The following result of Hormander [2] (see also Theorem 2.5.6 in [1]), plays a significant role in harmonic analysis since all convolution type operators are translation invariant.
Definition. We say that a bounded linear operator $T:L^p(\mathbb{R}^n)\to L^q(\mathbb{R}^n)$ is translation invariant if $T(\tau_y f)=\tau_y(Tf)$ for all $f\in L^p(\mathbb{R}^n)$ and all $y\in\mathbb{R}^n$, where $(\tau_y f)(x)=f(x+y)$.
Theorem (Hormander). If $T:L^p(\mathbb{R}^n)\to L^q(\mathbb{R}^n)$, $1\leq p<\infty$, $1\leq q\leq\infty$ is non-zero and translation invariant, then $q\geq p$.
The proof is simple and well known. The argument does not generalize to the case of $p=\infty$. However, the argument still works if we replace $L^\infty$ by $L^\infty_0$ which is the subspace of $L^\infty$ consisting of functions that converge to $0$ at infinity. In that case Hormander proved the following result:
Theorem (Hormander). If $T:L^\infty_0(\mathbb{R}^n)\to L^q(\mathbb{R}^n)$ is non-zero and translation invariant, then $q=\infty$.
Hormander (see p.97 in [2]), calls this result somewhat incomplete for $p=\infty$.
I was quite curious about the case $p=\infty$ and since I could not find an answer, I discussed it with several mathematicians. As a result of cooperation with M. Bownik, F. L. Nazarov and P. Wojtaszczyk we finally proved the following result:
Theorem.
If $T:L^\infty(\mathbb{R}^n)\to L^q(\mathbb{R}^n)$ is non-zero and translation invariant, then $q\geq 2$.
On the other hand, there is a non-zero translation invariant operator
$T_1:L^\infty(\mathbb{R}^n)\to L^2(\mathbb{R}^n)\cap L^\infty(\mathbb{R}^n)$. It follows that $T_1:L^2(\mathbb{R}^n)\to L^q(\mathbb{R}^n)$ is bounded for all $2\leq q\leq\infty$.
I was very excited about the result and our proof. However, a few days later I got an e-mail from M. Bownik who told me that that the result had already been proved by Liu and van Rooij [3]! Despite the fact that this paper solves a problem of Hormander, it has only one citation according to MathSciNet. Many textbooks in harmonic analysis quote the result of Hormander, but nobody mentions the beautiful result of Lui and van Rooij!
I cannot resist and I have to recall my e-mail conversation with Nazarov:
Dear Fedja, Bad news. The problem was solved in 1974 by Liu and Van Rooij.....
Dear Piotr, Actually, this is a wonderful news: instead of going through the painstaking and time consuming proofreading and submission process, we can just relax and think of something else :-).
[1] L. Grafakos, L.: Classical Fourier analysis. Second edition. Graduate Texts in Mathematics, 249. Springer, New York, 2008.
[2] L. Hormander, Estimates for translation invariant operators in $L^p$ spaces. Acta Math. 104 (1960), 93-140. https://doi.org/10.1007/BF02547187 https://link.springer.com/article/10.1007%2FBF02547187
[3] T. S. Liu, A. C. M. van Rooij, Translation invariant maps $L^\infty(G)\to L^p(G)$. Nederl. Akad. Wetensch. Proc. Ser. A 77 = Indag. Math. 36 (1974), 306-316. https://doi.org/10.1016/1385-7258(74)90021-3
