We know that convolution of an integrable function with an $p$-integrable is an $p$-integrable function. This follows from Young's inequality.
My question: Is it true that $L^p(\mathbb{R}^n)\subseteq L^p(\mathbb{R}^n)*L^1(\mathbb{R}^n)?$
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$\begingroup$ It’s certainly not true for $p=\infty$ since the convolution of an $L^p$ and $L^q$ function (where $p,q$ are Holder duals) is continuous but not every $L^\infty$ function is continuous. $\endgroup$– kieransquaredCommented Jul 8 at 14:23
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5$\begingroup$ This is discussed in Edwin Hewitt, The ranges of certain convolution operators, Math. Scand. 15 (1964), 147–155. $\endgroup$– Hjalmar RosengrenCommented Jul 8 at 14:29
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1$\begingroup$ Does $L^p(\mathbb R^n) * L^1(\mathbb R^n)$ mean the set of convolutions, or the span of the set of convolutions? $\endgroup$– LSpiceCommented Jul 8 at 19:51
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$\begingroup$ Thank you very much @Hjalmar Rosengren. It was very much helpful. $\endgroup$– user531870Commented Jul 9 at 5:36
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