The Method of Newton's Polyhedron in the Theory of Partial Differential Equations (Mathematics and its Applications)

f. With account of (56), relation (57) implies the

l(f,cp)l ~ llf,Hil(G)IIII'P,HYJliiIt follows that the form (f,
Hll(G) = (H;jll)',

YJl

and

H;jJL = (Hil(G))',

the norms of the left-hand spaces coinciding with the norms in the Banach conjugate spaces of the right-hand spaces. We come back to the proof of local solvability. Denote by H N(P),s the space of those distributions u E H( -oo) for which the norm ( 49) is finite. We have the following

45

Two-sided Estimates for Polynomials Related to Newton's Polygon

Theorem. Let operator (24) beN quasi-elliptic. If the region f! is sufficiently small, then Vs E R for any right-hand side f E HN(P),s(n) there is a distribution u E HN(P),s+ 1 (!l) which is a solution to Equation (53) in the sense of distributions (i.e. (54) is fulfilled). Proof. Consider the operator

P:

HN(P) ,s+l ~

HN(P),s(u

~-----+

P(x,y;Dx,Dy)u).

This operator is continuous. Indeed, in view of the definition of the norm ( 49) and Lemma 4.6, we have

I[Pu]IN(P),s ~

L

l[aa,aD;D~u]IN(P) ,s

(a,,B)EN(P)

~ const

L I[D; n:u] I

N(P) ,s

~ const I[u] IN(P),s+l.

(a,,B)EN(P)

Since the operator transforms the subspace (HN(P) ,s+l )JR2\fl into induces a continuous operator in the quotient spaces:

(H N (P),s)JR2\fl,

it

(58) In view of duality (57), the operator (59) is the adjoint operator of operator (58) while operator (58) is the adjoint of (59). According to Theorem 4.5, if the region n has a sufficiently small diameter, then the inequality

holds. Since V is dense in

(HN(P),-s)n

in the norm

j[

JIN( P),-s'

inequality (60)

is extended to the entire space (HN(P),-s)n, i.e. operator (59) has no kernel. By virtue of (60), this operator has a closed image. Indeed, if fi = tpVj, V j E (HN(P),-s)n, is a Cauchy sequence in (HN(P),- s -l)n, then, by virtue of (60), Vj is a Cauchy sequence in (HN(P) ,-s)n. According to the well-known theorem by Banach (e.g. see Yosida [1 ,Chapter 7, §5]), the image of the operator (58) coincides with HN(P),s(n). The above theorem can also b e deduced from the gener al results by Hormander [1, Chapter 3] on local solvability of operators of constant strength. According to Hormander, operator (24) is called an operator of constant strength in n if for any pairs ( x', y'), ( x", y") E !1 there is a constant c = c( x', x", y", !1) such that

46

Chapter 1

where

~

P(x,y;~,17) =

~ (a) 2) 1/2 L..t IP (x,y;~,7J)I ·

(

a~O

As a trivial consequence, it follows from Proposition 4.3 (ii) that anN quasi-elliptic operator is an operator of constant strength. 4. 7. Hypoellipticity of N quasi-elliptic operators. We shall say that a distribution u E 'D'(!l) belongs to Hfoc(n) if cpu E HI-L for all c.p E 'D(!l). Theorem. Let operator (24) be N quasi-elliptic and let tile region 0. be sufficiently small. Let u E D' ( 0.). Then

{P(x,y;Dx,Dy)u

E

Hfoc(0.)}

{D~D~u E Hfoc(0.),

=?

V(a,(3)

E

N(P)}.

Corollary. An N quasi-elliptic operator P is hypoelliptic, that is

{u

E

D'(!l),Pu

E

C 00 (0.)}

=?

{u

E

C 00 (0.)}.

The above-stated theorem follows from Hormander's results on hypoelliptic operators of constant strength (see Hormander [1, Chapter 13, Theorem 13.4.1]). Appendix We now show that function (52) satisfies condition (39') with constant c = c( R) --+ 0, R --+ +oo. It is convenient to obtain this estimate as a consequence of more general assertions. Denote by B0 the class of smoot h functions f-L( ~, R) depending on ~ E Rn and R > 1 and satisfying the following conditions. (a) There is p > 0 such that for any a > 0 we have

(b) There are Rt, R2, and Ct, c2 > 0 such that

Lemma 1. If f-L(e, R) E Bo, then the function f-L(e, R) satisfies condition (39') with a constant c = c(R) --+ 0, R--+ +oo . Proof. Expanding !-L( e', R) by Taylor's formula at the point remainder in Lagrange's form we obtain

f-L- 1 ((',R)(f-t((,R)- f-t((',R )) =

+

~ L..t iai=N

(e' -~")a a!

N-l

L

(t' I,

tff)a

~~

ial=l

f-L(a)( 1 , R) !-L( ~ R) ' '

e"

I

~

= c" ~,

e'

and taking the

(a)(t" R) f-L (e'~ ~) f-L '

+ t(t'~, -

t"),

~,

0 ,.,. t ,.,. 1 ~

~

.

47

Two-sided Estimates for Polynomials Related to Newton's Polygon

By virtue of condition (a), the first sum on the right-hand side can be estimated by means of N-1

L

cale'- e"llai(R + le"l)-plalja! ~ cR-P(1 + le'- e"I)N-1.

lol=l

In view of (a) and (b), the absolute value of the second sum does not exceed

(1) Using the elementary inequality

R +I(' I < 1 +I('- ("I R+l("l ' and setting (" =

R > 1,

e" and (' = ( we find

Substituting this inequality into (1) we estimate the expression under consideration by means of

Choosing N from the condition pN > IR1 - R 2 1 we prove the lemma. Thus, it suffices to prove that function (52) belongs to the class 8 0 . The proof is facilitated by the following lemma. Lemma 2. (i) If p, 11 p, 2 E 8 0 , then p, 1 p, 2 E 8 0 . (ii) If p,(e, R) E B0 and p,(e, R) > 0, then p, 8 (e,R) E Bo for any s E JR. Proof. The validity of condition (b) for the functions p, 1 p, 2 and p, 8 is obvious. We check the fulfilment of condition (a) for them. By Leibniz' formula, we have (

/-Ll, /-L2

) (o) _ """"'

-

'

a.

(o-13) (13)

L..J {3!( a _ {3)! f-L1

f-L2

'

13~0

and it remains to note that for p, 1 and p, 2 the condition (a) holds. By the chain rule, the derivative (p, 8 )< 0 ) is a linear combination of expressions

These expressions are readily estimated by means of (a).

Chapter 1

48

We now prove that VR,s E Bo. In view of Lemma 2 (ii), it suffices to verify this inclusion relation for s = 2, i.e. to prove that

By Lemma 2 (i), it suffices to prove the inclusion for each of the factors, i.e. to show that

(2)

For R

> 1 the condition (b) for the function

1ak pf ae I =

p(~, 7],

R) is obvious. Further, we have

+ 1 )l~l 2 m-k ::=;;2m .. . (2m- k + 1)(1~1 + I7Jinfm + Rlfm)2m-k = const(l~l + I7Jinfm + Rlfm)-k p,. 2m ... (2m- k

Similarly, Choosing p from the condition p function (2).

< m/n, 1/n, n/m, 1/m we obtain condition (a) for

CHAPTER II

PARABOLIC OPERATORS ASSOCIATED WITH NEWTON'S POLYGON Introduction This chapter is devoted to studying Cauchy's problem for differential operators in several variables. When studying Cauchy's problem, there naturally arises separation of variables into a temporal variable and spat ial variables. Taking into account separately the order of differentiation with respect to time and the total order of differentiation with respect to the spatial variables we associate with a differential operator a set of integral points in the plane that are used to construct Newton's polygon and the related set of principal quasi-homogeneous parts of the operator. In terms of these quasi-homogeneous parts we describe a new class of parabolic operators which relates to the classical operators parabolic in Petrovskil''s sense in the same way as N quasi-elliptic operators relate to quasi-elliptic operators. In Chapter 1, in relation to the problem of local solvability and local smoothness for differential operators on functions of two variables, we stated and solved the algebraic problem of describing polynomials in two variables P( 1J) that are estimated from below outside a large circle + 1] 2 = by means of the sum of the moduli of the monomials contained in them. A similar problem arises in the present chapter as well. Consider a polynomial in the variables = ( 6, ... , E JR. and

e

e,

c5

e

T

E

en)

n

C: (1)

Denote by v the set of integral points in the plane (lai,,B) such that aa 1 .. . anf3 =/=- 0, where lal = a1 +···+an. Adding to the points belonging to v the origin and the projections of points on the coordinate axes and taking the convex hull we obtain a polygon which we denote as !:!..(P). In the case n = 1 it coincides with Newton's polygon N(P). We are interested in finding necessary and sufficient conditions on polynomial (1) under which the inequality Im T ~ lo '

(

e

1 ' ... '

en' Re T) E R n+

1'

( 2)

(a,/3)E.6.(P)

holds. Here lei= (er+· · ·+e;) 112 • For n = 1 there arises an analog of the problem considered in §1.2, in which the range of variables is changed, namely instead of the exterior of a large circle in R 2 we consider the product of a straight line by the lower complex half-plane. The presentation of the material in this chapter follows a plan which is in many respects analogous to that of Chapter 1. §1 is of auxiliary character; it presents well-known Petrovski!'s definitions of correct and parabolic polynomials and describes correct polynomials in two variables in terms of Puiseux' series for their roots. 49

50

Chapter 2

In §2 we consider necessary and sufficient conditions for the validity of inequalities (2) in the case of two variables and introduce the classes of N -parabolic and N-stable correct polynomials. §3 deals with N-stable correct and N-parabolic symbols with variable coefficients and presents the proofs of theorems on correctness of Cauchy's problem for differential operators with such symbols. In §§4 and 5 the corresponding results of §§2 and 3 are generalized to the case of several variables. §1. Polynomials correct in Petrovski'l's sense 1.1. Definitions and some Classes of Polynomials Correct in Petrovski'l's sense. We consider Cauchy's problem for a differential operator on functions in JRn+l. As was mentioned above, one of the variables is separated as time t, th~ other variables being denoted x = (x 1 , ... , xn); the dual variables will be denoted as r E C and = (6, ... ,en) E !Rn. In the well-known paper by Petrovski'l [1], for the differential operator with constant coefficients

e

P(~-aa ,... ,~aa '~aa) z z Xn z t X1

corresponding to a polynomial m

P(6, ... 'en, r)

=

L Pj(6, ... 'en)rm-j'

(1)

j=O

Cauchy's problem

k=O, ... ,m-1,

(3)

was considered. Following Hadamard, Petrovski'l says that problem (2), (3) is correctly posed (well-posed) if for any right-hand sides J, u 0 , ••• , um-l bounded together with a finite number of derivatives there exists a unique solution u(x, t) bounded on the interval 0 ~ t ~ T together with the given finite number of derivatives, the operator {J, uo, ... , Um-d 1-4 u being continuous in the corresponding norms. Under the additional assumption that polynomial (1) is solved with respect to the highest power of r, i.e.

Po(6, ... , en)= const,

(4)

Petrovski1 proved that problem (2), (3) is well-posed (in the above sense) if and only if the following algebraic condition is fulfilled: there is a real number '"Yo such that Imr <'"Yo·

(5)

51

Parabolic Operators Associated with Newton's Polygon

Denoting by 'Tj( 6, ... , ~ n), j = 1, ... , n, the roots of polynomial ( 1) we can restate condition (5) as follows: there is a real number (o such that

j = 1, ... ,m.

(6)

The polynomials satisfying (5), ( 4) are said to be correct in Petrovski1 sense. Remarks. 1) Petrovski'l considered in fact general systems of differential operators with coefficients depending on time. However, it is the above-mentioned special case when Petrovski'l's result admits of an effective formulation. 2) A detailed analysis of problem (2), (3) was carried out by Volevich and Gindikin in [1]. In particular, it was shown that Petrovski1's result remains valid if (4) is replaced by a weaker condition:

If Cauchy's problem with zero initial data is considered, then one can discard condition ( 4') as well confining oneself to condition (5). According to Volevich and Gindikin [1], polynomial ( 1) is said to be exponentially correct if for any point w = (w1, ... ,wn) E IR.n the "translated" polynomial Pw(6, ... , ~n, r) = P(6

+ iw1, ... , ~n + iwn, r)

is correct in Petrovski'i''s sense. As was shown by Volevich and Gindikin [1], the exponential correctness is a necessary and sufficient condition for the correctness of Cauchy's problem (2), (3) in the class of functions of exponential growth (decrease). As will be seen in the end of this chapter, Cauchy's problem can be studied for exponentially correct equations with variable coefficients, which, however, requires an additional condition on the symbols (the condition of constant strength). In the proposition below some equivalent definitions of exponential correctness are collected. Proposition 1. For a polynomial P(6, ... ,~n,r) the following conditions are equivalent.

(i) The polynomial Pis exponentially correct, i.e. Vw E IR.n 3r(w) such that

Pw(6, · · ·, ~n, r) =/- 0, (i') VM

> 0 3r(M) such that

Pw(~I, ... ,~n,r) =/:- 0

for

lwl::;; M,

(ii) The distance dp(6, ... , ~n, r) from the point (6, ... , ~n, r) to the manifold of zeros of the polynomial P tends to infinity uniformly with respect to (~1, ... ,~n,Rer) E IR.n+l asimr -too. (iii) Vo: > 0 we have

p(a)(6, ... ,~n,r)P- 1 (ei,···,en,r)---+ 0

for Imr---+ -oo

uniformly with respect to (~ 1 , ... ,~n,ReT) E Rn+l.

(7)

Chapter 2

52

Proof. Conditions (i') and (ii) are tautologically equivalent. The equivalence of (ii) and (iii) follows from the well-known property of polynomials (see Hormander [3, formula ( 4.1.5))]:

p(a)(6, ... ,~n, 7 ) 1/lal P(6, ... 1 ~n 1 7 )

~

[ ( )] 1 const d p 6, ... , ~ n, 7 - .

The equivalence of (i) and ( i') is a consequence of some facts in multidimentionial complex analysis, namely the convexity of the maximal function 1( w) (for more detail see Volevich and Gindikin [1, Lemma 3.4.1 and Proposition 3.4.1]). An important sufficient condition for exponential correctness is contained in the following proposition. Proposition 2. If a hypoelliptic polynomial P satisfies Petrovski1's correctness condition (5), then it is exponentially correct.

For the proof see Volevich and Gindikin [1 (Proposition 3.4.2)]. LE:'mma. An exponentially correct polynomial is solved with respect to the highest power of 7 (i.e. ( 4) is fulfilled).

This assertion is deduced from Proposition 1 (iii) in the same way as Lemma 1 in Section 1.3.2 is deduced from the hypoellipticity condition. 1.2. Quasi-homogeneous polynomials correct in Petrovskil's sense. Let Po be a (1, ... , 1, q)-homogeneous polynomial, i.e.

(8) In view of the quasi-homogepeity condition (8), the conditions for correctness in Petrovskil''s sense (5), (6) for the polynomial Po take the form Po(~ll··· 1 ~n 1 7):f:O Im7oj(~1,···,~n) ~ 0

for

lm7<0, for

(6, ... 1 ~n,Re7)ERn+ 1 ,

(~1,···,~n) ERn,

j = 1, ... ,m,

(9) (10)

where 7oj(6, ... , ~n) are the roots of the polynomial P0 • Proposition. (i) For polynomial (8) solved with respect to the highest power of 7 condition (9) (or (10)) is fulfilled if and only if

(a) q is natural number; (b) ifq is odd (i.e. q = 2b+ 1), then lm7oj(6, ... ,~n) q = 2b), then

= 0;

ifq is even (i.e.

53

Parabolic Operators Associated with Newton's Polygon

(ii) For polynomial (8) solved with respect to the highest power ofT the strict conditions (9), (10), i.e.

P0 (6, ... ,En,T) =f. 0, Im Toj(6, ... , En)> 0,

lmT ( 0,

ITI +lEI-/= 0,

(6, ···,En) E lRn \ {0},

(9') (10')

are fulfilled if and only if q is even. The proof is based on the following

Lemma. Let k, p be coprime natural numbers and let z E C. Then

(i) the inequality -oo

< E< oo,

(11)

holds for all branches of etiP if and only if p = 1 and lmz?: 0

for even

lmz=O

forodd

k = 2b,

(12)

k=2b+1;

(13)

< E<

(11')

(ii) the strict inequality -oo

oo,

holds for all branches of etfp if and only if k = 2b,

lmz

> 0.

(12')

Proof. Let, for definiteness, E > 0. Then q = 0, 1, ... ,p- 1. If p ?: 2, then there is a natural number q0 lying between 0 and p- 1 such that the fractional part of the ratio q0 kjp is greater than 1/2. Therefore inequality (11) cannot hold simultaneously for the two branches corresponding to q = 0 and q · q0 (since the arguments in the left-hand side of (11) will differ by more than 1r). Now let p = 1 and let k be even. Then the sign of the left-hand side of (11) coincides with that of Im z, whence follows conditions (12) and (12'). If k is odd, then the left-hand side of (11) has opposite signs forE and -E, which implies (13).

The proof of the Proposition. 1) We first consider the case of a polynomial Po(E, 1J) in two variables E lR and T E c. In view of the quasi-homogeneit·y~ the roots Toj(E) have the form Toj = ajetfq, the number q being obviously rational. Applying the lemma we prove the desired asser tion.

e

Chapter 2

54

> 1 we associate with the polynomial Po(6, ... , ~n , r) the polynomial P0 (A~t, . . . , A~n, r) depending on 6, ... , ~n as parameters. Applying

In case n

Q(A, r)

=

the already proved proposition to this polynomial and setting ,\ = 1 we prove the proposition. If condition (i) of the proposition is fulfilled and q = 1, then Po is called a homogeneous hyperbolic polynomial. We can also consider polynomials corresponding to odd q > 1 (the so-called q -hyperbolic polynomials). If q is a positive even number, then a (1, ... , 1, q)-homogeneous polynomial satisfying condition (ii) is said to be q-parabolic in Petrovski1's sense. In this case we shall often put q = 2b, b E Z+, and, accordingly, say that the polynomial is 2b-parabolic in Petrovskil''s sense.

1.3. Inhomogeneous 2~parabolic polynomials. Polynomial (1) solved with respect to the highest power of T (i.e. condition ( 4) holds: P0 1) is said to be q-parabolic ( q is even) or q-parabolic in Petrovski'l's sense if its principal (1, .. . , 1, q)-homogeneous part Pq(~, r) possesses this property. As was already said in the Introduction, with a polynomial P( 6, ... , ~n , T ), a polygon fl(P) is associated in a natural manner, which for n = 1 goes into Newton's polygon N(P). We denote by (aj , {3j), j = 0,1, ... ,m+ 1, the vertices of this polygon, which are indexed counterclockwise beginning with the vertex ( a 0 , (30 ) = (0, 0). As in Chapter 1, put

=

m+l 3A(P)(6, ...

,~n,T) =

L

~~~ailrl 13i,

(14)

j=O Theorem. Polynomial (1) solved with respect to the highest power ofT (i.e. P(~, r) = Tm + O(lrlm-l )) is q-parabolic if and only if

(a) the polygon tl(P) is a triangle with vertices (0, 0), (0, m ), and ( mq, 0); (b) there are c > 0 and /o < 0 such that (15) (a,/3)EA(P)

Proof. Necessity. Let conditions (a) and (b) be fulfilled. Replace ( 6, ... , ~n, T) in (15) by (,\~J, .. . ,A~n,Aqr), divide both sides by ,\mq, and pass to the limits as

,.\ ~ +oo. This results in the inequality (16) (Under the passage to the limit inequality (15) goes into inequality (16) in the open half-plane Im T < 0. However, since both sides of the inequality are continuous for Im T ~ -0, the inequality remains valid for Im T = 0.) Therefore conditions (9') and (10') are sure to hold, i.e. the q-homogeneous polynomial Pq is q-parabolic.

55

parabolic Operators Associated with Newton's Polygon

Sufficiency. We show that for a q-parabolic polynomial conditions (a) and (b) are fulfilled. It suffices to verify condition (a) for the q-principal part Pq. Since for Po :=: Pq condition (9') is fulfilled, we have Pq(O, ... , 0, T) = CTm, c -f. 0, and

Pq(6, ... '~n, 0) = L aal···an~fl ial=mq

... ~~n

# 0,

Thus, the points (0, 0), (0, m), and (mq, 0) belong to !::l(Pq), whence follows that ~(Pq) is a triangle with the indicated vertices. We now note that, by virtue of condition (a), we have

L

l~ l alri,B ~ x(1 + lrlm

+ l~lmq).

(a,,B)EA(P)

If the principal part Pq(6, ... , ~n, r) satisfies (10'), then, in view of the quasihomogeneity, (16) holds, whence

L l~la l ri,B ~ cxjPq(6, ... , ~n, r)j +X ~ cxjP(6, ... , ~n, r)j

+ {x + cxjP(~t, ... , ~n, r)- Pq(6, ... , ~n, r)l}.

There is c > 0 and a constant c1 such that the expression in curly brackets does not exceed Ct (1 + lrl + l~lq)m-e ~ (1/2)~1~1alri,B provided that Im T < /o and -1 >'"1o(c). This implies inequality (15). 1.4. Expansion into Puiseux's series of the roots of polynomials in two variables correct in Petrovski'l's sense.

Proposition 1 (Borok [1]). A polynomial P(~,r) = Tm

+ LPi(Orm-j,

(17)

j=l

satisfies Petrovski1's condition ( 5) (or ( 6)) if and only if the expansions of the roots Tj(e), j = 1, ... , m, into Puiseux's series in the neighborhood of I~ I = oo have one of the following three forms:

(a) where

Xjt

> ··· >

Xjt

>

0 are integers,

Xjl

is even, and

Im c j 1 = · · · = Im cj, 1-1 = 0, (b)

Tj(O

, Im cj 1 > 0;

= Cjt~xil + · · · + Cjtexil + 0(1),

~~~---? oo,

Chapter 2-

56

wbere Xjt numbers·

> ·· · >

Xjl

are natural numbers and tbe coefficients Cjl, ... , Cjf are real

'

(c)

rj(O

= 0(1)

1~1 ~ oo.

for

Proof. Since conditions ( 6) obviously hold for the above types of roots, it only remains to prove the necessity of the indicated expansions. As was noted in Section 1.1.4, Puiseux's series for the roots rj(O have, in general, the form

(18) Let Xjt = kjp be the greatest power exponent in this expansion. If Xj 1 ~ 0, then rj(O is a root of type (c), and the proposition is proved. In case Xjt > 0, we have

If condition (6) is fulfilled for some /o, then there is / l such that -- 00

<~<

oo,

Xjl = kjp.

In view of the homogeneity, it follows that /1 = 0. Applying Lemma 1.2 we conclude that Xjt is an integer and one of the two conditions Imcj 1

= 0,

Imcj1

> 0,

(19) Xjt

IS

even,

(20)

is fulfilled. Under condition (20) the root Tj(O belongs to type (a), i.e. the proposition is proved. Let (19) be fulfilled and let Cj 2 ~-"'i 2 be the subsequent expansion term. Then, if xi 2 > 0, we have

and, by virtue of inequality (6), there is 1 2 such that

Applying the lemma OJJ.Ce again we conclude that xi 2 is a natural number and conditions (19) and (20) (where Cjt and Xjl should be replaced by Cj 2 and Xj 2 , respectively) hold. Continuing this process we complete the proof of the proposition.

57

parabolic Operators Associated with Newton's Polygon

Proposition 2. Polynomial (17) is exponent.ially correct if and only if the expansions of the roots 7j({), j = 1, ... , m, into Puiseux's series in the neighborhood of { = oo have one of the following three forms:

(a') where Im Cjl > 0 and Xjl is a positive even integer;

(b')

7j({) = Clj{ + o(l),

(c)

7j(0

= 0(1),

Cjl =/= 0,

Imcjl = 0,

1{1 --too,

1{1--t 00.

Proof. According to Proposition 1, the polynomial P({, 7) is exponentially correct if and only if for any real ry Puiseux' expansions of the roots 7i ({ + i 1]) of the polynomial P({ + iry,7) have the form (a), (b), or (c). Necessity. Since the roots 7j({ + iry) are holomorphic functions of { + iry in the neighborhood of oo, for large { and small 1J the expansions of the roots 7j({ + iT]) of the polynomial P({ + iry, 7) are obtained from the expansions of the roots 7j({) by replacing { by { + iry. Applying the binomial formula we find . 7j({ + iry) = Cjl~;xjl

+ iXjlCj(rJ{"'i + o({"'i 1 ) + o({xi 1-

2)

1-

+ Cj2 {xi for 1{1--t oo.

1

2

(21)

If x 11 > 1 and Cj 1 is a real number, then expansion (21) does not belong to the types (a), (b), and (c). Consequently, either Xjl is even and Imcj 1 > 0 or Xjl = 1 and Cjl is real. Thus, the root Tj({) can have Puiseux' expansion only of the form of (a'), ( b'), or ( c').

Sufficiency. As has been in fact proved, expansions (a'), (b'), and (c') go into expansions of the same type when {is replaced by { + iry. According to Proposition 1, the polynomial P({ + iry, 7) is correct in Petrovski!'s sense for any ry, i.e. IS an exponentially· correct polynomial. §2. Two-sided estimates for polynomials in two variables satisfying Petrovski'l's condition. N-parabolic polynomials In this section we shall construct a theory "parallel" to the one presented in §1.2. We shall state necessary and sufficient conditions on a polynomial P({,T), { E R, 7 E C, under which the inequalities

(a,fj) E N(P),

Im7

~

{o,

(1)

hold. These estimates differ from the analogous estimates in Chapter 1 in that the exterior of a large circle in JR2 is replaced by a region lR x { 7 E C, Im 7 < lo}, where l'o is a sufficiently large number.

Chapter 2

58

2.1. The main esthnate. The change in the region where adequate estimates are considered makes us somewhat modify the notion of minor monomials. The matter is that in the estimates under consideration there is in fact a large parameter, and the classification as senior and minor monomials is performed so that the ratios of the absolute values of minor monomials to the sum of the absolute values of all monomials tends to zero as the corresponding large parameters increases. In Chapter 1 the role of the large parameter was played by the expression +7J 2 , which is what implied the definition of minor monomials given there (see Section 1.1.1). In the present chapter the role of the large parameter in estimates ( 1) is played by - Im T. Therefore the conditions on minor monomials can be weakened as compared to Chapter 1. An (integral) point (a, {3) E N(P) is said to be minor if there is a point (o:',/3') E N(P) such that a'~ o:, {3' > {3. According to this more general definition, the minor points of N(P) are not only the points on the coordinate axes and the interior points of the polygon but also the points belonging to the vertical side not coinciding with the coordinate axis (of course, provided that the polygon N(P) possesses such a side). If (o:j, (3j), j = 0, ... , m + 1, are the vertices of N(P), then, as in Section 1.2.1, we define the function

e

m+l

3N(P)(e, r) ==

2.::: IC~j

T,Bj

I·

(2)

j=O

For function (2) an exact analog of Lemma 1.2.1 takes place. Lemma. (i) If (a, (3) E N(P), then

le<:Yr,Bj < SN(P)(e,T) V(o:,(3) E N(P). (ii) There is x

(3)

> 0 such that (3')

where b(P) is the polygon spanned to the minor integral points of the polygon N(P). Assertion (i) is proved in the same way as the analogous assertion in Lemma 1.2.1, and assertion (ii) is a trivial consequence of the definition of minor points. Theorem. For a polynomial P(e,TJ) the following conditions are equivalent. (I) There are constants c > O.and /o such that the inequality Imr

< /o ,

(4)

holds. (II) For any side rjl) that does not lie on the coordinate axes and is not vertical we have

pq(i)(e, r)

f.

0

for

e

f.

0,

lrl

f.

0,

Im T::;; 0.

(5)

59

parabolic Operators Associated with Newton's Polygon

(III) There exists a set of vectors q(j) = (q~j), q~j)), q~j) ?:: 0, q~j) > 0, and a set of q(j)_homogeneous polynomials pUl(e, ry), j = 1, ... , J.l, such that

pUl(e, 7)

i= o

lei+ ITI > o,

for

Im T

:(

0,

(6)

and an integer b ?:: 0 such that _,....

p

I"V

P,

p=

7 b p(l)

... p[~tJ.

(7)

Remark. In the statement of condition (II) of the theorem (in contrast to the analogous condition in Theorem 1.2.1) no vertical side not lying on the coordinate axis is involved. This is due to the fact (already mentioned above) that the corresponding monomials are minor. The proof is a simple modification of that of Theorem 1.2.1. A slight change must be made only when deriving (III) from (II). Indeed, according to Theorem 1.1.3, the polynomial P(e, T) differs from the polynomial Tb p[l] ... p[~t] (where p(j] are constructed using formulas (1.1.15)) only in terms that are minor in the sense of Chapt er 1 and, the more so, minor in the broader sense of the present chapter. If the polygon N( P) contains no vertical sides, then all polynomials p(jJ satisfy conditions (6). Moreover, if N(P) does not contain horizontal sides, then all polynomials p[i] are qULparabolic in Petrovski'l's sense. If the side is horizontal, then the polynomial p[l](~, ry) does not depend on T and is nonzero for all~ E R. Now let the side r~) be vertical so that p(mJ(e,T)- p[m](T) = aTb + O(Tb-l). Here the polynomials p[lJ(e,T) ... P[mJ(e,T) and P(e,T) = aTb.p[lJ ... P[m-lJ(e,T) differ in terms minor in the sense of the present chapter. The theorem is proved.

r;l)

2.2. Stability of polynomials admitting of estimate ( 4). Repeating the argument in Section 1.2.4 we can show that if a polynomial P( T) satisfies the conditions of Theorem 2.1, then the "perturbed" polynomial

e'

(8)

also satisfies these conditions for sufficiently small c and, the more so, Petrovski'l's condition (1.5). Hence, the polynomials satisfying the conditions of Theorem 2.1 are interior points of the finite-dimensional set of polynomials satisfying condition (1.5) and having a given Newton polygon. We have the following

Theorem. For a polynomial P(e, ry) conditions (I), (II), and (III) of Theorem 2.1 are equivalent to the following conditions. (IV) There are co > 0 and /o such that for lei Petrovski1's condition, i.e.

< co all polynomials (9) satisfy (9)

Chapter 2

60

(V) There are constants c1 > 0 and /'I such that (10)

e

where dr(O is the distance from the point to the manifold of complex roots of the polynomial P( z, T) for a fixed T. P~oof.

As has already been said, the implication (I)==?(IV) is proved in the same way as the analogous assertion in Theorem 1.2.4. (IV)==?(V). Each point z belonging to the circle jz- e1 < c:!el can be represented as z = e+ h~, lhl < c:. Therefore the polynomial P(z, T) can be written in the form (8) (cf. the proof of Theorem 1.2.4) and use can be made of (9). The proof of (V )==?(I) will be given in §4 where a generalization of this theorem to the case E IRn, n > 1, will be presented.

e

2.3. N-parabolic polynomials. In §1.3 we introduced the class of N quasielliptic polynomials that were hypoelliptic polynomials satisfying the two-sided estimate 1.2.1. We now study hypoelliptic polynomial satisfying the conditions of Theorem 2.1. We shall call them N -parabolic polynomials. We begin with an analog of Proposition 1.3.2. Proposition. For a polynomial P(e, T) satisfying the equivalent conditions in Th eorems 2.1 and 2.2 the following conditions are equivalent.

(i) P is a hypoelliptic polynomial. (ii) Newton's polygon N(P) is regular. (iii) There are x > 0, c > 0, and /'I such that (11) for any integral point (a,/3) E b0 (P) 1 ). Proof. (i)==?(ii) is true by virtue of Lemma 2 in Section 1.3.2. (ii)==?(iii) is proved like the analogous assertion in Proposition 1.3.2. (iii)==?(i). From (11) it follows that P,(Cry) = P(e,ry + il') is a hypoelliptic polynomial for I'< /'O· Since the polynomials P,(e, ry) and P(~, ry) differ by a linear combination of the derivatives of P1 , the polynomial P is also hypoelliptic.

Theorem 2.1 and the above proposition imply Theorem 2). The following definitions of anN -parabolic polynomial are equivalent . (I) There are c

> 0,

/'o such that

lm T ~ /'o, l )Recall (see Section 1.3.2) that a point (o-:,(3) belongs to t5 0 (P) if there is a point (o-: 1 ,(3 1 ) E N(P) such that o: ~ o:' and (3 ~ (3' , one of these inequalities b eing strict. 2 )Cf. Theorem 1.3.3 .

61

Parabolic Operators Associated .with Newton's Polygon

for any points (a,(3) E N(P). Moreover, there is x > 0 such that for all integral points (a,(3) E 80 (P) inequality (11) holds. (II) N(P) is a regular Newton polygon, and for its any side rjl) not lying on the coordinate axes the condition · (12)

holds, where q(j) is the outer normal to rjl). (III) Let Tj(e) = cjei be the principal parts of Puiseux' expansions for 1~1 ~ CX> of the roots of the polynomial P . Then bj is an even integer and Im Cj > 0. (IV) There exists a set of pairwise disproportionate vectors q<j), j = 1, ... , m, with positive components and a set of qU) -parabolic polynomials p[i] ( ~, r) such that P "' p[l] ... p[m], the polynomials pUJ being determined to within a constant factor. (V) The polynomial P is hypoelliptic and satisfies estimate ( 4 ). The proof of the theorem is a modification of that of Theorem 1.3.3. (I)~(II) follows from Theorem 2.1 and the above proposition. (II)~(III). As in Theorem 1.3.3, denote by Tjz(e) those roots of P whose principal parts are of the same degree, i.e. T'j[ = Cjzei + According to Theorem 1.1.4, Cjzei are the roots of the polynomial plil ( ~, r) related to Pqu), q<j) = ( 1, bi ), by (1.1.15). The polynomials plil(~, r) essentially depend on both~ and r (N(P) has neither vertical nor horizontal sides) and are solved with respect to the highest powers of both ~ and r. It follows that the condition

ocej ).

p[il(~,r)

f

0

for

~

f

0,

lrl f

0,

Imr ~ 0,

(13)

remains valid for I~ I + lrl > 0 and Im T ~ 0. Therefore the polynomial plil is qULparabolic. Applying Proposition 1.2 we obtain (13). (HI)~(IV)~(V) is proved according to the same scheme as the analogous assertion in Theorem 1.3.3. (V)~(I) follows from the proposition. 2.4. N-stable correct polynomials. As was already shown in Section 2.2, the class of polynomials satisfying the conditions in Theorem 2.1 is stable relative to the addition of any monomials ce)trP, (a,(3) E N(P) with sufficiently small coefficients c. By N -stable correct polynomials we shall mean the ones that satisfy the conditions of Theorem 2.1 and are solved with respect to the highest power of r. Lemma. For a polynomial P(~, r) satisfying the conditions in Theorem 2.1 the following conditions are equivalent:

(i) the polynomial P is solved with respect to the highest power of r (i.e. P is an N -stable correct polynomial); (ii) the polynomial P is exponentially correct.

