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Net S be a convolution operator in the space K_{M}' , M is an increasing continuous function, of distributions in R^{n} growing not faster than e^{M(kx)} for some positive integer k. We give necessary and sufficient condition on S to have S*K_{M}'=K_{M}' , then we use this condition to get necessary and sufficient conditions for solvability of systems of convolution equations in K_{M}' .

Solvability of convolution equations was investigated by L. Ehrenpreis [2] and L. Hrrmander [3] in the space of Schwartz distributions. In this paper we discuss solvability of systems of convolution equations in Beurling's distributions; the results of this paper extend some of Hrrmander's [3]. The first section contains notations and the preliminary results which will be used later in the work, in the second section we give an existence theorem in the case of one convolution equation, the third section contains existence theorems for systems of convolution equations. I. Preliminary resultsBjrrk [1] gave a thorough study of Beurling test function spaces and distributions. Here we give a very brief account of the definitions and the basic results which we are going to use later. In particular, we use the Paley Wiener theorem in a form slightly weaker than the one given by Bjrrk. For proofs of these results we refer the reader to [1].By d/we denote the set of all continuous real-valued functions co on R", satisfying the following conditions: If/2 is a subset of R", let Do>(f2)={r Supp ikcf2}. If K is a compact subset of R n, D~(K) is a Frrchet space when provided with the topology defined

In this note we give a structure theorem of the distributions in the space K p,k , k < 0, which is a subspace of the space of distributions which grow no faster than e |x| p , p > 1, and use this structure theorem to give a representation of the Fourier transform of the distributions in these spaces. The Fourier transform of members of several spaces of distributions has been studied by several authors. Gonzalez and Negrin [5] studied the Fourier transform over the spaces S k , k ∈ Z, k < 0 of tempered distributions introduced by Horvath [2]. They have shown that the Fourier transform maps each of the spaces S k , k ∈ Z, k < 0 onto itself, and proved a representation theorem for the usual Fourier transform of members of these spaces. Hayek, Gonzalez and Negrin [3] proved an inversion formula for the distributional Fourier transform on the spaces S k , k ∈ Z, k < 0. They applied their results to obtain a representation on S for any distribution of S k as limit of a sequence of ordinary functions. Gonzalez [4], established a structure theorem of the members of the spaces S k and gave a representation of the Fourier transform of these members. Sohn and Pahk [6] introduced the spaces K p,k , k ∈ Z, k < 0, p > 1, of distributions of exponential growth. Among other things they studied the Fourier transform of members of these spaces and gave an inversion formula for the elements of the spaces. In this work, along the lines of Barrose-Neto [1, proof of Theorem 6.2], we establish a structure theorem for the distributions in the spaces K p,k , k ∈ Z, k < 0, p > 1, then we use this structure theorem to get a representation of the Fourier transform of the elements of these spaces.

In this paper we give a necessary condition on the Fourier transform of a convolution operator S of the space D_{L^{p}}' ; 2\leq p<\infty , for the equation S*u=v to have a solution u in D',\lrcorner p for every v in D_{L^{p}}' . In the case p=2 , this condition with the additional assumption \hat{S}(\xi)\neq 0 for all \xi\in\Re^{n} , are sufficient for solvability of the convolution equation.