Let $S$ be an $L$ shaped region in the unit cube $Q:=[0,1]\times [0,1]$: $$ S:=Q\backslash C,\quad C:=\left[\frac 1 2,1\right]\times \left[\frac 1 2,1\right]. $$ Consider the multiplier operator $T$ defined by $$ \widehat {Tf}(\xi):=\hat f(\xi)1_{S}(\xi) $$ Then what can we say about the operator norm of $T$ on $L^p(\mathbb R^2)$, where $1<p<\infty$?
We have the following observations:
- If $p=2$, then the operator norm is 1, which follows trivially by the Plancherel theorem. Hence we consider $p\neq 2$ only.
- It is well known that any rectangle multiplier in $\mathbb R^2$ (by symmetries and the Hilbert transform) has operator norm $C_p$ which depends on $p$ only.
- Therefore, since $L$ can be partitioned into two disjoint rectangles (in any way), it is easy to see that $T$ has operator norm bounded by $2C_p$.
- Cordoba showed that for a polygon with $N$ sides, the polygon multiplier has norm bounded by $C(\log N)^{\alpha(p)}$. Moreover, this bound is sharp.
In the case of an $L$-shaped region, is it true that $\left\|T\right\|_p>C_p$ for any $p\neq 2$?
Cordoba, A., The multiplier problem for the polygon, Ann. Math. (2) 105, 581-588 (1977). ZBL0361.42005. showed that for any polygon