Chapter 2

62

Proof. (i)==>(ii). Generally speaking, the poly.o....._::.mial p(a)(C 1) is a linear combination of monomials C'1f3, (x, /3) E 8o(P). If the polynomial P is solved with

respect to the highest power of 7 1 then the polygon N(P) has no horizontal side not lying on the coordinate axis. Repeating the argument of Lemma 4 in Section 1.3.2 it is easy to prove that in this case 8o(P) = 8(P), where 8(P) is the convex hull of the minor integral points (in the sense of Section 2.1) of N(P). Applying Lemma 2.1 (ii) we conclude that for some x > 0 we have

(ii)==>(i) is true by virtue of Lemma 1.1. In view of the above results, we have the following Theorem. For a polynomial P the following conditions are equivalent.

(I) The polynomial P is N -stable correct. (II) The polygon N(P) bas no horizontal side not belonging to the coordinate axis, and for any side r}l), that does not lie on the coordinate axes and is not vertical condition (12) is fulfilled. (III) IfT.J(e) = ciei are the principal parts of Puiseux' expansions for lei-+ oo of the roots of the polynomial P, then either condition (13) is fulfilled or 7-i(O o (i.e. -rj(e) = 0(1)). (IV) There is an N -parabolic polynomial P 1 ( 1) and an integer b ~ 0 such that P rv 7b P1.

=

e,

Proof. The proof of (I):=>(II):=>(I) is based on Theorem 2.1 and the equivalence

of the conditions "the polynomial Pis solved with respect to the highest power of 7 11 and "Newton's polygon N(P) has no horizontal side not lying on the coordinate axis". (I):=>(III). By virtue of the lemma, the polynomial P is exponentially correct, and, according to Proposition 2 in Section 1.4, either the principal parts Tj(e) of its roots have the form (13) or 7-j(e) = 0 or 7-j(O = aje where aj is real. Hence, it only remains to show that an N -stable correct polynomial has no roots of the form of Imaj = 0. (13') Indeed, assume that there is a root having the form (13') . Consider :the polynomial Q(e, 7) =ice rrl¥/ 7-rz(e)). This polynomial is a linear combination of monomials corresponding to the points of the polygon N(P) . According to Theorem 2.2, the polynomial P + Q does not satisfy (1.5) for a sufficiently small c. And since the degree of Q relative to 7 is less than the degree of P relative to 7, the polynomial P + Q is correct in Petrovski'i''s sense and is solved with respect to the highest power of 7. The structure of the polynomial P + Q implies that one of its roots has Puiseux' expansion whose principal term is

63

parabolic Operators Associated with Newton's Polygon

(ai- ic)e. 1) If cis real and

c;

> 0, we arrive at a contradiction to Proposition

1 in

Section 1.4. (III)===>(IV) is easily proved using Theorems 1.3 and 1.4. (IV)===>(!) readily follows Theorem 2.3.

e'

Corollary 1. Let P( T) be an N -stable correct polynomial and let (a' /3) be a minor point of N(P). Then the polynomial

(14)

is N -stable correct for any (complex)

c;.

Proof. According to the results in Section 1.1.4, if the monomial ce):rf3 is added to the polynomial P, the principal parts of Puiseux' expansions of its roots do not change. Since f3 is less than the highest degree relative to r, the polynomial Pe remains being solved with respect to the highest power of r. It now remains to use definition (III) of N -stable correct polynomials. Thus, if the polynomial P is N -stable correct, then the polynomials Pe are correct in Petrovski1's sense for small lei and any (a, /3) E N(P), and for arbitrary c; and (a,/3) E 8(P). Let a polynomial P(e, r) be solved with respect to the highest power of r and let pUl(e, r) be the quasi-homogeneous polynomials related toP by (1.1.15). According to Section 1.1.4 the roots of these polynomials are the principal parts Tj(O of the expansions of the roots of P into Puiseux' series in decreasing powers of Therefore Proposition 1 in Section 1.4 implies that the conditions

e.

p[il(~, r) -=J 0

for

Im r

<0

(15)

are necessary for polynomial P to satisfy the correctness condition (1.5). If condition (III) of the theorem holds, then (15) should be replaced by the stronger condition p[il(e, r) -=J 0

for

lei+ lrl > 0,

Im T ~ 0.

(16)

Hence, we have

T

Corollary 2. A polynomial P((, r) solvecl·with respect to the highest power of is N -stable correct if and only if conditions (16) are fulfilled for j = 1, ... , n.

In view of what has been said, we state the following assertion that is of use for our further aims. we use the fact that the polynomials P and P = I1 (r - rt) differ only in minor monomials, and, in view of Sec~on 1.1.4, when determining the principal parts of the roots , the polynomial P can be replaced by P. 1 ) Here

Chapter 2

64

Proposition. Let a polynomial P( ~, ry) be solved with respect to the highest power Tm and let f3 < m in polynomial (14) . Then (i) if polynomial (14) satisfies the correctness C!Jndition (1.5) for all sufficiently small lei (c E C), then (a, (3) E N(P); (ii) if the polynomial remains correct for any c E C, tl1en (a, (3) is a minor point of N(P) . Proof. (i) Let the point (a, (3) lie outside N(P). Then it is one of the vertices

of N( Pe ), say the vertex fJ~ 1 , and let r~ = (a j, /3j ), /3j > (3, be the neighboring vertex of N(Pe) (which simultaneously is a vertex of N(P)). According to what was said above, the quasi-homogeneous polynomial PJjJ corresponding to the side r)0 )fJ~ 1 must satisfy condition (15) for sufficiently small c. However, as can easily be seen (see (1.1.15)), we have

pJjJ =

Tf3i-f3

+ (c/aaif3i )~a-ai,

and c can be selected so that the roots of pUl have negative imaginary parts, which contradicts (15), (ii) By what has been proved in (i), it suffices to consider the case when the point (a, (3) lies on the boundary of N(P), say on the side joining the vertices r}o) and

r)~l. Then the quasi-homogeneous polynomial pjjJ --: p[j] + c~ 0 Tf3 must satisfy a condition of type (15) for any c E C. However, this cannot hold since for arbitrary T 0 E C, ~ 0 E lR there is c such that PJjJ ( ~ 0 , T 0 ) = 0 ( c is determined from the linear equation p[j](~ 0 , T 0 ) + c(~ 0 ) 0 ( T 0 )f3 = 0).

§3. Cauchy's problem for N-stable correct and N-parabolic differential operators in the case of one spatial variable In this section we deal with differential operators

P(x, t, Dx, Dt)

=L

aap(x, t)D~ nf

m

=L

Pj(X, t, Dx)D';-j

(1)

j=O

on functions of the variables (x, t) E !R2 , where x is called a spatial variable, t is time, Dx ~ -i a;ax, and Dt = -i a;at. By T =a+ i, and~ we shall denote the dual variables: ((T,0,(t,x)) =tT+xC In what follows we shall assume, without special stipulation, that operator (1) is solved with respect to D~, i.e. Po(x,t;~)

Moreover, we shall assume that Po(x, t)

=Po(x,t).

= 1, i.e.

operator (1) is written as

m

P(x,t;Dx,Dt) =

nr: + l.:,Pj(x,t,Dx)Dr;- 1. j=O

(2)

65

Parabolic Operators Associated with Newton's Polygon

3.1. Differential operators with constant coefficients. A differential operator P(Dx, Dt) of the form of (1) with constant coefficients is said to be correct in Petrovski'l's sense or N -stable correct or N -parabolic if the polynomial P( ~, T) (the symbol of the operator P) possesses the corresponding property. When dealing with operators correct in Petrovski'l's sense it is convenient to pass from the spaces H(s) (see Section 1.4.1) to the spaces H[~f consisting of functions (distributions) u(x,t) such that exp(1t)u E H(s). The definition of the Fourier transform of a decreasing functions implies that

(exP"frt)u)(~, a)= (27r)-l

JJexp( -i~X- i(a + i1)t)u(x, t) dx dt = u(~, T).

By virtue of the definition of the spaces H(s), the inclusion u E H[~~ means that the integral

(j j (1 + e + a 2 ) 8 1u(~, a+ i1)l 2 d~ da)

112

def

II exp( ,t)ull<s)

is convergent. As a norm in HI~f it is more convenient to take the expression

(3) which is equivalent to the norm II exp( 1t)ul!(s), the equivalence constants depending on/· For positive integral values of s, applying Newton's binomial formula to the integrand:

(1

+

e + a2 + !2)s

=

sl

""'

. ea(a2

.

L,;

a+,B~s

od/3!( s - a - ,B)!

+ 1'2)2,8,

we conclude that norm (3) is equivalent to

!!exp(!t)D~Dfull 2 ) 112 ,

(L

(3')

a+,B~s

the equivalence constant not depending on 1 in this case. In the case of non-integral s, using the PDO with symbol (1 + + a 2 + 1 2)sf 2, we can rewrite norm (3) in the form

e

llull(s),/

= 11(1 + n; + n; + !

2 ) 1 12 (exp( ,t)u)ll·

(3")

By,. analogy with (1.4.15), with polynomial (1) we associate the norm !lull N(P),(s),/

= (

""' L._;

.

,B

liD~ Dt ull~s), 1

) 1/2 ·

(4)

(a,,B)EN(P)

Denote by H[~J(P),s the space of functions with finite norm ( 4). Let H[~~+ and

H[~]<;),s

denote the subspaces of functions equal to zero fort < 0 (and, respectively, the subspaces of distributions with support belonging to the half-space t :( 0).

Chapter 2

66

Theorem 1. For a differential operator (2) with constant coefficients the following conditions are equivalent. (I) The polynomial P( ~, T) is N -stable correct. (II) 3,0 , c > 0 such that \:Is E lR the inequality

~ cllP(Dx, Dt)ull(s),l'

lluiiN(P),(s),-y

(5)

holds. (II+) 3,0 , c > 0 such that Vs E lR the inequality lluiiN(P),(s),-y ~

cllP(Dx, Dt)ull(s),,,

holds. Proof. The equivalence of (I) and (II) follows from the definition of N -stable correct polynomials (see Section 2.4) and norms (3), (4) (cf. Theorem 1.4.2). Inequality (5+) is a special case of (5). It remains to show that (II+)===}(II). To prove this it should be noted that if v(x, t) = u(x- xo , t- t 0 ), then v( ~, T) = exp( -i~XO - iato + rto )u( ~, T), whence llvll(s),-y

= exp( rto)llull(s) ,/' llviiN(P),(s),-y = exp( rto)llui!N(P),(s),,-

By virtue of these relations, the function u( x, t) in ( 5+) can be replaced by the translated functions u( x, t - to). Since the set of the translated functions is dense in H[~J), we obtain (5). In an analogous way one proves Theorem 2. For a differential operator (2) the following conditions are equivalent: (I) the polynomial P( ~, T) is N -parabolic; (II) 3ro, c > 0, c' > 0, x > 0 such tl1at Vs E JR, along witl1 (5), the inequality llullo(P),(s+x),-y

~ c'IIP(Dx, Dt)ull(s),-y Vu

E

H[~J),

r ~ ro,

(6)

holds.

(II+) 3ro, c > 0, c' > 0, x > 0 such that \:Is E JR, along with (5+), the inequality (6) is fulfilled on the space H[~]l · Remark ( cf. Remark 1.4.2). In the left-hand norm in (6), in contrast to the analogous norm {1.3.4), the summation is extended over t he integral points of 8( P) and not of 8o(P). This is due to the fact at the very beginning we confined ourselves to symbols solved with respect to the highest power of T. As was already said in the foregoing section, in this case Newton's polygon contains no horizontal side, and, consequently, 80 (P) = 8(P). l) Here

for -y

11

<

we use the fact that for the spaces -y1 •

Hf;f+ the natural embeddings H[;/l+ C HG/'l+ hold

67

parabolic Operators Associated with Newton's Polygon

3.2. N-stable correct and N-parabolic differential operators with vari-

able coefficients. Consider operator (1), (2). At fixed (x , t) its symbol P(x, t; ~' T) is a polynomial in ~ and r, and it is possible to define the polygon N(P(x, t)) of the polynomial. As in Section 1.4.3, denote by N(P) the convex hull of the union of the polygons N(P(x, t)). Definition ( cf. Definition 1.4.3). Differential operator (2) is said to beN -stable correct ( N -parabolic) if (i) N(P(x, t)) = N(P) 'V(x, t) E JR2 ; (ii) for any fixed ( x 0 , t 0 ) E R 2 the polynomial P( x 0 , t 0 ; ~, T) is N -stable correct ( N -parabolic). Since the above-defined operators (in contrast to N quasi-elliptic operators in §1.4) will be considered globally, we shall have to make some additional assumptions concerning the behavior of the coefficients of operator (2) for lxl + !tl ~ 0. For simplicity, we shall assume that the functions aa(3(x, t) in (2) are constant outside a sufficiently large circle x 2 + t 2 = R 2 , i.e. a~,a(x, t) E

V.

(7)

Repeating almost literally the proof of Proposition 1.4.3 1 ) one can prove the following Proposition. Let a symbol P(x ,t;~,T) satisfy conditions (i) and (ii) of the definition and let condition (7) hold for the coefficients aa,a(x, t). Then

(i) the coefficients of the symbols p!il ( x, t; ~, T) defined for any ( x , t) by formulas (1.15) satisfy conditions of type (7); (ii) there are constants c > 0 and lo such that for all (x, t) E IR2 the inequality ~ E IR,

Im T :S; lo,

(8)

holds. Remarks. 1) The symbols P(x,t;~,T) for which (8) is fulfilled are said to be uniformly N -stable correct. The uniform N -stable correctness of a symbol implies conditions (i) and (ii) of the definition. In general, the converse assertion is false without some additional assumptions concerning the behavior of the symbol at infinity. ·. : However, all the results below on operators (2) remain valid if conditions (i) and (ii) of the definition and condition (7) are replaced by the condition of uniform N -stable correctness and the assumption of uniform boundedness on JR2 of all derivatives of the functions aa,a(x, t). l)Condition (7) allows us in fact to deal with functions on a compactum (as in the case of P roposition 1.4.3).

Chapter 2

68

2) When N(P) is a right triangle with vertices (0, 0), (0, m), and (mq, 0), where q is even, our definition goes into the definition of operators with variable coefficients q-parabolic (or, simply, parabolic) in Petrovski'i''s sense. For such symbols inequality (8) is equivalent to the following uniform estimate for the p rincipal q-homogeneous part of P0 (x, t; ~' -r):

c- 1 ~ IPo(x, t; ~' -r)l(l(lq + 1-rl)-m ~ c V(x,t)E~?,

V(~,Re-r)E~_2,

Im-r~O.

(9)

And this estimate is equivalent to the uniform positivity of the imaginary parts of the roots Toj(x, t) of the symbol P0 . As in §1.4, with symbol (1) we associate the space SLN(P) of the symbols

(10) for which N( Q) C N(P) and the coefficients bo:p(x, t) satisfy conditions of type (7). As in §1.4, we denote by 8(P) the convex polygon spanned to the minor integral points of N(P). And let S£N(P) denote the subspace of those Q E SL N (P) for which N(Q) C 8(P). Modifying the proof of Theorem 1.4.3 we prove the following Theorem. (i). Let operator (2) be N -parabolic and let its coefficients satisfy condition (7). Then there are

(a) a set of even numbers q1 > · · · > qJL > 0; (b) a set of operators p[il(x , t;Dx,Dt), j = 1, ... ,J.1, qj-parabo1ic in Petrovskii"s sense, with coefficients satisfying a condition of ty pe (7); (c) and a symbol Q(x, t; ~' -r) E S£JV(P) such that P(x, t; Dx, Dt)- p[Il(x, t; Dx , Dt) ... p[JL](x, t; Dx, Dt)

= Q(x , t; Dx, Dt)·

(11)

(i;.) Let operator (2) beN -stable correct and let its coefficients satisfy condition (7). Then there are (a) anN-parabolic operator P 1 (t,x ; Dx,Dy) with coefficients satisfy ing (7); (b) an integer b > 0; (c) and a symbol Q(x, t; ~' -r) E

SCN(P)

such that

(11')

3.3. A priory estimates for N-stable correct and N-parabolic operators with variable coefficients. In this section we shall prove analogs of Theorems 1 and 2 in Section 3.1 for the case of variable coefficients.

69

Parabolic Operators Associated with Newton's Polygon

Theorem 1. For an operator (2) with coefficients of the form of (7) the following conditions are equivalent. (I) The operator is N-stable correct (in the sense of Definition 3.2). (II) Vs E R there are c = c( s) > 0 and "'(o = "'(o ( s) such that V7 ~ "Yo the inequality

(12) balds. (II+) Vs E R there are c = c( s) inequality lluiiN(P),(s),-y ~

> 0 and "Yo

"Yo( s) such tl1at V7 ~ "Yo the

ciiP(x, t; Dx, Dt )u II (s),-y Vu

E

( oo) H [I]+

is fulfilled.

The proof of the theorem is preceded by two lemmas. Lemma 1. Let a(t, x) E

c=

and let

sup

ID~D~a(x,t)l <

Xc:x!J·

(t,x)EIR 2

Then Vs E R and V7

<

0 the inequality

holds, where the constant c8 (a) depends only on the constants Xa!J, where vis a fixed number not depending on a(x, t).

a+ f3 = Is I+ v , .

=

The proof is obvious for s = 0 (and in this case co( a) 0) . In the case of arbitrary real s it is based on the technique of pseudodifferential operators. For this lemma in the indicated form see Volevich and Gindikin [6, Lemma 3.2]. Lemma 2. L et (x 0 , t 0 ) be a point in R 2 , let P(Dx, Dt) = P(x 0 , t 0 ; Dx, Dt) , and let S( x 0 , t 0 , r ) be a circle of radius r with center at ( x 0 , t 0 ). Then for a sufficiently small r and a sufficiently large -70 the inequality (12) on the functions u E V with support in·S( x 0 , t 0 , r) is valid if and only if there holds an analogous inequality for the operator with constant coefficients P(Dx, Dt) .

Proof ( cf. the proof of Theorem 1.4.4.). Take a function lj;(x, t) E 1J equal to 1 in the circle S( x 0 , t 0 , r) and equal to zero outside the circle S ( x 0 , t 0 , 2r). If u( x, t) is a function with support lying in S( x 0 , t 0 , r ), then we have

Chapter 2

70

where

According to Lemma 1, we have IIIP(x, t; DXl Dt)ull(s),-y -IIP(Dx, Dt)ull(s),-rl

~

(L

max

(x,t)ES(x 0 ,t 0 ,2r)

~

L

lla~,a(x, t)D~ nfull(s),-y

la~,a(x,t)l+ci(a)IJI - 1 )11uiiN(P) ,(s)

For any £ < 1 there is a sufficiently small r for which the sum on the right-hand side does not exceed c/2. We then select /o such that c1 IJI- 1 < c/2 for 1 ~ /o. Hence, the right-hand side does not exceed cii!IIN(P); and therefore, if inequality (12) holds, then (4) is fulfilled, and if (4) takes place, then (12) also holds. The proof of Theorem 1. (I)==?(II). Using Theorem 1 in Section 3.1 and Lemma 2 we prove inequality (12) for functions with sufficiently small supports. And we now "glue" together the local estimates by means of a partition of unity. If the coefficients of operator (2) satisfy condition (7), then there is a circle S(O, 0, R) with center at the origin, outside which symbol (2) has constant coefficients. We select a set of functions Xj(x, t) possessing the following properties: (a) Xj(x, t);;:: 0, Xi E C 00 , j = 0, 1, ... , J; (b) LXj(x,t) 1 'v'(x,t) E IR2 ; (c) 'v'j E {1, ... ,1} there is (xi,ti) E IR2 such that the support of Xi belongs to a sufficiently small neighborhood of (xi, ti); the function xo(x, t) is equal to 0 outside the circle S(O, 0, R) and to 1 inside a circle S(O, 0, R'), R' > R. As was mentioned above, by virtue of Lemma 2 and Theorem 1 in Section 3.1, we have the inequalities

=

Adding together these estimates and applying Minkowski's inequality we obtain, in view of (b), lluiiN(P),(s),-y

= II L ~

L

XiuiiN(P),(s),-y

llxiuiiN(P),(s),-y

~c

L IIP(x, t; Dx, Dt)(xju)ll(s),-y(13)

According to Leibniz' formula, we have

P(xju) = XiPu +

L (a,,8)E6(P)

bja,aD~Dfu.

71

Parabolic Operators Associated with Newton's Polygon

With regard to this inequality, estimate (13) takes the form llttiiN(P),(s),/ ~ CtiiP( x, t; Dx, Dt)ttll(s),/

L

+ c2

liD~ D~ ttll(s),/l

(14)

(a,/1)E6(P) where the constants c1 and c2 do not depend on 'Y· Lemma 1.2 (ii) and the definition of the spaces H[~f imply the inequality

If -'Yo is sufficiently large, then for 1 ~'Yo the second term on the right-hand side of (14) does not exceed (1/2)llui!N(P),(s), 1 , whence follows (12). (II)==?(II+ )==?(I). The first implication is trivial, and we proceed to the proof of the second one. We have to show that (12+) implies conditions (i) and (ii) in Definition 3.2. If inequality (12+) holds, then, by virtue of Lemma 2, the operator P(Dx, Dt)P(x 0 , t 0 ; Dx, Dt) ((x 0 , t 0 ) is an arbitrary point in JR.2 ) satisfies an analogous inequality on the functions whose support is concentrated in the neighborhood of (x 0 , t 0 ). Therefore (cf. Theorem 1 in Section 3.1) an analogous inequality is retained for the translated functions. Since the set of the translated functions is dense in H[~f, we arrive at (5+) or (5), whence it follows that the symbol P(~,r) is N-stable correct. Let apoint (o:,{3) belong to the polygon N(P(xl,t 1 )) corresponding to the polynomial P(xl,t 1 ;~,r). Then

It follows (cf. Theorem 1 in Section 3.1) that

l~ar.BI ~ const IP(x 0 ,t 0 ;~,r)l

for

Imr ~'Yo·

A simple modification of the proof of Lemma 1.2.2 shows that ( o:, {3) E N(P(x 0 , t 0 )), i.e. N(P(x 1 , t 1 )) C N(P(x 0 , t 0 )). Since the points (x 0 , t 0 ) and (x 1 , t 1 ) are arbitrary, all the polygons N(P(x, t)) coincide. The theorem is proved. Theorem 2. For an operator (2) with coefficients oftl1e form of(7) the following conditions are equivalent.

(I) The operator is N -parabolic. (II) Vs E JR. there are c = c( s) > 0, x > 0, and 'Yo (1 2), the inequality lluii6(P),{s+x),-y ~ ciiP(x, t; Dx, Dt)uil(s), 1

= 'Yo( s) such that, along with VuE H[~l),

1 ~'Yo

(15)

holds. (II+ ) Under conditions (II), along with (12+ ), the inequality (15) is fulfilled for H (oo) U

E

bo]+"

72

Chapter 2

Proof. (!)==}(II). According to Theorem 1, inequality (12) holds. The definition

of parabolic polynomials and condition (i) in Definition 3.2 imply that llull6(P),(s+x), 1 ~

const llullN(P),(s),,. Using (12+) we obtain (15). Since the implication (II)===}(II+) is trivial, it remains to verify (II+ )==}(I). From what was proved in Theorem 1, it follows that the operator (2) i"> N -stable correct, and therefore we have to verify only the regularity of the polygon N(P). This property is proved by means of a literal repetition of the corresponding part of the proof of Theorem 1.4.4. 3.4. Cauchy's problem for N-stable correct differential operators with variable coefficients. By Cauchy's problem for operator (2) solved with respect to the highest derivative with respect tot is meant the problem of determining the function u(x, t) satisfying the equation

P(x, t; Dx, Dt)u(x, t) = f(x, t) ,

t > 0,

(16)

and the initial conditions

(D{- 1 u )(x, 0) = 'Pj(x ),

j = 1, .... m.

(17)

By means of the transformation m

u = v

+

L ti-l

(X) I (j - 1)!

j=l

problem (16), (17) can be reduced to an analogous problem with zero initial data:

(D{- 1 u)(x,0)=0,

j=1, ... ,m. (17') If conditions (17') are fulfilled, then Equation (16) remains valid if the functions u(x, t) and f(x, t) are extended as zero tot < 0, i.e. if u(x,t) = f(x,t) = 0 for t < 0. (18) Problem (16), (18) will be called the homogeneous Cauchy problem. It can be reformulated as the problem of inverting the continuous operator N(P),(s)) + ~ H(s) (Hb) b)+

( u ( x,t ) ~----+ P( x,t; D

x,Dt ) u ( x ,t )) .

(19)

Along with operator ( 19), we shall consider the operator HN(P),(s) ~ H(s) [I) hJ

(u

~----+ Pu)

(20)

defined throughout the space. The corresponding problem can be interpreted as Cauchy's problem with zero initial datafor t ~ -oo. Theorem. For differential operator (2) with coefficients of the form of (7) the conditions below are equivalent. (I) The operator is N -stable correct. (II) For any s E IR tl1ere is {o = lo(s) such that for 1 ~ /o the mapping (20) is bijective. 1 ) l )That is, operator (20) has no kernel and its im age coincides with the right-hand space.

73

Parabolic Operators Associated with Newton's Polygon

(II+) For any s E lR there is !o = !o(s) such that for 1' bijective.

~ l'o

the mapping (19) is

Proof. (II):=;.(I), (II+ )=;.(I). If operators (19) and (20) are bijective, then, by Banach's theorem, they possess continuous inverse operators

R.

•

H(s)

[-y] --+

HN(P),(s)

hl

'

R . +·

H(s)

hl+

--+

(HN(P),(s)) hl +·

(21)

The boundedness of these operators implies a priori estimates (12) and (12+ ). Applying Theorem 1 in Section 3.3 we prove that operator (2) is N -stable correct. (I)====?(II), (II+). The continuity of operators (19) and (20) follows from the definition of the space H~l(P),(s), and the absence of the kernel is implied by a priory estimates (12), (12+) in Theorem 1 in the foregoing section. Thus, the theorem reduces to the proof of the surjectivity of the operators (19) and (20). Since both the assertions are proved in a similar way, we shall consider only the first of the operators. So, we are going to prove that if -10 is sufficiently large, then Equation (16) possesses a solution u E (H~l(P),(s))+ for any right-hand side f E H[~f+' 1' ~ l'O· To prove this we make use of Theorem 3.2, according to which (22) where the symbols of the operator Q belongs to SLN(P) and the operator Po is a composition of operators parabolic in Petrovskil''s sense and the operator of differentiation with respect to t: (22') Lemma. There is 1'1 such that for 1' :::;; 1 1 there exists a continuous operator

R

.

0 +.

H(s)

hl+

(HN(P),(s)) --+

hl

(23)

+

serving as the right inverse operator of P0 , i.e. P0 R 0 f =

f

Proof. According to the well-known results for operators parabolic in Petrovskil''s sense (e.g. see Agranovich and Vishik (1]1)), the operators (w ~--t p[ilw)

possess continuous inverse operators

l) An independent proof of the solvability of Cauchy's problem for operators parabolic in Petrovskil's st:>nse will be in fact given in §5.

74

Chapter 2

for a sufficiently large -1. The ordinary differential operator p(o] = D~ possesses the same property, and we denote its right inverse operator as R(o] . We select N > deg P and consider the operator

(24)

HGt::>,

on the space 1 < 12. This operator transforms H[~]~N) into itself and is the right inverse operator of P0 on this space. We shall show that it can be extended to continuous operator (23). Indeed, the operator Po is N-stable correct and N(Po) = N(P). Therefore there is 'Yo such that for 1 < 'Yo the inequality

is fulfilled. Replacing v by Rog, g E H[~]~N), and using the property that P0 R 0 g we obtain

=g

i.e. operator (24) that was originally defined on H[~]~N) is continued by continuity to operator (23). We now prove the surjectivity of operator (19). To this end the solution u E (H[~(P),(s))+ to Equation (16) is sought in the form u = R 0 g, where g E H[~f+ and Ro is the operator in the lemma. Then for g the equation

g + Q(x, t; Dx, Dt)Rog = is obtained. Since Q E

SCN(P),

f,

(25)

we have

IIQRogll(s), 1 ~ const IIRogii8(P),(s), 1

~ c("f)IIRogiiN(P),(s),/ ~ cl('Y)IIYII(s),1 ,

Hence, Vc

c1(1)-+ 0,

1-+ -oo.

< 1 there is 1 such that

By virtue ofthis inequality, Equation (25) possesses a solution g E H[(s]) which is . I+ determmed by Neumann's series 00

g=

2) -QRo)kf. k=O

The theorem is proved.

75

parabolic Operators Associated with Newton's Polygon

§4. Stable-correct and parabolic polynomials in several variables In this section we extend the results of §2 to the case of polynomials in the variables~= (6, ... '~n) E IRn' T E C:

(1) In the Introduction we associated with each polynomial (1) a polygon D.(P). We now present necessary and sufficient conditions on polynomial (1) under which the inequality

(2) (a,,B)E~(P)

is fulfilled. Among the polynomials satisfying (2) we separate out analogs of Nstable correct and N -parabolic polynomials. In the multidimensional case the main idea is that the polynomial P( ~, T) is regarded as a polynomial in the two variables T and p = 1~1, depending on the parameter w = ~/1~1· Accordingly, we embed the class of polynomials in the class of pseudo-polynomials, i.e. polynomials in T and p depending smoothly on w. In this broader class we manage t o obtain an analog of the factorization in §2, in which t he role of polynomials parabolic in Petrovskil''s sense is played by pseudopolynomials parabolic in Petrovskil''s sense. 4.1. Polynomials and pseudo-polynomials. We introduce in IRn the polar coordinates ~ = pw, p = 1~1, w = ~/1~1; the angular variable w runs over the unit sphere sn-l. By a pseudo-polynomial will be meant a function

Q(~,T)

= Qw(I~I,T) =

L

bx,a(w)l~lxT.B,

(x,,B)Evq

where VQ is a finite set of nonnegative integral points and bxf3 are functions belonging to C 00 (sn-l ). Every polynomial (1) can be rewritten as

(3) where

C!: '."f3(w) = i.e. polynomial (1) is a pseudo-polynomial. For a fixed, w the pseudo-polynomial Qw is a polynomial in two variables, and therefore we can define Newton's polygon N(Qw) and the polygon 8(Qw) for Q. We set

D.(Q) =

U

wESn-1

N(Qw),

8(Q ) =

U

wESn -1

8(Qw )·

76

Chapter 2

In the case of polynomial (1) these definitions coincide with the definitions of the polygons ~( P) and 8( P) presented earlier. Obviously, we have

N(Pw)

c

~(P),

8(Pw)

C

8(P).

(4)

Let P be a polynomial (1). We retain the notation L~(P) and .C~(P) for the linear spaces of pseudo-polynomials Q satisfying (respectively) the conditions ~ ( Q) C ~(P) and 8(Q) C 8(P). All definitions of the foregoing sections are extended trivially to the pseudopolynomials, and therefore we shall speak, without any special stipulation, about pseudo-polynomials correct in Petrovski!'s sense, hyperbolic, parabolic in Petrovski!'s sense, N-stable correct, and N-parabolic.

4.2. Condition for the existence of estimate (2). The multidimensional analog of Theorem 2.1 is the following Theorem. For polynomial (1) the conditions below are equivalent. (I) There are c > 0 and 'Yo such that inequality (2) holds. (II) For each wE sn- 1 the pseudo-polynomial Pw(P, r) satisfies the equivalent conditions of Theorem 2.1, and we have

N(Pw) =

~(P)

\fw E

sn- 1 .

(5)

• t (III) TIL 11ere ex1s s a se t of vee t ors q(j) = ( q1(j) , q2(j)) , J. = 1, ... , p, q1(j) '.:;:; 0, q~i) > 0, and a set of qU) -homogeneous quasi-polynomials PYl (p, T) such

that

Imr

~

0,

(6)

and an integer b ~ 0 such that It

P(~, r)- Tb

IT PYl(l~l, r) = Qw(l~l, r) E .C~(P)·

(7)

i=l

Proof. (I)=:>(II). Replacing ~ in (2) by l~lw we obtain the inequality

(2') (x,fi)E~(P)

By virtue of (4), we have N(Pw) C ~(P), whence it follows that the polynomial Pw(P, r) with fixed w satisfies the conditions of Theorem 2.1. On the other hand, Lemma 1.2.2 remains valid in the case of estimates of type (2') (the trivial modification of the proof of the lemma is left to the reader). And therefore ~(P) C N(Pw)· Comparing this inclusion with (4) we obtain (5).

Parabolic Operators Associated with Newton's Polygon

77

(Il)==?(III) (cf. Section 3.2). Since for each w the polynomial Pw(p, T) satisfies the equivalent conditions of Theorem 2.1, for each w we can_ write relation (7) when:> in general, the number band the orders of homogeneity q(J) of the polynomials p[J] rnay depend on w while the coefficients of the yolynomial Qw even may not be continuous functions. However, the vectors q< 1 ) and the number b are uniquely determined by the polygon N(Pw), which, according to (5), depends on w. To prove that the coefficients of the polynomials PYJ are smooth functions (of w) we use explicit formulas ( cf. Section 1.1.3). Let ( x j, (J j) be the vertices of the polygon D,(P) . Then

(qU) ,( Ia! ,,B) }=mj X [ O"t

L + ...

(8)

O"n=Xj

Since (5) holds, we have

Consequently, the modulus of this function has a positive infinium on the unit sphere, whence follows that the coefficients of pseudo-polynomial (8) belong to c<sn-I ). And hence the right-hand side of (7) possesses this property. (III)===?(l). To prove this implication it is in fact necessary to repeat the argument in Theorem 1.2.1 (see the derivation of (III)===?(l)). Then for each w we obtain IPxT.B I ~ c!Pw(P, T)I, p E lR, lm T ~ /O·

L

(x,(3)EN(Pw)

Since the polygon N(Pw) is uniquely determined by the vectors q(j) and the number b (not depending on w), the polygon does not depend on w. Further, for at least one Wo E sn- 1 the polygon N(Pwo) coincides with l:l(P), and therefore N(Pw) = l:l(P) Vw E sn- 1 . If one examines carefully the proof of Theorem 1.2.1 (this was in fact done in the proof of Proposition 1.4.3), one sees that it is possible to select unified constants c and /o for which these inequalities hold for all w. Replacing pin these inequalities by 1~1 and recalling th'}t Pw(l~l, T) = P(~, T) we obtain (2). The theorem is proved. 4.3. Conditions for fulfilment of inequality (2) in terms of complex zeros of polynomial (1). We now state a multidimensional analog of Theorem 2.2. Let with each T E C a surface

L:T = {(

= ((1' ... '(n) E en' P( T, () = 0}

Chapter 2

78

be associated and let I:r:

dr(~)

denote the distance from the point~ E IRn to the surface

.

