A pointwise estimate for the kernel of a pseudo-differential operator, with applications

Revista de Ia Union Matematica Argentina Volumen 37, 1991. A POINTWISE ESTIMATE FOR THE KERNEL OF A PSEUDO-DIFFERENTIAL OPERATOR, WITH APPLICATIONS J. ALVAREZ, J. HOUNIE and C. PEREZ 1. Introduction 184 Given a pseudo-differential operator L in the Hormander class L:;-:6' it is a classical result, (cf. [8]) that the distribution kernel k(x, y) of L will be a Coo function away from the diagonal and will decay rapidly with all its derivatives as Ix -yl --t 00, if 0< p::; 1,0::; b < 1. Moreover, k(x, y) will coincide with a Cj function in all mn x mn, provided m+n+j < O. In [7], we completed the analysis, by proving a sharp estimate for k(x,y), when m+n+j ;::: O. \""Ie used, however, a non standard partition of unity, which forced us to consider separately the cases p = 1,0 < p < 1. We give here a much simpler proof of that estimate, using a standard partition in dyadic rings. The rest of the paper is devoted to show several new applications of the estimates, to other integral estimates, weak type (p, q), etc. Specially, in Section 6 we combine this estimates with a pointwise estimate for a modified sharp maximal function, to obtain LP' weighted estimates with Aoo weights for a class of oscillatory integrals closely related to those studied by D. Phong and E. Stein. Particularly, we have in this class pseudo-differential operators of order::;-{ n+l )(I-p), which generalizes the weighted estimates obtained in [14]. The organization of the paper is as follows. In Section 2, we precise the definition of the class L:;-:6 and state some classical properties for the distribution kernel. In Section 3 we complete the analysis by proving a sharp estimate when m+n+j ;::: 0 for some j. In Section 4 an integral estimate in the spirit of the one considered in [17] is proved. This generalizes [15], Lemma 2.1. In Section 5, we prove a weak type (l,q),q> 1 estimate in L:;-:o,m <--n(I-p), by estimating the operator in terms of a fractional integral Jo. Finally, Section 6 is mainly devoted to weighted estimates for a class of oscillatory integrals. The notation used in this paper is the standard in the subject. 2. The class L:;-:6 Following [3], we will denote by S;::6' mE m, 0::; p, b::; 1, the class of COO(mn x mn) functions p( x, e) such that 185 for every 0:, {3 E Il'r. Given p E 5;::6' we define the pseudo-differential operator L in the Hormander class L;::6 (cf. [8]), as The operator L is uniquely defined by p, which is called for that reason the symbol of L. L defines a linear and continuous operator from C�(JR") into COO(JR"). Let k(x,y) E D'(JRn x JR") be its distribution kernel. It is defined by (2.1) The following theorem summarizes several classical properties. We refer to [8] for proofs. Theorem 2.1 (cf. [8J). Let L E L;::6,m E JR, O 5:. <5, p 5:. 1, be a pseudo-differential operator with symbol p( x, e) and let k( x, y) be the distribution kernel of L. Assume that 0< p and <5 < 1. Then, a). (Pseudo-local property). The distribution k(x,y) coincides with a Coo function o'utside the diagonal in JRn X JRn. Moreover, given 0:, fJ E lNn, there exist., No E lN such that for' each N ? No, sup Ix-yINID�D:k(x, y)l<oo xi'y b). Suppose that the symbol p(x, () has compact support in e uniformly with respect to x. Then, the distribution k( x, y) is a Coo fundion in JR" x JR" and given 0:, (3 E INn, N E IN, we have c). Assume that m+n+M < 0, for some M E IN. Then, the distribution k(x, y) is a bounded continuous function with bounded continuous derivatives of order 5:. M. '-ci The case m+n+M = 0 for some M E IN is not considered in [8). When m+n+N = 0 a logarithmic estimate can be proved and when m + n + M > 0, we obtain the sharJ,: pointwise estimate we referred to in the introduction. Both cases can be obtained1 simultaneously as it will be proved in the next section. 