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The note concerns pseudo differential operators in the classes L 0, O < p < 1. These operators are pseudo-local, but they can displace the wave front set of distributions, as we show by means of an example in Lo -. 1. Classes L 6 and pseudo-local property. Let Q be an open set in R'. Denote by Sm(Q x Q x Rn), 0 S p < 1,0 < 8 < 1, the class of all a(x,y, ) E C '(Q x Q x R') such that for every compact set K c sa x sa and all multiorders a, ,B, y the estimate jD'Dy~Da(x,y, t)j < Cs aK( +IlI) Y+ 6a+fi (1) is valid in K x Rn for some constant C, P'ySK and write LZ6(2) for the class of pseudo differential operators A of the form Au(x) = (2wT) nfexp[i<x -y, 0]a(x,y, ()u(y) dy d (2) with symbol a(x, y, S) C x x Rn). In this note our aim will be to produce an example of a pseudo-local operator in Lo 1, which can displace the wave front set (WF) of distributions (for the definition of WF see Hormander [2]). Let us recall that a linear continuous map T of Co0(Q) into C(S2), and of $'(Q) into 6D'(Q), is said to be pseudo-local if sing supp Tu c sing supp u (3) is valid for each u E i '(Q) or, equivalently, the kernel H c 6D'(0i; X i) of T satisfies sing supp H c diagonal in Q x U. (4) We shall say that T is strictly pseudo-local if WFAu c WF u (5) for each u C $'(Q). Since for every f c 6D'(0) the projection in Q of WF f c T*(Q) \ 0 is equal to sing supp f, (5) implies (3). Concerning pseudo differential operators, Hormander has proved in [2] Received by the editors January 24, 1978. AMS (MOS) subject classifications (1970). Primary 35S05, 47G05.

A pseudo differential operator which shifts the wave front set