Theorem. For polynomial (1) the conditions (I), (II), and (III) in Theorem 4.1 are equivalent to the following conditions: (IV) There are co and

{o

such that all the polynomials

Po(~,r) = I~)aa 1 ... an.B +ba 1 ... an.B)~f 1 ···~~nT.B,

lbat···an.Bl < c,

(9)

satisfy simultaneously Petrovskil's condition; more precisely,

(V) There are constants

c1

> 0 and {o such that (11}

Remark. In the absence of the parameter r the inequality (11) is a condition for ellipticity. In the present context (11) can be interpreted as a condition for ellipticity with a parameter ( cf. Agranovich and Vishik [1 ]). We note that in condition (11) Newton's polygon is no longer involved. The proof of the theorem. (I)=?(IV). The derivation of (9) from (2) is carried out in the same way as in the case n = 1 (IV)=?(V). Let (10) be fulfilled and let T be a complex ( n x n) matrix, the sum of the moduli of its elements not exceeding c;. Then for the polynomial P( ~ + T~, r) Petrovskil's condition is fulfilled for Im r ~ {o and c < c 0 . Since each point of the complex ball!(-~~ < c:l~l can be represented as~ +T~, we have P(~, r) -/= 0 for the points ( belonging to this ball and Im 7· ~ 'Yo. Hence, inequality ( 11) with c 1 = c; holds.

The proof of the implication (V)=?(I) is based on the two lemmas below.

Lemma 1. Let V be a bounded set in IRn possessing the proper ty that any polynomial of degree no higher that 11 vanishing on V is identically equal to zero. Then there is a constant I< = I<(V, n, 11) such that for any polynomial Q(() of degree no higher than 11 ilhd for all ( E the inequality

en

IQ(a)( ()I ~ K sup IQ((

+ B)j,

8E'D

is fulfilled. l)!f the polynomial P(e, r) does not depend on~ (e.g. P

+oo.

=rb),

then we formally put dr (O ==

79

Parabolic Operators Associated with Newton's Polygon

Lemma 2. Let Q be a polynomial of degree J.t and let d( e) be tbe distance from tbe point to tbe (complex) surface of zeros {Q( () = 0}. Then

e

IQ(e + 01

~

21LIQ(01

for

1(1

~

d(e).

For the proof of these lemmas see Hormander [3(Lemma 3.1.5 and Lemma 4.1.1)]. We now complete the proof of the theorem, i.e. show that (11) implies (2). Set p = lei and write (12) (recall that W 0 = polynomial in en:

wf

1 • ••

w~n and lal

=

a1

+ · · · + an)· Consider an auxiliary (13)

Then (12) implies the following expressions for the coefficients of P a( T ):

We now apply Lemma 1 to polynomial (13) taking the spherical zone

as the set 'D. This yields

a! IPa(T)Iplal ~I< If 1J E 'D ,

sup IP(T,p1J)i. 1-e
(14)

e= pw, and lwl = 1, then

If c does not exceed the constant c1 in ( ll) and Im T we have

< /o, then, according to ( 11),

~nd therefore we can apply Lemma 2 to the right-hand side of (14), which results m

a! IPa(T)Iplal ~ 21lJ{IP(e,T)I,

eERn,

lmT ~ /O·

(15)

If ITI is sufficiently large, then IPa( T )I > const ITim"', where ma is the degree of the polynomial Pa( T ). Hence, inequality (15) is equivalent to (2). The theorem is proved.

Chapter 2

80

4.4. Parabolic polynomials. A hypoelliptic polynomial satisfying the equivalent conditions of Theorems 4.1 and 4.2 is said to be parabolic. According to Lemma 1 in Section 1.3.2, a parabolic polynomial is solved with respect to the highest power of any of the variables 6, ... , f.n, T. It follows that the polygon ~(P) is regular. Proposition 2.3 remains valid if f. E JR. in its formulation is replaced by f. E lR.n and N(P) is replaced by ~(P). Comparing this assertion with Theorems 4.1 and 4.2 we arrive at the following description of parabolic polynomials. Theorem. For polynomial (1) tl1e conditions below are equivalent.

(I) The polynomial P is parabolic (i.e. it is hypoelliptic and there are c > 0 and /"o such that (2) holds). (II) The pseudo-polynomials Pw(p,r) are N(Pw)-parabolic Vw E sn-t (Theorem 2.3) and condition (5) is fulfilled. (III) There are even numbers q1 , ... , q11 and pseudo-polynomials PYJ (p, T ), j = 1, ... , p,, q1-parabolic in Petrovskil's sense such that II

P(f.,r)-

IJ PYl(lcl,r) = Qw(lcl,r)

E .Cb.(P)·

j=l

Remark. The condition of hypoellipticity in (I) can be replaced by the existence of x > 0 and c1 > 0 such that

(lrl + l~l)x

2:::

l~laiTI{j ~ CtiP(~, T)l

for

Im T ~ /"O·

(16)

(a,jj)Eli(P)

4.5. Stable-correct polynomials. A polynomial (1) solved with respect to the highest power ofT and satisfying the equivalent conditions of Theorems 4.1 and 4.2 is said to be stable-correct. Proposition 2.4 remains valid in the multidimensional case as well, i.e. for polynomials satisfying the conditions of Theorems 4.1 and 4.2 the requirements that the polynomials should be solved with respect to the highest power of r and that they should be exponentially correct are equivalent. With regard to Theorem 4.1, there holds an analog Theorem 2.4. Theorem. For polynomial (1) the following c;onditions are equivalent.

(I) The polynomial P is stable-correct (i.e. Pis an exponentially correct polynomial and there are c > 0 and ')'o such tbat (2) holds). (II) The pseudo-polynomials Pw(P, r) are N(Pw)-stable correct (Theorem 2.4) Vw E sn-t and condition ( 5) is fulfilled. (III) There is an integer band a parabolic polynomial P 1 (f., r) such that P( f., r)rb Pt(f., r) E Lb.(P)·

81

Parabolic Operators Associated with Newton's Polygon

§5. Cauchy's problem for stable-correct differential operators with variable coefficients 5.1. We now extend the results obtained in §3 to the case of differential operators on functions of n + 1 variables, n > 1. Here attention is focused on the facts where the multidimensional situation differs from the one-dimensional case. As to the assertions that do not differ from those in §3 in either the statement or the proof, we shall not dwell on them. The spaces H[~{CIRn), n > 1, are defined in a natural way, and an analog of the norm (3.3) is introduced for them:

(1) where a = Re r and x = (x 1 , ... , xn) and~ = (~ 1 , ... , ~n) are the primal and dual variables relative to the form (x, e) = XI6 + ... + Xn~n· With a polynomial P(~, T) or, more precisely, with the polygon 6(P) (or t5(P)) the Banach norm

lluiiN(P),(s),/

=

(

L

liD~ D~ull(s),/

1/2 )

(2)

(a,/3)Et:..(P)

is associated, and let H~l(P),(s) denote the space of functions with finite norm (2). As in §3, we denote, respectively, by H~f+ and (H~l(P),(s))+ the spaces of functions and distributions vanishing fort < 0. In the case under consideration Theorems 1 and 2 in Section 3.3 remain true. 5.2. We now consider polynomial symbols (in ~ and r) with coefficients depending on the variables x and t. These symbols have the form (3.1) where a= (a 1 , ... , an) are multiindices, D~ = D~ 1 ••• D~n, Dj = -i 8j8xj, j = 1, ... , n, and Dt = -i 8j8t. Symbols (3.1) are said to be stable-correct (parabolic) if conditions (i) and (ii) inDefinition 3.2 are fulfilled. We shall assume that the coefficients of the symbols (3.1) are stabilized at infinity, i.e. have the form (3.7). Along with pseudo-polynomials in the foregoing section, one can consider pseudopolynomial symbol~, i.e. expressions of the form of w

= ~/1~1,

(3)

where qxf3( x, t, w) are C 00 functions of all the variables x, t, and w and do not depend on x and t for sufficiently large lxl +It!. Noting that

Chapter 2

.82

we can rewrite the pseudo-polynomial symbol (3) in the form (3') where the functions baf3( x, t, w) possess the same properties as the functions qxf3 in

(3). Let SL~(P) and sc~(P) denote the spaces of pseudo-polynomial symbols (3') such that the summation index (Ia!, f3) in them runs over /:).(P) and 8(P), respectively. The definitions of stable correctness, parabolicity, parabolicity in Petrovski1's sense, etc. are readily extended to pseudo-polynomial symbols.

Proposition. Let symbol (3.1) satisfy conditions (i) and (ii) in Definition 3.2 (in which N(P) should be replaces by b..(P)). Then the following assertions bold.

e,

(a) The pseudo-polynomials plil(x, t; 7) corresponding to the non-vertical sides of b..(P) and determined by formulas of type ( 4.8) are pseudo-polynomial symbols uniformly parabolic in Petrovski1's sense, i.e. there are qj, mj, and Cj > 0 such that

(b) The polynomial symbol (3.1) is uniformly stable-correct, 1.e. constants c > 0 and /'o such that ~ (t~,,7 ) .:::::..~(P)

< c IP( x, t ;~,,7 t )I

t E ~mn ~ ,

.ror

14

"'

lm7:::::: ~

"'~o I .

there are

(5)

(c) For the parabolic polynomial symbol (3.1) there are pseudo-polynomial symbols plil(x, t; 7 ), j = 1, ... 'J.L, parabolic in Petrovski1's sense, satisfying estimates of type ( 4), and such that

e,

Jl

P(x,t;e,T)-

IT plil(x,t;e,7) = Q(x,t;e,7) E sc~(P)·

(6)

j=l

(d) For the stable-correct polynomial symbol ( 3.1) there is a number b ~ 0 and a parabolic polynomial symbol P 1 (x, t ; 7) such that

e,

(7) Proof. (a). This assertion follows ft~m explicit formulas of type ( 4.8) ( cf. the proof of Theorem 4.2). Inequalities ( 4) make it possible in fact to repeat the argument of Theorem 1.2.1 and to obtain a uniform estimate (5). Assertions (c) and (d) are proved in the same way as their analogs for polynomials (Theorem 4.2). Assertion (b) in the proposition allows one to repeat literally the argument of Section 3.3 for the situation under consideration and to prove analogs of Theorems 1 and 2 of Section 3.3.

83

Parabolic Operators Associated with Newton's Polygon

5.3. • We now discuss the factorization of stable-correct differential operators with variable coefficients in the case of n+ 1 variables. As was shown in §3, for n = 1 the factorization (3.10), (3.11) takes place for the class of differential operators. We now show that for n > 1 to the factorization (6), (7) of the symbol there corresponds factorization in the class of pseudo-differential operators. We remind the reader of some well-known definitions. With each function a(x; w) belonging to C 00 (Rn X sn-l) and not depending on X for large lxl the classical pseudodifferential operator or, briefly, PDO (or, in a different terminology, singular integral operator) is associated:

(8) The function a( x; w) is called the symbol of operator ( 8). We note some properties of operator (8) important to our further aims. (i) For any s E R the operator

(f(x)

r-t

a(x;D)f(x))

is continuous. (ii) If ai(x,w), j = 1,2, are two symbols of the above-mentioned type, then Vs E Rand Vk E {1, ... , n} the operator [( a1a2)(x; D)- a1 (x, D)a2(x, D)]Dk: H(s)(Rn) ~ H(s)(Rn) is bounded. (iii) Vj E {1, ... , n} the relation

Dja(x; D)= a(x; D)Di

+ a(i)(x; D),

holds. For the proofs of properties (i) and (ii) see any textbook on PDO (for instance, Taylor [1]), and property (iii) follows directly from definition (8). With each symbol Q(x,t;e,T ) E SLA(P) of the form of (3') we associate the PDO

(9) where baf3( x, t, Dx) are operators of type (8) acting on the variables x and depending smoothly on the parameter t. Then, according to (i), Vs E R the operator HA(P),(s) ~ H(s)

bl

bl

is continuous. In case Q E

(u

r-t

SCA(P),

H8(P),(s) ~ H(s)

bl is continuous.

hl

Q(x,t;Dx,Dt)u),

the operator

(u

r-t

Qu),

(10)

Chapter 2

84

Theorem. (i) Let P(x, t; Dx, Dt) be a parabolic differential operator. there are PDO plil(x, t; Dx, Dt) parabolic in Petrovskil's sense such that

T

def

P(x, t; Dx, Dt)- p[Il(x, t; Dx, Dt) ... p[~Ll(x, t; Dx, Dt)

Then

(11}

is c_ontinuous operator from H[~P) ,(s) into HGf Vs E R and \:/1 <'Yo· (ii) For each stable-correct differential operator P(x, t; Dx, Dt) there is an integer b;::: 0 and a parabolic differential operator P1(x, t; Dx, Dt) such that the symbol of the operator P(x, t; Dx, Dt)- P1 (x, t; Dx, Dt)Df belongs to S.Ct::.(P)· Proof. (ii) follows trivially from Proposition 5.2 (d), and therefore we consider the proof of (i). Setting P(x, t; ,, T) = plil(x, t; ,, T) we rewrite operator (11) in the form

n

T

= [P(x, t; Dx, Dt)- P(x, t; Dx, Dt)] + [P(x, t; Dx, Dt) - p[Il(x, t; Dx, Dt) ... p[~Ll(x, t; Dx, Dt)] = Q(x, t; Dx, Dt) + T1.

According to Proposition 5.2 (c), the symbol of the operator Q(x, t; Dx, Dt) belongs to S.Ct::.(P), whence follows the continuity of operator (10). We now analyze the structure of T 1 • To simplify the notation, we assume that Jl = 2. We write p[il(t, x; (, T) = aia,a(t, z, w )(aT.B.

}-=

The symbol p[i] is solved with respect to the highest power Tm;. \Vithout loss of generality, we can assume that the monomial Tm; enters p[i] with a coefficient identically equal to 1. In view of (iii), the operator T1 can be rewritten

Tl

=

L {(alat,Ba2a' .B' )(x, t, Dx)- ala,B( x, t, Dx)a2a' .B' (x, t, Dx)} X

D xa+a'n.B+.B' t

+ "'"'B L · a" .B" na"n.B" x t

=

T'1 + TlII '

the summation in T{' extending over (a", /3") such that ( Ia" I, f3") E 8( P) and the operators Ba".B" being products of classical PDO with respect to the variables whose symbols are the functions aia,a(x, t; 0 and their derivatives with respect to x and t. In view of (i), to T{' there corresponds a continuous operator from nt-y)P),(s)

x,

.

H(s)

mto · bl" We now consider a typical term in the sum of operators T{. Since the highest derivatives with respect to t are involved in the operators p[i] with constant coefficients, the sum contains only those terms for which Jal + Ja' I > 0. Therefore for some k we have

D~+a' D~+.B' = DxkD~,D~+.B',

(Ia" I, f3 + /3') E 8(P).

Acccrding to (ii), the composition of the operator in curly brackets and the operator Dxk is a bounded operator in H[~f Vs E JR. It follows that the operator H(s) . . Th h T 1' : Ho(P),(s) hl ~ bJ IS contmuous. e t eorem is proved.

85

parabolic Operators Associated with Newton's Polygon

5.4. We now pass to the problem of solvability of the (homogeneous) Cauchy problem. As in §3, we consider the operators ( cf. (3.19) and (3.20))

(12) and

( H~(P),(s))

----+ H(s)

hl + bl+ (u(x,t) f-+ P(x,t;Dx,Dt)u). In the case under consideration Theorem 3.4 remains true. Theorem. For a differential operator whose coefficients satisfy conditions of type (3. 7) the following conditions are equivalent.

(I) The operator is stable-correct. (II) \Is E IR there is bijective. (II+) \Is E R there is bijective.

')'o

= '"Yo(s) such that

')'o

= ')'o(s) such that for')' ~

for')' ~

')'o ')'o

the mapping (12) is the operator (12+) is

The proof of this theorem differs from the proof of its one-dimensional analog (Theorem 3.4) in the part where the surjectivity of operators (12) and (12+) is proved. Indeed, owing to Theorem 5.3, this assert ion can also be proved following the scheme of Theorem 3.4. Recall that in this theorem we resorted to the assertion that the differential operator p(j] ( x, t; D x, Dt) parabolic in Petrovskil''s sense generates the bijective mappings

(H ~(p[il),(s)) bl +

H(s)

----+

hl+.

This result can be extended comparatively simply to differential operators whose coefficients are singular integral operators with respect to x. We thus arrive at the proof of the theorem in question. We now present another and more natural approach to the proof of the bijectivity of (12) and (12+) that does not use the factorization of the corresponding operator and is based on the apparatus of PDO. 5.5.

Definition. A symbol P(x, t; ~' 7) with smooth stabilized coefficients (satisfying (3. 7)) is called an exponentially correct symbol of constant strength if the following conditions are fulfilled: (i) V( x 0 , t 0 ) :E JRn+ 1 the polynomial P( C 7) = P( x 0 , t 0 ; ~, 7) is exponentially correct (see Section 1.1); (ii) there are constants A> 0 and ')'o such that V(x',t'), (x",t") E JRn+l the inequality I P(x',t';~,7)P- 1 (x",t";~,7)i
holds.

Im 7 ~'"Yo

Chapter 2

86

Obviously, by virtue of Proposition 5.2 (b), stable-correct symbols satisfy the conditions of this definition. Fix an arbitrary point (x 0 ,t0 ) E JR.n+I and set P(e,7) = P(x 0 ,t 0 ;~,7). We define the space H(s)P(D D ) H(s)} (13) H P,(s) { b]

=

u E

b] '

X'

t

u E

b]

and endow it with the natural norm

HGf+

Replacing H[~f in (13) by we define the space (H[~J(s))+· According to (ii), the space (13) does not depend on the choice of the point 0 (x ,, t 0 ). Its replacement by another point (x', t') results in an equivalent norm but the space continues to have the same set of elements. Further, in view of Proposition 5.2 (b), in case of stable-correct symbols the space (13) coincides with the space of functions (distributions) having a finite norm (2).

e,

Theorem. Let P(x, t; 7) be exponentially correct and have a constant strength. Then Vs E JR. there is lo = lo(s) such that (12) and (12+) are bijective mappings. We do not present the detailed proof and confine ourselves giving the main points of the scheme. For detailed proofs of this and more general theorems see, for instance, the monograph by Volevich and Gindikin [1, Chapter 5]. The proof of isomorphism (12+) is based on the calculus of PDO with holomorphic symbols. Denote by em the class of functions b(y; 7), y = (x, t) E JR.n+t, belonging to C 00 as functions of all their variables, holomorphic with respect to 7 for Im 7 ~ lo, not depending on y for sufficiently large IYI, and satisfying inequalities of the form of ID~b(y;e,7)1 ~ Ba(l)(1 + 1e1 + 171)m for lm7 ~ 1 ~ lo· (14)

e,

Let e be the union of all em, -oo < m < oo. With each function b(y; e, 7) E e the PDO b(y,D)=(27r)-(n+I)/ 2

j

exp(i(x,e}+it7)b(y;e,7)f(e,7)d~d7,

(15)

rn:n+l lm r=-y

can be associated. This operator possesses thefollowing properties: 1) the value of the integr~l on the right-hand side of ( 15) does not depend on the choice of 1 < lo; 2) if the support off E H[~f+ lies in the half-space t ~ T > 0, then the support of b(y; D)f also lies in that same half-space; 3) the inequality

llb(y; D)JII(s-m),-y ~ xB( I)!!J!I(s),-y

(16)

87

parabolic Operators Associated with Newton's Polygon

holds, where x does not depend on the symbol b, .a nd (17) We have to prove that the differential equation (18) has ::...t least one solution u E (H[~j(s))+· We shall seek the solution in the form u

= R(y; Dx, Dt)g,

(19)

where g E H[~f+ is a function to be determined and R(y; ~' T) = p- 1 (y; ~' T). Lemma 1. Let P(y; ~' T) be exponentially correct and have a constant strength. Then p-l(y; ~' T) E em form= 0. Assuming that the lemma has been proved, we come back to Equation (18). Since R E e, expression (19) makes sense. Apply the differential operator P(y; Dx, Dt) to the right-hand side of (19) and insert it under the sign of the integral determining the PDO R. On differentiating

P[R(y;f,,T)exp(i(x,e) +itT)] according to the Leibniz-Hormander rule we derive for g the pseudodifferential equation (20) where H(y;~,T) =

L

1

a>O

-1 P(a)(y;~,T)D~P- 1 (y;~,T). a.

Lemma 2. If P(y; ~' T) is an exponentially correct symbol of constant strength, then H(y; ~' T) E em form= 0 and, moreover,

ID:H(y;~,T)I < c,a(ImT),

c,a(ImT) ~ 0,

ImT ~ -oo.

(21)

Hence, if -/o is sufficiently large, then, according to (16), (17), and (21 ), for 1 :( /o the norm of the operator

H(y; Dx, Dt): H[~f+ ~ Hr~f+ is strictly less than 1, whence it follows that Equation (20) possesses a solution 9 E H[~f+ determining by Neumann's series 00

g

= ~( -H(y; Dx, Dt))i f. j=O

Chapter 2

88

It now remains to show that u E (H[~j(s))+· To this end we apply the differentia;i

operator P(Dx,Dt)

= P(x 0 ,t 0 ;~,r)

to the right-hand side of (19). This results in

where (22)

e,

Lemma 3. If P(y; r) is an exponentially correct symbol of constant strength, then H 0 (x, t; ~' r) E e;m form= 0. Lemma 3 and the properties of PDO with symbol belonging to 6 imply that (s) mto · · lf, 1.e. · P u E H(s) t h e operator H 0 trans£orms H hl+ 1tse bl+' and , consequent1y, P,(s)) (H [-y] u E +· It now remains to outline the proofs of Lemmas 1, 2 and 3. 1) We first of all show that there are c > 0 and /o such that

IP(y;e,r)i>c

for

Imr~lo·

(23)

Indeed, in view of the condition of constant strength, it suffices to verify inequality (23) at a fixed point y =Yo, i.e. for the polynomial P(e, T) = P(y 0 ; e' T ). The condition of exponential correctness (Lemma 1.1) implies that the polynomial P(~, r) is solved with respect to the highest power of r, i.e. P(e,r) =arm+ O(rm- 1 ). T~erefore 8~P(~,r) =am!. Since (8~P)P- 1 - t 0 as Imr - t - ( X ) (uniformly relative to the variables ~ and ReT), we arrive at inequality (23). 2) Consider the family of polynomials P(y; T) depending on the parameter y E R_n+ 1 • Since, by virtue of the condition of constant strength, the degrees of all polynomials P(y; T) are bounded by a unified constant, the whole family belongs to a finite-dimensional space of polynomials. Therefore a finite number of points y 1 , •.. , yJ can be chosen so that each of the polynomials P(y; e, r) is representable as a linear combination

e,

e,

J

P(y;Cr) = ~cj(y)P(yi,~ , r) .

(24)

j=l

Differentiating (24) with:respect toy and using the condition of constant strength be obtain the inequality

(25) 3) vVe now prove Lemma 3. The symbol P(y; ~' T) is obviously a smooth function of all its arguments and is holomorphic with respect to r for Im r ~ /o, where /O is taken from inequality (23). According to (23), for a = 0 the sytnbol b = R

Parabolic Operators Associated with Newton's Polygon

89

satisfies estimate (14) with m = 0. Let us show that a similar estimate also holds for the derivatives of R with respect to y. By Leibniz' formula, D~ p-I is a linear combination of expressions of the form of

By virtue of (23) and (25), these expressions do not exceed a constant. 4) We now prove Lemma 2. To this end we rewrite the symbol H in the form

where ha(y;~,T) = P(~,T)D~P- 1 (y;~,T)jo:!.

What has been said in 3) implies that the functions ha and their derivatives with respect to y do not exceed a constant. Further, differentiating H according to Leibniz' formula we find

Using ( 24) we derive

Thus, finally, with regard to the estimates for the derivatives of ha and the condition of cc.nstant strength, we have

Using the definition of exponentially correct polynomials we prove the assertion of Lemma 2. Lemma 3 is verified in a similar way. Hence, we have presented the proof of the continuity and surjectivity of the operat or

(26) The proof of the injectivity is based on the same ideas. Applying the PDO with symbol R(y; ~' T) = p- 1(y; ~' T) to relation (18) on the left we rewrite it as

whence

(27)

Chapter 2

90

We have already proved in fact that IIRJIIP,(s),/ ( const IIJII(s), 1 = const IIP(y; Dx, Dt)uii(s),,.

(28)

Combining the estimates for exponentially correct symbols of constant strength (Lemmas 1, 2, and 3) and the standard PDO technique (for more detail see Volevich and Gindikin [1], Chapter 5) we can prove the inequality

IIH1 uiiP,(s), 1

(

/(/o,

cs( !)lluiiP,(s), 1 t:1(!)-+0,

VuE

Hr\]) (29)

!-+-oo.

Substituting (28) and (29) into (27) we obtain

+ cs(!) lluiiP,(s),T

lluiiP,(s),/ ( ciiP(y; Dx, Dt)uii(s),/

Now, taking 1 1 such that cs( 1) < 1/2 for 1 (

iiuiiP,(s), 1

(

/1

2ciiP(y; Dx, Dt)uii(s), 1

we arrive at the inequality

Vu

E

H[\J),

(30)

1 (/I,

which implies the injectivity of the operator (26). 5.6. For our further aims it is advisable to note some a priori estimates contained in the above argument. Theorem. Let P(y;~,T), y = (x,t), be an exponentially correct symbol of constant strength, whose coefficients satisfy conditions of type (3.7). Then

(i) Va > 0 and Vs E IR there is lo and a constant .that the inequality

t: 80 (!)--+

0, 1--+ -oo, such VuE H(oo) b]

(31) is fulfilled; (ii) Va > 0 and Vr E IR there is Po and a constant that the inequality

e:;

0

(p)

--+

0, p

--+

oo, such

Vv E

p;;::: po,

H(oo) [p] .

(32) holds. Proof. (i) By virtue .of (30), we must first estimate the left-hand side of (31) with a small constant esc¥(/) by means of the left-hand side of (30). Using (24) we obtain " J

IIP(a)(y; DXl Dt)ull(s), 1

(

2: llcj(y)P(a)(yj; Dx, Dt)uil(s) ,

1

j=1

91

Parabolic Operators Associated with Newton's Polygon

Applying Fourier's transformation and the definition of exponentially correct polynomials we find

where ei('Y) -t 0, 'Y -t-oo. Substituting this into (33) and using the condition of constant strength we arrive at (31 ). (ii) We first of all note that if the polynomial P( T) is exponentially correct, then the polynomial P(e, -T) possesses the same property. Recalling that

e'

e,

-

e,

-(a)

e,

e,

P*(y; T) = P(y; T) + L:P(a)(y; T)ja! we conclude that P*(y; -T) is an exponentially correct symbol of constant strength. By what has been proved in (i), the operator P*(y; Dy, Dt) satisfies an inequality of type (31). Making change of variable t -t -t in this inequality we obtain (32). 5.7. Estimates (31) and (32) can be extended to spaces of functions of finite algebraic growth or decrease with respect to the variables y = (x, t). Denote by H{:/ the space of functions u E S' such that (1 + IYI 2 ) 1 12 u E H(s). Let II ll(s) be the norm in

H(s).

We define the

Hf:/

norm

which is equivalent to

(34') We can also consider the spaces Hf,S/,"Y+ and define the norms II II~;?,"Y and 'II II~;],"Y in them. Here we do not discuss this in detail referring the reader to Chapter 5 in the monograph by Volevich and Gindikin [1 ].

Theorem. Under the conditions of Theorem 5.6 Vs, l E R there is 'Yo(s, l) such that the inequality

L IIP(a)(y, Dy )u!I~:],"Y ~ est('Y )!IP(y, Dy )u!J~:],"Y

(35)

a>O

holds and an analogous inequality is fulfilled for P*v in the norms II

II ((-s) -1),-"Y·

Proof. Replacing u in (31) by (1 + IYI 2 ) 112 u and performing the summation of these inequalities over a > 0 we obtain

L I!P(a)(y, Dy)((1 + IYI a>O

2 ) 112 u)jj;s)

~ es(!')!IP(y, Dy)(1 + jyj 2 ) 1 1 2 ull~s).

(36)

Chapter 2

92

By Leibniz's formula, we have

P[(1 + lvl2)lf2u] = (1 + lvl2)l/2 Pu + :2:CD~'(1 + lvl2)l/2 I f3!)P(f')u. {J>O

Noting that the operator of multiplication by the function

is bounded operator in H~s) (see Volevich and Gindikin [1]) we conclude that

liP( (1 + IYI 2) 112 u )ll~s)

~ II Pull~;},,+ K1

L llp(f')ull~;],,.

(37)

{J>O

If we show that

L

!lp(l')uii~;],, ~ K2

f'>O

L IIP(o)(1 + IYI 2 ) 112 ull~),

(38)

6 >0

where J{2 does not depend on/, then substituting (37) and (38) into (36) we derive (35). To verify (38) it suffices to note that

(1 + lvl2)l/2 p(l')u = (1 + lvl2)l/2 p(l') [(1 + lvl2) - lf2(1 + lvl2)lf2u] =

I: [(1 + lvl2)lf2no (1 + lvl2)-l/2 I 8!] p(f'+o) [ (1 + lvl2)lf2uJ.

Remark. In the book by Volevich and Gindikin [1] estimates are also presented for exponentially correct operators of constant strength in spaces of functions of exponential decrease or growth.

CHAPTER III

DOMINANTI,Y CORRECT OPERATORS

Introduction In the foregoing chapter we studied stable-correct differential operators which are a natural generalization ofoperators parabolic in Petrovski!'s sense. Their symbols are polynomials P(~, T ), ~ E !Rn, T E C, solved with respect to the highest power of 7 and admitting of an adequate estimate from below by means of the sum of the moduli of all its monomials: Im T ~ {o,

(~,ReT) E !Rn+l.

(1)

(x,{3)E~(P)

By virtue of this inequality, all derivatives involved in the corresponding differential operator (even in the case of variable coefficients) are estimated by means of that operator in the norms of the spaces H[_;{. Inequality (1) implies a stronger estimate for minor monomials: c(Im T)---+ 0,

Im T

---+

-oo,

(2)

(x,{3)E6(P)

i.e. t he minor monomials of the polynomial P are estimated by means of the polynomial itself with an arbitrary large constant when the imaginary part of - T is sufficiently large. In the language of differential operators, inequality (2) means that in the spaces

H[_;{

the corresponding differential operator with an arbitrarily

large constant majorizes the H[~~ norms of all lower derivatives provided that - 1 is sufficiently large. As is known from the classical theory, along with parabolic operators, strictly hyperbolic operators (their definition is given in §1) possess this property. It turns out that the property is retained for the composition of hyperbolic and parabolic oper'itors. Therefore it seems natural to try to unify all these operators as a class of operators with a dominant (relative to Newton's polygon) principal p art.

Definition. A polynomial

T

P(~, T)

solved with respect to the highest power of is said to be dominantly correct if it satisfies inequality (2).

Remark. In this chapter we use the same definition of minor monomials of a polynomial as in the foregoing chapter. Since dominantly correct polynomials a re solved with respect to the highest power ofT, the polygon 8(P) coincides with 80 (P) , and (2) implies that dominantly correct polynomials are exponentially correct. The aim of the present chapter is to give an algebraic description of dominantly correct polynomials and to prove the correctness of Cauchy's problem for dmninantly correct opera tor s with variable coefficients. 93

Chapter 3

94

The, presentation of the material is organized in the following way. §1 is of auxiliary character and compiles some known facts relating to strictly hyperbolic polynomials and differential operators that are necessary for the further presentation. §2 provides the description of dominantly correct polynomials for the case of two variables. The main result consist in that, to within minor monomials, such a polynomial is a product of a strictly hyperbolic polynomial by a stable-correct polynomial. In §3 we consider dominantly correct differential operator with variable coefficients. An a priori estimate corresponding to (2) is obtained for them and an existence and uniqueness theorem is proved for the solution of Cauchy's problem in the spaces Hr~f. §4 is devoted to the extension of the results of §§3 and 4 to the case n > 1. It should be stressed that , in contrast to stable-correct operators, dominantly correct differential operators with variable coefficients are not operators of constant strength. Therefore the derivation of a priori estimates and the proof of the solvability of Cauchy's problem are substantially more complex than the derivation of the analogous assertions in the foregoing chapter. In Chapter 7 we shall return to the c-~nalytical problems presented in this chapter.

§1. Strictly hyperbolic operators As was already mentioned in the Introduction, stable-correct polynomials are dominantly correct. We now consider another class of polynomials for which inequality (2) is fulfilled. 1.1. A homogeneous polynomial Ho(e,r), hyperbolic if

eERn, T E
(i) the polynomial H 0 is solved with respect to the highest power of r , i.e. m

Ho(C r) = Tm

+L

L

aait~r m-j;

(1)

j=l lal=m-j

(ii) the roots Toj, j = 1, ... , m, of polynomial (1) are real; (iii) for 1e1 # 0 all the roots Toj( 0 are distinct . As was already mentioned in Section 2.1.2, homogeneous polynomials satisfying only conditions (i) and (ii) are said to be hyperbolic.

Proposition. Fora homogeneous polynomial H 0 of tbe mtb degree tbe conditions below are equivalent.

(I) Tbe p olynomial H 0 is strictly hyperbolic. (II) There is c > 0 such that

IImrl(lrl + lel)m- 1 ~ ciHo(e,r)l,

Im T ~ 0'

(

e' Re T) E 1Rn+

1.

(2)

95

Dominantly Correct Operators

Proof. (II)=>(I). Let us show that (2) implies conditions (i) to (iii). Putting ~ == 0 in (2) we find whence follows (i). Further, according to (2), H0 (~,7) =/= 0 for Im7 < 0. By virtue of the homogeneity, we have Ho(e, -7) = ( -l)m H0 ( -e, 7), whence Ho(e, 7) -=f. 0 for Im r > 0, i.e. the roots of the polynomial H 0 can only be real, i.e. (ii) is fulfilled. We now prove that for -=f. 0 all roots are distinct. Assume the contrary, i.e. let there be eo -=I- 0 such that Tol(e 0 ) = 7o2(e 0 ). Consider inequality (2) along the ray

e

e(t)=te 0 ,

t>O,

r(t)=7oi(e(t))-il,

1>0.

On the ray the left-hand side of (2) is estimated from below by means of const 1( 1

+ t)m-l.

Noting that roj(e(t)) = troj(e 0 ) we obtain the inequality m

j=l

= 12

IT lt(roi(e

0 )-

roj(e 0 ) ) - i1l ~ const1 2(1 + t)m- 2

j=3

for the right-hand side. For t --t oo we arrive at a contradiction. (I)~(II). The condition that the roots are not multiple means that there is c; > 0 and a covering {OI} of the unit sphere {(a-, 0 E Rn+I, o- 2 + lel 2 = 1} such that in every neighborhood the inequality Ia-- 7oj(01 < c: can hold for at most one J. To the covering {Oi} the covering of Rn+I \ {0} by the cones

n,

corresponds. Since the polynomial H 0 is solved with respect to 7m, inequality (2) is obvious for Re r = 0, = 0, and it suffices to establish (2) in each cone Vi. By what hM been said and the homogeneity of the roots (i.e. roj(te) = i7oj(e)), for a given l there can be only a single j for which Ia-- roj(OI < c:( o- 2 + lel 2) 1 12. If such j exists, we obtain the estimate

e

17- Toj(OI ~ llmrl If k

-=1-

j, then

for

(a-, e)

E

Vi (a-=

Re7).

(3)

Ia-- 7ok(01 > c:( o- 2 + lel 2) 1 12, whence 17 - rok(OI > c:(l lm71 2 +I Rerl 2 + lel 2) 112.