186 Under the hypothesis of Theorem 2.1, the distribution kernel k(x, y) can be written as a well defined oscillatory integral, if x I-y. This representation is obtained from (2.1) by repeated integrations by parts. 3. The pointwise estimate for the kernel Theorem 3.1: Let L E L;'6' m E JR, O:S; 8 < 1, 0 < p:S; 1 be a pseudo. differential operator with symbolp(x,e) and let k(x,y) be the distribution kernel of L. Then, a) if m+n+M = 0 for some M E IN, there exists C>O such that sup ID�D:k(x,y)I:S;Clloglx-yll, xl-y. lo+Pl=M b) if m+n+M>O for some M E IN, there exists C>O such that (3.1) sup ID�D�k(x,y)I:S;Clx-yrm+n+M)/p x I-y. Ia+PI=M Proof: We first observe that if k( x, y) is the kernel of a pseudo-differential operator in L;'6' it follows that D� D: k( x, y) is the kernel of a pseudo-differential operator of order:S;m+lal+I,8I. Hence, it is enough to prove the above inequalities with a =,8 = o. Also, according to Theorem 2.1, parts a) and b), it sufices to estimate k(x,y) for 0< Ix-yl < 1 assuming that the symbol p(x, e) vanishes for I{I:S; 1. let Now, let 'P E CO"(JR) be such that 'P >0, sUPP('P) C [1/2, l],i oOO, Pipdt = 1. Then, we can write k(X'Y)=--n. e,(z--y)·ep(x,O'P -de1 100 1 '. ( lei) dt (271") . 1 {lel2:1} t t k(x,y,t) = 1 ei(x-y)oep(x,O'P (ill) de· {lel2:1 } t Given ,8 E INn, we can write 187 The function e ---+ rt'{lfl) belongs to St�. Moreover, for each "'I E JNn, we have the estimate Thus, l(x-y)Pk(x, y, t)I::5 L C(a, P) 1 (1 + lelyro-plarHa-PI. or9 {le�l} ·X.uppe ".(Iel/t) (e)rla-Plde, where XA denotes the characteristic function of the set A. Since lei;::: 1 implies t;::: 1, we can estimate the expression above by C(p)tmin-pIPI. Thus, we obtain when IPI = N. Or, (3.2) Thus, Ix _yiN tpNlk(x, y, t)l;::: C(N)tmtn, tmtn Ik(x, y,t)1 ::5C(N) 1+lx-yINtPN· 100 tmifl-l Ik(x,y)1 ::5C(N) 1 1+(lx_yltP)Ndt. IT N;::: [7-] + 1, the integral converges. We can write C roo s�-l Ik(x,Y)I::5lx_yl(mtfl)/p 1 1 ;..,, 1 l+sN cIs. Thus, if Ix-yl ::/= 0 we have Ik(x,y)1 ::5 Ciloglx-yil when m+n = 0 and Ik(x,y)1 ::5 Clx-yl(mfn)/p when m+n>O. This completes the proof of the theorem. . It can be shown that these estimates are sharp, (d. [7]). IT m::5-{n+l)(I-p), we obtain from (3.1) the estimates, Ik(x, y)I::5 Clx_yl-n+�-l, IV' x,yk( x, y)I::5 Clx -yr-1• 1 p::/= n+l' Since we know that k is rapidly decreasing as Ix-yl ---+ 00, we deduce that the operators . in the class L�r;tl)(l-P) are associated to standard kernels in the sense of R. Coifman and Y. Meyer, (cf. [10]). Moreover, these kernels are bounded by integrable convolution kernels, when 0 < p < 1. This observation gives the LP continuity result, 1 < p < 00, (cf. [11]). 