On the convolution equations in the space of distributions of 𝐿^{𝑝}-growth

Pahk, D. H.

doi:10.1090/S0002-9939-1985-0781061-9

On the convolution equations in the space of distributions of $L^ p$-growth
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Proc. Amer. Math. Soc. 94 (1985), 81-86 Request permission

Abstract:

We consider convolution equations in the space ${D’_{{L^p}}},\;1 \leqslant p \leqslant \infty$, of distributions of ${L^p}$-growth, i.e. distributions which are finite sums of derivatives of ${L^p}$-functions (see [4, 7]). Our main results are to find a condition for convolution operators to be hypoelliptic in ${\mathcal {D}’_{{L^\infty }}}$ in terms of their Fourier transforms and to show that the same condition is working for the solvability of convolution operators in the tempered distribution space $\mathcal {S}’$ and ${\mathcal {D}’_{{L^p}}}$.

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On the space of convolution operators in

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