Multiplying (3) and (3') we obtain (2).

(3')

96

Chapter 3

1.2.

Definition. A polynomial H(~, T) is said to be strictly hyperbolic (hyperbolic in Pdrovski'l's sense) if its principal homogeneous part possesses this property. Theorem. A polynomial H ( ~, T) of degree m is strictly hyperbolic if and only if there are c > 0 and 'Yo < 0 such that

(4) Corollary. Every strictly hyperbolic polynomial is dominantly correct.

The proof of the theorem. Let (4) hold. If (~,T) in (4) is replaced by (t~,tT), the two sides of (4) are divided by tm, and the passage to the limit is performed as t---+ +oo, this results in (2), whence (Proposition 1.1) it follows that H is strictly hyperbolic. Conversely, if H 0 is a strictly hyperbolic polynomial, then, with account of (2), we have IH(~,

T)l > IHo(e, T)I-IH(e, T)- Ho(e, T)l ~ C- 1 Im Tl (IT I+ lei) m-l - CJ (1 + ITI + lel)m-l. 1

If I Im Tl is sufficiently large, we arrive at inequality (5). 1.3. We now consider a differential operator with variable coefficients

(5) and let H(x,t;~,T) be its symbol. Operator (5) is said to be strictly hyperbolic, if the principal homogeneous part Ho(x, t; T) of its symbol possesses the following properties:

e,

(i) the symbol Ho is solved with respect to Tm; subsequently we shall assume that the coefficient in Tm is identically equal to 1, i.e. m

H(x, t;

e, T) = Tm +I: hj(x, t, oTm-j; j=l

(ii) the roots Toj(x, t;0 are real and uniformly non-multiple in the sense that there is A > 0 such that

!Toj(x, t, 0- Tok(x, t, 01 > A V(x, t),

1e1 =

1,

j =I= k.

(6)

In the well-known works by Petrovski1 [2], Leray [1], etc. a priory estimates were obtained for strictly hyperbolic operators and the solvability of Cauchy's problem was proved. We have the following

Dominantly Correct Operators

97

Theorem. Let operator (5) be strictly hyperbplic and let (for simplicity) its coefficients satisfy conditions of type (2.3.7):

(i) Vs E lR 3-y0 ( s) such that the inequality I'YIIIull(.'l+m-1),-y ~ cliH(x, t; Dx, Dt ) u II (s),-y

(oo) Vu E H bl ,

1' ~

')'o(s),

is fulfilled. (ii) Vs E lR 3-y0 (s) such that Vf E H[~~ there is a single function u E satisfying the equation

(7)

HGtm-

1)

H(x, t; Dx, Dt)u = f in the sense of distribution theory. (ii+) All assertions in (ii) remain valid if the spaces

HGf are replaced by H[~f+·

For this theorem in the form given here see the survey paper by Volevich and Gindikin [6]; it is a special case of the more general results obtained in Chapter 7.

§2. Dominantly correct polynomials in two variables In this section we shall describe the structure of polynomials P( ~, r ), ~ E JR, satisfying inequality (0.2). The main result consists in that, to within minor monomials, each polynomial of this kind is a product of a strictly hyperbolic polynomial by a stable-correct polynomial. The definition ofdominantly correct polynomials involves the polygon 8(P). We begin with a detailed description of this polygon for polynomials in two variables correct in Petrovski'l's sense and, in particular, consider the problem of reconstructing N(P) from 8(P), which plays an important role in studying dominantly correct polynomials of several variables (§4) and dominantly correct differential operators with variable coefficients (§§3 and 4). .

2.1. The polygon 6(P). Let P(~, r) be a polynomial in two variables, let

(1) be the degrees of its roots Tj(O, and let /-lj, j = 1, ... , m be the number of the roots of degree bj. As was shown in §1.1.4, the vertices (a j, /3j ), j = 0, ... , m + 1, (o:o, f3o) = (0, 0), are uniquely determined by the numbers bj and /-lj (see formulas (1.1.23) and (1.1.23')) while the numbers bj and /-lj are reconstructed uniquely from the numbers (o:j,/3j) (see (1.1.25)). Thus the polygon N(P} is determined by the system of inequalities 0'. ~

0,

f3

~

0.

(2)

If a polynomial P satisfies Petrovski!'s correctness condition, the numbers (1) are integers (see Proposition 1 in Section 2.1.4), and consequently the right-hand sides of (2) involve integers.

Chapter 3

98

Lemma. Let P( ~, r) be a polynomial correct in Petrovskil's sense and let (a, ,B) be an integral minor point of N(P). Then (i) if bm > 0 (i.e. N(P) has no vertical side not lying on the coordinate axis), then (a+ 1, ,B) E N(P); (ii) in the general case either (a+ 1, ,8) E N(P) or (a, f3 + 1) E N(P). Proof. (i) If bm > 0, then the minor points cannot belong to the sides of N(P) not lying on the coordinate axes. Consequently, a+ bj/3 < ai + bj/3j, and, since we deal with integers, a+ bj,B:::;; ai + bj/3j- 1, i.e. (a+ 1, ,B) E N(P). (ii) If (a, ,B) belongs to the interior of N(P) or lies on a coordinate axis, then (a+ 1, (3) E N(P). In case the point (a, j3) belongs to the vertical side r~), it cannot coincide with the vertex (am, f3m)· And therefore ,8:::;; f3m -1, i.e. (a, /3+1) E N(P). The above lemma allows one to describe completely the correspondence between the polygons N(P) and 8(P). 1) If N(P) has no vertical side not lying on the coordinate axis (i.e. bm > 0), then 8( P) is determined by the system of inequalities

j = 1, . . . ,m.

(2')

We note that the straight line a+ b1j3 = a1 + b1j31 -1 = b1j31 - 1 = b1(f3I- 1/bi) intersects the axis {/3} at the point (0, /31 -1/b1) which is not integral when b1 > 1. In this case inequality (2') should be supplemented with the inequality

(2") and the vertices of the polygon 8( P) are the points

(0, 0), (0, ,81 - 1), (b1 - 1, ,81 - 1), (a2 - 1, ,82), ... , (ani+ I

-

1, 0).

(3)

2) If N(P) contains a vertical side (not lying on the coordinate axis), then 8(P) contains the points belonging to the line segment {a = am , 0 :::;; ,8 :::;; f3m - 1}. Hence, in this case 8( P) is the convex hull of the points

(0, 0), (0, /31 - 1), (b1- 1, ,81 - 1), (a2- 1, ,82), ... , (am -1, /3m),(am,f3m -1),(am , O).

(3')

All the points (3') except, possibly, (am-I -J, f3m) are vertices of 8(P). The latter point is a vertex in the case bm-l > 1 and is an interior point when bm-l = 1. 3) Our aim is to reconstruct N(P) from 8(P). Before indicating the method for reconstructing N(P) form 8(P) it is advisable to find out whether this procedure leads to a unique result. Let a polygon N be determined by a set of numbers

(4)

99

Dominantly Correct Operators

the case bm- 1 = 1 not being excluded here. Let a polygon N' be determined by the set of numbers

( 4') Then the polygons N and N' coincide to the left of the line a = am (see Figure 2), where the numbers a 1 , j3j, are found from the numbers (b1, ... ,bm-1,0),

(a. m,o) Figure 2 (/-lt,· .. , Jim-!, 1) by means of formulas (1.1.23), (1.1.23'). As can easily be seen,

the polygons of minor monomials 8 and 8' corresponding toN and N' coincide, the side of the polygon 8 which adjoins the axis {a} forming an angle with that axis equal to 7l" /4. ,· So, let ( /j, 81 ), j = 0, .. . , k, /o = 1 1 = 0, 8k = 80 = 0 be the given vertices of the polygon 8(P) corresponding to the original polygon N(P). In the reconstruction of N(P) from 8(P) one should distinguish between the following three cases. If 8( P) has a vertical side not lying on the coordinate axis, then N ( P) is sure to have a vertical side, not lying on the coordinate axis, with height no less than 2. The polygon N(P) is reconstructed uniquely, and its vertices are the points

Chapter 3

100

(cf. (4')) (0,0),(0,81

+ 1),(1'2 + 1,82), ... ,("Yk-2 + 1,8k-2),(1'k-1,8k-l + 1),(1'k-I,O).

(5')

If the angle between the side of 8( P) adjoining the axis {a} and that axis does not exceed rr /4, then N(P) is sure to have no vertical side not lying on the coordinate axis. The polygon N(P) is reconstructed uniquely and its vertices are the points

(0, 0), (0, 81

+ 1), (1'2 + 1, 82), .. . ' ( lk + 1, 8k)·

(5)

Ir.L case 8(P) has an angle equal torr /4, the two above-mentioned reconstruction methods result in two different polygons corresponding to the sets ( 4) and ( 4'). We have thus proved the following Theorem. Let P( ~, T) be a polynomial correct in Petrovskil's sense. If P bas no roots of the form ofT( 0 = 0( 1) (we call them "zero" roots) or if the multiplicity of such roots exceeds 1, then the polygon N(P) and, hence, the numbers bj , /lj are r econstructed uniquely from the polygon 8(P). In the general case all exponents bj > 1 and the corresponding multicities /lj are reconstructed uniquely from 8(P). The character of the possible non-uniqueness is indicated in ( 4) and ( 4'). 2.2. Product of a strictly hyperbolic polyno1nial by a stable-correct polynomial. As was proved above, stable-correct and strictly hyperbolic polynomials are dominantly correct. We now prove that in the case of two variables the product of such polynomials is also a dominantly correct polynomial. Proposition. Let H(~, T) be a homogeneous strictly hyperbolic polynomial of degree h and let S( ~, T) be a stable-correct polynomial, the polynomial H ( ~, T) having no zero roots in the case when S(~, T) has them. Then the polynomial P(~, T)

= H(~, T)S(~, T)

(6)

is dominantly correct. The proof is based on Lemma 1. Let polynomial (6) satisfy the conditions of the proposition and let (a , {3) E 8(P). Then there are points (a',{3') E N(S) and (a",{3") E 8(S), a polynomial q(~, T) of degree no higher than h- 1, and a constant c such tl1at the representation (7)

takes place. Assuming that the lemma is proved we prove the proposition. We have q( ~, T) ~o:' T/3' C~o:" T/3"

H(~,T) S(~,T) + S(~,T)

101

Dominantly Correct Operators

By virtue ofproposition 1.1 and inequality (0.2), the right-hand side tends to zero uniformly with respect to (~,Rer) as Imr-+ -oo. Remark. A. careful examination of the proof of inequality (0.2) for polynomial

(6) readily shows that

). > 0, can be taken as the constant e(Im r ). Indeed, according to Proposition 1.1, we have

If the polynomial S is N -stable correct, then

). > 0,

(a",(3") E 8(S).

Assuming that ). ~ 1 we prove the desired assertion. The proof of Lemma 1 is based on Lemma 2. Let a polynomial P have the form (7) and let h = deg H. Then for each point (a, (3) E /J(P) at least one of the three representations below holds:

(a, (3) = (a', (3') + (a/', (3"),

(a', (3') E N( S),

a" + (3" < h;

(8) (8')

(a,(3)=(a',(3')+(0,h),

(a'+1,{3')EN(S),

(a',f3'+1)rtN(S);

(a,(3) = (a',/3') + (h,O),

(a',(3' + 1) E N(S),

(a'+ 1,(3')

.

rt

N(S).

(8")

The proof of Lemma 1. Denote by 8°(P) the set of those (a,(3) E /J(P) which are representable as (8). If (a, (3) E 8°(P), then the monomial ~a'r/(3 is obviously representable in the form (7) with c = 0. If representation (8') takes place, then

~aTf3 = ~a'rf3'rh = ~a'rf3' (H(~,r)- LCjTh-j~j) j>O =

-(L:cjTh-j~j-l)~a'+lrf3' +H(~,r)~ 0 'r!3'. j>O

Now let (8") hold. This can be realized only when N(S) has a vertical side not lying on the coordinate axis, i.e. S has zero roots. In this case the polynomial H has no zero roots, and it can be solved with respect to ~h:

e

=doH+ Ldi~h-jTj. j>O

Chapter 3

102

Therefore

ear/3 = ea' r/3' (doH+

2..:: dieh-iri) j>O

=

(:I: djeh-jTj-l )ea' r,B'+l + doH(e, r)ea' r.B'. j>O

The proof of Lemma 2. We can assume (without loss of generality) that S = rb R(e, r), where R is anN-parabolic polynomial. Denote by b1 , ... , bk the degrees of the roots of Rand by ( aj, (3 j), j = 0, ... , k, the vertices of the polygon N(R). If (a,(3) E t5(P), then, according to Lemma 1.1, either (a+ 1,(3) E t5(P) or

(a, {J + 1) E t5(P). Consider the first case. We begin with assuming that (3 this case we set (a, (3) = (a, (3- h)+ (0, h) and show that

(a+ 1,(3- h) E N(S).

~

h. In

(9)

According to the description of the polygon N(P) presented in Sectionl.1, if(~, jj) E N(P), then

0:, jj ~ 0. Putting 0:

= a+ 1, jj = (3- h we obtain

i.e. (9) holds. Let b ~ (3 < h. If a~ ak, then we set

(a,(3) = (a,b)+(0,(3- h), and in case a> ak we put

(a, (3) =(a, b)+ (0, (3- b). If (3

< b for a= ak, (a, (3)

E N(S), then for a

> ak we put

If (a, (3 + 1) E N(P) and (a+ 1, (3) ¢:. N(P), then the point (a, (3) belongs to the vertical side, i.e. a= ak + h, (3
103

Dominantly Correct Operators

2.3. Description on dominantly correct polynomials. In the foregoing section it was shown that the compositions of strictly hyperbolic and stable-correct polynomials are dominantly correct polynomials. We now show that the polynomials of that kind exhaust (to within minor monomials) the dass of dominantly correct polynomials in two variables. We have the following Theorem. For a polynomial P( ~, 7) the following conditions are equivalent: (I) the polynomial Pis dominantly correct, i.e. it is solved with respect to the highest power of 7 and inequality (0.2) is fulfilled; (II) the polynomial P is correct in Petrovski1's sense; moreover, for any polynomial Q(~, 7) E LN(P) there is f'o = f'o(Q) such that IP(~,7) + Q(~,7)1

=f. 0

for

lm7:::;; ')'o ,

(~ , Re 7 ) E 1R2 ,

i.e. the polynomial P + Q is correct in Petrovskil's sense; (III) the principal parts Tj (0 of Puiseux' expansions of the roots of P belong to one of the following types (see Proposition 2 in Section 2.1.4): (a' ) Tj(O = Cjl~xil , Imcj1 > 0, Xjl is even; (b') Tj(O = Cjt~, Imcj1 = 0; (c) 7(~) • 0, the roots of type (b') being distinct , i.e. Cjk = ckl , j =f. k; (IV) there is a strictly hyperbolic polynomial H(~, 7) having no zero roots and a stable-correct polynomial S( ~, 7) such that P(~, 7)- H(~, 7)S(~, 7) E LN(P) ·

(10)

Proof. For the proof of (IV)===}(I) see Proposition 2.2. (I)==}(Il). If Q E £ N(P), then, according to (0.2), there is I'( Q) such that IQ(~,7)1

<

IP(~,7)1/2

for

Im7 < I'(Q),

whence IP(~,7)+Q(~,7)1

>

1 2IP(~,r)l

for

Imr:::;; ')'o,')'(Q).

(II)===>(III). Without loss of generality we can assume that Tj( O = Cj ei since discarding the lower t erms in Puiseux' expansions of the roots we change only minor monomials of the polynomial (see Section 1).4). By P roposition 1 in Section 2.1.4, ~he numbers bj are integers, and we have Im Cj = 0 if bj is odd and Im Cj ~ 0 if bj 1s even. We have to show that there are no roots belonging to the following type:

Chapter 3

104

Indeed, if the polynomial P has a root of the form of ( 11), then the polynomial Q(~, r) = i~ fl(r- Tk(O) belongs to LN(P), i.e. the polynomial

P(~,r) + Q(~,r) = (r- Cj~qi

+ iO IT(r- Tk(O) k#j

is correct in Petrovski'l's sense. However, this contradicts Proposition 1 in Section 2.1.4 according to which a polynomial correct in Petrovski!'s sense cannot have roots of the form T = Cj(qi + i~, qj > 1, where Cj is real. Hence, with account of Proposition 1 in Section 2.1.4, we have proved that the polynomial under consideration can have only roots of types (a'), ( b') and (c). It remains to show that the roots of type ( b') must necessarily be distinct. Assume the contrary, i.e. let the polynomial P have the form ( T - a0 2 P', where a is real, and let the polynomial P' be correct in Petrovski'l's sense. Then Q(~, r) = ~P' belongs to LN(P), and consequently the polynomial P + Q is correct in Petrovski1's sense, which contradicts Proposition 1 in Section 2.1.4 since P + Q possesses roots T =a~+ i~. (III)::=;.(IV). Let cjei be the principal parts of the roots of P. Then the polynomial P(~,r) = IJ(r- cjei) differs from P only in minor monomials. Denoting by H( ~, T) the product of all factors of the form ofT- Cj~ and by S( ~, T) the product of the other factors we obtain, in view of Theorem 2.2.4 and 1.1, relation (10). §3. Dominantly correct differential operators with variable coefficients (the case of two variables) In this section we consider differential operators

(1) whose symbols P(x, t; ~' r) are dominantly correct polynomials for all x and t. Under the additional assumption that the polygon of minor monomials 8(P) does not depend on x, t we shall prove that, to within lower terms, operator (1) is a product of a strictly hyperbolic differential operator by a stable-correct differential operator. Combining the results of §2.3 with the results on the solvability of Cauchy's problem for strictly hyperbolic operators (Theorem 1.3) we prove the unique solvability of Cauchy's problem for dominantly correct differential operators with variable coefficients. 3.1. Structure of dominantly correct differential operators. A differentialoperator (1) is said to be dominantly correct if its symbol satisfies the following conditions:

(i) the polygons of minor monomials 8(P(x, t)) of the symbols P(x, t; ~' r) do not dep.e nd on (x, t), i.e.

8(P(x, t))

= 8(P)

V(x, t);

105

Dominantly Correct Operators

(ii) for any fixed (x 0 , t 0 ) the polynomial P(~, r) = P(x 0 , t 0 ; ~' r) is dominantly correct. Since dominantly correct polynomials are solved with respect to the highest power of r, the symbol P(x, t; ~' r) possesses the same property. In what follows we shall additionally assume that the coefficient in the highest power ofT is identically equal to 1, i.e.

P(x, t; ~' r) = TM

+ L Pj(x, t; e)rM-j.

(2)

j~l

As in §2.3, we shall assume that the coefficients of operator (1) belong to c= and do not depend on (x,t) for sufficiently large (x,t), i.e. condition (2.3.7) holds. As a rule, in what follows we shall not stipulate this condition.

Proposition. Let symbol (2) be dominantly correct. Then it is representable as

P(x,t;Cr)

=

H(x,t;~,r)S(x,t;~,r)

+ Q(x,t;~,r),

(3)

where the symbols H, S, and Q possess the following properties: 1) S(x, t; ~' r) is a stable-correct symbol (in the sense of Section 2.3.2), its coefficient in the highest power of T is equal to 1, and the coefficients of S satisfy condition (2.3.7); 2) For any (x, t) the symbol H(x, t; ~' T) is a homogeneous strictly hyperbolic polynomial, its coefficient in the highest power ofT is equal to 1, and the coefficients of H satisfy condition (2.3. 7). If the symbol S( x, t; ~, T) has a zero root of multiplicity k for some x = x 0 , t = t 0 , then it has a zero root of multiplicity k for all (x, t), and in this case the symbol H(x, t; ~' T) can be solved with respect to the highest power of ~; 3) Q(x,t;~,r) E SCN(P)· be the degrees of the roots Tk(x, t; 0 ofpolynomial (2) indexed in the decreasing order and let J.-Lj(x, t) be the number of roots of degree bj(x, t). We first assume that one of the three conditions below is fulfilled: (a) symbol (2) has no zero roots, i.e. bm(x 0 ,t 0 ) > 0 V(x 0 ,t 0 ); (b) bm(x 0 ,t0 ) = 0, J.-Lm(x 0 ,t0 ) = 1 V(x 0 ,t 0 ); (c) 3(x 0 ,t0 ) such that bm(x 0 ,t0 ) = 0, J.-Lm(x 0 ,t0 ) > 1. As is seen from Theorem 2.1, under these assumptions the polygons N(P(x 0 , t 0 )) are reconstructed uniquely from the polygons 8(P(x 0 , t 0 )). Since, by the definition of dominantly correct symbols, the latter polygons do not depend on (x,t), Newton's polygons N(P(x 0 , .t 0 )) and, consequently, the numbers bj(x 0 , t 0 ) and f.1j(x 0 , t 0 ) do not depend on (x 0 ,t 0 ) either. In particular, the quantifier 3 in condition (c) should be replaced by V. We now follow the same argument as in the proof of Theorem 1.4.3. Accordin~ to Theorem 2.3, for each (x, t) there is a number b, a set of polynomials p[Jl(x, t; C r) parabolic in Petrovskil''s sense, a homogeneous strictly hyperbolic Proof. Let bj(x, t), j

= 1, ... , m,

106

Chapter 3

polynomial H(x,t;e,r), H(x,t;f,O) -=f 0, and a polynomial Q(x,t;e,r) such that

E LN(P)

P(x, t; e,r)- rb(x, t; e, r)

11 p[il(x, t; e, r) = Q(x, t; e, r),

(3')

where b = 0 if the polygon N(P) has no vertical side not lying on the coordinate axis. We now show that the coefficients of all polynomials involved in (3') satisfy conditions of type (2_.3.7) . Indeed, according to formulas (1.1.15), .the coefficients of the polynomials H and p[i] are products of coefficients of the polynomial P and the functions a-;;~ 13 .(x,t) where (aj,/3j) are the vertices of N(P(x,t)) distinct from (0, 0) and from the vertex ( O:m+l, 0) lying on the vertical side (if bm = 0). Since, by what has been proved, the polygon N(P(x, t)) does not depend on x and t, we have a 0 if3i(x,t) -=f 0. )

)

Since, according to condition (2.3.7), the functions a0 i /3i (x, t) are in fact defined on a compactum, there are Aj > 0 such that laoi/3i (x, t)l > Aj, whence it follows that the functions a-;;~13 . (x, t) satisfy condition (2.3.7). Therefore the coefficients of Q(x, t; e, r) also satisfy this condition, i.e. Q E S£N(P)· We also note that, in view of the compactness, the polynomials pUJ uniformly satisfy Petrovski'l's parabolicity condition, and H has distinct roots uniformly with respect to (x, t) E IR 2 • Thus, under above assumptions (a), (b), and (c), the proposition is completely proved. )

)

Assume now that at some separate points x 0 , t 0 the symbol (2) has a zero root of multiplicity 1, i.e. bm(x 0 , t 0 ) = 0, J..t(x 0 , t 0 ) = 1. This relates to the property that the polynomial P(x, t; r) possesses roots of the first degree,one of them vanishing at some separate points. In this case factorization (3') is not unique at the point x = x 0 , t = t 0 • To eliminate this ambiguity, we include the zero root in the hyperbolic symbol, i.e. we write (3') with b = 0. Repeating literally the above argument we prove the proposition. Arguing in the same way as in Theorem 1.4.3 we deduce from (3) the following

e,

Theorem. A dominantly correct differential operator with symbol (2) is represented in one of the following two forms:

P(x, t; Dx, Dt) = H(x, t; Dx, Dt)S(x, t; Dx, Dt) + Ql (x, t; Dx, Dt),

(4)

P(x, t; Dx, Dt) = S(x, t; Dx, Dt)H(x, t; Dx, Dt) + Q2(x, t; Dx, Dt),

( 4')

where the symbols H and S satisfy the conditions of the propositiqn, and we have Qj(x, t; r) E S£N(P), j = 1, 2.

e,

3.2. A priori estimate. We have the f~llowing

e,

Theorem. Let P(x, t; r) be a dominantly correct symbol with coefficients satisfying condition (2.3.7). Then Vs E IR 3/'o = l'o(s) such that the inequality

(5)

107

Dominantly Correct Operators

holds, where

(5') Proof. 1) By virtue of (5'), for large -"(the inequality (5) remains valid when an expression cs('Y)KIIull.s(P) ,(s),-y is added to the right-hand side, where K does not depend on 'Y. Replacing P on the right-hand side by expressions ( 4) and ( 4') we reduce ( 5) to the system of inequalities

IID~D~ull(s),-y ~ cs('Y)(IIH · Sull(s) ,-y +liS· Hull(s),-y + (oo) V (a, f3 ) E 8( P ) , Vu E H [-y] .

llull.s(P),(s),-y),

(6)

2) As in Section 2.2, denote by 8°(P) the set of points (a, (3) E 8(P) representable in the form (2.8). With the set 8°(P) we associate in a natural way the norm II ll.so(P),(s),-y and the class of symbols S£<Jv(P) that are linear combinations of monomials r;,arf3, (a, (3) E 8°(P). We estimate II ll.so(P) ,(s),-y by means of the righthand side of (6). By (2.8), we have

a' +/3' ~h-1

(a" ,/311 )EN(S)

~

I:

IIDxa" Dt/3

11

2

2

ll(s+h-1),-y = lluiiN(S),(s+h-1),-y·

(a" ,j3")EN(S)

Using Theorem 2.3.3 for large-"( we can estimate lluiiN(S) ,(s+h- 1),-y from above by means of I!Sull(s+h-1),-y, i.e. we obtain the inequality

(7) We now apply the estimate in Theorem 1.3 (i) to the right-hand side of (7). This finally results in

(8) 3) We now derive inequality (6) for (a,f3) E 8(P) \ 8°(P) i.e. for (a,f3) representable as (2.8') or (2.8"). 4) Let (2.8') hold. Then the operator D~ D~ can be rewritten in the form

The second operator on the right-hand side has a symbol belonging to S£<Jv(P)' i.e. the norm of its value on the function u can be estimated by m eans of the right-hand

Chapter 3

108

side of (8). Let us proceed to the estimation of the first operator on the ..right-hand side. According to Lemma 2.2.1, there is x > 0 such that

L

~~a"r.B"j ~ const lrl-x

l~ar.Bj,

(a,,B)EN(S)

whence it follows that

liD~" D(' wll(s),'"t ~ const 111-xllwiiN(S),(s),'"t· Applying Theorem 2.3.3 once again we coincide that

.B"

liD~ Dt wll(s),'"t ~ const 111-xiiSwll(s),T 1/

Replacing w in this inequality by H u we finally obtain the estimate liD~

1/

.B" . Dt Hull(s),'"t ~ const lfl-xiiS · Hull(s),T

5) It now remains to consider the case (2.8"). It can take place only when N(S) has a vertical side and, consequently the symbol H can be solved with respect to the highest power of~' i.e. H(x,t;~,r) = qh(x,t)~h +O(I~Ih- 1 ).

Then the operator D~ D~ can be rewritten as

a .B DxDt

h = Dxa" Dt,8" qh-1 H + Dxa" Dt.B" (Dx--:qh-1 H) .

The symbol of the second operator on the right-hand side belongs to S.C~(P)· The first operator is estimated in just the same way as in the case qh = 1. Cauchy's problem. With a dominantly correct differential operator P(x, t; Dx , Dt), along with the space 3.3.

Hr,l(P),(s) = {u E H[~~,D~D~u E HbJ,V(a,{3) E N(P)}, a broader space will also be associated:

1-{N(P),(s) = {u E H(s) Da D.Bu E H(s) V(a r.J) E 8(P)} bl

bl

l

X

t

bl

l

l

fJ

with the natural norm II llo(P),(s),T Denote by (H[~](P) ,(s))+ and (H~}P),(s))+ the f.ubspaces consisting, respectively, of functions and distributions vanishing for t > 0. By the generalized solution, to Cauchy's problem for the differential equation

(9) will be meant a function (distribution) u( x, t) E (H~l(P),(s))+ satisfying (9) in the sense of distribution theory:

(u,tP
V
(9')

109

Dominantly Correct Operators

Theorem. If P(x, t ; Dx, Dt) is a dominantly correct operator, tben Vs E lR :l/o( s) sucb tbat for 1 < 'Yo( s) tbere exists a unique solution u E (H~l(P),(s))+ for

any rigbt-band side

f

E

H[~/+.

Proof. We associate unbounded closed operator with Equation (9). By virtue of Theorem 3.2, it will be injective. And its surjectivity is proved using the theorem on the invertibility of stable-correct and strictly hyperbolic operators. We split the proof into several stages. 1) We associate with Equation (9) the unbounded operator

p.

.

(HN(P),(s))

hl

with domain Dp = (H[~{P) , (s))+·

+---+

H(s)

hl+

(u ~ Pu)

(10)

Operator (10) can be closed.

Indeed, let a

sequence Uj be given such that Uj ---+ 0, j ---+ oo, in the norm of the space H~fP),(s)

HG.{.

Let Pu j ---+ f in the norm of We have to prove that Indeed, by virtue of the obvious embedding

f

= 0.

HN(P),(s) C 1-{N(P),(s) C HN(P),(s-1)

hl

-

hl

hl

'

the operator (10) can be regarded as a restriction of the continuous operator ( HN(P),(s-1))

hl

+

---+

H(s-1)

hl+

(u ~ Pu).

Therefore Puj ---+ 0 in H[~]-1), and f = 0. In what follows we assume that operator (10) has already been closed, i.e. (10) is a closed operator. 2) The operator (10) is injective. Indeed, let u E Vp and let Pu = 0. By the definition of an unbounded closed operator, there is a sequence Uj E (H[~{P) , {s))+ such that Uj ---+ u in (H~fP),(s))+ and Puj---+ 0 in H[~( The a priori estimate (5) is continued by continuity to the space H(~](P),{s), whence lluillo(P),(s),-y ~

Since Uj---+ u, we have u

const IIPujll(s),-y---+ 0

as

J---+

=0.

3) The image of operator (10) is closed. Indeed, let

f

E

oo.

H[~f+ be a limit point

in the range of operator (10), i.e. there be a sequen~e Uj E (H[~(P),(s))+ such that Pui ---+

f

in Hr~f. By virt ue of inequality (5), {Uj} is a Cauchy sequence

in the space 1-{~](P) , (s) and, consequently, it converges in this space to an element u E

(H~l{P),(s))+· The closedness of operator (10) implies that

Pu

=f.

Chapter 3

110

4) To prove the theorem it remains to show that the image of operator (10) coincides with the entire space Hr~f+· To this end (see Section 2.3.4) we consider the operator -71. ('1.../N(P),(s)) (H(s)) (11) r · " l rl + ---+ bl + (v ~---+ H · Sv). Since the operator H · S differs from P by an operator with symbol belonging to SCN(P), Theorem 3.2 holds for this operator as well. Therefore we can repeat literally the above argument and prove that f3 is a closed injective operator with a closed image. 5) We now prove that the image of (11) coincides with Hr~f+· In view of the closedness of the image, it suffices to show that the image of the operator contains a dense set in Hr~/+' say H[~]+N), where N is sufficiently large. To this end we consider the equation (12) By Theorem 1.3, if -"'( is sufficiently large, then the equation

H g =

f

E

H[~tN)

possesses a unique solution g E H[~]~N+h-l). Further, according to the results of §2.3, the equation (s+N+h-1) S( x, t; D x, D t ) u = g E H bl+ , is sure to have a solution u E H[~]~N+h- 1 ). If N is sufficiently large, then H(s+N+h-1) C (HN(P),(s))

bl+

bl

+'

I.e. Equation (12) possesses a solution belonging to the domain of operator P. 6) We now complete the proof of the theorem ( cf. the end of the proof of Theorem 2.3.4). Since the closed operator (11) is injective and surjective, by Banach's theorem it has a bounded inverse operator R: H[~f+ ---+ (1fN(P),(s))+, and, by virtue of (5), the norm of the operator tends to zero as "Y---+ -oo. With regard to expansion (4), the operator equation Pu = f can be rewritten as (13) Since the operator Q1 : (1fN(P),(s))+ ---+ H[~f+ is continuous and its norm does not depend on"'(, for large I"YI the norm of the operator RQ 1 is less than unity, and hence Equation (13) has a unique solution u E (1i~l(P),(s))+· The theorem is proved.

111

Dominantly Correct Operators

§4. Dominantly correct polynomials and the corresponding differential operators (the case of several spatial variables) 4.1. Description of dominantly correct polynomials. As in §2.4, with each polynomial P(~,T), ~ E !Rn, we associate a family of pseudo-polynomials

Pw(l~i, T)

def

P(l~lw, T), i.e. polynomials on two variables depending on the pa-

sn-

1 . We shall describe dominantly correct polynomials in several rameter w E variables in terms of such families; the definition of these polynomials was given in the Introduction to the present chapter.

Theorem. For a polynomial

P(~,T) = Tm

+I: aal···anfi~fl ... ~~nTfi,

(1)

,B<m

the following conditions are equivalent:

(I) polynomial (1) is dominantly correct, i.e. inequality (2) in the Introduction holds; (II) polynomial (1) is correct in Petrovskil's sense, and for every polynomial Q(~,T), ~(Q) C 8(P), there is /o = lo(Q) such that IP(~,T)

+ Q(~,T)I # 0

for

ImT ~ /o,

(~,ReT) E 1Rn+ 1 ;

sn-

1 the polynomials Pw(P, T) satisfy the equivalent condi(III) for all w E tions of Theorem 2.3, the polygons 8(Pw) not depending on w and hence coinciding with 8(P); (rv) there are an integer b?:: 0, a parabolic polynomial R(~, T), and a homogeneous strictly hyperbolic pseudo-polynomial Hw(l~l, T) such that (2) and if b > 0, then the pseudo-polynomial Hw bas no zero roots, i.e.

(3) Proof. (I)==}(II) is proved in the same way as the analogous implication in the case n = 1 (see Theorem 2.3). (II)=::>(III) . Consider the polynomial Pw(p,T) = P(pw,T). We have 8(Pw) C 8(P), and therefore the polynomial Pw satisfies the equivalent conditions in Theorem 2.3. Moreover, this polynomial remains correct under the addition of monomials cpaTfi, (a,/3) E 8(P), with ,an arbitrary coefficient c E C. Therefore, according l)Recall that, as in Section 2.4.1, LN(P) denotes the space of pseudo-polynomials Qw(l~l, 1) such t h a t N(Qw) C 6(P) .

