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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 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 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 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 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
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,tP0
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
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, (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 {tqlog-.
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
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 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 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 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 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,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 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,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
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
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 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 ( 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 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 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 k containing f(k) .
Operators of Principal Type Associated with Newton's Polyhedron
(ii) If the face
r,e). The basic estimate. Let r 0 such that (16)
It follows from (15) and (16) ( cf. the derivation of (10)) that for a sufficiently small c;
we have
3y~l(r(kl)(e)::::; cl ~~(e) I V~ E U.
(17)
Performing the summation of inequalities (17) over l = 1, ... , s we find
(18) where
s
v(f(k))
is the convex hull of
UT~)(
(19)
l(k)).
1=1 1 lThe
extension of the normal cone can appear only in the case when nate hyperplane which does not intersect the face f(k) .
-y(k)
intersects a coordi-
Chapter 6
192
We now show that
(20) where rCk) is an arbitrary essential face of N(P). Inclusion (20) and inequality (18) imply that (21) 36(?)(0 ~ const Igrad P(OI v~ E u. 4. We proceed to the proof of inclusion (20). The definition of the polyhedrons implies that
T~1\·yC k))
where J = J(l) U · · · U J(s). We begin with the case when the face f(k) do_!s not lie in the coordinate planes. Here, according to Lemma 1.3, we have Vk = Vk and rp.inqj = m~n qj. Using the definition of the function dp(q) we see that in this case 1EJ
1~1~77
we have v(f(k)) = {a E IR(a)+, (a, q) = dp( q), Vq E
It is clear that the polyhedron
V(k) }.
"6( P) determined by the inequalities
is contained in (22'). The case of faces lying in coordinate planes reduces to the above. Indeed, let a face r log for f3 E 1 j, q E 1J j. £
Then 1J = n1Ji is the desired V( 1) semi-cylinder. To construct Dj we take an arbitrary element qo E 7r~ \ 81r~. According to the definition of the half-space 1r~, we have
We note that (a- /3, q0 } does not depend on a (since (a, q0 } = const for a E 1r(f(l)), q0 E 1r~ C 7r(V(l))· By virtue of the compactness of /j, 3b > 0 such that (a-
/3, qo) > b
Va E 1r(f(l)),
V/3 E 'Yi.
Let Cj C C be an arbitrary 1r~ semi-cylinder and let Dj = Cj + Lq 0 • If L = L(c:) is sufficiently large, then inequality (6') holds for q E 1Jj. The proposition is proved. 3.3. Some additional remarks. 1) Let V( l) be determined by ( 5) and let 7?~ be the extension of the half-subspace
1r~ to a half-space of IR" transversal to 7r(V(l))· Let R+ be a translation of 7?~ . ......
The region 1J = 1J n R+ is called the truncation of the V(t) semi-cylinder 1J in the direction of 7r~. As can easily be seen, f> is a V{t) semi-cylinder. When proving Proposition 3.2 we have in fact established that by means of the operation of truncation in the directions of 1r~ it is possible to obtain from an arbitrary V(l) semi-cylinder V 0 C C (where C is a 7r(V(l))-cylinder) a V( 1) semicylinder 1J possessing the property g(f(l), £ ). 2) Let r. Then as the above half-subspace 1r~ we can take the half-space bounded by a hyperplane Q j orthogonal 2 ) We note that property 2) makes it necessary to take the points {3 E T(f(l)) lying outside the positive coordinate n-hedron.
197
Operators of Principal Type Associated with Newton's Polyhedron
to r 0 there is r = r(t:) such that the region ~n \ i_(r) 1 ) admits of a covering by a finite set of regions V>.. each of which satisfies the condition g(f~k),t:) for a certain senior facef~k) C N(P). The theorem will be proved by induction. We shall proceed from the covering of ~n \ i_ (O) by the polyhedral an!~les of normals V({) corresponding to the vertices of
N(P). At thelthstepweshallobtainacoveringof~n by "final" V({) semi-cylinders, k ~ l - 1 (they do not change in the further rearrangements) and "preliminary"
V({)
semi-cylinders appearing at the foregoing step.
truncation in the directions corresponding to the faces we cut out of the "preliminary"
V({)
V({)
of maximum dimension
semi-cylinders the "final" V{t) semi-cylinders
possessing the property g(r)1), c). 1 )For
Applying the operation of
the definition of the region i_(r) see (2).
198
Chapter 6
We now show (and this is the geometrical basis for the induction) that the part of a V(z) semi-cylinder that remains after a certain truncation of that semi-cylinder is removed is covered by the V({+l) semi-cylinders corresponding to the (l +I)dimensional faces on whose boundary
r.. constructed in the theorem are polyhedral, I.e. they are determined by a finite systems of inequalities of the form of
V{t)·
j = 1, ... 'J>.,,
(10)
where the vectors z(>..,j) and the numbers R>..j are determined up to within a normalizing factor, and R>..j = R>...i(c:). In view of the remarks in Section 3.3, as the vectors zC>..,j) c IR(a) one can take only the direction vectors of the various one-dimensional faces (both senior and minor) of N(P). If 'D>.. corresponds to a facer~) of dimension l > 0, then among inequalities (10) there must be such that differ in sign. In other words, V>.. C g(f~) , £)is determined by a system of inequalities i = 1, ... , I>..,
(11)
j = 1, . . . 'J>.,.
(11')
The vectors zC>..,i) correspond to the one-dimensional faces belonging to f~), and zC>..,i) correspond to the one-dimensional faces adjoining r~). 4) Since N(P) is Newton's polyhedron of a polynomial, it has only integral vertices. Therefore as the direction vectors of one-dimensional faces of N ( P) one can also take vectors with integral and, which is more, even components. Here, we can assume, without loss of generality, that the vectors z(>. ,j ) in (10) (or in (11)) h ave only even components.
§4. Differential operators of N-principal type with variable coefficients In this section we consider differential operators with coefficients belonging to
c=: P(x,D) = Laa(x)Da,
(1)
whose symbols P(:x; 0 are polynomials of N -principal type at every fixed point x E IRn . Under the additional assumption that the coefficients in senior monomials of the polynomial P( x ; 0 are real we shall prove a multidimensional analog of L 2 estimate ( 4.3.3) . A simple modification of the argument in §4.4 makes it possible to extend the estimate to the scales of HI-L and to prove an analog of estimate (4.4.3) and a local solvability theorem generalizing Theorem 4.4.1. We shall not dwell on these questions and leave the proof of the indicated theorems to the reader as an exerc1se.
Operators of Principal Type Associated. with Newton's Polyhedron
201
4.1. The statement of the basic result. As in §4.3, with a symbol P(x; 0 we associate the polyhedra N(P(x)), 5(P(x)), and 8(P(x)) at each point x E Rn and denote by N(P), 5(P), and 8(P) the convex hulls of the unions of the indicated polyhedra over x E Rn. Definition 4.3.1 is extended trivially to the case n > 2, i.e. operator (1) is called an operator of N -principal type if (i) N(P(x )) = N(P) Vx ERn; (ii) P(x 0 ;0 is a polynomial of .N-principal type Vx 0 ERn. Recall that condition (i) means, in particular, that if c/i)(x), j = 1, ... , J, are the senior vertices of N(P), then a 0 ci)(x) i=- 0, x ERn. Condition (ii) means, in particular, that the polyhedron N(P(x 0 )) is regul~r. By virtue of (i), the polyhedron N( P) also possesses this property. As in Chapter 4, we impose on the symbol P(x; 0 an additional condition, namely Condition (R). If an integral point a E N(P) is not minorl), then aa( x) is a real function. Theorem. Let (1) be an operator of N -principal type and let the additional condition (R) be fulfilled. Then Vc > 0 3w(c) such that in a region n of a sufficiently small diameter, diamn::::; w(c:), the inequality (cf. (4.3.3))
lluii6(P) ~ c:llP(x, D)ull Vu E V(f!) holds, where
lluliF 0 3w(c) such that Vj = 1, ... , J the inequality
(8) holds in a region n, diamn
< w(c), where t > n/2.
