (Vj ands are as in the definition of SZ having a resolved bondary.)

Let the ball B c fl. We choose m E N and the ball B0 in such a way that B0 C B fl Cl fl V,n. By step 3 and Holder's inequality it follows that II!IIL,(Bm) <- M7 (Ilf IIL,(Bo) + Ilf IIW,(nnvm)) <- Ms (11/ IIL,(B) + Ilf 11t;(n))

4.4. ESTIMATES FOR INTERMEDIATE DERIVATIVES

163

Let j 96 m. We choose a chain of parallelepipeds U1, ..., U, of the covering which are such that U1 = V3, Uk n Uk+1 54 0, k = 1, ..., or - 1, and 1, and set U, = Vm. Next we consider balls Bk c Uk n Uk+1, k Bj, B, := Bm. Then by step 3 Bo IIf II L,(Bk) < M9 (IIf IIL,(Bk+,) + IIf IIW;(nM/k)) I

k = 01 ..., or

-1.

Consequently, for each j = 1, ..., s IIfIIL,(B,) 5 M1o (IIIIIL,(B) + IIlIIw,(n)),

and inequality (4.106) for bounded domains having a resolved boundary follows from (4.109). (We note that M6, ..., Mlo are independent of f .) The argument for a bounded domain Sl having a quasi-resolved boundary is similar. 5. By Theorem 8 embedding (4.118) is compact. hence, by Lemma 13 inequality (4.121) holds where q = p and inequality (4.117) follows.

Next we give some examples showing that assumptions on St in Theorem 6 are essential. The first two examples show that for open sets 0, which do not have a resolved boundary inequality (4.105) does not always hold. 00

Example 8 Let l E N, 0 E K, 0 < 10 1 < 1, 1 < p:5 oo and 0 = U B(xk, rk), k=1

where rk > 0 and B(xk, rk) are disjoint balls. Suppose that rk - 0 as k -+ 00 and 00 rklpi- )° < oo if p < oo. We set f (x) := rk ° (x - xk)" on Qk, k E N. k=1

Then f E W,(1) but D' Of ¢ Lp(St). Example 9 Let 1 E N, (01, 02) E No, 01 96 0, /31 + 02 < 1, 1 < p < oo and let 0 be the domain considered in Example 3. We set f (x1 i X2) 4' x22 where a2 0 No. Then, for 1