188 If m �--n(l-p), the estimates we obtain from (3.1) are (3.3) Ik(x,Y)I�Clx-yr 1'\7 z,yk(x; y)1 � Clx_yl-fl-l/P• Thus, operators in the class L-;:P-p) are associated to weakly strongly singular kernels, (cf. [12]), that is to say, kernels more singular at the diagonal than standard kernels, but not too singular to prevent the operators to be continuous in LP, 1 < p < 00. C. Fefferman has proved, (cf. [13]), that pseudo-differential operators in the class L�;/2)(1-P) 0 � D < p � 1 are bounded in LP, for 1 < p < 00, but this is not longer true if m >-(n/2)(1-p). Thus, -(n/2)(1-p) is the maximum order for which a pseudo differential operator can be considered weakly strongly singular. For this class the estimates on the kernel will be Ik(x, y) �Clx-yr(;+1)/2 1'\7 z,yk(x, y)1 � Clx-yr-n[( ;+1)/2)-1/P. These are the weakly strongly estimates (3.3) only when p = 1. However, it is possible to prove for these kernels integral conditions resembling those considered by J.L. Rubio de Francia, F. Ruiz and J.L. Torrea, (cf. [7]). This will be shown in the next section, using the pointwise estimate already ob tained for the kernel k( x, y ). 4. An integral estimate for the kernel Theorem 4.1: Let L E L;:6,m 2:-(n+1)(1 -p),m+n+1 > O. Then, gwen 1<p<00,3 {dj} E I1,O</:k1, such that: Ij O<r<l, (4.1) ( )1h sup 1 (Ik(x, y)-k(z, y)iP+lk(y, x)-k(y, z)iP)dy � I z-z I<r 2i rB <I y-z l<2if,l r9 d· <c }IJ /. -IB(z,2Jr )11-1 p (4.2) 189 Ifr?l, sup (1 (lk(X,y)-k(Z,y)IP+lk(y,X)-k(y,Z)IP)dy)I/P � 1 x-z l<r 2i r<1 y-z 1<2;+1 r d· <C .1IJ /. -IB(z,2 1r )11-1 P Proof: Using (3.1) with M = 1, we obtain by the mean value theorem that (4.3) Ix-zl Ik(x, y)-k(z, y)l+ Ik(y,x)-k(y, z)l5: C Iy-zl(mffl+l) / p' if 2lx-zl < Iy-zl. Let us first prove (4.1). It suffices to consider one of the terms in the left hand side. Using (4.3), we can write Since -(n+1)(1-p) >-n-1+in for p> 0, 1 <p, we have that n-;(m+n+1) <0. Thus, the integral above converges to C r(2irIJ)[n-{p/p)(mtn+1)lIP. Since-{n+1)(1-p» -n(l-p)-l, the condition m?-{n+1)(1-p) implies that the exponent of 2i is < O. On the other hand, we want the exponent of r to be ?O. Or, 1-8 (�-n) ?O. This is equivalent to the condition 1 8 � !!!±!!±l . -n p Let us now consider (4.2). As before, it is enough to look at the first term in the left hand side. We obtain 190 Once. again, the exponent of 2; is n - � < O. This time we want the exponent of r to be :S 0, which it is, since m �-{n+1)(1-p). Finally, we have showed that (4.1) and (4.2) hold with (4.4) O<(}:Smin {I, �-n}' This completes the proof of Theorem 4.1. Let us observe that when m = -{n+ 1)(1-p), we can choose in (4.4) (} = 1. When m = -n(1-p) (4.4) holds for () = p. Finally, when m = --¥(1-p), we can choose () = 2tn�i-p)' In [15], S. Chanillo and A. Torchinsky proved an estimate similar to (4.1) with m= -¥-{1-p) and p = 2 and asked whether a similar estimate would hold for p>2. 5. A weak type estimate in the class £1;:'6 We will now obtain as an application of Theorem 3.1, that operators in L';:,6' -n < m<-n(l-p) can be estimated, pointwisely, in terms of a fractional integral la. This estimate will immediately lead to weak type estimates. Theorem 5.1: Let L E L';:,6,-n < m < -n(1-p), 0 < p:S 1,0:S 6' < 1. Then, there exists C>O such that fo r each f E Cgo(IRn),x E IRn, ILf(x)I:SC la(lfl)(x), m+n a=n---·. p Proof: According to Theorem 2.1 b), it suffices to assume that the symbol p(x,�) vanishes for I�I:S 1. Using the same notation as in Theorem 3.1, we have i .. 