Remark. As in the remarks in Section 4.3.3, we assume that the coefficients of the operator p belong to c= and are uniformly bounded on ]Rn together with all their derivatives.
203
Operators of Principal Type Associated with Newton's Polyhedron
The reduction of Theorem 4 . 1 to the above theorem is carried out by means of an almost literal repetition of the argument in Section 4.3.3, and we do not dwell on this here. The proof of estimate ( 8) under condition ( 3) encounters no serious difficulties and is a simple modification of the argument in Section 4.3.4. Thus, the proof of the main Theorem 4.1 reduces to proving inequality (8) on condition that the symbol P( x, 0 satisfies inequality (4) in the region Uj. 4.3. The proof of estimate (8) under condition (4). To contract the notation we shall write '1/J instead of '1/Ji and also v = '1/J(D)u. The general scheme for the derivation of the estimate is in many respects analogous to the case of real operators of principal type (see Hormander (2] or Egorov [2]) and is based on an identity that will be derived below. Lemma 1. We have the identity n
'E IIP(j)(x,
n
D)vll 2
j=l
=
L Im[xjP(x, D)v, p(i)(x, D)v] j=l n
n
j=l
j=l
- L ImJxip(j)*(x, D)v, P*(x, D)v] + L Re[P(i)*(j)v, P*v] n
+ Llm[xjv,[PU),P*]v],
(9)
j=l
where pU) is the operator with symbol 8P(x; 0/Bej, p(i)* is the adjoint operator of pU), and the symbol p(i)*(j) is obtained by differentiating the symbol pU)* with respect to ei; [, ] designates the Hermitean scalar product in L 2 . Proof. For any differential operator we have P(x, D)(xjv) = XjP(x, D)v- iP(j)(x, D)v. Multiplying scalarly by p(j)(x; D)v and taking the imaginary part we obtain
IIP(i)(x, D)vjj 2
= Im[xjP(x, D)v, pU)(x, D)v] - Im[P(x,D)(xjv),P(i)(x,D)v].
(10) ~·
!,-·
Integrating by parts we transform the second term on the right-hand side in the following way:
- Im[P(xjv ), pU>v] = - lm[xjv, P* pU>v]
== -
+ Im(x jV, (P(j), P*]v] Im[x jpU>*v, P*v] + Re[P(i)*U>v, P*v] + Im[x jV, [P(j) , P*]v]. Im(P(j)*(x jV ), P*v]
204
Chapter 6
Substituting into (10) and performing the summation over j we arrive at identity
(9). We now estimate separately the various terms in identity (9). The most difficult problem is to estimate the last term on the right-hand side of (9). We begin with estimating the left-hand side. Lemma 2. Let v = '1/;(D)u, u E 'D(O), and let inequality(~) be fulfilled on the support of the function '1/J(O. Then \It > n/2 \/w, I n/2. Substituting these estimates into (12) we arrive at (11).
205
Operotors of Principal Type Associated with Newton's Polyhedron
Lemma 3. Let the conditions of Theorem 4.2 bold. Then Vc
> 0 3w(c) such
that in a region n, diamn::::; w(c), the right-hand side of identity (9) is estimated from above by means of
ciiP(x, D)vll 2 + Ccllvii}(P) + c(c)llull~-t)'
(14)
where t > n/2, v = 1/J(D)u, and the function 1/J satisfies conditions of the type (6) and (7). Comparing Lemmas 1, 2 and 3 and assuming that constant in (11)) we arrive at (8).
Cc < KI/2 (where
K 1 is the
The proof of the Lemma 3. 1) Estimation of the first term on the right-hand side of (9). As in the proof of Lemma 1' we take the truncating function x( X). We have
Im[xjP(x,D)v,P(j)(x,D)v] ( llpU>vllllxjPvll
( c3llvii6(P) llxxiPvll + c3llvllg(P) 11(1- x)xjP~(D)vii·
(15)
Take 280 , the diameter of the support of X, so that lx j XI ( c. Then the first term on the right-hand side of inequality (15) does not exceed (15') By virtue of the pseudo-locality of the operator Xj(1- x)Pl/J(D), the last term on the right-hand side of (15) does not exceed (15") Substituting (15') and (15") into (15) we estimate the left-hand side of (15) by means of expression ( 14). 2) Estimation of the second term on the right-hand side of (9). Repeating literally the above estimation we show that
If condition (R) 1 ) is fulfilled, then
P*(x, 0
= P(x, 0 + L P~:~(x, 0/a! = P(x, 0 + Q(x, e),
(17)
a>O
where Q is a polynomial symbol in Therefore
e, and we have N(Q(x; 0) c
IIP*vll ( IIPvll + const llvii6(P)" l)Note that up to now we have not used this condition.
8(P) for all
X.
(17')
Chapter 6
206
Substituting (17') into (16) we estimate the left-hand side of (16) by means of expr;o;ssion ( 14). 3) The estimation of the third term on the right-hand side of (9). By Schwarz' inequality, we have (18) The symbol of the operator pU)*(j) is a linear combination of the monomials corresponding to the minor points of the polyhedron N(PU>*) = N(PU>) C E{P). It follows that IIP(j)*(j)vll ~ c4 IID 11 vll. (19) fjE6(PCi))
L
By virtue of the regularity of the polyhedron b(P), Vc and V/3 E 8(P(j)) there is a constant c;1( c) such that
It follows that
Substituting these inequalities into (19) we see that (19') Substituting (19') into (18) and estimating IIP*vll by means of (17' ) we obtain
Re[P(j)*(j)v, P*v] ~ cscllvii6(P) I!Pvll
+ C7cjjvii}(P) + c6(c )IIPvllllull(-t) + cs( c)jlvii6(P) llull( -t) ~ ciiPvll 2 + c(c7 + c~/2 + 1)jjvii}(P) + (c - 1 c~(c) + c- 1 c~(c ))llu11(-t) ' which proves the desired estimate. . 4) Thus, to complete the proof of the lemma we have to establish the inequality
4.4. The proof of estimate {20). To begin with, we consider in detail the structure of the commutator of the operators p(i) and P*.
207
Operators of Principal Type Associated with Newton's Polyhedron
Lemma 1. The symbol Hj(x; 0 of the commutator [P(j), P*] can be represented as (21) Hj(x, 0 = Q(x, 0 + R(x, ~),
where the symbol Q can be written in the form
Q(x,O =
L
qap(x)~a+P.
(22)
aEli(P) f3Eli(P)
The symbol R is written
(23) where f3 E N(P) and the (integral) multiindices a have the form a=a I - aII - aIll ,
a I E N(P) . ,
a II ,aIll
> 0.
(23')
Proof. We first of all note that if the condition (R) is fulfilled, then
P*(x; D)= P(x; D)+ Q(x; D), where Q is a linear combination of the operators Da, a E [P(i), P*] = [pU), P]
~(P).
Therefore we have
+ [pU), Q].