Chapter 3

112

to Proposition 2.2.4 (ii), we have b(P) C b(Pw), when.ce b(Pw) = b(P) for any wE

sn- 1 •

(III)==:;.(IV). Since, by the hypothesis, each polynomial Pw(P, r) = P(pw, r) is·. dominantly correct, Theorem 2.3 implies that for any w E sn- 1 there is an integer b(w ), an N -parabolic polynomial R~(p, T ), and a strictly hyperbolic polynomial H~(p, T ), H~(1, 0) =f 0, such that

(2') where R~ is the product of polynomials PYl(p, r), j = 1, ... , k(w), bj(w)-parabolic in Petrovski'i''s sense. By Theorem 2.1, the number k(w), the (even) numbers bj(w), and the numbers xi(w) (the degrees of PYl with respect tor) are uniquely determined by the polygon b(Pw) of minor monomials. Since, by the hypothesis, these polygons do not depend on w (and, consequently, coincide with b(P)), the numbers k, bj, Xj, j = 1, ... , k, do not depend on w either. Further, if the vertices of the polygon N(Pw) are denoted as (0,0), (a(j)(w),,Bu)(w)), j = 1, ... ,m(w) ~ k, then the vertices (a(j), ,B(j)), j = 1, ... , k + 1, do not depend on wand coincide with the vertices of the polygon fl(P). Denote by R( ~, T) the sum of the monomials of the form of

where the points ( a 1 + · · · +an, ,8) belong to the broken line joining the vertices (a( 1),,B(l)), ... , (a(k+l),,B(k+ 1))· The method of defining the polynomials PYl (see Section 1.1.3) implies that R~(p, r) in (2') can be replaced by Rw(P, r) = R(pw, r).l) To this end, in order to examine the factors Tb(w) H~ we consider the three cases below. 1) Assume first that there is w = w0 such that b(w0 ) > 1, i.e. the polygon N ( Pw) contains a vertical side of height no less than 2 not lying on the coordinate axis. By Theorem 2.1, in this case the polygon N(Pw) is determined uniquely by the polygon b(Pw), and, consequently, the numbers b(w) do not depend on w. Thus, all the polygons N(Pw) and, hence, fl(P) have the vertices (a(j),f3(j)), j = 1, ... , k + 3, the side joining (a(k+2),f3k+2) and (a(k+ 3), ,B(k+ 3)) being vertical, i.e. a(k+2) = a(k+3), ,B{k+ 3) = 0. Denote by Po(~,r) the sum of the monomials corresponding to the side joining (ak+1,,8k+1) and (a(k+ 2),,8k+ 2) (this side relates tohyperbolic roots of the polynomial P), and let

l)We stress that, generally speaking, the polynomials R~(p, r) depending on w in an uncontrollable way. As to the polynomial Rw(P, r), its coefficients are smooth functions of w.

113

Dominantly Correct Operators

be the sum of the monomials corresponding to the vertex (a( k+I), ,8( k+ 1)). Since this a common vertex of all N(Pw), we have a(m+ 1 )(w) f= 0 Vw E sn- 1 . It follows that the function

(4) is a pseudo-polynomial, and we have Hw = H~ for each wE sn- 1 . We thus arrive at (2) with b, R(e, r), and Hw(!e!, T) determined above, and since we have Hw = H~, condition (3) is fulfilled. 2) Now let b(w) ~ 1 and let there be w = w0 such that b(w0 ) = 0. In this case we have Tb(w) H~ = Hw, where the function Hw is determined by means of (5). We thus obtain factorization (2) with b = 0. 3) Finally, if b(w) 1, then we can again write factorization (2) with b = 1 and Hw in ( 4). The function Hw must satisfy condition (3) since otherwise we would arriv2 at a contradiction to the condition that E(Pw) does not depend on w. (IV)===}(!). The proof of this implication is analogous to the proof of Proposition 2.2. It is only necessary to establish, instead of expansion (2. 7), its multidimensional analog

=

(5) where the degree of the pseudo-polynomial q is less than that of the pseudopolynomial Hw, and (a',,B') E 6.(P), (a",,B") E E(P). Expansion (5) is proved following the plan of the proof of Lemma 1 in Section 2.3, and we leave this to the reader. 4.2. Dominantly correct differential operators. A differential operator

nr; +

L aal···amf3(x, t)Df

1 • ••

D~n Df'

f3<m

where x E Rn, is said to be dominantly correct if its symbol

P(x,t;e,r) = Tm

+L

aal···anf3(x,t)ef 1

•••

e~n,

(6)

f3<m satisfies the following conditions ( cf. Section 3.1): (i) the polygons of minor monomials E(P(x, t)) do not depend on (x, t), i.e.

E(P(x, t)) = E(P) = UE(P( x, t)); (ii) the polynomial P( JRn+1.

e'

T)

=

P( x 0 ' t 0 '

e' 7) is dominantly correct V( x

0'

t0 )

E

The multidimensional analog of Proposition 3.1 is proved by means of a literal repetition of the proof of the implication (IIl)===}(lV) in Theorem 3.1.

Chapter 3

114

Proposition. A dominantly correct symbol (6) is representable as

(7) wlwre the polynomial symbol S(x, t; ~' r) and the pseudo-polynomial symbols Hw and Qw possess the following properties: 1) S(x, t; ~' r) is a stable-correct symbol (in the sense of Section 3.1 where N(P) should be replaced by 6.( P) ), the coefficient in the highest power of T is equal to 1, and the coefficients satisfy condition (2.3.7); 2) Hw(x, t; 1~1, r) is a strictly hyperbolic pseudo-polynomial, its coefficient in the highest power ofT is equal to 1, and the other coefficients satisfy condition (2.3. 7); 3) if the symbolS is not parabolic (i.e. S"' Tb pb b > 0 where pl is a parabolic symbol), then the symbol Hw has no zero roots, i.e. Hw(x,t; 1,0) -f. 0 V(x,t) E IR; 4) the pseudo-polynomial Qw belongs to S£A.(P) (see §2.5), i.e. is a linear combination of monomials of the form of a(x, t;w)i~iar.B, (a, {3) E b(P). Theorem. A dominantly correct differential operator with symbol ( 6) is represented in the form

P(x, t; Dx, Dt)

= S(x, t; Dx, Dt) · H(x, t; Dx, Dt) + Q1,

P(x, t; Dx, Dt) = H(x, t; Dx, Dt) · S(x, t; D x, Dt)

+ Qz,

(8) (8')

wl1ere Q 1 and Q 2 are bounded operator from H~fP),(sh) into H[~f for any s E JR. and 1 ~ lo· Proof. The composition of the differential operator S(x, t; Dx, Dt) and the pseudodifferential operator H(x, t; Dx, Dt) is a pseudodifferential operator with symbol S(x, t; ~' r)H(x, t; ~' r)

+ L s< 8)(x, t; ~' r)H(8)(x, t; ~' r)/b!. 8>0

With regard to relation (7), Q 1 is a pseudodifferential operator with symbol Q1(x, t; ~' r) = -

I)s< 8) H( 6))(x, t; ~' r)/8! + Q(x, t; ~' r). 6>0

.,From the description of the polygon 6.( P) that was in fact presented in §2 it follows that the symbol Q1 is a linear combination of expressions of the form of

qa,a(x, t; w )D~ 1)

nf,

For the definition of this space see Section 3 .3 .

(Ia!, {3) E 8(P),

115

Dominantly Correct Operators

.. PDO £ '1..JN(P),(s) . t i.e. Q 1 E S£~(P), and t h e correspond mg trans orms 'Lhl m o (8) is proved. Comparing (8) and (8') we see that

H(s)

.

hl, I.e.

We now show that the commutator on the right-hand side is a linear combination of operators of the form of

(Ia!, {3) E 6(P), where Qa/3 are bounded operators on

S=

L

(9)

HGi for any s and/· Writing

Saf3(x , t)D~D~,

L

H=

Hp8(x, t; Dx)D~D~

IPI+8~h

( lai,/3)Eil(S)

we represent S · H - H · S as a linear combination of operators

where the dots symbolize the terms appearing under the commutation of the operators of differ entiation with the operator of multiplication by the function Saf3(x , t) and the PDO Hp8(x, t ; Dx); these operators are obviously written in the form (9). Since t he coefficients in the highest powers of Dt in the operators S and H are equal to 1, the terms in (10) corresponding to the senior points (lad+ IPI, {3 + 6) of the polygon ~( P) involve differentiation with respect to the variables x 1 , ... , x n. Since [Saf3, Hp8]Dj are bounded operators on H[~f, the right-hand operators in (10) are readily represented in the form (9). The theorem is proved.

4.3. Cauchy's problem for dominantly correct differential operators. We now extend Theorems 3.2 and 3.3 to the case of several variables Theorem. Let P(x, t; Dx, Dt) be a dominantly correct operator. Then Vs E R 3/o = / o( s ) such that the assertions below hold:

(i) the estimate

where £ 8 ( 1) - t 0, 1 - t --oo , is fulfilled; (1i) Vf E H[~f+ th ere is a function u E (Ht,)P),(s))+ satisfying the equation P(x, t; Dx , Dt)u = in sense of distribution theory.

J,

(12)

Chapter 3

116

The proof of (i). This assertion is proved following the scheme for the onedimensional case (Theorem 3.2) based on analogs of expansions (2.8), (2.8'), and (2.8"); if (a, j3) E 6(P), then

(a,{l)

= (a',/3') + (a",/3"),

(a,(l)

= (a',/3') + (O,h),

(a,j3)

= (a',/3') + (a",O), !a"l =

(la'l,/3') E ~(S), !a"l + (3" ~ h -1, (13) (la'l + 1,/3') E ~(S), (!a'!,fl' + 1) t/:. ~(S), (13') h,

(la'!,/3' + 1) E ~(S), (la'l + 1,/3') ¢:. ~(S).

(13")

As in Theorem 3.2, denote by 6°(P) the set of points (a, (3) E 6(P) representable as (13) and associate with this space the class of symbols S.CCJ~.(P) and the corresponding norm. Combining the estimates for stable-correct operators with the Petrovski!-Leray inequality (see Theorem 1.3) we obtain, as in Theorem 3.2, the inequality (14) llulloo(P),(s),-y ~ canst lri- 1 IIH · Sull(s),.,. In the case (13') the norm of DC: D~ is estimated as in Theorem 3.2. The operator DC: D~ is rewritten in the form

The second operator on the right-hand side is a PDO with symbol belonging to S£ 6 ~p), and therefore the norm of its value on the function u can be estimated by means of the right-hand side of (14). As in the case n = 1, we have

A more intricate problem is the estimation of the norm liD~ D~ull(s) ,-y in the case (13") since it is this point where the specificity of the multidimensional case manifests itself. The case (13") is possible only when the symbol S possesses zero roots, i.e. (Proposition 4.2) the symbol Hw(x, t; 1~1, 0) def H(x, t; ~' 0) is nonzero for 1~1 =/: 0 and is an elliptic symbol with respect to ( x, 0 of order h depending on the parameter t. We make use of the well-known estimate for elliptic PDO (e.g. see Eskin [1]):

I

I is the ordinary Lz norm in !Rn. DC:D~u(x, t) we find

where

Replacing w in this estimate by

(15)

117

Dominantly Correct Operators

e:x"

Since we have T/3 E ..c~( P)' the second term on the right-hand side has already been estimated. As to the first· term, we have ,

/3

H(x, t; Dx, O)D~ Dt =

D~

,

/3

Dt H(x, t; Dx, Dt)

+ D~" D~(H(x, t; Dz, 0)- H(x, t; DX) Dt)) Expressions of the type ! 1 were already estimated, 12 is a PDQ with a symbol belonging to S..C~{P)' and the commutator his a linear combination of PDQ with symbols of the form of 1111

+ lJ > 0.

(16)

If Ill I > 0, then symbol (16) belongs to s..c~(P)' i.e. we have to consider the case ll = 0, v > 0. Then (16) has the form (16') where the dots designate symbols belonging to s.c~(P)• Symbol (16') is a linear combination of polynomial symbols EILrv with coefficients depending on x, t, and w, and 11 and v satisfy conditions of type (13'), (15'). We now summarize the results. We have shown that if conditions (13") are fulfilled, then liD~ Df ul!(s),l' can be estimated by means of the right-hand side of (11) and a finite set of norms IIDIL nrull(s),/'' where J1 and I satisfy conditions analogous to those for a and f3 with the additional requirement that v < f3. Repeating these estimates a finite number of times we prove inequality (11).

Th e proof of (ii) . The existence theorem for the solution to Cauchy's problem_ for strictly hyperbolic differential operators is extended to strictly hyperbolic PDQ (e.g. see Eskin [2]). In view of this, we can literally repeat, based on inequality (11) , the argument in Theorem 3.3 and prove the solvability of Equation (12). A direct proof of this assertion will be given in Chapter 7.

CHAPTER IV

OPERATORS OF PRINCIPAL TYPE ASSOCIATED WITH NEWTON'S POLYGON

§1. Introduction. Operators of principal and quasi-principal type 1.1. The present chapter is devoted to studying differential operator that majorize locally all lower (in the sense of Newton's polygon) derivatives. The corresponding class of operators is an extension of the class of N quasi-elliptic operators (Chapter 1) to the same degree as the class of dominantly correct operators (Chapter 3) is an extension of the class of stable-correct operators (Chapter 2). Recall that in Chapter 1 we studied a class of differential operators on functions of the variables (x, y) E R 2 , for which in any region n C R 2 of a sufficiently small diameter the inequality

(1) ( a ,{J)EN(P )

holded. In the case of constant coefficients, by virtue of the well-known result of Hormander (see (1.4.20)), inequality (1) is equivalent to the corresponding inequality for the symbol P( (, 1J):

(2) (a,{J)EN(P)

If the polygon N ( P) is regular (as in Chapter 1, only in this case we consider est imates for variable coefficients), then inequality (2) is equivalent to the estimate considered in the first half of Chapter 1, namely 3c, c0 > 0 such that

(3) (a,f3)EN(P)

Necessary and sufficient conditions for the validity of (3) are stated in t erms of quasi-homogeneous parts of the polynomial P corresponding to the sides of the polygon N(P). In this chapt_e r we shall study differential operators P( x , y; Dx, Dy) for which in an arbitrary bounded region of a sufficiently small diameter diam ~ A ~ Ao the inequality

n

L

n

IID~D:ull ~ c(A)iiP(x,y;Dx,Dy)uii

(a,{J)E6(P)

VuED(n) c(A) ---t O, 11 8

A---tO,

(4)

Operators of Principal Type Associated with Newton's Polygon

119

holds. Inequality (1) implies (4). This is not an automatic reduction since the lower derivatives must be estimated by means of a small constant. The constant c(..\) in inequality ( 4) being small, the inequality remains true under any perturbations of the coefficients in lower derivatives in P. More precisely, with any symbol Q(x, y; ~' ry) E SCN(P) a constant ..\(Q) can be assoCiated, such that for ,\ ~ ..\( Q), diam ~ we have

n ..\,

(4') (a,jJ)Eo(P)

In the case of constant coefficients this implies the algebraic condition

(5) (a,jJ)Eo(P)

on the symbol. In the first part of this chapter, under some additional conditions on Newton's polygon, we shall find necessary and sufficient conditions on the polynomial P under which inequality (5) is fulfilled. As in the case of inequality (4) (or inequalities in Chapter 3), these conditions will be stated in terms of the principal quasi-homogeneous parts of P corresponding to the sides of Newton's polygon. These parts either do not vanish (the condition of N quasi-ellipticity) or have simple (in the above-mentioned sense) real zeros. The second part of the chapter is devoted to the proof of inequality ( 4) for the case of variable coefficients. Here we impose an important additional condition that some of the coefficients of the symbol P should be real. These conditions are such that the symbols P( x, y; ~, TJ) and P* ( x, y; ~, TJ) satisfy them simulatneously. It follows that inequality (4) remains valid when Pis replaced by P* or tp, whence the local solvability of the operator P is derived. 1.2. The results of this chapter (as well as those in Chapter 6) are a generalization and further development of the well-known result by Hormander [2], according to which a polynomial P(~ 1 , •.• , ~n) of degree miscalled a polynomial of principal type if I grad P(m)(6, · · ·, ~n)l =/- 0, (~I, ... , ~n) E !Rn \ {0}, (6) where P(m) is the principal homogeneous part of P. Hormander proved that condition ( 6) is fulfilled if and only if

(7) where Q is an arbitrary polynomial of degree no higher than m- 1. Similarly, a differential operator P(x; D) with variable coefficients is called an operator of principal type if the symbol of its principal homogeneous part satisfies the condition

Chapter 4

120

For these operators, under the additional assumption that the symbol the a priori estimate

llull<m- 1) :::;; c:(A)IIPull, VuE V(n),

diamn:::;; A,

c:(A)

-t

P(m)

is real;

A-t 0,

0,

(8)

was proved. 1.3. We now pass to the quasi-homogeneous case corresponding to a positive vector q = (q1 , ... , qn) whose components are normalized by the condition min qi ::::: 1. The question is posed as to what are the necessary and sufficient conditions on the polynomial P(O and the symbol P(x; 0 under which inequalities (7) and (8) hold if in the left-hand sides of the inequalities the sum 161 + · · ·+ l~n I and the norm II ll(m- 1) are replaced by 16 jlfq1 + · · · + l~nl 1 fqn and by the quasi-homogeneous norm

respectively. The corresponding polynomials and operators are called polynomials and operators of q-principal or quasi-principal type. Hormander's results were generalized to operators of quasi-principal type by Shananin [1, 2] and Lascar [1, 2]. In the quasi-homogeneous case, if we want to state all results in terms of principal quasi-homogeneous parts, we have to impose on the vector q the following additional arithmetical condition. The numbers q 1 , •.. , qn are integers. The meaning of this condition will be eludicated below (see Remark 3). Thus, let a polynomial

(9) of q-degree m be given, i.e. m = degq P =

max (q1a1

(a1, ... an)

+ · · · + qnan),

and let Pq = P(m;q) be its principal q-homogeneous part. When generalizing condition (7) to the quasi-homogeneous case it should be taken into account that, according to Euler's formula, we have

and therefore condition (6) is significant only where polynomial Hence, condition (6) can be replaced by

{ P( m) (~1' · · · , ~ n) = 0}

=} {

P(m)

Igrad Pm(~ 1 , ... , ~n) I =f. 0}.

vanishes.

121

Operators of Principal Type Associated with Newton's Polygon

Further , the polynomials 8P/8ej have q~orders m- qj, i.e. if qi > 1, then the derivative 8PI aej must not affect the estimates for monomials of q-degree m- 1. In view of what has been said, a variable ei will be called an essential variable of polynomial (9) if qi = 1. The collection of essential variables will be denoted e'. Polynomial (9) of q-degree m is called a polynomial of q-principal type if

(10) Similarly, if a symbol P(xt, ... 'Xni el, ... 'en) is given, then the variables Xj, ej corresponding to qi = 1 are said to be essential, and their collections are denoted (respectively) as x' and e'. An analog of (6') is the condition

Under the additional assumption that the symbol Pq is real, in the above-mentioned works by Shananin and Lascar analogs of Hormander's results I) were obtained. Our aim is t o extend these results from the quasi-homogeneous case to the case of polynomials or operators whose principal part is determined using Newton's polygon (polynomials and operators of N -principal type). To gain a better understanding of these questions, we now present a simple algebraic result relating to the quasi-homogeneous case.

Theorem. Let q = (q 1 , ... ,qn) E zn and let minqi = 1. Then for polynomial (9) the following conditions are equivalent: (i) P is a polynomial of q-principal type (i.e. (10) holds) (ii) for any polynomial Q(e), degq Q < m, there is a constant cq > 0 such that (11) Proof. (i)===}(ii). Select on the "sphere" p(e) 1 a finite covering {Uj}, the covering being so fine that either Pq(e) # 0, ~ E Uj, or (see (10)) I grade, Pq(OI # 0, E Ui . These conditions remain valid in the q-cones

e

as well, which, obviously cover Rn \ {0}. Assume first that ~ E KUj and Pq( 0 # 0. Since among the polynomials Q(a) there is a constant, say a, we have · ·· (P

+ Q)(~) >

lal

+

IP(e)

p(a)

+ Q(OI >a+ ciPq(OI ·~ ciP(e)- Pq(e) + Q(e)l.

+

(12)

I) Using Carleman's estimation technique, Lascar replaced condition (10') by the weaker condition {Pq(x;~)

= 0} => {lgrad(x',(') Pq(x,~)i =P 0}.

Chapter 4

122

In view of the quasi-homogeneity, there is x such that

The polynomial P - Pq + Q is a linear combination of monomials of q-degree less than m. When estimating these monomials the following elementary lemma is of use. Lemma. Let q1 a 1 + · · · + qnan < m. Then V>.. > 0 tl1ere is a constant cA sucb that

(13} Proof. Since

lEi I :s; pqi (E), we have IE~l

... e~n I :s; p( Ettqt+···+anqn.

Setting

in the elementary inequality a, b > 0,

1/p + 1/p' = 1,

we obtain (13). By virtue of the lemma, the third term on the right-hand side of (12) can be estimated from below by means of

Substituting this into (12) we obtain

Taking>.. :s; x/2 and c < a/2c(x/2) we derive inequality (11) forE E I
(P + Q)(O >

Ia! +I grade(P + Q)l

>a+ cl grade Pq(OI- cl grade,(P(O- Pq(E) + Q(O)I. Since I grade, Pq(OI > x' pm-I(E) .and grade,(P- Pq + Q) is a polynomial whose q-degree is less than m -1 (here we use the fact that qj ~ 1), it remains to repeat the above estimation.

123

Operators of Principal Type Associated with Newton's Polygon

(ii)=>(i). Assume that (10) is not fulfilled, i.e. there is a point ~ 0 = ( ~~' ... , ~~) E !Rn such that (10o) Consider inequality (11) on the curve ~(t)

for t

--+ CXJ.

= (~~tq 1

1 ••• 1

~~tqn)

Then we have consttm- 1 ~

(1 +

p(~(t)))

m-1

-

~ c(P + Q)(~(t)).

(14)

We write where P~ and Qq are q-homogeneous polynomials of degree m-1, and the q-degrees of P" and Q' do not exceed m -- 2. Then inequality (14) takes the form const tm-1 ~ c(IPq(~o)ltm + IP~(~o) + Qq(~o)ltm-1 +I grade Pg(~ 0 )1tm- 1 + o(tm- 1)).

(14')

By virtue of (10 0 ), the first and third terms on the right-hand side are equal to zero. The polynomial Q can always be chosen so that P;(~ 0 )

+ Qq(~ 0 ) =

0.

(15)

/:.

Indeed, let, for definiteness, ~J 0 for j ~ s and ~J = 0 for j > s. If there are integers G't, . . . , as such that a1q1 + · · · + a 8 qs = m- 1, then setting Q(O = c~fl . .. where c = -P~(~ 0 )/(~f' 1 ••• ~~a·), we attain the fulfilment of condition (15). In case there are no integers a1, ... , a 8 for which a1 q1 + · · · + a 8 q 8 = m- 1, we hav~ P~(~ 0 ) = Qq(~ 0 ) = 0, i.e. (15) remains valid. In view of what had been said, we have o(tm- 1) on the right-hand side of (14'), and we arrive at a contradiction proving the desired assertion.

e;·,

Remarks. 1) If q = (1, ... , 1), then inequality (11) goes into (7). We note that inequality ( 6) is equivalent to

(16) lal~m-1

In the quasi-homogeneous case inequality (11) is stronger than the quasi-homogene0us analog of (16): l~fl

... ~~n I ~ c(P + Q)(O .

(17)

Indeed, according to inequality (11), the function (P+Q) majorizes the expressions leil(m- 1)/qi. If qi > 1, the number (m- 1)/qi can be non-integral, and then the left-hand side of (17) contains only l~ilki, kj = [(m -1)/qj]. 2) We present two examples of quasi-homogeneous polynomials for which condition (10) is violated and, consequently, inequality (11) may not hold. Nevertheless, these polynomials satisfy inequality (17).

Chapter 4

124

Example 1. n = 2, P(O = ef variables is 6. It is obvious that

+ ie?6.

IP(OI + IBP/861 = 0

In this case q

for

6 =

0,

= (1, 2), and the essential 6

E JR.

On the other hand, it is clear that

L

~~lla 1 1~2la 2

:(

P(O.

a1+2a2~3

This inequality (with another constant) also remains true under replacement of P by P + Q, where Q is an arbitrary polynomial of (1, 2)-degree no higher than 3. Example 2. n = 3,

This polynomial coincides with its (1, 2, 2)-homogeneous part. The essential variables is 6 ', and we have ·

IP(OI + IBP/861 = 0

for

6 = 0, 6 = 6

E

JR.

On the other hand, the inequalities t6 + t2(t2 + t2) lp( ~t)l > ~1 ~1 ~2 ~3 '

1!;}2 u Pj!::lt2 u~ 1 1> 2(t ~ 2 + ~t 32),

readily imply (17). 3) In the case of non-integral q1 , ... , qn there can exist monomials

~a

such that

m- 1 < (a, q) < m. These monomials can majorize the left-hand side of inequality (11) and guarantee its fulfilment when the q-principal part does not satisfy condition (10). In other words, without the assumption that qj are integers (10) is not a necessary condition for the validity of (11). On the other hand, as is seen from the above proof, (10) is a sufficient condition for the validity of inequality (11) for any values of qb . . . , qn. §2. Polynotnials of N-principal type In this s~ction we shall elaborate a complete description of polynomials in two variables for which inequality (15) holds. In the special case when Newton's polygon is a triangle the established result goes into Theorem 1.3. In other words, the polynomials satisfying inequality (15) are a natural generalization of polynomials of quasi-principal type to the case of arbitrary Newton's polygon. The main result in this section will be proved under some additional arithmetical conditions on Newton's polygon, and we begin the presentation of the material with the description of these conditions.

125

Operators of Principal Type Associated with Newton's Polygon

2.1. Remarks on Newton's polygon. We shall deal with polynomials in two variables

(1)

r)o) =

As above, N(P) is Newton's polygon,

(aj,/3j), j = 0, ... ,m+ 1, are the ver-

tices of N(P) (where (a 0 , j30 ) = (0, 0)), and 1 ) are the sides joining 0 ) and r)~ 1 . All the further statements are essentially simplified if it is assumed that Newton's polygon N(P) is regular. In what follows we do not stipulate this condition. Let q(j) = (q~j),q~j)) be the outer normal vector to N(P). If j =/:- 0, m -1 (i.e. the side r} does not lie on the coordinate axes), then, by virtue of the regularity condition, the vector q(j) has positive components. Since the vector is determined up to within a positive factor, we normalize its components by the condition

r)

min(q(j) q(i)) - 1 1

'

2

-

'

r)

max(q(j) q(j)) 1

'

2

= bJ'·

(2)

i.e. either q(j) = (1,bj) or qU ) = (bj,1). We also remind the reader that the polygon N ( P) can be determined by the system of inequalities

((a, j3), q(j)) ~

Cj,

j = 1, ... , m,

a;::: 0, j3;::: 0.

(3)

If t he regula~ity condition is fulfilled, then all the earlier presented definitions of minor points are equivalent. Therefore (a, j3) E N(P) is a minor point if 3( a', j3') E N(P) such that a < a', j3 ~ j3' or a ~ a', j3 < j3'. An integral point (a, j3) will be called a Z-minor point if there is an integral point (a' j3') possessing the indicated properties. The convex hull of all Z-minor points will be denoted as 5z(P). Obviously, we have 8z(P) C 5(P). We present an example of a polynomial for which this inclusion relation is strict.

e

Example. P(e,Tf) = +ry 2 +eery. In this case N(P) is a triangle with vertices (0, 0), (0, 3), and (2, 0). The point (1, 1) is an interior point of the triangle and, consequently, is minor point. However it is not a Z-minor point. On the other hand, in the case of polynomials correct in Petrovski1's sense the sets 5(P) and 8z(P) coincide (see Lemma 2 in Section 3.2.2). We state necessary and sufficient conditions for the coincidence of the polygons 5(P) and 8z(P). Lemma. For polynomial (1) with regular Newton's polygon the conditions below are equivalent. (i) The number bj, j = 1, ... , m, in (2) are integers. (ii) If the polygon N(P) is determined by inequalities (3) and the vector q(j) is normalized by condition (2), then

((a,j3),q(j)) ~

Cj

(iii) 8(P) = 8z(P).

-1,

j = 1, ... ,m,

V(a,j3) E (N(P) \

rj1 )) c Z 2 .

(4)

Chapter 4

126

Proof. (i)==:=;.(ii). Recall that

c·((a·) l if-l.) q(j)) • }- . -'}l Since the vertices (a j, fii) have integral coordinates, under condition (i) the numbers in (3) are integers. If a point (a, fi) does not lie on the straight line passing along r} 1), then ((a,fj),q(j)) < Cj. The numbers being integral, we arrive at (4). (i:i)==:=;.(iii) . Take an arbitrary point (a 0 ,fj 0 ) E b(P). In view of the regularity of N(P), the point cannot lie on the sides rjl), j = 1, ... , m, and therefore it satisfies inequalities ( 4). We now show that either for the point (a, fi) . ( a 0 + 1, fj 0 ) or for the point (a, fi) = (a 0 , (i 0 + 1) inequalities (3) hold, i.e. the corresponding point belongs to N(P) and ( a 0 , fj 0 ) is a Z-minor point. If all vectors qU) have the form q(j) = ( 1, bj), then inequalities ( 3) are fulfilled for (a,(i) = (a 0 + 1,fj 0 ), and if qU) = (bj, 1) for all j, then (3) holds for (a,fj) = (ao,r;o + 1). We now consider the general case when

Cj

q (j)

= (1' bj)'

j = 1' ... ' h'

q(j)

= (bj, 1),

j

> h.

Denote by N' the intersection of N(P) with the half-plane (i ;:? fih and by N" the

N'u

Figure 3 intersection of N(P) with the half-plane a ;:? ah, and let N"' = { (a, (i), a < ah, (i ~ fih}, (see Figure 3) . By what has been said, if (a 0 ,fj 0 ) EN', then (a 0 + 1,fj 0 ) E N(P), and if (a 0 ,fj 0 ) EN", then (a 0 ,(i0 + 1) E N(P). In case (a 0 ,fj 0 ) EN"', we

127

Operators of Principal Type Associated with Newton's Polygon

have either a 0

< ah or {3° < f3h, i.e. either ( a 0 + 1, {3°) E N(P) or ( a 0 , {3° + 1) E

N(P).

(iii)==?(i). Let qU) = (1, bj) where bj = m/n > 1, the numbers m and n being natural and coprime. Then the point (aj+l,{Jj + [m/n]) belongs to b(P) (as an interior point of N(P)) but is not a Z-minor point, which contradicts (iii). In what follows we impose on the polygon N(P) the arithmetical

Condition {A). The polygon N(P) satisfies the equivalent conditions of the lemma. ......

Denote by b(P) the polygon determined by the inequalities

h(P)

= {(a, {3)

E IR~, ((a, {3), q(j)) ~

Cj-

1,j

= 1, ... , m }.

If Condition (A) is fulfilled, then, according to Lemma (ii), the polygon h(P) is an extension of b(P), i.e.

b(P)

c h(P) c N(P),

and (in contrast to b(P)), the polygon h(P) is regular (of course, if N(P) possesses this property). The example of polynomials whose Newton's polygon is a triangle (Theorem 1.3 and Remarks 1) and 2) in Section 1.3) demonstrates the natural character of the introduction of the polygon 8(P).

2.2. Statement of the main result. Let q(j) be the outer normal vector to the side r}l) normalized by condition (2) and let pq(i) ( 11) be the principal q(j)homogeneous part of P. The variable (accordingly, 17) is said to be an essential variable of the side if q~j) = 1 (accordingly, q~j) = 1). Essential variables will be denoted

e.

r)l)

e'

e

Definition. Polynomial (1) is called a polynomial of N -principal type if for any side 1), j = 1, ... ,m, with essential variable (or variables) ewe have

r)

{Pq
# 0},

e#

o, 11 # o.

(5)

Theorem. Let N(P) be regular Newton's polygon and let condition (A) be fulfilled. Then the following conditions are equivalent.

(i) P is a polynomial of N -principal type, i.e. for j = 1, ... , m condition ( 5) holds. (ii) For any polynomial Q, N(Q) C b(P), there is a constant

CQ

such that

(6)

Chapter

128

4

Proof. (ii)==}(i). Let condition (5) be not fulfilled for some j, i.e. let there be

eo =I 0, TJ

0

f:. 0 such that (5')

Considering inequality (6) on the curye e(t) = tq~j)eo, ry(t) = tq~j)TJ 0 and repeating literally the argument in Theorem 1.3 we show that (5') contradicts (6). The implication (i)==}(ii) is derived from the following Proposition. Let P( equality

e, TJ) be a polynomial of N -principal type.

36{P) ( e' TJ)

Then the in-

~ c( IP( e' TJ) I + I grad P( e' TJ) I + 1)

(7)

holds. Since the definition of a polynomial of N -principal type involves only those terms of the polynomial that correspond to the sides 1), j = 1, ... , m, of Newton's polygon, inequality (7) remains valid under replacement of P by P + Q, N( Q) C b(P). Hence, the proposition implies (i)==}(ii).

r}

Remarks. 1) If N(P) is a right triangle, then the theorem goes into Theorem 1.3. 2) Inequality ( 6) implies (1.5). However, as is shown by Example 1 in Section 1.3, conditions (5) are not necessary and sufficient conditions for the fulfilment of (6). 3) We note that the proposition does not use the arithmetical condition (A). Thus, for every polynomial of N-principal type inequality (1.5) holds. In general, the converse assertion is not true. The remaining part of this section is devoted to the proof of the proposition, which splits into two stages. Inequality (7) is first proved for special regions relating to the vertices or sides r}k), k = 0, 1. These regions are selected so that the corresponding principal part Pr~"') dominates the other terms. We prove that the }

specially selected regions cover the entire plane JR2 , and therefore inequality (6) results from "pasting together" the inequalities in these special regions. We note that incidentally we obtain a new proof of Theorem 1.3.3. When proving (7) we can confine ourselves to the case 17 > 0, (the general case reduces to the former by means of the transformations e --+ -e or TJ --+ - TJ).

e,

2.3. Estimates in the corner regions G(r~o) ,Eo) relating to the vertices

of the polygon N(J?). Let Denote by

V({)

r}o)

= (ai,(Ji),

r}o)

=I (0,0), be a vertex of N(P).

the angle between the normals at the vertex, i.e. the region lying

b etween the rays {tq
= (-1,0)

and q<m+l)

= (0,-1)).

As can easily be seen, q

= (q1,q2 )

and only if q 2(j-l)q 1 -

q(j-l)q ~ 0 1

2:;---.

E

V({)

if

129

Operators of Principal Type Associated with Newton's Polygon

Denote by G(rj0 ), co) the set of ( e, 1J) such that the vector (loge, log 1J) 1 ) belongs to the translated angle

V{{):

-q~i) log~+ q~j) log 1J >log~' co

( . 1) 1 ( . 1) q/log~- q/- log 1J >log-.

co

(8)

We now show that in the region G(rjo), co) the monomial ecxi ryf3i corresponding to the vertex r}O) dominates the other monomials. Lemma. Vc > 0 and for any closed set I< C N(P) \ r}o) there is a constant co(K,c) such that for co= co(K,c) the inequality (~,1J ) E

(0)

G ( ri ,co),

(9)

holds. Since the polygon lf(P) does not contain the vertices of N(P) distinct from the origin, an immediate consequence of the lemma is the following

e'

Proposition. If co is sufficiently small, then for ( 1J) E G(r}O)' co) inequality (7) holds: (7') Proof. We choose the set K C N(P)\fJ 0 ) so that it contains lf(P) and all integral

r}

0 ). Then, according to the lemma, for points of N(P) distinct from (ai,/3i) = co = co(K, c) the left-hand side of (7) does not exceed cx~cxi ryf3i, where x is the number of vertices in lf(P). On the other hand, in view of the lemma, we have

~cxiryf3i = !acxif3i ~-liP(e,7J)- (P(e,1J)- acxif3i~cxi1Jf3i)j

~

!acxif3i l- 1 (1P(~,7J)I

+

L

!acxf3ecx1Jf31)

(cx,(3) EN( P) \r) 0 )

~ lacxi f3i I- 1 IP( ~, 1]) I + c ( L !acxf3a~i1f3i I) ~cxi ryf3i . Selecting c so that the coefficient in ~cxi ryf3i on the right-hand side does not exceed 1/2 we estimate ~cxi ryf3i by means of P( ~' 1J) and thus prove inequality (7'). The proof of the lemma. Denote by Lj and L j - l the rays passing along the sides and r)~ 1 and issuing from the vertex r)o). Then

r)l)

t)we once again remind the reader that under consideration are only nonnegative~ and ry.