100 dt A Lf(x) = e""'�p(x, 0 'P(I�I/t)-f(�)d� = IRn 1 t 1M i . e . A dt = lim e'X' p(x, Ocp(I�I/t)f(O-d� = M-+oo 1 IRn t 1 h lMi . dt = lim - (2 ) e,(x-y).ep(x, O'P(I�I/t)d� -f(y)dy. M-+oo 7r n IRn 1 IRn t 191 For M fixed, let From the inequality (3.2), there exists C > ° depending on M such that Thus, 1M tm+n-l IkM(X,y)I :::; C 1 l+(lx_yltP)Ndt :::; <c dt. 100 tm+n-l -1 l+(lx-yjtP)N 2: r kM(X, y)f(y)dy :::; jE7J, J2i+1>jX-y1>2i C '" r If(y)1 dy = j 7lz J2 i +1>\X-y1>2i Ix _yl(mtfl)/ P C r If(y)1 dy = C lar(lfl)(x), a = n-m+n lInn Ix-yln-o p Finally, Lf(x) = limM_oo J kM(x, y)f(y)dy. Thus, ILf(x)I:::;C lar(lfl)(x). This completes the proof of the theorem. Corollary 5.2: Let L E L;;:s,-n<m<-n(l-p),O<p:::; l,O :::; S<l. Then L is of weak type (1, �), for some 0< e < 1. Proof: This result is an immediate consequence of the fact that lar is of weak type (l,q), where � = 1-�, (d. [9], p.120). Thus e = !!!:f11. , np This completes the proof of the corollary. 6. LP weighted estimates for a a class of oscillatory integrals It is true that, for most operators in Harmonic Analysis, boundedness on Y(IRn), 1 <p< 00, will imply boundedness on Y(IRnj w), 1 <p< 00, wE Ap. However, there is a remarkable theorem, due to R. Coifman, (cf. [1]), which states .,,� that for a classical singular· integral operator, Tf(x) = p.v f k(x-y) fey) dy 192 a deeper result holds. Namely, (6.1) { ITf(y)IPw(y)dy�c ( [Mf(y)]Pw(y)dy, f E Cgo(IRn), lIR" . lIR" where O<p<oo,w E Aoo, C = C(p, w)>0 and where M denotes de Hardy-Littlewood maximal operator. Coifman's proof of (6.1) is based on a difficult good A inequality involving the maximal operator T*, defined as T*f(x) = sup I ( k(X,Y)f(Y)dyl, fEC go(IRn), £>0 llv-o:tx and the operator M. We describe in [2] a different proof of(6.1), under conditions that allow for con sideration of a wider class of operators. Our approach combines the following two ingredients. First, we prove a pointwise estimate. Indeed, we show that there exist 0 < 8 < 1,C = C(8» 0, such that (6.2) where M! is the s-sharp maximal operator, defined as (6.3) M# being the sharp maximal operator of C. Fefferman and E. Stein, (d. [3]), (6.4) M#(g)(x) = sup inf (IB 11 (Ig(y) - c l dy ) B(z) cE(J;' lB where sUPB(z) means that the supremum is taken over all the balls centered at x and IBI, as usual, is the volume of B = B(x). Second, we use an estimate due to C. Fefferman and E. Stein, (cf. [3]). Namely, (6.5) { [Mf(y)]Pw(y)dy�C ( [M# f(y)]Pw(y)dy, f E cgo(IRn) lIR" l IR n where O<p<oo,w E Aoo, C = C(P,w) >0. A simple proof of (6.5) can be found in [4], p.42. Then, in order to conclude (6.1), we combine (6.2) and (6.5) in the following way. 199 [12] J. Alvarez and M. Milman: "H' continuity properties of Calderon-Zygmund type operators" J. Math. An. Appl. 118 (1986), 63"'"79. [13] M. de Guzman: "Real variable methods in Fourier analysis", North Holland, (1981). [14] N. Miller: "Weighted Sobolev spaces and pseudo-differential operators with smooth symbols". n-ans� AMS 269 (1982),91-109. [15] S. Chanillo and A. Torchinsky: "Sharp functions and weighted LP estimates for a class of pseudo-differential operators" Arkiv fOr Matematik 24 (1986), 1-25. [16] S. Chanillo and M. Christ: "Weak (1,1) bounds for oscillatory singular integrals". Duke Math. J. 55 (1987), 141-155. [17] J.L. Rubio de Francia, F. Ruiz and J.L. Torrea: "Calderon-Zygmund theory for , operator-valued kernels", Advances in Mathematics 62 (1986), 7-48. • J. Alvarez Department of Mathematics, New Mexico State University, Las Cruces, NM 880030001, USA, e-mail: [email protected] • J. Hounie Departamento de Matematica-UFPE, Cidade Universitaria, 51730 -Recife-PE, Brasil, e-mail: [email protected] • C. Perez Department of Mathematics, University of Kentucky, Lexington, KY 405060, USA, e-mail:[email protected]