The symbol of the commutator [pU), Q] has the form
I)p(j)('y)Q 0, we can use in the case the same
Lemma 3. Under the conditiolJS of Theorem 4.2 Vc: > 0 3w( c) such that in a region n, diam n < w( c), we lwve
(25) where the multiindices a and (3 are of the form of (23').
Proof. If the vector a+ {J can be represented as a sum a' +{3', a', {3' E 8( P), then the desired assertion reduces to the foregoing lemma. Therefore we shall consider the case when no such representation is possible and the ordinary integration by parts does not help. 1) Put
(26) It is clear that
h(0 ~ const36(P)(~)
V~ ERn.
(27)
On the other hand, according to ( 4), we have h( ~)
> const 36(P) ( 0
for
~ E supp 'lj;.
(27')
Operators of Principal Type Associated with Newton's Polyhedron
Let us show that there is p, 0 < p ~ 1, such that V'Y
209
> 0 we have (28)
To prove (28) we note that the regularity of the polyhedron 28(P) implies that there always exists p > 0 such that
With account of inequality (27'), we conclude that
It follows that the function h( 0 = .jh2{Jj satisfies the inequality ( cf. the proof of Lemma 2 in the Appendix to §1.4)
By virtue of (27), we obtain (28). 2) Take a function O
Comparing (28) and property (iii) of the function 0 and Vc > 0 we have
This inequality implies that
(34) Comparing again (28) and property (iii) of the function R, and therefore the first factor in (2) will be dropped. 2. As in Item 1 in Appendix to §4.3, we first verify inequality ( 4. 7) for (3 = 1 i.e. the inequality
(3) Its derivation is a literal repetition of the derivation of inequality (1) in the indicated Appendix. It follows from (3) that if there are w > 0 and p, 0 < p ~ 1, such that
(4) then (3) implies (4.6). 3. Consider a region Uj corresponding to a vertex r~o) not lying on the coordinate axe~. The normal cone of this vertex consists of positive vectors q E IR+, i.e. there is p such that
Since we have supp 1fi E Uj and the image of Uj under the logarithmic mapping ~ ~ (log 1~11, ... , log l~nl) is contained in the translation of V(~)' we have log ~~kl
> plog l~d -logwo
V~ E supp1{1j,
k, l = 1, ... , n.
(4')
Chapter 6
212
Inequalities (4') imply ( 4) and hence condition ( 4.6) for the functions 'lj;j. We now verify condition ( 4. 7). According to the hypothesis, we have. supp 'lj;i E Uj, where the region Uj possesses the property G(f~0 ),c:), which means (see Sec-
tion 2.3) that if a(A) is a coordinate of the vertex f~O), then
Hence, it suffices to verify ( 4. 7) for a = a(A). Since r~O) does not lie on the coordinate hyperplanes, all the coordinates ( a(A)l, ... , a(A)n) are positive, and consequently are no less than 1. Using inequality (3) for {3 = ({3 1 , ... , f3n), f31 ~ 1, we obtain By virtue of condition (4 }, the expression in the parenthesis does not exceed a constant. Multiplying both side of the inequality by le}:p..J I we obtain
4. We now consider the case when r~o) lies in a coordinate hyperplane of codimension n- m. Expand lR(o) as a direct sum: a= ({3, 1), {3 E IR(.B)' 1 E IR(1)m, and assume, to simplify the notation, that if a= (a1, ... , an), then {3 =(all ... , am) and 1 = ( am+l, ... , an)· Accordingly, the variables are divided into two groups:
e
e= (ry,(), "7 E Rm, ( E Rn-m.
Thus, let the vertex r~O) belong to the plane {am+l = ... = an = 0} and not belong to a coordinate plane of a higher dimension. Let us divide the onedimensional faces passing through f~O) into two groups, namely
r~l)' f1 = 1, ... '/11' belonging to {am+l = .. . =an = 0};
(5)
r~l)' f1 = /11 + 1, ... '/12, transversal to {am+l = ... =an= 0}.
(5')
In accordance with this division of one-dimensional faces we write: J.l.l
'1/Jjco =
'I/J]l)co'I/J)2 )co
=
II c1ejl' 1- RilL) II c1eil' 1- RilL).
(6)
iL=l
We note that the function 'lj;Y) depends only on the variables 1J (and does noi depend on the variables(). Denote by Nm the section of N(P) by the coordinate plane {am+l = · · · = an = 0}. Obviously, Nm is a regular polyhedron in Rm and r~o) is a senior vertex of that polyhedron not lying in the coordinate planes. Therefore the argument in Item 3 implies that
ITJd > w1l"lk IP 1 ,
1]
E supp 'lj;J 1),
l, k
= 1, ... , m,
0
. ), 0) be the coordinates of the vertex
r~o) and let (0, 'Y(J.t)), J1 = Jll + 1, ... ~ Jll, be the coordinates of the intersections of the straight lines passing through the one-dimensional faces (5') with the subspace {a1 =···=am= 0}. Denote by Nn-m the convex hull of 'Y(J.t)' J1 = J.11 + 1, ... , J.12, and the origin. The polyhedron Nn-m is the section of the regular polyhedron T+(f~)) (see (2.6)) by the plane {a1 =···=am= 0}. It is clear that Nn-m is a regular polyhedron in Rn-m. For an appropriate normalization of the direction vectors [J.t of the one-dimensional faces (5') we have J1 = Jll
+ 1, ... ,f.12·
Consequently, the function ~J2 )(0 in (6) has the form
~?\ry,() =
ll2
IT
8(ry2Pp..)(-21(1')- RJ.tj)·
(8)
J.t=J.tl+l
It follows from the definition of the function for
( TJ,
Among the vertices 'Y(J.t) there are vertices coordinate axes. Hence, (9) implies
(J
that (2)
( ) E supp'lj;j , 'Y(J.tt)
= 1"1e1, l
1= m
Jll
0 and \18 > 0 there is lo(M, 5) such that for lsi ~ M and { l < 12 < 1o(M,5) tbe inequalities llull(s),"Y2,"Yl ~ Ksi1Pull(s),"Y2 "Yl'
(25)
1
II vii( -s),-"Yl ,-"Y2
~
K; II Pull< -s),-"Yl ,-"Y2
(25')
bold. In view of the duality, inequality (25') implies the existence of a solution belonging to H[(s) "Yl, "Y2] . Therefore the assertion of the theorem follows from what has been said 1n 4). 6) The proof of the lemma. To simplify the notation, we confine ourselves to the proof of (25) (inequality (25') is proved in a similar way). By definition, for 12 > 1 1 we have By Leibniz's formula, m
xPu = P(xu)-
L Xlp(l)u,
Xl =
D!x/a!
l=l
On writing down an analogous relation for (1- x)Pu we obtain
m
1=1 m
~I: [(,o- ll) 1IIP(l)(xu)ll<s)m + (l'o -1'2)'11p(l)(l- x)ull<s),"'f2 1=1
(26) We have
Using ,the fact that Xt(t), l ~ 1, is a function of compact, suppo~t we show that IIXlp(l)(l- x)ull(s),"Yl
Comparing (27) and (28) we find
~ c(s,/'2- ldiiP(l)(l- x )ull(s),"Y2'
(28)
223
The Method of Energy Estimates in Cauchy's Problem
It is shown in a similar way that llxzp(l)ull(s),"Y2 is also estimated by means of the right-hand side of (29). Substituting these inequalities into (27) and taking a sufficiently large -"(2 < -"(I we estimate the right-hand side of (26) from below by means of m
~I: [('Yo- 'YI) 1IIP(l)(xu)ll<s) ,"Yl
+('Yo- 'Y2) 1IIP(l)(l- x)ull(s) ,"Y2]
l=I
~
ml m! -ic 'Yo- 'Y2)m [llxull<s),"Yl + 11c1- x)ull<s),"Y2J = 2 c 'Yo- 'Y2)mllull<s),"Y2,"Yl,
i.e. we arrive at inequality (25).