Chapter 4

130

and, according to the first inequality (8), for (a, j3) E Lj we have ~Otr,P

= exp( a log~+ j3log 7J) = exp( aj log~+ /3i log 1J) x exp( t( q~j) log~ - q~j) log 1J)) ~ c~~Otj 1J/3j,

where t that

= t(a, j3).

Similarly, for any point (a, j3) E Lj-1 there is 8

= 8( a, j3) such

~Ot1]/3 ~ cg~Otj 1]/3j.

It remains to note that the points (a,j3), (a',j3') E Lj, and (1,6), (1',8') E Lj- 1 , t( a', j3') > t( a, j3) > x > 0, 8(1', 8') > 8( 1, 8) > x > 0, can be chosen so that the set J( lies in the convex hull of (a,j3), (a',j3'), (1,8), and (1',8'). Then for c 0 < c the inequality (7) holds. 2.4. Estimation in half-strips G(r~t) ,£0 ,s 1 ) relating to the sides of the

polygon N(P). If

r)

1)

is a side of N(P) not lying on the coordinate axes, i.e.

j = 1 ... , m, and q(j) = (q~j), q~j)) is the outer normal vector, then we denote by G( r)l)' co' Cl) the half-strip determined (in the plane (log~' log 1])) by the inequalities log co <

-q~j) log~+ qij) log 1J < log_!_, co

Lemma 1. (i) For any pair of points (a, j3), (a', j3') E on a, j3, a', j3') such that cg ~ ~Ot1J/3~-0t'1J-!1' ~ c 08

for

r)

1)

there is 8 (depending

(~,7J) E G(r)l) ,co,ci).

r)

(ii) V(a,j3) E 1), Vc > 0, co> 0 and for any closed set I< is c 1(c, co, K) such that

c

N(P) \

r}l)

there

(a'j3') E K, for c1 = c1 (c, co, I<).

Proof. (i) If (a,j3), (a',j3') E

r)l), then qij)(a- a')+ q~j)(j3- j3')

= 0 since for

some 8 we have

a - a' - -8q(j) -

2

'

j3 - j3' = 8qij).

Therefore

With account of the first inequality (10), the right-hand side is estimated from below by means of cg and from above by means of c 08 •

131

Operators of Principal Type Associated with Newton's Polygon

(ii). For any point (a',/3') E N(P) \ r)l) there is a point (1,8) E r)l) and t > 0 such that a I = 1- t ql(j) , Then, in view of (i) and the second inequality (10), we have

In this inequality the constants t and () depend continuously of the point (a', (3'), and therefore for (a', f3') E K C N(P) \ r)l) there are unified t and () for which the indicated inequality is fulfilled. Selecting c 1 from the condition ci c;;-IBI < c we prove the desired assertion. Lemma 2. Let

r)

1)

be a side of N(P), let q(i) be the outer normal vector to

r)l), and let e be the essential variables of r)1 ) .

Let

Then, if condition (5) is fulfilled and the difference b- a is sufficiently small, we have either (12) or

Igrad~, Pq
V(~, 77) E W(a, b)\ (0, 0).

(12')

Proof. The definition (11) of the region W(a, b) implies that this region can intersect the axes {~ = 0} and {7] = 0} only at the origin. It follows that at all points beloaging to W (a, b)\ ( 0, 0) one of the expressions Pq(i) ( ~, 7]) and Igrade Pq(i) ( ~, 17) I is nonzero. Further, the q-homogeneity of the polynomial Pqul implies that the values of these expressions are reconstructed from their values on the arc

Therefore, if b- a is sufficiently small, then, in view of the continuity, at all points in W 0 either PqU) or Igrade Pqul I is nonzero, whence it follows that either (12) or (12') is fulfilled. Proposition. Let

r)1 )

this side. Then for any co inequality

be a side of N(P) and let

e be the essential variable of

> 0 there is c 1 such that in the region G(rC.l), co, c1 ) the

:=:S(P)(~, 1J) ~

1

c(IP(e, )I+ Igrade P(~, ry)l)

(7")

holds. Proof. 1) We first of all note that, according to definition (11) and the first inequality (10), we have G(r)l),co,c 1 ) E W(logc 0 ,log(l/co)). Dividing the closed

Chapter 4

132

interval (log co, log(1/ co)] into sufficiently small closed subintervals [a.x, a.x+d and replacing the first inequality in (10) by the set of inequalities

a.x ~ -q~j) loge+ qij) log 1] ~ a.x+l we split G(r}1), co, ci) into a finite number of regions in each of which either (12) or (12') holds. To simplify the notation, assume that the assertion of Lemma 2 holds throughout the region G(r}l), co, c1 ). 2) Let first Pq
(13) Using Lemma 1 (ii) we conclude that for an appropriately chosen c 1 we have

(~,17) E G(r}l),co,ci),

for

(a',/3')

E

N(P) \ fJl) ·

(14)

By virtue of these inequalities, for (e,7J) E G(fJ1),co,ci) we obtain IP(~,7J)- Pq
<

cc1IPqu)(~,7J)I

<

1Pqu)(~,7J)I/2

provided that c is sufficiently small. It follows that 1Pqu)(~,7J)I

<

2IP(~,7J)I.

Substituting this inequality into the right-hand sides of (13) and (14) we prove (7") (without the term Igrade, PI on the right-hand side). 3) The estimate established in 2) can somewhat be strengthened. Denote by T(rjl)) the triangle formed by the straight line passing along rjl) and the coordinate axes, i.e. T(r}l))

= {(a,/3)

E 1R2 ,aqii)

+ f3q~j) ~

cj,a ~ 0,(3 ~ 0}.

We have in fact proved that

~a1Jf3 ~ const IP(~,7J)I 4) Now let Igrade,

PqU)

for

(a,/3) E T(fJ1)),

(~,17) E G(r}1),c 0 ,ci).

(15)

I =f. 0 in the region in question. Let, for definiteness,

e = ~ , l) i.e. 8P(~,1J)/8~ =I OA~,7J) E G(r}l),co,cl)· ThenforBP/8~ an inequality of the type (15) is fulfilled if the triangle T(r}l)) is replaced by the smaller triangle T'(r}l)) = {(a,/3) E R2 ,aqii)

e

+ f3q~j)

~ ci -1,a ~ 0,(3? 0}.

l)If consists of the two variables~ and 77, then, narrowing (if necessary) the region under consideration, we can assume that the derivative with respect to one of the essential variables is nonzero.

133

Ope·rators of Principal Type Associated with Newton's Polygon

Thus, we have

It now remains to note that 8(P) C T'(rj1 )).

2.5. The proof of proposition 2.2. We first of all note that since P(C ry) > const, inequality (6) is obviously fulfilled in any circle + ry 2 ~ R 2 • Therefore, by virtue of Proposition 2.3 and 2.4, we have to show that the complement of the set

e

m

UG(rJ

m+l 1),

c:o, c1) u

j:I

U G(r;o), co)

j=l

in the positive quadrant ~~ is covered by a circle of a sufficiently large radius. We pass to the plane q = (q1 ,q2) , q1 =log~, q2 = logry, and denote by g(rj0),co) and

g(rjo), co, c 1 ) the corresponding angles and half-strips:

Hence, we have to show that the complement

is contained in the translation of the negative quadrant

To prove this fact recall that the regions g(rJo), c: 0 ) result from the translation of the angles v(~) between the normals at the vertices r;o)' i.e. g(r;o)' 1) = V(~). Since N(P) is a convex polygon, we have m+l

~2

=

uV(~)'

j=O

Chapter 4

134

the angle between the normals at the vertex r~o) (0, 0) coinciding with the negative quadrant. Thus, m+I

JR 2

= IR:_ u

Ug(fJ

0 ),

co),

co= 1.

j=l

Decreasing co we translate the angles g(f}0 ), co) inside themselves so that "gaps"

Figure 4 are f-Jrmed between them (see Figure 4 ). Outside the neighborhood of the origin these "gaps" are covered by the half-strips g(fJ1 ), co, c 1 ). As to the neighborhood of the origin, it can be covered by the translated negative quadrant. Remark. Comparing Proposition 2.4 with Item 2) in the proof of the proposition in the present section we obtain a new proof of Theorem 1.2.1 for the case of regular Newton's polygon. This method makes it possible to prove the theorem in the general formulation as well.

Operators of Principal Type Associated with Newton's Polygon

§3. The

main~

135

estimate for operators of N-principal type

3.1. Statement of the results. We consider a differential operator with coefficients belonging to c=:

(1) As above, with the symbol P(x, y; Cry) the polygons N(P(x, y)), 8(P(x, y)), and 8(P(x, y)) are associated, and by N(P), 8(P), and 'i(P) the convex hulls of the unions of these polygons over all (;r, y) E IR2 1 ) are denoted.

Definition. A differential operator (1) is called an operator of N-principal type if (i) the polygons N(P(x,y)) do not in fact depend on (x,y), i.e. N(P(x,y)) =

N(P); (ii) at every fixed point (x 0 ,y0 ) the polynomial P(x 0 ,y 0 ;~,TJ) is a polynomial of N -principal type. The definition implies that the polygon N(P) is regular. Using the considerations in Section 3.2.1 one can weaken condition (i) by replacing the condition of independence of the polygons N(P(x, y )) on (x, y) by the condition of independence of the polygons 8(P(x,y)) or 8(P(x,y)). We do not dwell on these questions. According to Definition 2.1, condition (ii) can be rewritten as (ii') the polygon N(P) is regular, and for its any side (not lying on the coordinate axes) we have

{Pq
eis the essential variable of rjl)' ~ =J 0, T/ =J 0.

(2)

To the symbols satisfying the conditions of the above definition all estimates in §2 are extended. More precisely, for any compactum J( C !R2 there is a constant c = c( K) such that

36(P)(~,)

< c(IP(x,y;~,TJ)I + lgrad(.; , 17 )P(x, t;~,TJ)I + 1) for

( ~, ) E IR2 ,

( x,

y) E K.

The main result in this section is

Theorem 1. Let ( 1) be an operator of N -principal type and let tbe following additional condition bold: l)In what follows we shall prove for operator (1) only results of local character, and it can be assumed that the variables (x, y) range in a fixed region 0 C IR 2 . For the sake of brevity of the statements, we put n = JR2.

Chapter 4

136

Condition (R). If an (integral) point (a, [J) E N(P) is not integrally minor, tl1en the function a 01 r;( x, y) is real. Then Vc > 0 :Jw( c) such that in a region n of a sufficiently small diameter (diam n ~ w( c)) the inequality llui1;5(P) ~ ciiP(x, y; Dx, Dy)ull

holds, where lluiiFcP>

=

11

(3)

VuE V(r!)

e,

3F(P) (e, 11 )lu( 11 )1 2 de dry.

Remark. When N(P) is an isosceles right triangle, operator (1) is an operator of principal type and (3) goes into Hormander's inequality. In the case when N(P) is a right triangle with integral leg ratio the estimate (3) was established by Shananin [1] and Lascar [1]. As was already mentioned (see Footnote on page 121 ), Lascar used a weaker condition than (2).

e,

Corollary 1. In the conditions of the theorem, for any symbol Q(x, y; 17) E S£-g(P) there is w(c:, Q) such that under the condition diam < w(c: , Q) the inequality lluii6(P) ~ cii(P + Q)(x, y; Dx, Dy)ull

\!u E V(r!)

( 4)

is fulfilled. Corollary 2. If the conditions of Theorem 1 hold, then Vc > 0 :Jw*(c:) such that if diamr! < w*(c:), then we have lluii6(P) ~ell tP(x, y; Dx, Dy)ull VuE V(r!)

(3')

Proof. If condition (R) holds, then the symbols P(x,y;e,ry)and

P*(x,y;~·,ry) = P(x,y;e,ry)+ LP~:j(x,y;e,ry)ja! differ by a symbol Q(x,y;e,1J) E S£N(P)· Therefore, by virtue of (4), we have lluii6(P) ~ ciiP*(x, y; Dx, Dy)ull

\lu E V(r!).

Using (1.4.6) we obtain (3'). If the arithmetical condition (A) is imposed on the polygon N(P), then condition (R) means that the coefficients a 01 r;( x, y) corresponding to the senior points of N ( P) assume real values. This condition can somewhat be be weakened. Introduce the following notation: if q E ~_2, then charqP == {(x,y;e,1J) E IR2

X

IR2 ,Pq(x,y;e,1J)

= 0}.

Let M 0 = (x 0 ,:y0 ; eo, ry 0 ) E charq P and let eo # 0, ry 0 # 0. Let q = qU> be the outer normal to r}l) and let, for definiteness, be an essential variable. If condition (2) is fulfilled at the point M 0 , then, by the implicit function theorem, there exists a smooth function A(x, y, 17) such that in the neighborhood of M 0 the set charq P is determined by the equation = .\(x, y, ry). In other words, charq P is a smooth real manifold of codimension 1.

e

e

The theorem below is a version of Theorem 1.

Operators of Principal Type Associated with Newton's Polygon

137

Theorem 2. Let (1) be an operator of N -principal type and let tbe polygon N(P) satisfy condition (A). Let tbe following additional condition bold: Condition (R'). Let r}l), j = 1, ... , m, be an arbitrary side of N(P) not lying on the coordinate axes, let M 0 = ( x 0 , y 0 ; ~ 0 , r/) be an arbitrary point belonging to charqU) P, wbere ~ 0 =f:. 0, ry 0 =f:. 0, and let q
G(rjl),a,b) (see Sections 2.3 and 2.4) is considered. A function u(x,y) ED is represented as a sum of functions whose spectra belong to the indicated regions. In these regions the orders of the derivatives involved in the operator are lower relative to those contained in the principal quasi-homogeneous part corresponding to a certain side or vertex. This property substantially simplifies the derivation of estimates. The plan of our further presentation is the following . We first construct (Section 3.2) a partition of unity in JR~e,!J) relating to the partition into the regions

G(rjo), ... ) and G(fJ 1), . • . ). After that (Section 3.3) estimate (3) is microlocolized; Sections 3.4 and 3.6 are devoted to the proof of microlocal estimates. 3.2. Partition of unity. Let a finite covering { Uj}, j = 0, ... , J, of the plane of the variables (~,"7) be given. A system of functions {~j(~,7])} is called a generalized partition of unity subordinate to the covering {Uj} if (i) ~j(~, TJ) E C 00 ; (ii) ~j(~,TJ);;?: 0 and supp~j C Uj; (iii) there is a constant J( > 0 such that

JR 2

J

J(-1

~ I:~j(~,rJ) ~ J(.

(5)

j=O

For

J(

= 1 we obtain an ordinary partition of unity: J

2:::: ~j(~, rJ)- 1.

(5')

j=O

We note that from every generalized partition of unity an ordinary partition of unity is obtained by means of the transformation ~j - t ~j('L-~k)- 1 . However, for our aims it is more convenient not to perform this transformation and to deal with condition (5). When estimating the commutators of the PDQ ~j(Dx,Dy)

Chapter 4

138

corresponding to the functions in (i)-(iii), Hormander's condition proves extremely useful: (iv) there is p, 0 < p < 1, such that Vj the inequalities

(6) hold, where

V'Ja,f3)(~,77) = aa+f31/Ji(~,77)/8~aa77f3· Proposition. Let P be a polynomial with regular Newton's polygon N(P). Consider the covering {Uj} by means of the regions in §2, i.e. Ui

= G(r}0 ),aj),

j

= 1, ... ,m+1,

(7)

G(r)l), aj, bj), j = m + 2, ... , 2m+ 1, Uo = {(~, 77) E IR2 , + 772 ~-- R2 }.

(7')

Uj =

e

(7")

Then thereexists a system of functions {1/Ji(~, 77)} satisfying conditions (i)-(iv) and the additional condition (v) for any a ~ 0, (J ~ 0, a+ (J > 0, \fj and any integral point ( -y, 8) E N(P) there is a constant Cjcxf3r6 such that

(8) Proof. Recall that regions (7) are determined by the inequalities ( - 1) 1 ( - 1) q/log 1~1 - q/- log 1771 > log-.

co

In the case of regions (7') a third inequality

q~j) log I~ I +q~j) log 1771

>log 2_ c1

should be added to the former two. In other words, each of the regions Uj, j > 0, is determined by the set of inequalities

(9) 1 and fJ = 1,2,3 for j > m + 1. The numbers qki,tL) are the components of the outer normals q(j) to the sides of N(P) so that under an appropriate normalization (which is inessential) they can be assumed to be integral or, which is more, even numbers.

where

{l

= 1,2 for j = 1, ... ,m +

139

Operators of Principal Type Associated with Newton's Polygon

Fix a sufficiently small x > 0 and take a function B(t) E B(t) = 0 fort~ 0 and B(t) = 1 fort;;::: x. Set

I'

1/Jo(~,77) = B(R~-

e -77

coo, B(t) ;;::: 0, such that

(10')

2 ).

The expressions for these functions immediately imply that (i') 1/;j(~,T/) E coo if~=/= 0,17 :f. 0. It follows from the definition of the function B(t) that for functions (10) and (10') condition (ii) is fulfilled. Let us verify condition (iii) Since the number of the functions 1/; j is equal to 2m + 2 and each of the functions does not exceed 1, we see that the right-hand inequality (5) with I< = 2m+ 2 holds. Further, the constructions in the foregoing section make it possible to select the numbers Rj" in (9) so that for any point (~, 7J), + 7] 2 > (Ro- x) 2 , there should exist j such that inequality (9) is fulfilled when log Rift is replaced by log(Rj" + x). In this case 1/Jj( ~' 17) = 1, and hence inequality (5) with J{- 1 = 1 holds. The verification of the smoothness of the functions 1/; j on the coordinate axes and the conditions (iv) and (v) requires a rather detailed analysis of the structure of the regions Uj and functions (10); to simplify the presentation of the material we placed the proofs of these facts in the Appendix.

e

3.3. Microlocalization of estimate {3). An important step in the proof of Theorem 1 or 2 is the following

Proposition. Let a differential operator (1) possess the following property: if {Uj} is the covering (7), (7') , (7") and {1/;j} is its subordinate generalized partition of unity satisfying all conditions of Proposition 3.2, then Ve, > 0 3w( c) such that in an arbitrary region n c R 2 ' diarn n < w( c), the inequalities

111/Ji( Dx, Dy )ull;s(P) ~ cliP( x, y; Dx, Dy )( 1/Ji( Dx, Dy )u )II VuE 'D(r!), t > 1, j > 0,

+ c( c)llullc -t), (11)

are fulfilled. Then estimate (3) takes place.

Remark. Since inequality (3) was considered on functions u(x, y) of compact support, the only condition imposed on the coefficients aaf3( x, y) of the operator P was that they should be smooth. As to inequality (11), here the differential operator is applied to the function 1/;j(Dx, Dy)u which, in general, may not b e a funct ion of compact support. Therefore in what follows we assume that the coefficients aaf3(x, y) are uniformly bounded throughout the plane together with their derivatives of any order, i.e. for any k1 and k 2 there is a constant I
n;

2a af3 ( x , Y)l

~

I
Chapter

140

4

Under this condition on the coefficients the inequality (11) makes sense, and we can apply Hormander's results on PDQ (see Lemma 2 below). The proof of the proposition. According to (5), the function a(~, 17) is bounded from above and below. Therefore

=

E 'l/;j(~, 17)

J

!lullh'(P) = lla-l(DXl Dy)

L '1/Jj(Dx, Dy)ullh(P) ]=0

J

J

~ KL 11'1/Jjullh'(P) ~ L(c:KIIP('l/;ju)ll + Kc(c:)llull(-t)) + J(!I'I/Joullh'(P)" j=O

j=l

Since 'l/; 0 (~, ry) is a function of compact support, the term 11'1/Joullh'(P) does not exceed c(t)llull(-t) for any t E R. Thus, the inequality can be rewritten as J'

llullh'(P) ~ Kc:

L IIP('I/Jju)ll + c'(c:)llull(-t) j=l

J

~ KJc: II Pull

+ c'(c:)llull(-t) + Kc: L

II(P'l/;j- 'l/;jP)ull·

(12)

j=l

The estimates for the second and third terms on the right-hand side will be stated as separate assertions.

Lemma 1. If t > 1/2, then

u Lemma 2. If the function tion 3.2, then

'1/Jj(~,

E 'D(!l), D- = diamn.

(13)

ry), j > 0, satisfies the conditions of Proposi(14)

with some constants K 1 and K 2 . If inequalities (13) and (14) are assumed to be proved, then estimating the righthand side of (12) by.means of these inequalities we find

Fix a sufficiently small c: so that K1K2 Jc: < 1/2. Then take w(c:) such that be"( c)< 1/4 for 8 < w(c:). As a result, we obtain

141

Operato1'S of Principal Type Associated with Newton's Polygon

Replacing 41{ J £by c we arrive at (3). The proof of Lemma 1. By the definition of Fourier's transforms of functions belonging to V, we have

e

Squaring both sides of this inequality, multiplying by (1 + + 17 2 )-t/ 2 , integrating with respect to ' and TJ, and noting that J Area n ~ const diam n we derive inequality (13) . The proof of Lemma 2. We write the commutator of P and '1/Jj in the form

(a,{3)EN(P)

=

L

lja{3·

a,{3,j If condition (iv) in the foregoing section holds, then the commutator of aaf3(x,y) and '1/Jj(Dx, Dy) can be written as

L

'l/;j(Dx,Dy)aaf3(x,y)-aaf3(x,y)'I/Jj(Dx,Dy) =

(D~D~aa{3)'1/JJp,q) jp!q!+RN,

O
where, according to Hormander [4), for any t E IR there is N such, that r

IIRND~ D~ull ~ const llull(-t)

V(a, /3) E N(P ).

Further, if condition ( v) in the foregoing section is fulfilled, then

II(D~D~aa{3)'1/JJp,q) D~ D~ull ~ const 111/JJp,q) D~ D~ull

j j 1'1/J)p,q) ('' 1J )'a1Jf31zlu('' 1J )12 de d1J) ~ const (jj l'28(P)(e,1J)u(e ,TJ)I de dry) const liuii8(P)'

= const (

I/2

2

112

=

3.4. Estimate (3) in regions relating to the vertices of the polygon N(P). Let r;o) be a vertex of Newton's polygon N(P) (distinct from the origin) and let G(r;o), co) be the corresponding region with a sufficiently small c: 0 . We first of all refine the estimate for the symbol in this region.

Chapter 4

142

Lemma 1. Let P( x, y; ~, 17) be the symbol of an operator of N -principal type and let !{ be a compactum in ~2 . Then there is Eo( I<) such that for Eo = Eo(K) there is a constant c = c( I<) > 0 such that

(15) Proof. The right-hand inequality is obvious; the proof of the left-hand inequality

is a trivial modification of the proof of Proposition 2.3. Lemma 2. If the diameter of the region equality

n

is sufficiently small, then the in-

(16) with an arbitrary t E IR holds, where '!/Ji is the function in Proposition 3.2 relating to the region G(r}o), £0 ), where Eo = £o(n). Proof. Inequality (15) allows one to follow the proof of Theorem 1.4.4. Let, for definiteness, the origin belong to n, let P(Dx, Dy) = P(O, 0; Dx, Dy), and let P'(x, y; Dx, Dy) = P(x, y; Dx, Dy)- P(Dx, Dy)· By virtue of (15), for any function v we obtain

whence, putting v = '!/Jju, we derive

(17) To estimate tP.e second term on the right-hand we consider a truncating function x(x,y) E 'D, x(x,y) = 1 for x 2. + y 2 ~ x(x,y) = 0 for x 2 + y 2 > 285, where 80 > diam n. Then

85,

Since the coefficients of the operator P' vanish at the point x = y = 0, they do not exceed const 8o for x 2 + y 2 ~ 286, and therefore

proved that 80 is sufficiently small. We now note that (1- x)u 0 for u E V(n). If the symbol '!/Jj(~, 17) satisfies condition (iv) in Section 3.2, then, according to Hormander [4], the PDQ '!/Jj(Dx, Dy) possesses the property of pseudo-locality, whence

=

11(1- x)P'('!/Jiu)ll

~ Ctllull(-t)

\!t E IR.

143

Operators of Principal Type Associated with Newton's Polygon

Substituting the resulting estimates into (17) we arrive at (16). Since N(P) is regular Newton's polygon, we have

whence Substit uting the last inequality into (16) and using Lemma 1 in Section 3.3 we obtain inequality (3) in the region in question G(r;o),co) with diam!l < w(c) and £o = £o(c,diam!l). 3.5 . The structure of the operators in regions relating to the sides of Newton's polygon. In what follows it will be convenient to deal with narrower regions as compared to G(r;l), co, c 1 ). For a < b we denote by G(r;l), a, b, c) the region in the plane R~e, 11 ) determined by the inequalities

-q~j) log 1e1 + q~j) log I7JI > log a,

q~j) log 1e1 + q~j) log I7JI > log b,

Using the argument in Lemma 2 of Section 2.4 we shall prove the following Le1nma. Let the difference b-abe sufficiently small, namely b-a ~ x( diamf!) . Tl1en in the region G(r;l), a, b, c) (j = 1, ... , m) one of the following two conditions holds: (a)

(b)

Pqcn(x, y; C 7J) # 0 for (x, y) En, there is a variable, say esuch that 8Pqcn(x,y;e,1J)/8e

#0

for

(e, 1J) E G(r;l), a, b, c);

(x,y;e,1J) En

X

G(r;1)a,b,c).

Proposition. Let conditions (A) and (R') of Theorem 2 be fulfilled and let the region n and the difference b - a be such that the assertions of the lemma are true for the region n X G(r;l)' a, b, c), the variable being essential. Let the intersection

e

charpq(j)

= {(x, y; e, 1J), Pqcn(x, y; e, 1J) = 0} n (n X

G(r;l)' a, b, c)),

be non-empty. Then the factorization

(x,y;e,1J) En

X

G(r;l),a,b,c), (18)

Chapter

144

4

takes place, where Aj and Qj are q(j)_bomogeneous functions in(~, 77) of orders 1 and Cj -1, respectively, (wl1ere q(j) = (1,bj), bj) 1) depending smoothly on tbe parameters (x, y) E f!, and tbe conditions (1) ) =J 0, (x,y;~,ry) En X G ( rj ,a,b,c' Aj(x,y;(,ry) = ~- Aj(x,y;ry),

Qj(x,y;~,ry)

(19) (20)

bold, where Aj(x, y, ry) is a real hi-homogeneous function of order 1 of the variable "7·

P·roof. According to Section 1.1.3, the symbol Pqu l is represented in the form

where the symbol p[i] is a polynomial in~ and 7] solved with respect to the highest .powers of ~ and 7] and having the coefficient 1 in the highest power of ~. Since Newton's polygon does not depend on (x, y), we have

Further, as was already mentioned, the region G(r}1), a, b) does not intersect the coordinate axes, and therefore ~ 0-i rybi+ 1 =J 0, whence

Aj(x,y;~,ry) =J 0

for

(x,y;~,77)

E f! x

G(rjl),a,b,c).

By the hypothesis, the symbol Pqul and, consequently, the symbol p[il(x, y; ~' 17) have a real zero (xo, Yo; ~o, "7o), i.e. the point ~o is a real root of the polynomial 71"j ( 0 = p[i] ( x o, Yo; ~, 7]o). According to condition (b) of the lemma, this root is simple. Therefore in a neighborhood of (xo, yo, 7]o) there exists a smooth branch ).. j ( x, y, 77) of this root. Hence, we have obtained the factorization

(181) where the functions Aj and hj are q(j)_homogeneous and smooth in the neighborhood of the point (xo, Yo , ~o, 7]o), and we have

in the neighborhood. In view of the q(jLhomogeneity, the functions Aj and hj can be defined for all (~,77) E G(r}1 ),a,b,c) provided that b- a is sufficiently small. Further, if the diameter of n is sufficiently small, we can assume that factorization (18') holds for all (x,y) E f!. Setting Qi = Aihi we arrive at factorization (18).

Operators of Principal Type Associated with Newton's Polygon

145

3.6. Estimate (3) in regions relating to the sides of the polygon N(P). 1) We are going to prove inequalities ( 11) in region (7') relating to the side rjl). Dividing the closed interval [log c 0 , log 1/ co] into small subintervals [a.x, a.x+ 1] we partition the region in question into the subregions G(rj1), a, b, c) in each of which either condition (a) or condition (b) of Proposition 3.5 is fulfilled . As can easily be seen, in the microlocalization of estimate (3) the regions (7') can be replaced by the indicated smaller regions. Indeed, each of the regions G(r}1 ) , a, b, c) is determined by inequalities (9) where it is only necessary to replace the constants Rill on the right-hand side by the constants Rill.X depending on the additional parameter .X. Replacing Rill in (10) by Rjll.X we obtain functions 1/Jj.x whose supports lie in G(rj1), a.x, a.x+I, c 1). To simplify the notation, assume that one of the conditions of Lemma 3.5 holds throughout the region (7'). If condition (a) holds, then, by virtue of the results in Section 2.4, inequality (15) is fulfilled. Since this case has already been considered in Section 3.4, in what follows we shall assume that condition (b) holds, i.e. the q(j)_homogeneous part of Pqul has the form (18) (q(j) = (1, bj)). 2) By the definition of the region

G(r?), a, b, c), we have

It follows that in this region we have ~

We associate with the vector

q(j)

c,

(22)

= (1,bj) the norm

Then (22) implies Lemma 1. Let U be a region of the type of G(r} 1 ), a, b, c) and let supp 1/J( ~, 1J) U. Th en there is a constant I<= I<(U) such that

c

(23) 3) In view of Lemma 1, instead of (11) it suffices to prove the inequality

(24) If condition (A) is fulfilled, then

Chapter

146

4

Thus, in (24) the operator P can be replaced by its qULhomogeneous principal part, and hence the inequality

(24') has to be proved. 4) We associate with the symbols Aj and Qj in (18) the PDO Aj(x, y; Dx, Dy) and Qj(x, y; Dx, Dy)· Using the standard technique of PDO with quasi-homogeneous symbols it is proved that if Rj = P- AjQj, then

It follows that (24') is equivalent to the inequality (25) 5) The main point in the proof of (25) is the estimation of the operator Aj with real symbol ( 20). Lemma 2. Let the symbol Aj(x, y; ~' 77) possess the properties indicated in Proposition 3.5. Then the inequality

(26) is fulfilled. Proof. We have

( Taw) i ax + Im(xw, .Ai(x, y, Dy)w)

- lm(xw, Ajw) = - Im xw, 1 =-2

jj xa:;;8lwl dx dy + jj x (x~-i .Xi) ww dx dy. 2

1

2

Since the symbol Aj(x, y; ry) is real, the difference .Aj- Aj is a bounded operator in L 2 . On integrating by parts in the first term on the right-hand side we obtain 1

211wll 2

~- lm(w,xAjw)

+ cllxwllllwll

~ llxAiwllllwll

+ cllxwllllwll·

Dividing both sides of this inequality by llwll we obtain (26). 6) We now complete the derivation of inequality (11 ). To this end we substitute w · Qj(x,y;Dx,Dy)('!f;u) into (26). As in the proof of Lemma 2 in Section 3.4, taking a truncating function x( x, y) equal to 1 in a (small) neighborhood of the region n we find

IIQj('l/;u)ll ~ 2llxAjQj('l/;u)ll + cllxQj('l/;u)ll ~ 2llxxAiQi('!f;u)ll + 211(1- x)xAiQi('!f;u)ll + cllxxQj('l/;u)ll + cllx(1- x)Qj('l/;u)ii·

(27)

147

Operators of Principal Type Associated with Newton's Polygon

If the diameter of n is sufficiently small, then 2xx < c, whence

2llxxAiQi(1jJu )II ~ ciiAiQ j( wu )II, cllxxQ j( 1jJu )II ~ cciiQi( 1jJu )II· As in Section 3.4, in view of the pseudo-locality, the second and fourth terms on the right-hand side of (27) are estimated by means of !lull< -t) with any t. Consequently,

Taking c satisfying the condition cc

< 1/2 and replacing 2c by c we obtain (28)

The PDO Qi is q(j) quasi-elliptic. Applying the technique of such PDO (which does not practically differ from the technique of elliptic PDO) it is easy to derive the inequality ll1jJujj(qU>,ci-1) ~ const IIQi(1jJu)ll + c"llull(-t)· Substituting this into (28) we arrive at (25). Theorem 2 is proved completely.

Appendix Below we present a complete proof of Proposition 3.2, i.e. we shall prove the smoothness of functions (3.10) in the neighborhood of the coordinate axes and verify properties (iv) and (v) playing a decisive role (see the proof of Proposition 3.3) in the microlocalization of the estimates in question. We shall assume that + ry 2 is sufficiently large, and therefore the first factor in (3.10) will be omitted. 1) We begin with the verification of (3.8) in the special case when 1 = a and b = f3, i.e. we shall prove the inequality

e

(1) For the sake of simplicity, consider the case a = 1, f3 --:- 0 since the other derivatives are estimated in an analogous manner. Differentiating in (3.10) we obtain

By the definition of the function 8(t), its derivative 8'(t) is nonzero for 0 ~ t ~ x, i.e. the right-hand side of the above relation is nonzero when

Chapter

148

4

whence trivially follows (1) for a= 1, {3 = 0. The proof of the proposition will be presented in the following order. We first consider functions (3.10) with supports in the regions corresponding the vertices r<.o) not lying on the coordinate axes. Then we consider the case of vertices lying o~ coordinate axes and the case of regions corresponding to the sides. 2) Thus, let r}o) = (aj,{3j); aj,{3j > 0, be a vertex of N(P ). In this case for large + ry 2 the function (3.10) has the form

e

and is nonzero when -1

Itl-q~i) I'rf lq~n '

co .~ ~ Putting Xj

= q~j) jq~j)

and Pi= c;;IJq
(3) whence (IXj-1-Xj);: I~ /"'

p·p· } J-l·

By virtue of the convexity of the polygon N(P), we have x 1 > · · · > with account of inequalities (3), on the support of (2) we find

Xm,

whence,

(4) Thus, the support of '1/Jj does not intersect the coordinate lines, and therefore '1/J j E

c=. Inequalities (3) imply that there are p > 0, w > 0 such that logl~l,loglryl>plog(1+l~l+lryl)

for

(~ , ry)Esupp'!/Jj·

(5)

In view of (5), it trivially follows from (1 ) that condition (iv) holds. We now show that the function (2) satisfies (v). In view of Lemma 2.3, we can confine ourselves to the case 1 = aj, b = {3j. Let, for definiteness, a> 0 in condition (v). Then, with account of (1) and (4), we obtain 1 '!/J?~,p\~, ry)l ~ const 1~ 1 -al'rfi-P ~ const l~l- 1 . Multiplying this inequality by l~lai l'rf!Pi we obtain ( v ). 3) Now let the vertex rjo) lie on a coordinate axis, say j ( -1, 0) in (2), and the function (2) is nonzero when

=

1. Then

q(j-I)

=

(6)

Operators of Principal Type Associated with Newton's Polygon

149

Hence, (2) is a smooth function in a neighborhood of the axis rJ = 0. Further, ( 1)

( 1)

since ITJI > c:;;- 1 , we have 1e1-q2 lrJiq1 ~ oo as lei ~ 0, and therefore the second function on the right-hand side of (2) is identically equal to 1 for small lei, that is the smoothness of (2) is completely proved. We proceed to condition (iv). It was noted in the foregoing section that if inequalities (5) hold on the support of the function (2), then condition (iv) is a consequence of (5). Conditions (5) can in fact be replaced by more delicate conditions: log 1e1 > plog(1

+ 1e1 + ITJI) -logw, log ITJI > plog(1 + 1e1 + ITJI) -logw,

I

ce, TJ) E a'lj;jfae, ce,rJ) E a'l/J j 1a'r} .

