On the boundedness of pseudo-differential operators

Abstract

In this note, we use the sequence version of Cotlar's lemma and a partition of unity to give a proof of the L2-boundedness of a class of pseudo-differential operators. Introduction and results. Let p(x, 5) be a continuous function on RI x RI. Then, a pseudo-differential operator P with the symbol p(x, 5) is a linear map of CO (Rn) into CO(Rn), defined by p = ((27r)1/2)-n eixtp(x )2(4) d4 for u E CO (Rn), where u is the Fourier transform of u. In [1] A. Calderon and R. Vaillencourt prove the following THEOREM A. Let p(x, $) be a function defined on R' x R' such that 1(1 + aw)3 . . (I + ax1)3(I + a )3 . . (1 + at1)3p(x, t)I ? C for all (x, t) e Rn x Rn. Then the pseudo-diferential operator associated with the symbolp(x, $) can be extended to a bounded operator from L2(Rn) to L2(Rn). Using the sequence version of Cotlar's lemma (cf. [2]) and a partition of unity, we can prove the following result analogous to Theorem A. THEOREM. Let p(x, $) be a function defined on Rn x RL such that (1) J a,tp(x, dx _ C, for O _ oi < 2, 0 < f3j < 3, and all (x, t) E Q x Rt, where Q is any cube with edges of length two and parallel to the axes. Then the associated operator can be extended to a bounded operator from L2(Rn) to L2(R'). Received by the editors June 26, 1972 and, in revised form, December 30, 1972. AMS (MOS) subject classifications (1970). Primary 35S05.