Anisotropic Pseudodifferential Operators o 5 Type 1

The author studies boundedness and microlocal properties for a general class of pseudodifferential operators of type 1, 1 in the frame of the anisotropic Sobolev spaces. I n t r o d u c t i o n . Moving from an essentially formal point of view, we say pseudodifferential operator on R ~ the linear map from 8(R ~) to 8' (R~), defined by (0.0.1) a(x, D) u(x) = ( 2 z ) ~ f e i(x' ~) a(x, ~]) u(~7) dy , where ~ represents the Fourier transform of u e 8(R ~) and a(x, ~) ~ 8' (R ~) is the operator symbol. In the literature of the last 25 years, many general classes of symbols have been introduced and the problem of their boundedness on suitable weighted Sobolev spaces widely studied. Particularly in his doctoral thesis [3] and in more recent works [4], [5] G. BOURDAUD shows that the pseudodifferential operators in (OpS~176 are bounded from Hc~omp(Q) to Hl~(t~), s e R , Y2 open subset of R ~. For better arguing the pathologies of the pseudodifferential operators of type 1, 1 H(~RMANDER [11], 1988, points out that (o.o.2) F ~* (~, ~]) = ~ o T(~ , ~/), T (~ , ~]) = ( ~ , ~ + y), where the Fourier transform acts only on the x variable and a*(x, D) is the adjoint of the operator a(x, D). After observing that the map T changes the horizontal space {(~, 0); ~ e R ~} into the twisted diagonal {(~, -~ ) ; ~eR~}, HORMANDER [11] identi(*) Entrata in Redazione il 6 maggio 1996. Indirizzo delrA. : Dipartimento di Matematica, Universit~ di Torino, via Carlo Alberto 10, 10123 Torino. E-mail: garello@dm.unito. it 136 G. GARELLO: Anisotropic pseudodifferential operators, etc. fies the class of all operators in Op S ~ 1 ( ~ ) which are H~-bounded for every s ~R, the set (OpS ~ ~(tg)) A (OpS ~ 1(t9))* and OpS ~ 1(~). This last operator class is given by all the pseudodifferential operators in OpS ~ t(t9) whose symbol a(x, ~]) ,~vanishesin a suitable way in a neighborhood of the twisted diagonal. In addition, under more restrictive conditions on the symbols, H()RMANDER [11] gives a suitable symbolic calculus for adjoint and composition. In a recent work [8], starting from the results described above, I introduce a new selfadjoint operator algebra contained in OpS~,l(t2) where I consider a somewhat more precise symbolic calculus, which leads me to study suitable microlocal properties for Sobolev singularities. Always suggested by the arguments in H~RMANDER [11], the aim of this paper is to investigate the properties of the anisotropic pseudodifferential operators of type 1,1, whose symbol is a function in G ~ (t2 • RS), which satisfies, for some anisotropic weight function [~]M, M = (M1, ., Ms) e N s, Mj >I 1: (0.0.3) sup 1~3~a(x, ~])[ ~< C~,Z,K(1 + [~]M) "~+(M'~-"), x e K where K is any compact subset of ~ , a, fi are multi-indeces in Z~ and (M, a} = = M l a l + . + M s a n . More precisely using some properties of the anisotropic Littlewood-Paley decomposition, introduced in w 1, in w 2 I consider some restrictive conditions on the symbols, which are necessary and sufficient for the continuity of the respective operators, in the frame of the anisotropic Sobolev spaces. In w 3 I set up a suitable symbolic calculus which leads me to show in w 4 a microlocal property for anisotropic Sobolev singularities. In the second part of this last section, in the more general framework of the inhomogeneous pseudodifferential calculus, I perform a counter-example, which shows as the previous result is in some sense optimal. Following closely the arguments in HORMANDER [11, w 7], in the last part of w 3 I state an inequality of ,,sharp G~rding, type. This result generalizes in some way the classic ,,sharp G~rdinginequality, see [10], and its anisotropic version as stated in SEGhLA[16]; it may be useful, I hope, in future studies on the propagation of singularities. At the end let me notice that boundedness and microlocal properties for anisotropic paradifferential operators, which are widely studied, in the more general frame of quasi-homogeneous Triebel spaces in the works of YAMAZAKI [18], [19], are someway confirmed in the present paper. 1. An i so trop ic L i t t l e w o o d P a l e y d e c o m p o s i t i o n . Let M = (M1, ., Ms) be an n-tuple of integer positive numbers, each one greater or equal than one, and define, for ~ R s, [~] = [~]M the unique positive root of the equation t 2 M ~ + . ~ + t -2M~2 n ~--1, [0] = 0. The quasi-homogeneous (anisotropic) weight function [~] satisfies the following properties, which are widely proved in literature, see for example LASCAR [13]. G. GARELL0: Anisotropic pseudodifferential operators, etc. 137 With standard vectorial notation: (1.0.1) [~ + ~] < [~] + [~]; (1.0.2) min{ l~ l , I~I1/M~ ~< [~] ~< I~l, where M0= m axM~; (1.0.3) [tM~] = [ ( t M ~ , . . . , tM~n)] = t[~], for t > 0 ; (1.0.4) [ ~] = [~]; (1.0.5) [~ ~] /> I[~] [~] I ; (1.0.6) [~] e e ~ (R ~ \0) and a~[~] = ~ a ( [ ~ 0 ] ) [ ~ ] 1 -(M, a) , where ~o = [~]-M~= ( [ ~ ] l ) M ~ e z n 1 and Z~, is a continuous function of TeR+. Generally speaking we shall say that a function A(O) is M-homogeneous of degree )[ e Z if A(tMo) = t ' ~ A ( 0 ) , for every t > 0. We can easily see that the derivatives of order a e Z ~ of A(O) are M-homogeneous of degree ~ (M, a / and moreover A(O) <~ c[0] ~, for some constant c > 0. Lastly we shall say that WcR~ is M-conic if for every t > 0 , tM~ew when ~eW. Let us consider now c; ~ C ~ (R), 0 ~< of(t) ~< 1, such that, for some K i> 1, supp cp c c { t~R; Itl < K } , ~ ( t ) = 1when Itl < 1/2K, anddefine, forp >10: Fp(~) = cpp§ q~(~), where c;p(~) = c;([~]/2P). Since