7) Thus, to complete the proof of the theorem we have to establish (28). We write
llxtp(l)(l - x)ull(s),"Yl
def
II exp( 'YI t)As(Dx, Dt)Xlp(l) (1 - x)ull
= ll.\s(Dx, Dt + i"(I) exp( "(It)xzP(l)(1- x)ull
As(Dx, Dt
+ i'YI)
= II As(Dx, Dt +
( . )[ ( ) ) i'Y20 As Dx, Dt + Z"f2 exp 'YI t- 'Y2t XI
(30)
x exp( "(2t)P(l)(l- x)ull· By virtue of the elementary inequality l.\s(e,a + i'YI).\;I(e,a + i'Y2)I :s;; (1 + b2- 'YIIY,
the expression A8 (Dx, Dt + i"(I)A_ 8 (Dx, Dt + i'Y2) is a bounded operator in L2, and therefore the right-hand side of (30) is estimated from above by means of c('Y2- 'YI)II.\s(Dx,Dt + i"(2)[exp('Y2t- 'Y!t)xz]exp("f2t)P(l)(l- x)ull·
(31)
The expression in square brackets is a function of compact support, and its derivatives are estimated by means of constants depending on the difference "(2 - "(1 solely. It can be shown that (31) is estimated by means of
c' ('Y2- 'YI )liAs( Dx, Dt + iJ2) exp( 'Y2t)P(l) (1- x)ull
def
c' ({2 -{I )llp(l) (1- x)ull (s),"Y2
(this inequality is proved particularly simply in the case of integral values of s
~
0 ).
§2. Sufficient conditions for the existence of energy estimates In this section we present a set of rather cumbersome conditions making it possible to estimate from above and below the forms
- Im[Q(y; D)u, Q( 1 )(y; D)u](s) ,"Y'
Q = P, P*,
(1)
by means of analogous for.m s corresponding to an operator Q with constant coefficients (frozen at a point y = y 0 ). From the estimate for forms ( 1) we derive inequalities (1.3) and (1.3') and thus establish the solvability of Cauchy's problem. As consequences of the conditions in the present section, we shall obtain in §3 some easily verifiable conditions guaranteeing the solvability of Cauchy's problem. The results in this section are taken from the paper by Volevich [1].
Chapter 7
224
2.1. Formulation of the main results. We shall deal with differential operators P(y; D), y = (t, x) E JRn+I, solved with respect to the highest derivative with respect tot, i.e. the symbol P(y; ~' 7) has the form
P(y;~,7)
= 7m + LP;(Y;07m-j.
(2)
j>O
As in Chapters 2 and 3, we shall assume, without a special stipulation, that the coefficients of the polynomial symbol (2) belong to c= and do not depend on y for large IYI· It will also be assumed that conditions (I) and (II) in the foregoing section are fulfilled. When estimating the forms ( 1) we shall use the norm
{u}(s),-y=
(jj Hp(yo;~, 0,
(7) (8)
(9)
where cp(/), cap(!)~ 0 as 1 ~ -oo. Then VM > 0 3/o(M) such that Vs E R, lsi~ M, and 1 ~ !o(M) the two-sided estimate
c:;- 1{ u Hs),-y holds.
~ - Im +[P(y; D)u, p(l)(y; D)u](s),-y ~ c8 { u Hs) ,-y
Vu E
H[\J)
(10)
225
The Method of Energy Estimates in Cauchy's Problem
Proposition 2. Let conditions (I) and (II) in §1 be fulfilled, let condition (7) bold, and, besides, let
I{P,Pca)}(P)(y;C0",/)1 < cp(!)Hp(y;(,O",/), I{P,P(a+f3)}(f3)(y;~,0" 1 /)I
f3 > o,
(11)
< c 0 p(!)Hp(y;~,0", /) (1 + l ~l)lal, f3 ~ 0, a > 0.
Tben VM > 0 3/~(M) such that \Is E llt, estimate
lsi ~
M, and 1 ~ !~(M) the two-sided
c:-l {v }(-s),--y ~ Im -[P*(y; D)v, p*(l)(y; D)v]( -s),--y ~ c:{v }( -s),--y Vv E
(12)
1
H(oo)
[--y]
(13)
bolds. Remark. Formally, (9) and (12) are infinite sets of conditions. However, if
lal >
x + deg H p + 1, where xis the constant in Corollary 1.2, then inequalities (9) and (12) are fulfilled automatically.
2.2. Theorem. Let symbol (2) satisfy all conditions of Propositions 1 and 2 in the foregoing section. Then inequalities (1.3) and (1.3') take place. Proof. If conditions (I) and (II) hold, then, by virtue of Proposition 1.2 (II), Vy' E JRn+I we can write the inequality m
c: L)
/O -
!) 21 - 1 IIP(l)(y' i D)ullts),-y ~ {U }(s),-y 1
(14)
1=1
where the constant /o can be selected so that inequalities (14) are fulfilled for any y' E JRn+I and 'Y ~ lo· These inequalities readily imply an analogous inequality for the operator with variable coefficients: m
c~
L)/o -
!) 21 -lllp(l)(y; D)ull(s),-y ~ { u }(s),-y·
(15)
l=l
Indeed, since t4e degrees of all polynomials P(y ; ~' T) are uniformly bounded with respect to y E Jltn+I, they form a finite-dimensional space, and a~ong them there are a finite number of linearly independent elements. Therefore we can write J
P(y;e.,T) = "Lcj(y)P(yi;e,T), j=l
(16)
Chapter 7
226
where the functions Cj(Y) possess the same smoothness properties as the coefficients of the original polynomial (2). Differentiating (16) with respect to T we obtain J
p(l>(y; e, T) =
2: Cj(y)P(yi; e, T).
(16')
j=1
Applying inequality (14) with y' = yi to each of the operators on the right-hand side of (16') we obtain (15). Comparing (15) and (10) we find ( cf. Section 1.3) m
c~ I:Cio- !)21 - 1 IIP( 1)ull(s),-y ~- Im +[!u, p( 1 )u)(s),-y ~ IIPullcs),-yiiP(l)ull(s) ,-y 1=1
~ ( lo -
m
!)- 112 11Pullcs),-y (
L(lo -
1/2
!)21 - 1 IIP( 1)ull(s),-y)
,
1=1
whence follows inequality (1.3). Similarly, to prove (1.3') it suffices to show that m
c~ * I:Cio- 1?1- 1 IIP*(l)v11(-s),--y ~ {v H-s),- -y·
(15')
l-1 According to (16), we have
1-"'
1
. -"'-(,B) . -(,B) i. P * (y,e,T)LtP(p)(y,e,T)/(3.Lt (3!D ,B c1.(y)P (y ,e,r).
Hence, to prove (15') it suffices to establish the inequalities m
const
L(!o -
1)21 -
L
~
{ v }(-s),--y,
(y';e,<J +i!) l ~ Hp(y';e,<J,!),
I::::; lo·
1
IIP(l)(,B) (yi; D)vll(-s),--y
,8~0
l=l
which are equivalent to the set of inequalities 21
1
,a·-( l)
const(!o- 1) - 18 P We put 17
= (e,
(J ).