(7)

(7')

Inequality (7') holds by virtue of the first inequality (6). Further, we have

1.e. the function in question is nonzero when

(9) i.e. condition (7) is fulfilled. When verifying condition (v) we can confine ourselves (Lemma 2.3) to the case 1 = 0, 8 = {31, i.e. to proving the inequality

If {3 > 0, then this inequality readily follows from (1 ). Thus, it remains to consider the case {3 = 0. To simplify the notation, assume that a= 1. By virtue of (8) and (9), we have ·

l/J . 1 ~ const lrJif3llel-1 'r}!J1Ra'a[

I

We note that

~ const

lrJif31-ql< >1q2< >. 1

1

ri1 > is determined by the equation (q?> /q~ 1 ))a + f3 = {31,

i.e. the point (1, {31 - q?) fq~l)) belongs to N(P). Therefore the point (0, {31 q~l) /q~ 1 )) belongs to "i(P), and thus (7) is proved for the region G(ri0 >,c; 0 ,c:I). 4) We now proceed to the case of the function (3.10) corresponding the region (1) • ( ") ( G rj ,£o,£t). Settmg q 1 =(a, b), where a and bare natural numbers, we write

Chapter 4

150

This function is nonzero when log co ( -b log 1e1

+a log 1171

1

(log- , co

1

a log lei+ blog 1171 >log-. ·

c1

(10)

These inequalities imply that log 1e1 > -(blogco + alogc 1 )/(a2 + b2 ),

(11)

log 1171 > (alogco + blogc 1 )/(a2 + b2 ),

(11')

i.e. the support of '1/Jj does not intersect the coordinate axes, and therefore 1/Jj is a smooth function. Inequality (10) implies (5) and, consequently, condition (iv) for the function '1/J j. By virtue of Lemma 1 in Section 2.4, when verifying condition ( v) we may confine ourselves to the points (1, 8) E 1 ). We have to estimate the derivative of '1/Ji( o:, (3).

fJ

r}l)

Let, for definiteness, o: > 0. On the side there always exists a point ( "f, 8) such that 'Y ~ 1. By virtue of the inequalities (1 ), (11 ), and (11'), we have 11/J)a,fi>(e, 17)1

< const lel- 1 ,

whence

IC117 6 '1/JJa,,6')(e,17)l::::; const le...,- 1 17 6 1 (

const3;s(P)(e,17)·

Proposition 3.2 is proved completely.

§4. Local solvability of differential operators of N-principal type In Section 1.4.6 the definition of local solvability was stated for a differential operator, and the local solvability was established for N quasi-elliptic differential operators with variable coefficients. We now extend this result to those operators of N-principal type for which inequality (3.3) holds. 4.1. We associate with an operator P(x, y; Dx, Dy) of N-principal type the family of norms

(1)

and denote by
As in Section 1.4.6, we denote by <5~) the set of those distributions belonging to whose supports belong to n and by


151

Operators of Principal Type Associated with Newton's Polygon

Theorem. Let a differential operator P(x, y; Dx, Dy) of N -principal type satisfy the additional conditions guaranteeing the fulfilment of inequality (3.3). Then \:fs E R there is w( s) such that in a region n, diam !1 < w( s ), the equation

P(x,y;Dx,Dy)u = J, possesses a solution u E ~(s+I)(f2)

(2)

\:If E ~(s)(Q).

The major part of this section is devoted to the proof of the a priori estimate (cf. Section 1.4.6)

[v]-s :::; Cs,n[P(x, y; Dx, Dy)v]-s-1

(3)

\:lv E 'D(!l).

The theorem is derived from (3) using some general concepts offunctional analysis. 4.2. We begin with extending (3.3) to the scales of norms II

llw

Proposition. Let inequality (3.3) hold. Let the function J-L(e, ry) satisfy (1.4.39) and the additional conditions ( cf. Proposition 3.2) laa+PJ-L(e,ry)fae~ary~'l

< capJ-L(e,ry)(1 + 1e1 + lryl)-p{a+P),

IC1ry 6aa+J' J-L( e, 17 )/ aeaa17~' I < Cap,638(P)(e, 17 )J-L(e, 17 ), Then there are t(J-L) and w(J-L) such that in the region inequality

o<

\:/( ,, 8)

n,

E

diam n

IIJ-L(D)uii6(P) ~ Cpn(IIJ-L(D)Pull + llull(-t)),

p ~ 1,

(4)

N( P).

(5)

< w(J-L ), the (6)

is fulfilled for any t < t(J-L ). Proof. Take w 0 so small that in a region of diameter no great than w 0 the inequality (3.3) holds, and let diam!l < w < w0 • Select a truncating function 1/J(x,y) equal to 1 on n with support lying in the ball x 2 + y 2 < 2w~. Applying inequality (3.3) to the function 1/J(x, y)J-L(Dx, Dy)'u we obtain

IIJ-L(Dx, Dy)uii8(P) ~; II1/JJ-Luii6(P)

+ 11(1- 1/J )J-Luii8(P)

:::; c:(!l)IIP(1/JJ-Lu)ll + By Leibniz' formula, we have

whence

11(1- 1/J)J-Luii6(P)"

(7)

152

Chapter4

Substituting this into (7) and using the smallness of the constant c(f!) we obtain

llp;uii6(P) ~

E1 (f!)IIP(p;u )II

~ EIIIPuiiiL

+ 11(1 -

~ )p;uii6(P)

+ c1II(P ll- p;P)ull + 11(1 -

~ )puiiF(P) ·

(7')

We now prove that

II(Pp;- p;P)ull ~ c11llluii6(P)

+ Czllull(-t),

11(1 - ~ )p;uii6(P) ~ c31lull< -t) ·

(8) (9)

Substituting (8) and (9) into (7') and taking into account the smallness of c 1 we arrive at (6). The proof of (8) is obtained by means of a trivial modification of the proof of Lemma 2 in Section 3.3. The proof of (9) is based in the property ofpseudolocality of the PDO p(Dx, D9 ) whose symbol satisfies condition ( 4). Take a function c.p( x, y) E 1J equal to 1 on the support of the function ~( x, y) and, consequently, in the region n. Then

(1- ~)p;(Dx,Dy)u = (1-

~)p;(Dx,Dy)(c.pu)

N

=

I:

(1- ~)D~D~c.pp;(a,f3)(Dx, Dy)uja!f3!

+ (1- ~)RNu = (1- ~)RNu.

a+f3=0 Taking sufficiently large N one can attain the fulfilment of (9) for any t E JR. 4.3. Proposition. For any s E IR there are w( s) and t( s) such that in a region n, diamn < w(s), for any t < t(s) the inequality

holds.

As is shown in the Appendix, Vs E IR there is a smooth function Ms(C ry) that satisfies conditions (1.4.39), ( 4), and (5) with p; = M 8 , and for some c8 > 0 we have

(ll) By virtue of these inequalities, the inequality for the norms

holds. Therefore IIMs(Dx, Dy)ull can be taken as the norm (1), and inequality (10) is obtained if we set p;(~,ry) = Ms(~,17) in (6).

153

Operators of Principal Type Associated with Newton's Polygon

4.4. We now show that in the case of a region n of a small diameter the second term on the right-hand side of (10) can be dropped.

Proposition. Let inequality (10) be fulfilled. Tben Vs E R tbere is w( s) sucb that in tbe region n, diam n < (.V( s ), tbe inequality

(12) holds. Proof. Since [u]s+l ~ const llull for s ~ -1, in this case llull(-t) can be estimated by means of (1/2)c( s, !l)[u]s+l if the diameter of n is sufficiently small (see Lemma 1 in Section 3.3). To prove (12) for s < -1 we use the argument in the paper by Nirenberg and Treves [1]. Assume that in an arbitrary small region n the inequality (12) does not hold. Then there is a sequence Uj E 1J possessing the following properties:

(13) By virtue of (10), the norms [uj]s+l are uniformly bounded. If tis sufficiently large, then the set of functions

{u,[u]s+I::;; const,suppu E n,diamn

~

oo},

is compact in H(- t). Therefore t he sequence {uj} contains a subsequence (we again denote it as {u j}) possessing the following properties: Uj

~

U

.

Ill

• H< -t) ,

P(x,y;Dx,Dy)u

= 0,

suppu

= (0,0).

Hence, if (12) does not hold, then for a sufficiently large t there is a distribution u in the space H(-t) with support at the point (0,0) such that P(x,y; Dx,Dy)u = 0. We shall prove that u _ 0. Since, by virtue of the first condition (13), we have llull< -t) = 1, the resulting contradiction proves inequality (12). If the support of a distribution u E H( -t) is at the origin, there is a differential operator Q(Dx, Dy) such that u = Q(Dx, Dy)8(x, y), where 8(x, y) is Dirac's delta function. Expanding the coefficients of P(x, y; Dx, Dy) into Taylor's series at (0, 0) we write the operator in question in the form

P(x,y;Dx,Dy) = P(Dx,Dy) +

L

x'YlP-yo(Dx,Dy)·

-y+8>0

Since the degrees of the differential operators P-yo do not exceed the degree of the operator P(x,y;Dx,Dy), there is an integer N > 0 such that x'Yy 8 P-yo(Dx,Dy)8(x,y) is equal to zero for 1 + 8 > N. Thus, the condition Pu = 0 means that N

(PQ)(Dx, Dy)8 +

L -y+8=1

x'Yy 8 (P-yoQ)(Dx, Dy)8 = 0.

Chapter 4

154

Passing to the Fourier transform .we find

(14) Take a positive vector q = (q1 , q2 ), for example, q = (1, 1). Then the q-degree of the sum in (14) is less than that of PQ. Therefore (14) can be rewritten as

=

=

It follows that Pq(e,rt)Qq(e,rt) 0, H(e,TJ) 0. Since Pq(e,rt) =/=. 0, we have Qq(e, rt) 0, and, consequently, Q(e, rt) = 0, whence u = 0.

=

4.5. The proof of Theorem 4.1. 1) We first of all note that inequality (12) is a consequence of (3.3), i.e. it applies to those operators for which (3.3) is fulfilled. Since the operator P(x, y; DXJ Dy) and the transposed operator tP(x, y; Dx, Dy) simultaneously satisfy the conditions of Theorem 1 or 2 in Section 3.1, inequality (3.3) holds for the former operator, and therefore the inequality v E v(n),

(3')

is fulfilled. 2) We consider the unbounded operator 11l;( -s) ____,. 11l;( -s-1)

\Jn

___, \Jn

(

v

~---+

tp( x, y; D x, D Y ) v ) '

(15)

with domain H~-s). This operator is a restriction of a continuous operator, and, consequently ( cf. the proof of Theorem 2.2.3) is a closable operator. Indeed, if s ~ 0, then, according to Lemma 3 in the Appendix, Q;( -s) C H N(P),-s, Q;( -s- 1 ) c H N(P),-s-l, and (15) is a restriction of the continuous operator

In case s

~

0, we have

(5(-s)

C

H N (P),-sf 8 , Q;(-s- 1)

C

HN(P),(-s- 1 )/ 8

-1 ), and it is necessary to consider a continuous operator from H N(P ),-s/8-1/8.

.

(for s

HN(P),-s/ 8

n

~

into

3) Let us assume that the closure of operator (15) has been performed. Since in the case of a region n with a smooth boundary the space (5( -s ) is the closure n . of 'D(fl), inequality (3') is extended throughout the domain of operator (15). This implies the injectivity of operator (15). Repeating the argument in the proof of Theorem 2.2.3 we conclude from inequality (13) that the image of operator (15) is closed.

155

Operators of Principal Type Associated with Newton's Polygon

4) The domain of operator (15) is dense in <5~-s), and therefore this operator has an adjoint operator which is also a closed operator with closed image. The injectivity of (15) implies the surjectivity of the adjoint operator. The duality of (!;t's) and (5(s)(D,) (cf. Section 1.4.6) and the definition of the operator tp (see n . (1.4.54)) 1mply that the operator (5(s)(D,) ~ (5(s-l)(D,)

(u

f-+

P( x, y; Dx, Dy)u ),

with domain containing HN(P),s(n) is surjective, which proves Theorem 4.1.

Appendix Lemma 1. There is x > 0 such that

The proof is based in the fact that the polygon N(P) is regular and the polygon b(P) is the convex hull of a finite number of minor (not necessarily integral) points of N(P).

Lemma 2. There is()

> 1 such that (2)

Proof. Let (!j,bj) be the vertices ofb(P). Instead of (2) it suffices to prove the inequality (2') This inequality follows from the purely geometrical fact that there is () that _...._

N(P)

c ()8(P).

Indeed, the polygon N( P) is determined by the system of inequalities 1 )

(q<j),(a,/3)) ~ Cj,

j = 1, ... ,m,

a~ 0,

f3 ~ 0,

and the polygon b(P) is determined by the system

(q(j), (a, /3)) ~ Cj - 1. Putting()= maxcj/(cj -1) we prove the desired assertion. As a consequence of Lemmas 1 and 2 we have l)

Under an appropriate normalizat ion of the outer normals (see Section 2.1).

> 1 such

Chapter 4

156

Le1nma 3. Let, as in §1.4, norm

H N(P),s

denote the space of distributions with finite

Then the following embeddings take place: HN(P),s C Q;(s) C HN(P),s/0

for

s ;::: 0,

C Q;(s) C HN(P),s

for

s

HN(P),s/8

<

0.

Proposition. Vs E R there is a smooth function Ms(~, rJ) satisfying conditions (1.4.39), (4.4), and (4.5), for which inequality (4.11) holds.

We first of all reduce the proof of the proposition to the case of a single fixed value of s, say s = 1. Lemma 4. Let a smooth function

M(~, 'rJ)

satisfy the conditions

c- 1 ::;; 3-g(P)(~,rJ)M- 1 (~,'rJ):::; c,

1aa+f3 M(~, rJ)/8~aarJ 13 1 < cap(1

(3)

+ 1~1 + ITJI)-p(a+f3) M(~, TJ), 0 < p < 1, (4)

1~-y r] 0aa+/3 M( ~' TJ )/a~ a arJ 13 1 < Caf3-yoM 2( ~' 'rJ ),

V("f,8) E N(P),

V(~,TJ) E Il~?.

(5)

Then the function Ms ( ~, TJ) = M s( ~, TJ) satisfies the conditions of the proposition. Proof. Inequality (4.11) for Ms

= Ms

is a trivial consequence of (3). Let us show that (4) implies (4.4) for f-£ = Ms. For the sake of brevity, we shall write f-t(a,/3) = aa+/3 f-t/a~aary/3. By the chain rule, (Ms)(a,/3) is a linear combination of expressions of the form of

whence it follows that (4.4) is fulfilled for f-£ =Ms. Similarly, ~'Yry 0 (Ms)(a,f3) is a linear combination of expressions ((YTJO M(a1 ,/3t))Ms-1

II M(ak>f3k) M-1.

(6)

k>1

By virtue of ( 4 ), the factor is bounded, and, by virtue of ( 5), it does not exceed const M 2 , i.e. all expressions (6) do not exceed constMs+I ::;; const3-g{P)Ms and hence ( 4.5) with f-£ = Ms holds.

Operators of Principal Type Associated with Newton's Polygon

157

The proof of the proposition. We first assume that 'i(P) has integral vertices ('Yj, 8j) and set

In view of the lemma, it suffices to verify conditions ( 4.4) and ( 4.5) for the polynomial M 2 • Since only a finite number of derivatives (M 2 )(0i,{3) are nonzero, condition (4.4) follows directly from the regularity of the polygon N(M 2 ) = 28(P). To verify (4.5) we note that ~~ ry 6 ( M 2 ) ( 01 ,{3) is a linear combination of monomials of the form of a+ {3 ~ 1, (1,8) E N(P), (7) where the q(j)_degree of the monomial ~ 1 ry 6 does not exceed Cj and the q(j)_degree of elk -OifJ 20 k -{3 does not exceed 2(q(j) 1 ( Tk, 8k)}- {q(j) 1 (a, ,B)} ~ 2( Cj -1) -1 ~ 2Cj -3. Hence, the qULdegree of (7) does not exceed 3( Cj - 1 ). This shows that (7) can be estimated by means of canst M 3 • In the case when some of the numbers lj, 8j are not integers we take a positive function ltl E c= such that ltl = ltl for It! ~ 2 and ltl = 1 for ltl < 1. Setting

we obtain a function satisfying all the desired conditions.

CHAPTER V

TWO-SIDED ESTIMATES IN SEVERAL VARIABLES RELATING TO .N EWTON'S POLYHEDRA

Introduction Beginning with this chapter we develop the method of Newton's polyhedron in the theory of differential equations. The passage from Newton's polygon to Newton's polyhedron is quite natural. For polynomials we consider the convex hull of the set of monomial multiexponents completed in a certain way. In the present chapter we are interested in conditions under which the polynomials admit of an adequate estimate by means of the sum of the moduli of the constituent monomials. Of course, this relates to adequate estimates for differential operators whose symbols possess the indicated property. In the case of polygons the main technique was the factorization of symbols, up to within lower terms, as a product of quasi-homogeneous symbols. This technique is not generalized to the multidimensional case. Nevertheless, it turns out that, as before, the essential properties of polynomials can be determined by considering simultaneously all quasi-homogeneous parts of the polynomial (corresponding to the faces of Newton's polyhedron). As was shown by Mikhallov [1, 2], in these terms conditions for the existence of an adequate estimate are stated. In the presentation of Mikhallov's theorem we follow the paper by Gindikin [1]. Some other condit ions for the existence of adequate estimates are also retained, namely the stability condition for the property that the polynomial does not vanish and the absence of zeros in a complex neighborhood of the real space (see Volevich and Gindikin [2]). In the analytical aspect no new considerations are required as compared to the two-dimensional case.

§1. Estimates for Polynomials in Rn relating to Newton's polyhedra In this section we derive conditions under which a polynomial admits of a twosided estimate throughout the space IRn by means of the sum of the moduli of the constituent monomials. This is the basic estimate for deriving other estimates in various types of regions. Among them there are regions that go into those considered in the foregoing chapters in the case n = 2. 1.1. Newton's polyhedron of a polynomial. ~ = (6, · · · ,~n) E JRn:

P(O =

L

aat~,

Let P(O be a polynomial in

(1)

aEv(P)

where ~a= ~ 01 • • • ~~n and a= (a1, ... , an) E !R~. Here v(P) is a finite set in !Rf. (the closure of the positive coordinate n-hedron in JR~ ). It is convenient to admit the value aa = 0 for some a E v(P) and it is only important that v(P) is finite. 158

Two-sided Estimates in Several Variables Relating to Newton's Polyhedra

159

Thus, we shall sometimes regard v( P) as being broader than the set of multiindices corresponding to the nonzero coefficients. The statement of the problem. To find out for what polynomials there is c > 0 such that the following estimates hold:

(2) Naturally, the answer to the question is given in terms of Newton's polyhedron, a multidimensional generalization of Newton's polygon. By Newton's polyhedron N(P) of a polynomial P will be means the convex hull of the set v(P). We note that it is not assumed that dimN(P) = n. Let V(N) be the set of vertices of the polyhedron N; V(P) def V(N(P)) Consider the space R; dual to the space of exponents, and let the duality be determined by the form

(3) For a polyhedron

NCR~

we put

d(N, q) = max(q, a), o:EN

(4)

I.e.

(q, a)

= d(N, q),

(5)

is a supporting hyperplane to N. In the case of Newton's polyhedron N(P) we shall call the expression

dp(q)

def

d(N(P), q),

the q-degree of the polynomial P. Let Nq be the face of N lying in the supporting hyperplane (5). By the q-principal part of P will be meant the polynomial

Pq(O =

L o:ENq(P)

ao:~o: =

L

ao:~o:,

(6)

(q,o:)=dp(q)

It is clear that Pq is a quasi-homogeneous ( q-homogeneous) polynomial:

(7) We obviously have the following generalizations of Lemmas 1 and 2 in Section 1.1.2.

Chapter 5

160

Lemma 1. We have P( tq 0 = tdp(q) Pq(e) (PQ)q

= PqQq,

Pq (P+Q)q= { p

+ o(tdp(q) ), degpq(q)

+ Qq

q

t ~ oo ,

(8)

= degp(q) + degq (q),

(9)

for

dp(q) = dq(q),

.r lOT

dp(q) > dq(q).

(10)

Lemma 2. If two polynomials P and Q satisfy the conditions

dimN(P) = n, for all q' then p - Q is expanded into monomials of N(P) = N(Q).

ex' where a is an interior point

For some reasons that will be eludicated in what follows it is convenient to set

(11) 1.2. Basic estimate. Generalizing definition (1.2.2) we set

:=:N(O =

L

leal,

(12)

aEV(N)

Recall that V(N) is the set of vertices of the polyhedron N. The problem stated in the foregoing section (estimate (2)) is equivalent to the question of exist ence of c > 0 such that

(2') The equivalence follows from ( cf. Lemma 1.2.1) Lemma 1. For a EN we have

(13) The proof is carried out as in the case of n = 2 (Lemma 1.2.1). For c > 0 we denote by !1( c) the set determined by t he conditions

c- 1 ~

leil ~ c

Vj E {1 , ... ,n}.

(14)

Hence, !1( c) is a union of nonintersecting parallelepipeds lying in coordinate n hedrons.

161

Two-sided Estimates in Several Variables Relating to Newton's Polyhedra

Lemma 2. (i) Let dim N == n and let a be an interior point of N. Then for any C1 > 0 there is C2 such that outside the set n( c2) the inequality

(15) holds. (ii) If

(13') for some c > 0, then a 0 E N. And if for any c > 0 there is c1 such that inequality (15) is fulfilled outside n( C} ), then a is an interior point of N. Proof. (i) For a given j = 1, ... , m we denote by f3(j) and 8(j) the points of intersection of the straight line passing through a E N and parallel to the j th coordinate axis with the boundary of the polyhedron N. By virtue of Lemma 1, there is a > 0 such that n

a 'L)I~.Bu) I+

1ecn l) ~ 3N(0

for

~ E lltn.

j=l

Hence, n

n

a L)I~.Bcnj

+ ICScnl) =a I)l~iiAj + l~ij-ILi)j~al (

j=l

3N(0,

j=l

~

E lltn,

Aj,J.lj

> 0, 1 ~ j

~

n.

Select a number c2 so that the inequalities for

l~il>c2,

for

l~il

1

> -, c2

j = 1, . . . ,n, j = 1, ... , n,

hold simultaneously. Then in the complement to n(c 2 ) the inequality (15) holds. (ii) The first part of the assertion is proved by repeating literally the argument in Lemma 1.2.2. To prove the second part of the assertion we note that if the point a belongs to the boundary of N, then the supporting hyperplane (q, a) = (q, a 0 ) passes through it. Therefore along the curve {~i( t) = tqi j = 1, ... , n} inequality (13') is violated for t -+ +oo or t -+ 0 for any c.

eJ,

Lemma 3. Let P(e) and Q(O be arbitrary polynomials. Then there is c c(P, Q) sucl1 that

=

The proof is a literal repetition of that in the case n = 2 (see Lemmas 1 and 2 in Section 1.2.3).

Chapter 5

162

Theorem. A polynomial P(O admits of estimate (2'), for all~ E lRn if and only if ,t he principal quasi-homogeneous parts Pq( 0 are nonzero for all q E lRn outside tl1e coordinate planes, i.e. (16) Remark. By the accepted convention, conditions (16) include the condition

(16') corresponding to q = (0, ... , 0). Proof. Necessity. Let ~( 1 ) =/= 0. Then 3p(0 > 0, and hence, by virtue of (2'), we have IP(OI > 0. Further, for ~( 1 ) =/= 0 we have c' = c'(~)

> 0.

Indeed, to prove this inequalities it suffices to retain in sum (12) only those terms corresponding to a E Nq(P) for which j(pq~y:~l = pdp(q)le~l. It now follows from (S) that

S'ufficiency. To begin with, we make two preliminary simplifications. First, in-

stead of the case of the entire space lRn we can consider the case of the positive coordinate n-hedron. The matter is that the case of JRn obviously reduces to considering all coordinate n-hedrons and the case of an arbitrary coordinate n-hedron is reduced by means of a trivial transformation to the case of the positive n-hedron. Second, one can confine oneself to the case when the polynomial P( 0 has real coefficients ( cf. Mikhallov (1]). Indeed, if P(O is a polynomial with complex coefficients for which Pq(O =/= 0 for ~( 1 ) =/= 0, then the polynomial Q(O = IP(01 2 with real coefficients possesses the same property since, according to (9), we have

If the theorem is proved for polynomials with real coefficients, then for c have

> 0 we

By virtue of Lemma 3, we conclude that for some c1 > 0 we have c 1 (3p( ~) )2 ~ 3q(0, whence for some Cz

> 0.

Two-sided Estimates in Several Variables Relating to Newton's Polyhedra

163

Thus, we consider a polynomial P(e) with real coefficients on R+ for which (16) is fulfilled. Therefore P(e) retains sign, and it can be assumed without loss of generality that P(~) > 0 for ~j > 0, j = l, ... ,n. Then, by virtue of (8), we have

Pq(e) > 0

for all

q ERn

and

ei > 0,

j = 1, . . . ,n.

We shall prove the validity of the desired estimate by double induction, namely on the dimension n and on the number of (integral) points in N(P). For n = 1 we have the following obvious assertion: if a polynomial m

P(e) =

L aaea, a=k

is nonzero for

e=I= 0, then it satisfies an inequality of the form of

Assume that the desired assertion is true for polynomials inn- 1 variables. We first of all prove it for quasi-homogeneous polynomials in n variables. So, let

N(P) = Nq(P),

q =I= (0).

Let, for definiteness, q1 =/= 0. We set Q('r/2, · · •, fJn) = Pq(l, 'r/2, · · ·, fJn)·

Then

P(e)

= Pq(e) = ~:p(q)fq 1 Pq(l , fJ2, ... ,fJn) = ~dp(q)fq 1 Q(fJ) , Q(rJ) = e;dp(q)/ql Pq(e).

Newton's polyhedron N(Q) is obtained from the polyhedron N(Pq) by projecting the latter on the subspace {a 1 = 0}. Here the faces of N(Q) are obtained by projecting the faces Nr:(Pq) lying on the boundary of N(Pg) (i.e. t he faces of dimension no higher n- 2), and we have

Qr(fJ) = Pr:(l, 'r/2, .... , fJn), It is obvious that

Qr( fJ) =I= 0

for

fJ(l) =I= 0;

Chapter 5

164

including r = (0) and with account of

we obtain (2') for Pq. We now begin the induction on the number of points. A subset 'D of integral points belonging to N(P) is said to be regular if (i) 'D contains all the vertices of N(P), i.e. 'D :> V(P); (ii) if 'D contains an interior point of a face Nq, then 'D contains all integral points of that face. Further, for the sake of brevity, we shall say that 'D contains Nq if 'D contains all integral points belonging to this face. The above definition implies that 'D is the union of a set of faces of N(P) (including all the vertices) and a number of interior points of N(P). We put

Pv(O =

L aat\

oE'D

wl1ere a 0 are the coefficients of the polynomial P. It follows from ( i) that

V(Pv) = V(P), for all regular sets 'D. By means of induction on the number of points in 'D we now prove that for some c > 0 and c1 we have

(17) We remind the reader that we have Pq(O ~ 0 for all q. In particular, the coefficients a0 corresponding to the vertices a E V(P) are positive. Hence, inequality (17) holds for 'D = V(P). Lemma 2 (i) allows one to add to V(P) an arbitrary set of interior points (provided that c1 in ( 17) is sufficiently large). Further, since (17) has already been proved for quasi-homogeneous polynomials in n variables, the estimate we are interested in remains valid when 'D = 'D(q), where 'D(q) is the set obtained by adding to the points belonging to the face Nq(P) the other vertices of N(P) not belonging to that face. This estimate is taken as the basis of the induction hypothesis. ' ,Now let 'D be a regular set, and let us assume that estimate (17) has already been proved for all regular sets with a smaller number of points. We shall prove the estimate for 'D. It is natural to assume that 'D contains some boundary points distinct from the vertices. Then, according to (ii), the set 'D contains a face Nr(P), r E Rn, of maximum dimension. We associate with this face two regular subsets of 'D, namely the subset 'D( r) obtained as the union of N r and the other vertices of N(P) not belonging to Nr and the subset 'D{r} obtained by removing from 'D the interior points of N r· By the induction hypothesis, for the sets 'D( r) and 'D{ r} '

Two-sided Estimates in Several Variables Relating to Newton's Polyhedra

165

inequalities of the type (17) hold. Multiplying these inequalities we obtain the estimate for the polynomial Q(O = Pv(r)(OPv{r}(e): (17') We now perform the main rearrangement of the introduced polynomials. Denote by V(r) the regular set obtained by adding all faces lying on the boundary of the face N r( P) to the set of vertices. We put

Let us show that the polynomials Q( 0 and Q( 0 differ by a linear combination of monomials corresponding to the interior points of the duplicated polyhedron 2N(P). Then, by Lemma 2 (i), Vc > 0 there is c1 = c1 (c) such that

ea

According to Lemma 3, Sp2(0 ~ (3p(0) 2 • Taking c = c/2 and substituting into (17' ) we see that for E !Rn \ !1( ci) we have

e

Cancelling by 3p(e) we arrive at (17). Thus, with account of Lemma 2 in Section 1.1, we have to show that

If vector q is not parallel to r (i.e. q D(r), D{r}, and V{r} it follows that

#

Ar ), then from the definitions of the sets

Multiplying these relations we obtain Qq =Qq. In case q = Ar we have

Hence, inequalities of the type (17) have been proved for all regular sets 1) and, in par ticular, for the boundary of the polyhedron N(P). Therefore, by virtue of Lemma 2 (ii), inequality (17) holds for 1) = N(P) as well. Consequently, we have proved inequality (2') outside the compact set n( ci). However, since we have P(O # 0 for # 0, we see that (2') with some constant c > 0 is also fulfilled everywhere on n( ci) (we have not yet used this condition). The proof is completed.

el)

Remark. As is seen from the proof, if the condition P( 0 =J 0 for dropp ed, then inequality (2') is fulfilled for some

CJ

e(I)

outside the set !J( CJ ).

=J 0 is

Chapter 5

166

1.3. Some other conditions for the existence of an adequate estimate (2'). In this subsection we shall derive an analog of Theorem 1.2.4, namely we shall prove that the existence of estimate (2') is equivalent to the property that the polynomial P( 0 remains nonzero. for any small perturbations of the coefficients or that it has no zeros in a complex neighborhood of a special form of the real space

JRn. Theorem. For a polynomial (1) the following conditions are equivalent:

(I) inequality (2') holds for all ~ E !Rn; (II) there is c > 0 such that all the polynomials

P8(0 =

I:

(aQ + bQ)~Q'

(18)

QEN(P)

are nonzero for ~( 1 ) =/: 0; (III) there is c > 0 such that for all the polycylinder

~ E

!Rn the polynomial P( 0 has no zeros in

1

~

j

~

n, ( E

en.

(19)

Proof. (I)===*(II). By virtue of the lemma in Section 1.2, inequality (2') is equivalent to the existence of c' > 0 such that

c'

L

I~QI ~ IP(OI

v~ E !Rn.

QEN(P)

Therefore it suffices to take c = c' /2 for the condition P8(~)

=/: 0

to be fulfilled for

lbQI
We have P( () = :EaQ(Q = L.:( aQ +b'Q)~Q, where b'Q = aQ[(1 +d)Q -1]. We take c > 0 satisfying the condition (1 + c)m - 1 < c where m is the degree of the polynomial P and cis the constant involved in (II). For this choice of c we have lbQI < c, and, by virtue of (II), P( () =/: 0 for ( E 'D€( c). (III)===*(I). In view of Theorem 1.2, it suffices to show that (III) implies that Pq(O =/: 0 for all q E !Rn, ~(I)=/: 0. Let Pq(~) = 0 for some q, ~'and ~( 1 ) =/: 0. We fix these values of q and ~. It follows from (III) that for some c > 0 we have

P(pq ()

=/: 0

for

i(j - ~j I < c,

1 ~ j ~ n, p > 0.

(20)

Two-sided Estimates in Several Variables Relating to Newton's Polyhedra

167

It suffices to take c = cmin l~jl, where cis the constant in (III). We select an integral vector 1r = (7rb···,7rn) such that the face N1r(P) is a vertex belonging to the face Nq(P). This can be done since the vectors 7r E !Rn for which a certain vertex is the face N1r(P) form a polyhedral cone of maximum dimension. Therefore P1r( () is a monomial with a nonzero coefficient, and we have P1r(() =f. 0 for ( E (<1) =f. 0. Consider the expression P(z'Tr0 which is a linear combination of integral powers of z with exponents lying between -dp( -?T) and dp(tr), the coefficient in the highest power of z being equal to P1r(O and nonzero =f. 0). If we set r = max{dp(-1r),O}, then zrP(z'TrO is a polynomial in z of degrees= r + dp(1r). We put

en,

eel)

Qp(z) = p-dp(q)zr P(pqz1rO· This is a polynomial in z whose coefficients are continuous functions of the parameter p for p --+ +oo (see(8)). The coefficient in the highest power Z 8 is equal to Here we make use of the fact that the vertex N 7r belongs to the face N q. So, we havy found that the leading coefficient in the polynomial Q p( z) does not depend on p, and hence its roots depend on p continuously in the neighborhood of p = +oo. For p = +oo we have

Qoo(z) = zr Pq(z1r0, i.e. z = 1 is a root of Q00 (z). Therefore for any 6 1 > 0 there is M such that for p > ..\1 t he polynomial Qp( z) has a root z(p) for which

lz(p)-

11 < 6

1

•

We have Selecting c' > 0 such that lz'Tr~j- ~jl < c, 1 :s; j :s; n, we arrive at a contradiction to (20) for lz- 11 < c1 • The proof is completed. 1.4. Estimates with account of summarized degrees with respect to groups of variables. When applying Newton's polygons to polynomials in several variables in Chapters 2 and 3 we have already taken into account the summarized degree with respect to all spatial variables. We now consider this technique in a more general situation. Let the space !Rn be represented as a direct product of subspaces JR 1i:

We set

Chapter 5

168

. where leu> I is the Euclidean norm of the vector the set of monomial exponents a E Z+:

e(j)

E JR1i. Accordingly, we split

-[.

a=(a(I), ...

