§0. INTRODUCTION The purpose of this work is to study certain operators of the form, T^(x) = jK(x,y)^(y)dy, acting on spaces of functions or distributions in Kn. In general, the way in which the above integrals have to be interpreted, depends on the properties of the kernel K, and these expressions are, a priori, only known to make sense for some particular class of functions usually 3 or #". Actually, in a general sense, all linear operators defined on 3 are of the above form. In fact, the Schwartz kernel theorem (see [H2]) tell us that each linear and continuous operator T: 3 (Kn)—.0/(IRn), is determined by a distribution K -0'(IRnx fRn), so that 1,^ = K,^8 £», for all (p, ip e 3(Un). Thus, for example, if K is a continuous function (or just locally integrable), then, T(p,t/» = JJK(x,y)^(x)^(y)dxdy, or T^(x)-jK(x,y)^(y)dy, 1
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