(17)
For any polynomial Q( 17) we have the inequality c1Q(,B)(7J)I ~ sup IQ(1J + B)l,
(18)
181~1
where the constant c depends only on the degree of Q and the dimension of the space. In view of inequality (18) (applied to the polynomial Q(17) = pU>(y'; (J + i1)), we have
e,
c1(1o -!) 21 - 1IP( 1)(,B)(y';e,<J+if)l 2
::::;
sup Hp(y';1J+B,,). IBI~l
227
The Method of Energy Estimates in Cauchy's Problem
We now show that if -1 is sufficiently large, then
Indeed, exp anding the left-hand side by Taylor's formula we obtain
It now remains to apply condition (7). 2.3. The plan of the proof of the propositions in Section 2.1. We set w = exp( lt).A't(D)u. If u runs over H[\]J, then the function w runs over H~=). Setting D 1 = (Dx, Dt + i1) we can rewrite the middle term in (10) in the form
- Im[.A~(D 1 )P(y; D-r ).A~s(D, )w, .A~(D-r )P(l) (y; D, ).A~s(D, )w] =
(~.A: 8 (D- 1 ){ p(I)*(y; D_ 1 ).A~(D,).A;(D- 1 )P(y; D 1 )
(19)
- P*(y; D_-y).A~(D-y).A;(D_ 1 )P< 1 )(y; D 1 )} .A~s(D,)w, w). Here we used the fact that
If the coefficients of P were constant ( cf. Section 1.3), we would obtain the quadratic form (Hp(D,1)w,w). In the case of variable coefficients we separate out the Hermitian form corresponding to the differential operator H p(y; D, 1), i.e. rewrite ( 19) in the form Re(Hp(y;D,1)w,w) + (Q 81 w,w), (20)
where
Qs 1 = ~.A:s(D--r){ p(I)*(y; D_,)Xt(D, )X;(D- 1 )P(y; D 1 ) - P*(y; D_ 1 ).A~(D-y).A;(D--r)P< 1 )(y; D 1 ) }.A~ 8 (D 1 ) -
~Hp(y;D,1)- ~Hf,(y;D,1).
(21)
We shall prove the following inequalities:
c-; 1 (Hp(y 0 ;D,1)w,w ) ~ Re(Hp(y; D,1)w,w) ~ c1 (Hp(y 0 ,D)w,w) I(Qs 1 w,w)l ~ c(I)(Hp(y 0 ;D,I)w,w),
c(l)----+ 0,
I-+ -oo.
(22) (23)
Chapter 7
228
Noting that
we obtain the proof of Proposition 1 in Section 2.1. We note that (22) is an analog of Garding's inequality for inhomogeneous quadratic forms. As to Proposition 2 in Section 2.1, after the substitution of
z = exp( -!t)>..;(D)v the middle term in (13) can be rewritten as
Re(Hp(y;D,1)z,z) + (Rs'Yz,z),
(24)
where
Rs"( =
;i)...~s(D'Y){ p(l)(y; D'Y)>..~s(D'Y)>..;(D_'Y)P*(y; D_'Y) - P(y; D'Y )>..~s(D'Y )>..=s(D-"( )P( 1)* ( y; D_'Y)} >..; (D_'Y) -
~Hp(y; D, !) - ~Hp(y; D, 1).
(25)
Proposition 2 in Section 2.1 follows from (22) and the inequality
2.4. The proof of Inequality (23}. If Q1(y; D) and Q 2 (y; D) are two PDO's then for any natural N we set
This relation is usually called the commutation formula. Setting
we write
(28)
The Method of Energy Estimates in Cauchy's Problem
229
The expression under the summation sign in (28) is a PDO. To calculate its symbol we note that if Q 1 and Q 2 are two differential operators, then, in view of Leibnitz's formula, the symbol of Qi Q2 is equal to
Therefore the symbol of the PDO under the summation sign in (28) is equal to
App1ying the commutation formula once again we write operator (21) in the form
Q S"f
= Q s"{N + Ts"{N,
(29)
where Q s'YN is a PDO with symbol """""
1
~ od ,8! 8!
{P.
(a)'
p(f3)}
(f3+0)
~(a),\-(0),\+ (. 2s -s -s- Hp y, 1], I
)
-
a,{3,0
1""""" ({3) ( . 2 ~ H P(f3) y, 1], I
)
.
{3>0
(30) Noting that {P, P} = Hp we rewrite the symbol in the form
1 {P. ({3)} (a),\-(0)+ 1"' ({3) a! ,8! 8! (a)l p (f3+8)~2s -s ,\_s- 2 ~ H P(f3)•
""""" ~ la+f3+8I>O
(30')
{3>0
The operator Ts'YN is written
~,\=sCp(l) RN(~2s, P)- P* RN(~2s, p(l)Y~s)
+
1 R (,\- {P. p(f3)} )~(a),\-(0) + a! ,8! N -Sl (a)' (f3+0) 2s -s ,\_s•
"""""
~
(31)
!a+f3+0I>O
Lemma 1. The symbol of the operator Q s'YN is represented a; Qs'YN(Y;1J) = Lci(y)bj(1J,I), j=l
where
(32)
Chapter 7
230
Proof. We now show that, by virtue of Proposition 1 in Section 1.2, we have
1Qs 1 N(Y;TJ)I < e(!)Hp(y 0 ;TJ,/),
e(!)--+ 0,
1--+ -oo.
(34)
Writing the symbol P(y; ~' r) in the form of (16) we arrive at representation (32). The inequalities
IH~~)(y; 7], !)I < ep( !)Hp(y 0 ; TJ, !) follow from (7) and the condition of constant strength. We have to estimate the first sum in (30'). As can easily be seen, for 1 < /o we have
~~~~\TJ,/).A::::~8 )(TJ, !)X~s(TJ,!)I ~ I 0 there is a
IR(1Jj,lj)Q-1(1Jj,lj)l ~c. We set aj = R(1Jj,''lj)/Q(1Jj,lj)· Then !ail < c-I, and, by virtue of (ii), the sequence lo( aj) is bounded from below by a constant lo. Since lj --+ -oo, we have 1 i < 1o for sufficiently large j, and we hence
lj < IO· The resulting contradiction proves the lemma. Lemma 2. Let polynomials Pj(~, T ), j = 1, 2, satisfy Petrovskil's condition, that is ::!10 such that ·
_..,. lo, I rnT::::::::
.,t E
l!l>n, ~
(8)
Then
Proof. If T1k(0 and T2k(0 are the roots of the polynomials P1 and P 2 , then replacing P1 and P2 by their factorizations we derive
The Method of Energy Estimates in Cauchy's Problem
239
If Im7 ~ lo, then, according to (8), we have Pt(e,1)P2(e,1) =I 0. Further, since Im( 7- 7j k( e)) < 0, j = 1, 2, the two expressions in the square brackets have positive imaginary parts, and consequently ( 9) holds. If two functions Q( 1], 1) and R( 1], 1) are related by conditions of Lemma 1, we shall write R -< Q. The same notation will be retained for Q( 1) and R( 1) regarded as functions of the variables 1J = ( Re 7) and 1 = Im 7.
e,
e,
e,
Lemma 3. If P is a polynomial correct in Petrovski'l's sense and R( P(e, 1 ), then {R,P}(e,a,l)-< Hp(e,a,l)·
e, 7)
-
0, and Q 1 (~,r) =
P(y" ;Cr) we obtain (7). To prove (3) we write Pin the form n
P
= Tm + L
~jPj +Po,
N(Pj) C 8°(P),
j
= 0, ... , n.
j=l
Differentiating this relation with respect to y (it is this place where use is made of the fact that the coefficient in the highest power of r :is identically equal to a constant) we find
I{ P, P(o:)} I =
I_L ~i {P,
Pj(o:)}
+ {P, Po(o:)}
~ ca(!)Hp(y;~,a,!)(l
+
1~1),
I c(!) ~ 0,
'Y ~ -oo.