,O'(k)),

O'(j)EZ-f.,

j=1, ... ,k,

aPl = ( la(l) I, ... , la(k) I) E R~, where Iau) I denotes the sum of the components of the vector O'(j). Proceeding from polynomial ( 1) we construct the set

v(ll(P) ={f) E R~,j) = al 1l,a E v(P)}, and a reduced polyhedron N(ll(P), the convex hull of v(ll(P). Let y[ll(P) be the set of vertices of the polyhedron Nl1l(P). For q E JRk we put max (q, f)) = max (q, a(ll).

d[l](q) =

,8Ev(ll(P)

p

oEv(P)

Let NJ1l(P) be the face of N(ll(P) lying in the supporting plane (q,j)) = dp(q) and let aa~o. Pfl(O =

2:

o(ll E N!11( P)

Finally, let

'2~(0

=

Clel(l1),8 .

I:

,8EV[ 1l(P)

Theorem. For polynomial (1) tbe following conditions are equivalent. (I) Tbere is c > 0 sucb tbat

c'2~(0 ~ IP(O( v~

(21)

ERn.

(II) For all q E JRk we bave PJlJ ( 0

-I- 0

for

1~(1) I· . ·l~(k) I -f. 0.

(22)

In particular,

IP[l] I def IP(e)l -f. 0 (0, ... ,0)

for

1~(1) I·· ·l~(k) I -I-

0.

(22')

(III) Tbere is c > 0 such tbat all the polynomials (24) are nonzero for 1~(1) I·. ·l~(k) I -I- 0. (IV) Tbere is c > 0 sucb that for eacb ~ E R the polynomial P( () is nonzero in tbe complex polyspbere (a direct product of complex spheres)

l([j]- ~uJI < cl~uJI,

1 ~ j ~ k, ( E en.

Two-sided Estimates in Several Variables Relating to Newton's Polyhedra

169

This theorem is derived in a simple way from the corresponding assertions in Sectipns 1.2 and 1.3. We eludicate the proof of the implication (Il)==::}(l). In each group of the variables ~(j) we pass to polar coordinates:

Then To the polynomial Pw ink variables with coefficients depending continuously on w we apply Theorem 1.2, which results in

c(w )3Pw(0

~

P(O

for

~ E

IRn.

It remains to note that N(Pw) = N[ij(P), the constant c(w) can be regarded as a continuous function of w, and w is a point belonging to a compact set.

Remark. We note that the set {1~(1) I· .. j~(k) I = 0} is contained in the set {1~1 ... ~nl = 0} and the conditions of the theorem are stronger as compared to those in Sections 1.2 and 1.3 guaranteeing the existence of estimate (2). Let us discuss the corresponding geometrical pattern. Let N(P) be the set of those a E IR+ for· ~Bich a[ 1) E N[ll(P). It is clear that

N(P)

c

N(P),

and, generally, this is a strict inclusion relation. The conditions of the theorem correspond to the existence of the estimate

(21') which is equivalent to (21). According to Lemma 1.2, to this end it is necessary that N(P) = N(P). This is a rather strong condition on Newton's polyhedron N(P).

§2. Two-sided estimates in some regions in R• relating to Newton's polyhedron. Special classes of polynomials and differential operators in several variables The results of the foregoing section cannot be applied directly to differential operators. Such applications require estimates holding not throughout the space but for part (or all) of the variables tending to infinity in some way. Moreover, in the case of differential operators Newton's polyhedron is usually completed by adding some "minor" points that are selected proceeding from the character of the desired analytical estimates.

Chapter 5

170

2.1. Estimates on condition that part of the variables tend to infinity. We divide the variables in JRn into two groups E JRk and TJ E 1R1, k + l = n, and let (1) P(e,TJ) = aOif3eOiTJf3,

e

2:::

(Ot,(J)Ev(P)

We shall prove the following generalization of Theorem 1.2.

Theorem. For a polynomial (1) the conditions below are equivalent: (i) there are c 1 > 0 and c 2 such that (2) (ii) if at least one of the components of a vector r E JR 1 is positive, then

(3) For l = 0 the theorem goes into Theorem 1.2. In the situation under consideration there naturally appears the notion of minor points.

Definition. A point (a, (3) E Z+ is said to be minor for a polyhedron N if for some A> 0 and allj ~ l we have

where e j is a vector belonging to JR 1, whose j th component is 1 and the other components are zeros. The points belonging to N that are not minor will be called senior points. Of course, the notion of minor points depends on the way the variables are divided into groups. Denote by Nu the polyhedron obtained from N by adding all the minor points, by 8u the set of minor points of N, by rru N the set of senior points, and by vu the set of senior vertices of N. When N = N(P) is Newton's polyhedron of polynomial (1 ), we shall add (P) to these symbols.

Lemma. (i) If (a, (3) is a minor point of N, then for every c > 0 there is a> 0 such that

(ii) rru N is the union of all faces r j is positive.

N(q,r)

such that at least one of the components

The assertions of the lemma are verified directly. To prove (i) we t ake c satisfying the condition c(a/1)>.. ~ 1, where ..\is the number involved in the definition of a

171

Two-sided Estimates in Several Variables Relating to Newton's Polyhedra

minor point (ex,(3). Then cl~o:II1J,B+>-eil?:: l~o:1J,BI for l1Jil > ajl for all j, whence follows (4). As to (ii), in this case if(ex,(3) E N(q,r) where rj > 0, then (ex,f3+Aej) tf_ N for all>. > 0. In case (ex,f3) E N(q,r), where all rj :s; 0 <;tnd (ex,f3) does not belong to the faces with rj > 0, the expression (ex,f3 + Aej) satisfies all inequalities determining N for all j and a sufficiently small >., namely if (ex, (3) E N(q,r), then all rj :s; 0, and the inequality holds for all >. > 0; and if (ex, (3) tf. N(q,r), then for (ex, (3) the inequality is strict and is retained for small >.. We now proceed to the proof of the theorem. The implication (i)==*(ii) is proved as in Theorem 1.2. Consider the implication (ii)==*(i). Let N be the convex hull of the set of vertices V"" ( P). Then the union of the senior faces 1r"" N ( P) is a regular set for N (the definition is contained in the proof of Theorem 1.2). Therefore for some a1, a2 > 0 we have

outside the compact set f2(a 2 ) (with respect to the variables ~ and 1]; see (1.14)). Using assertion (i) of the lemma we can replace this inequality by the inequality

(4') for some other a 1 , a2 > 0. For any fixed ~ 0 E f2( a 2 ), considering P(o,r)(~ 0 , 17) (i.e. q = (0, ... , 0)) and applying Theorem 1.2 and part (i) of the lemma we conclude that for some b1 and b2 we have

(5) and from the proof it is clearly seen that the constants b1 and b2 can be regarded as being independent of the point ~ 0 belonging to the compact set f2(a 2 ). The combination of ( 4') and (5) yields (2). We note that in the theorem the expression 3p can be replaced by Sf, where the summation extends only over senior vertices

V""(P) . Remarks. Following the same idea it is possible to construct many versions of these estimates. Below are several examples (for more detail see Gindikin [1]). 1) If the existential quantifier with respect to c2 in the statement of the theorem is replaced by the universal quantifier (Vc2 3c1 ), then in condition (3) the existence of a positive component of the vector r should be replaced by the existence of a nonnegative component. 2) The cases indicated in 1) can be unified by, dividing the set of indices into three groups :E1, :E2, ~3. For any a > 0 there exist c 1 , c 2 > 0 such that

c1SP(O :s; P(O for all j E ~1: l~il >a, j E :E2: l~il > c2, j E :E3, if and only if

Pq(O =f. 0

for

~(I)

=f 0,

qi?:: 0, i E

E2,

and

qi

> 0, i E ~3·

Chapter 5

172

2.2. Estimates in the case of complete polyhedra. In Chapter 1 we meant by Newton's polygon the convex hull of not only monomial exponents but also their projections on the coordinate axes. Let us consider a multidimensional generalization of this construction. We shall write 1 ~ a; a, 1 E !Rn, if ( j ~ a j for all j ~ n. By the completion of a polyhedron N C IR+ will be meant the polyhedron N obtained from N by adding those 1 E !Rf. for which 1 ~ a for some a E N. A polyhedron N is said to be complete if N = N. Lemma. 1) A polyhedron N is complete if and only if it is determined in IR+ by a finite system of inequalities of the form of

(q, a)

~

d(q),

(6)

2) For a complete polyhedron N all the points not belonging to the faces Nq, where qj ~ 0, j ~ n, are minor (in the sense of the definition in Section 2.1 for k = 0). 3) Fbr a complete polyhedron N a face Nq, where qj < 0, lies in the coordinate plane {aj = 0} and coincides with the intersection of the face Nq(j) with this plane, where q(j) is obta.ined from q by replacing qj < 0 by qi = 0. In particular, if N = N(P), then

(7) If qi < 0 in the vector q for j E J and if q( J) is obtained from q by replacing these qj by zeros, then j E J.

In particular, if qj

< 0 for

j E J and qj

= 0 for j f/:. J,

(7')

then

j E J.

(8)

Recall that P(o, ... ,o)(O = P(O. Proof. It is clear that, along with a, all 1 ~ a satisfy system (6), i.e. system (6) determines a complete polyhedron. Now let N be a complete polyhedron. Consider Nq where the+e,is a component qj < 0. If N contains a point a with a j > 0, then the point obtained from a by decreasing a i does not belong to N but we have a~ a. Consequently, Nq lies in the plane {a1 = 0}. By virtue of the completeness of N, either Nq(j) lies in {aj = 0} or Nq(j) has a non-empty intersection with {aj = 0} (the projections of thepoints of Nq(j))· In both cases this intersection coincides with Nq. We simultaneously conclude that the inequality (q, a) < d( q) is inessential in the determination of N. It can easily be seen that this argument remains valid when q has no positive components. Relations (7), (7'), and (8) are a

a

173

Two-sided Estimates in Several Variables Relating to Newton's Polyhedra

direct expression for these geometrical properties for the case N = N(P). Finally, if a point a E N belongs to none of those faces Nq for which all qj ~ 0, then all inequalities in (6) are strict, and the point obviously remains in the complete polyhedron under a small positive perturbation of any coordinate a j, i.e. is minor. It is often convenient to think of a complete polyhedron as the union of the parailelepipeds {I~ a,/ E R+} over the senior points a EN.

Theorem. For a polynomial (1) the following conditions are equivalent.

(i) There are c1 , c2 > 0 such that (9) where N(P) is the completion of Newton's polyhedron N(P). (ii) If qj ~ 0, ri ~ 0, j ~ k, i ~ l, and the vector r has a positive component, then

(10)

f

We note that the condition ~(q)T/(r) # 0 means that ~j # 0 if qj 0 and 1Ji i= 0 if r .i # 0. Without loss of generality, we can assume that N(P) = N(P) regarding some ao:/3 in (1) as being equal to zero. Then this theorem is merely a specialization of Theorem 2.1 for the case of complete Newton's polyhedron N(P). To prove the theorem use should be made of the description (7), (7'), (8) of t he principal parts P(q, r)(~,ry) when among qj, ri there are negative components. The description alloV~:s one to confine oneself in the case of complete Newton's polyhedra to ( q, r) with nonnegative components under the requirement that P(q,r) should be nonzero on some coordinate planes. We note that for l = 0, along with the conditions P(q)(O # 0 for ~(q) # 0, for the faces N(q)(P) the condition P(O # 0 also takes part for all~ ERn. Among the complete polyhedra we separate out a class of regular polyhedra. This notion generalizes the notion of regular Newton's polygons considered in Section 1.1.

Definition. A complete polyhedron is said to be regular if it contains no faces parallel to the coordinate planes and not belonging to them. In view of the lemma, this condition is equivalent to the property that N is determined in R+ by a finite system of inequalitie~ . of the form of

(q, a)

~

d(q),

(11)

R emarks. 1) Regular polyhedra can also be separated out among the complet e polyhedra by means of the following condition: if a E N and 1 = a- e1 E Z+., then 1 is a minor point of N . . 2) For a regular polyhedron the face Nq, where q1 = 0, j E J, and q1 > 0, J ~ 1, lies in the coordinate plane {a j = O,j E J} , and it is the intersection of a

Chapter 5

174

face Nq{J} with this plane, where all the components of q{ J} are positive and the cornponents with indices j E J coincide with qj . Therefore

(12) Here Nq = Nq where 'i}j = qj for j ~ J and qj = £ > 0, j E J, the number£ > 0 being sufficiently small. This means that whereas in the case of arbitrary complete polyhedra one can confine oneself to considering the principal parts Pq with qi ): 0, j ~ n, in the case of regular polyhedra it is possible to confine oneself to the consideration of Pq with qi > 0 for all j. In particular, if in the conditions of the theorem it is additionally required that the polyhedron N(P) should be regular, then (ii) can be replaced by the condition (ii') if qi > 0, 0 ~ i ~ k, and Tj > 0, 1 ~ j ~ l, then

P(q,r)(~,7J) =J 0

for

(~,7])( 1 ) =J 0,

P(o,r)(~,7J) =J 0

for

7]( 1 )

=J 0,

~ E IRk.

(13)

We remind the reader that last the group of conditions guarantees that P(q,r)(~, 7]) =J 0 for(~, 7J)(l) =f. 0 if qi ~ 0, j ~ k. Using Remark 2) we can combine some of the conditions, namely if N((j,T) is the intersection of the face N(q,r) with a coordinate plane, then the condition on P((j,T) can be added to the condition on P(q,r)· 2.3. N quasi-elliptic polynomials and operators.

Definition. A polynomial P(e), ~ E !Rn, is said to be N quasi-elliptic if its Newton's polyhedron N(P) is regular and the equivalent condit ions of Theorem 2.2 for k = 0 hold, i.e. if qj > 0, 1 ~ j ~ n, then

(13') Under these conditions for some c1 and c2 we have

(3') To N quasi-elliptic differential operators (i.e. operators with N quasi-elliptic symbols) the result of §1.4, which were obtained for the case of two variables, are automatically extended. In particular, N quasi-elliptic differential operators are hypoelliptic, and an analog of Theorem 1.4.2 holds for them. Similarly, if the symbol P(x, 0 of an operator P(x, D) is N quasi-elliptic for each x and Newton's polyhedron N(P(x)) does not depend on x, then an analog of the results in Section 1.4 .4 takes place. In this case we have hypoelliptic differential operators of constant strength.

175

Two-sided Estimates in Several Variables Relating to Newton's Polyhedra

2.4. N-parabolic polynomials and operators. t ) P( ~:,,7 =

~ ~

tal tan aa/31:.1 .. ·l:.n 7

f3

Consider a polynomial

(14)

'

(a,f3)Ev(P)

By its Newton's polyhedron will be meant the conve~ hull of the point ( c:.; (3) E v( P) and t hose minor points (a, (3) for which ( a 1 , . .. , anf3) E v(P) for some (3 > (3. This corresponds to the notion of minor points in the sense of Section 2.1 if 7 is included in the second group of variables. In this section the completed polyhedron will simply be called Newton's polyhedron and denoted as N(P).

Definition. A polynomial (14) is said to beN stable-correct if (i) P(~, ry) is solved with respect to the highest power of 7; (ii) the polyhedron N(P) is complete and regular with respect to a, 1.e. determined by a finite system of inequalities of the form of

(q, a)+ rf3

~

dp(q, r ),

qj

> 0, j

~ n, r ~

IS

(15)

0;

(iii) there are c 1 > 0 and c2 such that

(16) If instead of (ii) the stronger condition holds: (ii') the polyhedron N(P) is regular; then the polynomial is said to be N -parabolic.

Theorem. For a polynomial satisfying the conditions (i) and (ii) the condition (iii) is fulfilled if and only if P(q,r)(~,7)

#0

for

qj

> 0, j

~ n, r

> 0, 76 .. -~n

# 0,

Im7 ~ 0.

(17)

This theorem is derived directly from Theorem 2.2 with account of (13') and the following circumstances. Instead of 7 E
176

Chapter 5

Remark. If the above argument is continued, we conclude that for a polynomial P(~, r) satisfying condition (ii) and not obeying (i) the condition (iii) holds if the following condition is added to (17): Q(~) =J 0 for ~ E IRn, where Q(O is the coefficient in the highest power of T in the expansion of P in powers of r. N -stable polynomials are exponentially correct, and N -parabolic polynomials are, in addition, hypoelliptic. If the polyhedron N(P(x, t)) of a symbol P(x, t; ~' r) does not depend on the point (x, t), then the differential operator P(x, t; Dx, Dt) corresponding to this symbol is of constant strength. Therefore the results on Cauchy's problem for N stable-correct and N quasi-parabolic operators are direct consequences of Theorem 2.5.5.

CHAPTER VI

OPERATORS OF PRINCIPAL TYPE ASSOCIATED WITH NEWTON'S POLYHEDRON

Introduction In this chapter the results of Chapter 4 are extended to the case of polynomials in n > 2 variables, and accordingly, to differential operators in n variables. The general plan of the presentation of the material is the same as in Chapter 4. However, the passage to the case of n variables involves some additional difficulties, mainly of geometrical character. The thing is that a polygon in the plane has faces of only two types, namely faces of zero dimension (vertices) and faces of maximum dimension (sides). For n > 2 there appear faces of intermediate dimension between the zero and maximum dimensions. The notion of essential variables of a side in Section 4.2.2 is readily generalized to the notion of essential variables of a face of maximum dimension whereas the definition of essential variables of faces of lower dimension is rather intricate. Similar difficulties arise when constructing a covering by regions corresponding to faces of different dimensions, which serves as an analog of the covering by the regions G(r;o), co) and G(r;l), co, c 1 ) in §4.2. Briefly, the plan of the presentation of the material in this chapter is the following. In §1 we define polynomials of N-principal type and state a multidimensional generalization of Theorem 4.2.2. Irl §2 special regions are defined and analogs of estimates ( 4.2. 7) are proved for them. §3 presents the main geometrical construction of a covering of Rn by regions associated with the faces of Newton's polyhedron. This construction allows us to complete the proof of the basic theorem in §1. In §4 differential operators with variable coefficients are considered whose symbols P( x; 0 are polynomials of Nprincipal type with respect to ' . For these operators a multidimensional analog of estimate ( 4.3.3) is proved. The resulting estimate makes it possible to reproduce almost literally the argument in §4.4 and to prove a local solvability theorem for differential operators of N -principal type with variable coefficients. In the Appendix to the chapter we present the construction of a partition of unity subordinate to the covering in §3. §1. Polynomials of N-principal type

In this section all ne.c essary notions involved in the definition of polynomials of N -prin cipal type are introduced and a multidimensional analog of Theorem 4.2.2 is stated. 1.1. Newton's polyhedron and polyhedra of minor terms. Consider a polynomial

P(O =

L aE1(P)

177

aaea.

(1)

Chapter 6

178

If the coefficient acx corresponding to a = 0 is equal to zero, then we shall add the origin to the set 'Y(P). Let ::Y(P) = 'Y(P) U {0}. In this chapter by Newton's polyhedron N(P) of polynomial (1) will be meant the convex hull of the finite set ::Y(P). Moreover, we shall deal only with those polynomials for which

the polyhedron N(P) is regular.

(2)

This means that the polyhedron N(P) is complete (i.e. along with every point a E N(P) it contains all its projections on the faces of the various dimensions lying in the coordinate planes) and is determined by the systems of inequalities

(q, a)

~

dp(q),

q E R~,

a

E R+.

(3)

This description implies that N ( P) has vertices at {0} and on each of the coordinate axes and that the (n-1)-dimensional faces of N(P) lie in the coordinate hyperplanes { aj = 0, j = 1, ... , n }, and in the hyperplanes

j = 1, ... 'J,

(3')

where all components of the vectors q(j) are positive. As in Chapter 5, a point a E N(P) is said to be minor if there is j, 1 :::;; j :::;; n, such that a+ ej E N(P), where ej = (0, ... , 1, 0, .. . , 0) 1 ). The points of N(P) that are not minor are said to be senior. Denote by 8(P) the convex hull of the set of minor points of N(P). As was noted in Chapter 4 (for the case n = 2), as a rule, the polyhedron 8(P) is not regular (although it is of course complete). In what follows we shall deal not with the polyhedron 8(P) but with its extension 8(P). We set

(4) Let "i(P) denote the convex solid determined by the inequalities ........

(a, q) :::;; dp(q),

(5)

Lemma. If the polyhedron N(P) is regular, then the solid 8(P) possesses the following properties: (i) "i(P) is a convex polyhedron; (ii) 8(P) is a regular polyhedron; (iii) the inclusion

8(P)

c 8(P) c N(P),

(6)

takes place. The proof of the lemma will be given in the next section and will be preceded by some general remarks on polyhedra. l)In the terminology of Chapter 4, the point a should be called a Z-minor point. Since no minor points in the sense of Chapters 1 and 4 are involved in our further presentation, the term Z -minor points will not be used .

Operators of Principal Type Associated with Newton's Polyhedron

179

1.2. The properties of the polyhedron 6(P). We remind the reader that if N is a convex polyhedron, then (a, q) ~ d (7)

is called a supporting half-space to N if (7) is fulfilled for all a E N, and for at least one point a E N the equality is attained. By the convexity of N, to every supporting half-space there corresponds a single face rCk) C N ( k = dim rCk)) of maximum dimension belonging to it . On the other hand, in general, to every face there correspond many supporting half-spaces containing it. The vector q in (7) is called the direction vector of the half-space. The set of direction vectors of all sup-porting half-spaces containing a face rCk) C N is called the normal cone of the face rCk) and is denoted V(k)· We note that V(k) is a closed convex polyhedral angle (cone), and we have dim Vc k) = n - k. It should be noted that the interior points of V(k) are direction vectors of the half-spaces for which the face of N of maximum dimension contained in them is the face r< k). The vectors belonging to the boundary 8V( k) are direction vectors of supporting half-spaces containing the faces r k, the face lying on the boundary of the face rCk') being rCk). Thus, with a convex polyhedron N finite set of closed conv~x polyhedral angles V/k)' k = 0, ... , n- 1, j = 1, ... , J is associated, the angles V(~) (the normal cones of the vertices of N) covering the whole space: J

Rn =

uV(~)' j=l

The boundary

8V(t)

of each of the angles in this set is a union of a finite number

of angles V(~~l) of lower dimensions. Fork= n -1 we obtain one-dimensional rays

{>..q(i)}, where q(j) are the direction vectors of the supporting half-spaces containing the ( n - 1 )-dimensional faces of N. The function

d(N, q) =max( a, q) a EN

is called the supporting function to the polyhedron N. It is a positively homogeneous function of degree 1, i.e. d(N, >..q) = >..q(N, q), >.. > 0. If q E Rn and rCk) is the face of maximum dimension lying in the supporting half-space for which q is the direction vector, then obviously

d(N, q) = max (a, q). aEf(k)

And it follows that if V{k) is one of the normal cones of the polyhedron N, then

d(N, q' + q") = d(N, q')

+ d(N, q"),

q', q" E V{k) \ oV(k).

(8)

Chapter 6

180

Remark. We presented above a dual description of a polyhedron: More precisely, given the above-mentioned finite set of convex polyhedral angles V(1k)' 0 ~ k ~ n-1, j = 1, ... , J 0 , and a homogeneous function d(N, q) satisfying conditions (8), the convex solid (a, q) :S; d(N, q) \/q E JRn

is a convex polyhedron and the angles V(~) are the normal cones of its faces of the various dimensions. The proof of Lemma 1.1. (i) Let us show that all inequalities (5) are consequences of a finite number of inequalities among them. Since the polyhedral angles V(1k) of

the faces rjk) C N(P) cover the whole space Rn, it suffices to separate out a finite number of inequalities among

so that the remaining inequalities are their consequences. We introduce the following notation. Let S be a subset of Rn and let I ( i1, ... , in), s :S; n, be a set of indices assuming the values 1, ... , n. Then

We divide the system of inequalities (5') into a finite set of subsystems corresponding to the various sets I of indices:

(a, q) ~ d(q), It now remains to note that on the set (V(1k) \BV({))I the function dp(q) is positively homogeneous and additive. (ii) As has been proved, one can select a finite number of vectors qU) E R+., j

=

1, ... , J, such that the polyhedron

8(P) is determined by the inequalities j = 1, ... , J,

a E R+..

According to Section 5.2.3, such polyhedra are said to be regular. (iii) The right-hand inclusion (6) is obvious. Since the polyhedron 8(P) is the convex hull of a finite set of integral points, it suffices to show that all integral minor points a E N(P) belong to 8(P). According to the definition of minor points, for every such point a there is j such that f3 =a+ e1 E N(P). Therefore

(a,q)

= (/3,q}- qj

:S; dp(q)- mjn qi

For our further aims we need the following

l~z~n

= dp(q).

Operators of Principal Type Associated with Newton's Polyhedron

181

Lemma. If a hyperplane (a, q) = dp(q), q E JR.+, contains an integral point ao E N(P), then (a, q) = dp(q) is a supporting hyperplane to both b(P) and 8(P). Proof. According to Lemma 1.1 (i), 8(P) lies in the half-space (a,q) ~ dp(q). If minqj = qj 0 , then the point a 0 - ej 0 belongs simultaneously to 8(P), b(P), and

the hyperplane (a,q) = dp(q). 1.3. Essential variables corresponding to the faces of the polyhedron

N(P). When defining polynomials of N -principal type in two variables in §4.2 we associate with each face of N(P) of nonzero dimension certain variables that were called the essential variables of that face. We now extend this definition to the multidimensional case. A vector q E R+ is said to be essential if (a, q} ~ d(q) is a supporting half-space to b(P); the set of essential vectors will be denoted as lR+. If V(k) is the normal cone of a face r
V(k) = V(k) n i+. Lemma. (i) If a face r
then V(k) = V(k) · (ii) Let a face f(k) lie in the coordinate hyperplane {ai = 0, i E I}, where I is a subset of the set {1, ... , n }, and not lie in the coordinate hyperplanes of lower dimensions. Then

Proof. (i) follows automatically from Lemma 1.2. The same lemma implies that if q E V(k) n JR.+, and there is an index j 0 tJ_ I such that minqi = qj 0 , then q E V(k)·

Therefore the vectors q E V(k) \ V(k) belong to the right-hand set in (8). We note that in the situation (ii) the cone V(k) has faces of codimension 1 in the

hy~erplanes {qi = 0, i E I}. It follows (see (8)) that dim(V(k) \ V{k)) =dim V(k)· As to V( k), the dimension of this closed set can be less than that of V( k). In particular, if the smallest components of the vectors q E V( k) have indices i E I, then V( k) is an empty set. The face which V(k) = 0 will be called an inessential face of N(P), and the other principal faces will be called essential faces. Let J = (]t, ... ,js), s ~ n, be a set of indices assuming the values 1, ... , n. Definition. eJ = (eil' ... , (7.) is called the set of essential variables of a face if

r
a) i/(k),J = {q E i/(k),minqi = qi:::} j E J} is a relatively open set in f(k); b) the set .......J is maximal in the sense that there is no broader set J' :) J for which V(k),J' satisfies condition a).

Chapter 6

182

Let rn- 1 be a face of N(P) of maximum dimension and let q0 be the outer normal to rn- 1 . Then (according to the lemma) q0 is an essential vector, and the essential variables are those ~j for which qJ = min q0 • In the case of faces f(k) of dimension k < n- 1 there can appear several sets of essential variables. For the sake of visuality, we present another, less formal definition of essential variables. Let f(k) be a face of N(P) and let 1r(f(k)) be the k-dimensional hyperplane passing through it:

Assume that for some j the intersection 1r(f(k) - ej) n S( P) is non-empty and belongs to the boundary of S(P). Then there is a face b.(k') C S(P) lying in the plane 1r(f(k)- ej)· The faces b.(k') and f(k) are called an associated pair of faces. Let J = (j 1 , ... , j s), j 1 = j, be the maximal set of indices such that the planes 1r(f(k) - ej,J, X = 1, ... , s, coincide with 7r(f(k) - ej ); the set is called the set of essential variables of the associated pair f(k) and b.(k'). A face f(k) is said to be inessential if there are no faces _6.(k') C S(P) associated with it (f(k) must lie in a coordinate plane). We note that every face of N(P) of maximum dimension has exactly one face of 8(P) associated with it. However, as is shown by simple examples (even in the case of the plane), the polyhedron S(P) may have faces of maximum dimension that have no faces of N(P) of maximum dimension associated with them. We demonstrate the above definitions by an example of a quasi-homogeneous polynomial P for which N(P) is determined by the inequality

eJ

(9) where it is assumed (cf. Theorem 4.1.3) that minqJ = 1 and that the numbers

q~, ... , q~ are integers. We shall show that the polyhedron ~

n

0

8( P) = {a E IR+, (a, q )

~

m - 1}.

E( P)

has the form

( 10)

To give an accurate proof of this fact we calculate the supporting function dp(q). As can easily be seen,

(11) Indeed, set aU) = (m/qJ)ej, j = 1, .. . , n. Then each point a E N(P) is represented as n

a=

2: j=1

Xja(j),

Operators of Principal Type Associated with Newton's Polyhedron

183

i.e. c/i) is a vertex of N(P). Further, Va E N(P) we have

(a,q) = Lxi(a<j),q) ~ mmaxqifqj. If the maximum on the right-hand side is attained for j = k, then for a = a(k) this inequality goes into an equality, which proves (11 ). Since dp(q 0 ) = dp(q 0 ) - minqj = m- 1, the polyhedron b(P) is contained in the polyhedron .6. determined by the right-hand side of (10). Let us check the opposite inclusion, i.e . .6. C b(P). To this end it suffices to show that the points &(j) = ((m-1)/qj)ej, the vertices of .6., belong to b(P), i.e. to verify the inequalities

(&(i)' q) ~ dp( q), or, assuming that min qj

= 1,

the inequality

The latter inequality is equivalent to

(m- 1) ( m~x _q·t t q?

-

q.) qj

_]_

q· + ( m~x -' t

q?

1) ~ 0.

The first term on the left-hand side is obviously nonnegative. If q and q0 are normalized by the conditions min qi = min q? = 1, then the second term is also nonnegative. We now describe the essential variables of the faces of N(P). Polyhedron (9) has exactly one face of maximum dimension:

r
E

JRq., (a, l) = m }.

If J(q 0 ) is a set of indices j E {1, ... , n} for which qj = minqj of variables ~ J( qo) is the set of essential variables of that face.

def

1, then the set

We now consider the face r~n- 2 ) = r
8(P)

= {a

E

R+, (a, q

0)

~ m- 1, aj ~ ~

- 1,j

= 1, ... , n }·

Chapter 6

184

If qJ > 1 then (m-1)/qJ > mjqJ -1 so that 8(P) is obtained from 8(P) by "cutting off", using planes parallel to coordinate planes, those vertices ( ( m - 1) / qj )e j for which qj > 1. We note that in the general case as well 8(P) is obtained from 8(P) by cutting off, using planes parallel to coordinate planes, neighborhoods of some of the vertices of 8( P) lying in the coordinate planes. 1.4. An additional arithmetical condition. We present a multidimensional generalization of condition (A) in Section 4.2.1. It relates to a face f(k) C N(P) with normal cone V(k)·

Condition (A). (i) Let rCk) be an essential face; this means that the set V(k) is non-empty and decomposes into non-empty components V(k),J · Then in each of the components there is a vector q, q ¢:. 8V(k) 1 ), for which

(12) (ii) Let rCk) be an inessential face. Then there is a vector q E V(k) \ 8V(k) such tl1at ((3, q) ~ m_.?-X (a, q) V(l E (N(P) \ rCk)) n (13)

zn.

aE8(P)

This condition is employed when proving the necessity in the main theorem stated below ( cf. Theorem 4.2.2). It is rather difficult to verify condition (i). We note that an obvious sufficient condition for the validity of (i) is the condition (i') if rCk) c N(P) is an essential face, then in every component of V(k),J there is an integral vector q normalized by the condition min qi = 1. Remarks. 1) Condition (i') means, in particular, that the outer normals q(j) to t he faces of maximum dimension are integral vectors if they are normalized by the condition minqkj) = 1. k

2) According to Lemma 1.3, for n = 2 all the faces of the polygon N(P) not lying on the coordinate axes are essential, i.e. condition (A) reduces to (i). It is easy to see that in this case condition (i) coincides with condition (i') and condition (ii) in Lemma 2.1. 1.5. Polynomials of N-principal type.

Definition. A polynomial P( 0 is called a polynomial of N -principal type if for each senior face rCk) of the polynomial the following condition hold. ( i) If r< k) is an essential face, then for any set ~ J of essential variables of this face we have

~( 1 ) = ~1

· · ·

~n

-=/=

0;

(14)

l)That is, in the supporting half-space (a,q} ~ dp(q) there are no faces f(k) C N(P) of dimension k' > k containing f(k) .

Operators of Principal Type Associated with Newton's Polyhedron

(ii) If the face

r
185

is inessential, then (15)

Theorem. Let N ( P) be a regular polyhedron and let condition (A) bold. Then tbe conditions below are equivalent.

(i) Tbe polynomial Pis a polynomial of N -principal type, i.e. tbe conditions of tbe above definition are fulfilled. (ii) For any polynomial Q(O, N(Q) C b(P), there is a constant c = c(Q) such that (16) The necessity of conditions (14), (15) for the fullfilment of inequality (16), i.e. the implication (ii)===?(i), is proved following the scheme that has already been used many times. Let for some TJ E !Rn, ry 1 , .•. , TJn -=J. 0, the relations (17) hold, where r
36(P)(~(t)) > consttdp(q).

We shall show that if ( 17) is fulfilled, then for an appropriate choice of the polynomial Q we have

(18) The resulting contradiction will prove the desired assertion. According to (A), we have

Let us take the monomial c~f3 as Q(O, where . ((3, q) = dp(q). Selecting c from the condition w + CTJ{J = 0 (recall that TJ 1 , •• • , 1]n -=J. 0) and using the first condition ( 17) we obtain P(~(t)) + Q(~(t)) = o(tdp(q)) for t-+ +oo .

Chapter 6

186

Further, by the definition of the set V{k),J, we have qj 1 = · · · = qia therefore

< q1, l t/:. J, and

Using the second relation in (17) we prove (18). We now assume that f(k) is an inessential face and that d( q) =

m11x (a, q). aE6(P)

According to condition (A), there is a vector q such that on the curve ~(t) defined above we have

If Pr(k)(TJ) = 0, then fort~ oo we arrive at a contradiction. In §§2 and 3 we shall prove the following

Proposition. Let a polynomial P( 0 satisfy the conditions of the definition. Tl1en tbe inequality :=:6