Let $\mathbb T$ be a circle group.
In 1939, Salem, has shown that, every member of $L^{1}(\mathbb T)$ can written as a product(convolution) some other two members of $L^{1}(\mathbb T),$ that is, $L^{1}(\mathbb T)\subset L^{1}(\mathbb T)\ast L^{1}(\mathbb T);$ other way one can get by Fubini's theorem, so we have, $L^{1}(\mathbb T)=L^{1}(\mathbb T)\ast L^{1}(\mathbb T).$
I have learn from the book(R. E. Edwards, Fourier Series A Modern Introduction Volume 1, p.124), $L^{p}(\mathbb T)=L^{1}(\mathbb T)\ast L^{p}(\mathbb T), (1\leq p < \infty).$
In 1959, Walter Rudin has shown for the real line that, $L^{1}(\mathbb R)=L^{1}(\mathbb R)\ast L^{1}(\mathbb R).$
By Young's inequality, we have, $L^{1}(\mathbb R)\ast L^{p}(\mathbb R) \subset L^{p}(\mathbb R), (1\leq p < \infty).$
My Question is: Is it well-known in a literature that, $L^{p}(\mathbb R)\subset L^{1}(\mathbb R) \ast L^{p}(\mathbb R), (1< p< \infty)$ ? If answer is yes, please let me know some references; If not, what one can expect ?
Thanks,