Chapter 7
242
3.4. Strictly pluriparabolic differential operators. Here we shall present a class of differential operators whose symbols satisfy the conditions of Theorem B. These operators include as special cases the strictly hyperbolic and q-parabolic operators. We first give the definition and description of these operators for the case of constant coefficients, i.e. for polynomials. R~present the space Rn+ 1 as a direct sum of the subspace Rk of the variables o- = ( o- 1, ... , O"k) and the subspace R 1 of the variables ( = ( (1 , . .. , (l ), l + k = n + 1. We separate out the variable O"t, and let o-1 = (o-2, ... , o-k)i in Cauchy's problem o-1 plays the role of a variable dual to time. Let q = 2b be an even positive integer. In what follows we shall assign the weights q and I to the variables o- and (, respectively.
Definition 1. A (q, ... ., q, I, ... , I)-homogeneous polynomial P0 (a, 0 is said to be strictly pluriparabolic (see Gindikin [2), and Volevich and Gindikin [5]) if (i) the polynomial P0 ( o-, 0) is strictly hyperbolic; (ii) there is .X > 0 such that ImToj(o-',0 ~ .XI(Iq,
j = I, ... ,m,
where Toj( o- 1 , 0 are the roots of the polynomial P with respect to o-1.
Definition 2. A polynomial P( o-, 0 is said to be strictly pluriparabolic if its principal (q, ... , q, I, ... , I)-homogeneous part possesses this property. The strict hyperbolicity of P( o-, 0) implies that this polynomial and, conse~ quently, the polynomial P(o-, 0 as well can be solved with respect to the highest power o-i (the coefficient in o-i is assumed to be equal to I):
P(o-, ()=o-r+ Laix.ao-~-ia'"(.B .
(II)
j~1
Proposition. Let q > 0 be even. Then for polynomial (II) the following conditions are equivalent: (I) polynomial (11) is strictly pluriparabolic; (II) there are ro and c > 0 such that for r:::;; ro we have
where the notation ry = ( o-', 0
Hp((,a,r) = -Im(P(o-1 +ir,o-',08P(o-1 + ir,o-',()/8a 1 )
(I3)
is used; (III) there are ro and c1 > 0 such that
(I4)
243
The Method of Energy Estimates in Cauchy's Problem
Proof. (!)===}(II). 1) We first assume that Pis a (q, . .. , q, I, ... , I)-homogeneous polynomial, and let roj(a',(), j =I, ... ,m, be its roots. According to (1.11), for 1 < 0 we have m
2:C -~+1m rok( a',()) II Iat + i1- rok( a', ~)1 2
Hp( (,a, 1) =
k=l
~
j#k
(lfl + .\j(jq)H( (,a, 'Y),
where n
H( (,a, 1) =
2: II [( a1 k=l
Re rok( a', ~)) 2
+ (-~ + Im rok( a', 0) 2 ]
(I5)
i#k
is a ( q, ... , q, I, ... , I)-homogeneous function of degree 2( m - I )q. To prove ( I2) in the quasi-homogeneous case it suffices to show that (I6) Since -1 + lm rok( a',() ~ Ill+ .\l(lq, it suffices to verify (I6) only for 1 = 0, ( = 0 (i.e. for the strictly hyperbolic polynomial P( a, 0) ). Since the roots rok( a', 0) are real and are distinct for ja'l # 0, one of the numbers a1- Rerok(a',O) is nonzero, whence follows (16). In case u' = 0, we have
2) Now let P(a, () = Po(a, () + Q( a,(), where Po is a (q, ... , q, I , .. . , I)-homogeneous polynomial and the (q, .. . , q, 1, ... , I )-degree of Q does not exceed mq - I. By v:hat was proved, for P = P0 inequality (I2) has already been proved. To prove it in the general case we show that
To prove ( 17) we note that H p - H p 0 = {Po, Q} + {Q, P 0 } + {Q, Q} is a polynomial in ~, a, and 1 of ( q, 1, ... , I )-degree no higher than 2mq - q - I, i.e. is a linear combination of monomials of the form of
These monomials can be represented as 'the expressions
which are obviously estimated by means of the right-hand side of (17) with c(1) = const 111-l/q.
Chapter 7
244
(II)=?(III). Since 8Pf8a 1 is a polynomial of (q, ... , q, I, ... , I)-degree no higher than (m- I)q, we have
whence for large -1 follows (14). (III)=?(I). Inequality (14) for the polynomial P implies an analogous inequality with 'Yo = 0 for its (q, ... ,q, I, ... , I)-homogeneous part. As was already done many times, to show this one should replace (a,() by (tqa, t() and pass to the limit for t ~ +oo. In what follows we assume that the polynomial P is (q, I, ... , I)homogeneous. If we set ( = 0 in (14), this results in
whence it follows that P( a, 0) is strictly hyperbolic. It now remains to verify condition (ii) in Definition 1. In view of the quasihomogeneity, it suffices to show that there is >. > 0 such that
P(a1
+ i1,a',() =I- 0
for
I~>.,
1(1 =I,
a E IRk.
(18)
By virtue of (I4), we have to consider only the case I ;;:::: 0. Setting I = 0 in (I4) and assuming that 1(1 = I we find
whence
IP(a1
Taki>.1g >.
+ i1, a', ()I > IP(a, ()1-IP(ai + i1, a',()- P(a, ()I ;;::: CJ (1 +a )m-1 - C2"!((1 + lal)m-1 + 'Ym-1 ].
< cJ/4c 2 and
1 ~
>.
~ I we obtain (I8).
Remark 1. Let P( a,() be a strictly pluriparabolic polynomial and let Toj( a',() be the roots of its principal (q, ... , q, I, ... , I)-homogeneous part. Then, by virtue of condition (i), there is b > 0 such that
!Toj(a',O)- Tok(a',O)I > bla'l,
j =I- k.
(19)
A careful examination of the proof of the proposition shows that the constant c in (12) depends on b (this follows form (19)), >. (by the condition (II)), and the maximum of the moduli of the coefficients of the polynomial P. We now consider a symbol P(y; ~' () with smooth stabilized coefficients, solved with respect to a1 (the highest power of a!), the coefficient in a! being equal to 1.
245
The Method of Energy Estimates in Cauchy's Problem
Definition 3. A symbol P(y; ~' () is said to be strictly pluriparabolic if the polynomial P(y 0 ; a,() is strictly pluriparabolic for each y0 and, moreover, the roots Toj(y'; a',() of the principal ( q, ... , q, 1, ... , 1)-homogeneous part of P satisfy for some ..\, 8 > 0 the inequalities
lroj(y;a',O ) - Tok(y;a',O)I > 8ja'j, Im Toj(y; a',() > ..\j(jq ·
j
=/=-
k,
(20) (21)
Theorem. A symbol P satisfying the conditions of Definition 3 satisfies conditions of Theorem B. Proof. With account of Remark 1, inequality (12) holds for P = P(y; a,() with a unified constant c. It follows that the condition of constant strength holds for Hp. Condition (1.15) is a direct consequence of inequality (14) for P = P(y; a,() (recall that, by virtue of Remark 1, the constant c 1 in (14) does not depend on y ). To prove (3) we note that the symbol P(a), a > 0, does not contain the highest power of a 1 and is represented as k
P(a)(y;a,() = 'LajPaj(y;a,() j=l
l
+ L(jPai(y;a,z) + Pao, i=l
where the ( q, 1, ... , 1 )-degrees of the symbols Paj, Pai, and Pao do not exceed (m- 1)q, mq- 1, and mq- 1, respectively. Relation (3) now follows immediately from (14). Remark 2. Applying the argument used in the proof of the proposition one can easily show that the symbol in Definition 3 satisfies the conditions of Theorem 3.3. Remark 9. In case of pluriparabolic operators the method in §2 can be specified to obtain energy estimates in norms that take into account the quasi-homogeneity of the principal part of the operator, and a rather accurate result on the smoothness of the solution to Cauchy's problem for these equations (see Volevich and Gindikin [5]).
