I am interested in the following somewhat obscure question:
Is there some $n \in \Bbb{N}$, and a set $A \subset \Bbb{R}^n$ of finite measure such that the Fourier transform $\widehat{1_A}$ of its indicator function satisfies $\widehat{1_A} \in \bigcap_{q > 1} L^q$, but such that there is some $p \in (1,\infty)$ such that $1_A$ is not a Fourier multiplier for $L^p$, i.e., such that the map $L^p \to L^p, g \mapsto g \ast \widehat{1_A}$ is not bounded (and thus not even well-defined).
The motivation for looking at this question is the following: We know that if $f \in L^1$, then (by Young's inequality) the map $g \mapsto g \ast f$ is bounded on every $L^p$ space, $1 \leq p \leq \infty$. Furthermore, even if one is only intersted in boundedness on the $L^p$ spaces with $1<p<\infty$, the assumption $f \in L^1$ cannot be weakened in general to the assumption $f \in \bigcap_{q > 1} L^q$, as can be seen by considering $f(x) = 1_{[1,\infty)}(x) \cdot x^{-1} \cdot \ln x$, $p=2$, and $g(x) = 1_{[e,\infty)} (x) \cdot x^{-1/2} \cdot (\ln x)^{-3/2}$.
However, to provide a counterexample to a certain claim (see below), I need a similar counterexample, but for the case where the convolution $g \mapsto g \ast f$ is a projection, which would be satisfied if $f = \widehat{1_A}$.
Sadly, the simplest example (i.e., the ball multiplier) does not work, since it is an $L^p$ multiplier for all $1 < p < \infty$ in dimension $n=1$, and does not satisfy $\widehat{1_B} \in \bigcap_{q > 1} L^q$ for $n > 1$.
I have found the question What can be said about the Fourier transforms of characteristic functions?, where a paper by Lebedev is cited which shows the following (see Corollary 4 of the paper):
There exists a bounded domain $D \subset \Bbb{R}^2$ with $C^1$ boundary, and with $\widehat{1_D} \in \bigcap_{q>1}L^q$. The boundary of $D$ does not contain line intervals.
I have no idea, however, how I could show (or if it is true) that $1_D$ is not a Fourier multiplier on some $L^p$ space, $1 < p < \infty$.
Also, Corollary 2 of the same paper shows
Let $D \subset \Bbb{R}^n$ ($n \geq 2$) be a bounded domain with $\partial D \in C^{1,1}$. Then $\widehat{1_D} \notin L^p$ for $p \leq 2n/(n+1)$.
Thus, at least for dimension $n \geq 2$, this shows that if a set with the desired properties exist, then it cannot have a $C^{1,1}$ boundary.
Finally, a brief remark on the claim I want to find a counterexample for: A continuous Parseval frame is a family $(\psi_x)_{x \in \Omega}$ in a Hilbert space $\mathcal{H}$, indexed by a measure space $(\Omega, \mu)$, such that $x \mapsto \psi_x$ is weakly measurable, and such that $$ \int_\Omega |\langle f, \psi_x \rangle|^2 d \mu(x) = \| f \|_{\mathcal{H}}^2 \qquad \forall f \in \mathcal{H}. $$ I would like to find such a Parseval frame such that the Gramian kernel $K(x,y) = \langle \psi_y, \psi_x \rangle$ does not yield a bounded operator on some $L^p (\Omega)$ space, $1<p<\infty$, but such that it satisfies $$ \sup_x \int_\Omega |K(x,y)|^q d\mu(y) < \infty $$ for all $1 < q <\infty$. If there exists a set $A$ as in my question above, then I can simply take $\mathcal{H} = L^2(A)$, and $\psi_\xi (x) = e^{2\pi i \xi x}$, $\xi \in \Bbb{R}$.
