Growth of $L^{p}$ Lebesgue constants for convex polyhedra and other regions

Abstract:

For any convex polyhedron $W$ in $\mathbb {R}^{m}$, $p\in \left (1,\infty \right )$, and $N\geq 1$, there are constants $\gamma _{1}\left (W,p,m\right )$ and $\gamma _{2}\left (W,p,m\right )$ such that \[ \gamma _{1}N^{m\left (p-1\right ) }\leq \int _{\mathbb {T}^{m}}\left \vert \sum _{k\in NW}e\left (k\cdot x\right ) \right \vert ^{p}dx\leq \gamma _{2}N^{m\left (p-1\right )}. \] Similar results hold for more general regions. These results are various special cases of the inequalities \[ \gamma _{1}N^{m\left (p-1\right ) }\leq \int _{\mathbb {T}^{m}}\left \vert \sum _{k\in NB}e\left (k\cdot x\right ) \right \vert ^{p}dx\leq \gamma _{2} \phi \left (N\right ), \] where $\phi \left (N\right )=N^{p\left (m-1\right ) /2}$ when $p\in \left ( 1,\frac {2m}{m+1}\right )$, $\phi \left (N\right )=N^{p\left (m-1\right ) /2}\log$ $N$ when $p=\frac {2m}{m+1}$, and $\phi \left (N\right )=N^{m\left ( p-1\right ) }$ when $p>\frac {2m}{m+1}$, where $B$ is a bounded subset of $\mathbb {R}^{m}$ with non-empty interior.