3.5. Remarks on Cauchy's problem in spaces of increasing and decreasing functions. It was noted in Section 2.5. 7 that for exponentially correct symbols of constant strength we have estimates in the spaces H((s)) . Similar estiu ,"'{ mates also take place under the condition ofTheorem B. Moreover, it is possible to generalize the Propositions 1 '.a nd 2 in Section 2.1 to the case of the II II~:)),"Y n 0 and c(1') ~ 0 as 1 ~ -oo. In view of this, the second term on the right-hand side of (:26) can be estimate from above by means of c( 1 ){ v )2, whence follows inequality (24). The theorem is proved. Remark. Under the conditions of Theorem B it is possible to derive estimates in norms involving exponentially increasing (decreasing) weights. For more detail see Volevich [1] .
The Method of Energy Estimates in Cauchy's Problem
247
§4. Cauchy's problem for dominantly correct differential operators 4.1. In this section we shall prove that a dominantly correct symbol P(y; ~' r) solved with respect to the highest power of r:
P(y; ~' r) =
Tm
+L
Pa 1 .•• anf3(Y)~~ 1
• ••
~~nrf3
(1)
f3<m
satisfies all conditions of Theorem B, and hence Cauchy's problem is uniquely solvable for the corresponding differential operator P(y; D). Recall that, according to the definitions in Section 3.4.2, symbol (1) is said to be dominantly correct if the following conditions hold: (i) the polygons 8(P(y)) do not depend on y; (ii) Vy 0 E Rn+l the polynomial P(~,r) = P(y 0 ;~,r) is dominantly correct. For symbols satisfying (i) and (ii) we shall establish a strengthened version of Theorem 3.3, in which the conditions of the theorem are supplemented with a condition of "equivalence" of the variables 6, ... , ~n· We state the necessary definitions. Consider a symbol
(2) We denote by tl.(H(P(y))) the polyhedron in R 3 spanned on the triples (Ia I, /3, r) for which ha 1 ... anf3r(Y) =/= 0 for a1 +···+an = Ia I and on their projections on the coordinate axes. As usual, let fl.( H p) denote the convex hull of the union of all f'l.(Hp(y)) over y E Rn+I. We have
Theorem C. For a dominantly correct symbol (1) the following assertions bold: (a) f'l.(Hp(y)) = tl.(Hp) Vy E Rn+I; (b) :3/o, c > 0 such that tl1e estimate from below
Hp(y; 7J,/) > c
(3) (a,f3,r)Efl(Hp)
balds; (c) Vy E Rn+I and for any polynomials Q 1 (~,r), Q 2 (~,r), f'l.(Q 1 ) C f'l.(P),l) f'l.(Q2) C 8(P), there .is a function c:(1) such that c:(1)-+ 0 as 1-+ -oo and
(4) It is clear that symbol (1) satisfying the ccmditions (a), (b), and (c) satisfies the conditions of Theorem 3.3 and, consequently, the conditions of Theorem B. The proof of Theorem C is based on an equivalent description of dominantly correct polynomials in terms of the functions H p, and it is this question that is treated in the present section. l ) For
the notation 6.(P) see the Introduction to Chapter 2.
Chapter 7
248
4.2. The description of dominantly correct and stable-correct polynomials in terms of Hp. We have Theorem 1. For a polynomial P(~, T) solved with respect to the highest power ofT the following conditions are equivalent.
(I)
P(~, T)
is a dominantly correct polynomial (i.e. the equivalent conditions of Theorem 3.4.1 lwld for it). (II) The following conditions are fulfilled: (a) the polyhedron ~(Hp) C IR3 is reconstructed uniquely from the polygon h(P). (b) 3/o, c > 0 such that the estimate from below ( cf. ( 3))
(5) (a,fi,r)ED..(Hp)
holds; (c) For any polynomials Q 1 (~,T), Q2(~,T), ~(Ql) C ~(P), ~(Q2) C h(P), there is a function c(/), c('Y)--+ 0 as{--+ -oo, such that
(6)
The implication (Il)==?(I) is an immediate consequence of Theorem 3.4.1. Indeed, (5) implies that the polynomial P is correct in Petrovskil's sense. Further, let Q( ~, T) be a polynomial and let ~( Q) C h( P). In view of the relation
HP+Q = {P
+ Q,P + Q} =
Hp
+ {P,Q} + {Q,P} + {Q,Q}
(7)
and condition (c), there is 1( Q) such that
Hence, the polynomial P + Q is also correct in Petrovskil's sense, whence it follows that the original polynomial is dominantly correct. The proof of the implication (I)==?(II) is rather cumbersome and occupies the entire remaining part of this section. As in the case of Theorem 3.4.1, the central point here is the proof of the corresponding assertion for the case n = 1, while the proof for the case of n > 1 is obtained from the former by passing to polar coordinates ( ~ = pv.;). In the course of the proof of Theorem 1 we shall obtain an analogous description for the stable-correct polynomials as well. We shall prove
The Method of Energy Estimates in Cauchy's Problem
249
Theorem 2. For a polynomial P(C 7) solved with respect to the highest power of 7 the conditions below are equivalent.
(I) P(e, 7) is a stable-correct polynomial (i.e. it satisfies the equivalent conditions of Theorem 2.4.5; also see Theorem 3.4.3). (II) Conditions (a) and(b) of Theorem 1 are fulfilled as well as the strengthened version of condition (e): (cmax) VQi(e,7),~(Qi) C ~(P), i = 1,2, 3/'I ')' ~ /'I ·
(8)
Condition (cmax) and relation (7) imply that all polynomials P + c:Q, where ~(Q) C D.(P), are correct in Petrovski!'s sense for sufficiently small c. Therefore the polynomial P is stable-correct, i.e. the implication (II)===?(I) has been proved. Relations (c) and ( Cmax) are purely geometric conditions. The second of them is equivalent to the condition ( c~ax) if D.( Qi) C ~(P), i = 1, 2, then
D.({Q1,Q2})
C D.({P,P}).
As to the first of these conditions, to investigate it we need the following Definition. A point (a, /3, r) E D.(Hp) is said to be minor if there is a point (a', f3', r') E D.(H p) such that a ~ a', f3 ~ f3', and r < r'. The convex hull of the minor (integral) points of D.(Hp) will be denoted 8(Hp ). A geometrical equivalent of condition (c) is the condition (c') If Q1 and Q2 are polynomials and ~(Qt) C D.(P), ~(Q 2 ) C D.(P), then
The equivalence of (c) and ( c') follows from a simple lemma that will be presented below. Let Q(zo, z1, ... , zk) be a polynomial in k+1 variables with Newton's polyhedron N( Q) . A point (ao, ... , ak) E N( Q) is said to be minor if 3( a~, ... , ak) E N( Q), ao < a~, CXj ~ aj, j = 1 . .. , k. The points of N(Q) that are not minor are called senior. The set of the senior points will be denoted as 1r N( Q). The senior points belong to those faces of N(Q) which do not lie in coordinate planes and do not cont ain segments of straight lines parallel to the axis {a 0 }. Lemma. The condition
lzal ~ ca(zo)
L
lzPI,
ca(zo) ~ 0
;3EN(Q)
is fulfilled if and only if a E
N(Q) \ 1rN(Q).
as
lzo l ~ oo,
(9)
Chapter 7
250
Proof. Sufficiency. If a~ 1rN(Q), then a straight line parallel to the axis {ao} can be drawn through the point a, and let a = ( ao, ... , a k), ao > ao, be the point of intersection of this line with the boundary of the polyhedron N( Q). Then we have lzal/3(z):::::; const jzal/lzal = const lzolao-ao. Necessity. A point a E N(Q) \ 1rN(Q) is characterized by the property that no supporting plane {q, {J} = c > 0, q = eqo, . .. , qk), qo > 0, lq1l + · · · + lqk I > 0, can be drawn through it. It now remains to note that if such plane passes through a, then e9) cannot hold. Indeed, if there is q = eqo, ... , qk), qo > 0, such that
(q, a}
~
(q, {J} V{J
E
Ne Q),
then condition e9) is violated along the curve Zjet) = tqi, j = 0, ... , k. 4.3. The general scheme of the proof of the Theorems in Section 4.2.
As has already been mentioned, the most laborious part of the proof of the assertions stated in Section 4.2 is their verification for n = 1. According to Theorem 3.2.3, a dominantly correct polynomial Pee, r), e E IR, has the form
Pee,r) = .Pee,r) + Q(e,r),
NeQ) c h(P),
e1o) e11)
.Pee, r) = rb Ree, r)cee, r), where p.
R(e,r) =ITer- aiek),
Imai > 0,
bk are even numbers,
e12)
k=l
h
cee,r) =ITer -cje),
Imc·0 J- '
for
j
#
k.
e13)
k=l
It is obvious that if all assertions in Theorem 1 are proved for the polynomial
P,
then they will also be true for polynomial (10) with any Q, and the relation
N(Hp)
= N(H?)
(14)
will hold. Theorem 2 in Section 4 . 2 with n = 1 corresponds to the simpler case G Thus, we shall prove the following assertions. Proposition 1. Let a polynomial c > 0 such that
P h~ve
the form en)-e13). Then :3/o and
I:::::; /o,
HpefJ,!)>c
=1. e15)
(a,f3,r)EN(H j>)
where the constants /o and c in e15) depend on max lai I, max lei I, max elm aj) - l , ~ax lei - Ck l- 1 eon b and the numbers b1 , ••• , bk) solely. rf.k
251
The Method of Energy Estimates in Cauchy's Problem
Proposition 2. (i) Let a polynomial P have the fonn (11)-(13) and let C 1 and C 2 be polynomials such that N(C1 ) c N(P) and N(C2) c 8(P). Then
(ii) If G
=1 in (13) then i = 1, 2.
Proposition 3. If a polynomial P has the form (11)-(13), then the polygon N(H p) C ~ 3 is reconstructed uniquely from the polygon 8(P). Assuming that Propositions 1, 2, and 3 have already been proved we shall complete the proof of Theorems 1 and 2 in the foregoing section. As in Chapters 2 and 3, we put ~ = pw, p ~ 0, lwl = 1, and associate with the polynomial P( ~, T), ~ E ~ n, the set of polynomials
Pw(P, r) = P(pw, r).
(16)
According to Theorem 3.4.1, if the polynomial P is dominantly correct, then all polynomials (16) (in the variables p and r) are also dominantly correct, and we have (17) In case the polynomial Pis stable-correct, polynomial (16) is N -stable correct, and we have N(Pw) = fl(P) Vw E sn- 1. (17') Using (17) and Proposition 3 we conclude that the polyhedra N(Hpw) do not depend on w and are uniquely determined by the polygon 8(P):
(18) By virtue of Proposition 1, for each w E
Hpw(p, a, I)> c(w)
sn-I
L
(a,,B,r)
we can write the estimate from below
IPI 1ai,Bhlr, 0
(:(lo(w).
(19)
We can select unified constants c(w) and lo(w) for all w E sn-1 . Indeed, if P is a dominantly correct polynomial, then the polynomials Pw(P, T) have the roots Tj(p,w)
= aj(w)pb;(w) + o(pb;(w)), Tj(p,w)
= Cj(w)p + o(1),
Tj ( p, W) = Q ( 1),
p ~ oo,
j
j = 1, ... , k(w),
= 1, ... , h(w),
j = 1, ... , b(W),
Chapter 7
252
where the numbers bj(w), k(w), h(w), and b(w) are reconstructed uniquely from the polygon 8(Pw) and, according to (17), do not depend on w. The coefficients aj(W) and cj ( w) are determined from the polynomials PH] in §1.1. Since the coefficients of these polynomials are smooth functions of w, it can be shown that the numbers iaj(w)i and jcj(w)l are uniformly bounded from above and the numbers Imaj(w) and !aj(w)-ak(w)l, j -f. k, are uniformly bounded from below by nonzero constants. Thus, it can be assumed that "Yo and c in (19) do not depend on w. Noting that
we obtain assertion (b) in Theorem 1. Assertion (c) and, the more so, assertion ( Cmax) readily follow from Proposition 3. 4.4. The proof of Proposition 1 in Section 4.3. According to Theorem 5.2.2, the fulfilment of (15) is equivalent to the property that for any nonnegative vector q = (q 1, q2, q3), q3 > 0, we have
(20) provided that
")' < 0,
~
-f. 0
(if q1
> 0);
a
-f. 0
(if q2
> 0).
(21)
However, it is difficult to determine from Theorem 5.2.2 the character of the dependence of the constants c and "Yo in (15) on the coefficients of polynomials (11 ). In this connection we shall prove inequality (15) in two stages. We first estimate H P from above and below via a positive function T(~, a, 1') not depending on the coefficients aj and Cj in (12) and (13) and after that show that under conditions (21) we have
(22) Proposition. There are constants c, c', and 'Yo depending on the same parameters as c and "Yo in (15), such that the inequality
(23) holds, where IL
T( ~'a,")')
=
11'1( a2
+ 1'2)b-1 ( a2 + 1'2 + e)h II (a2 + 1'2 + ebj) j=l IL
+ ca2 + '1'2)b+h I:)I"YI + 1 ~ 1 bj) II ca2 + 1 2 + ebk) j=l
k-:f:.j
c24)
253
The Method of Energy Estimates in Cauchy's Problem forb~
1 and IL
T(e,a,,) =
bl(a2 +'2 +e)h-1 IIca2 +'2 +ebi) j=1 IL
+ (a2 + ,2)h l::cbl + ej) II (a2 + '2 + ebk) j=1 k=f.j forb= 0. Tbe constants c, and maxjc1· - ckl- 1 .
c',
(24')
and lo depend on max !ail, max lei!, max(Imaj)- 1 ,
j=f.k
Proof. Replacing the derivative 8PI 8T in the expression H P = - Im P8P I 8T by
we represent H P as a sum of three nonnegative terms:
(25) Letnma 1. If a polynomial R bas tbe form (12), tben for 1 ~ 0 we bave
,.,
d1 ~ IR(e, 7)1 21II (o- 2 + , 2 + ebj) ~ d~, j=1
(26)
a2 ~ HR(~,o-,,)II)I,I +ei) II
