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

For any convex polyhedron $W$ in $\mathbb{R}^{m}$, $p\in\left(1,\infty \right)$, and $N\geq1$, there are constants $\gamma_{1}\left(W,p,m\right)$ and $\gamma_{2}\left(W,p,m\right)$ such that

$$\gamma_{1}N^{m\left(p-1\right) }\leq\int_{\mathbb{T}^{m}}\left\ve... ...e\left(k\cdot x\right) \right\vert ^{p}dx\leq\gamma _{2}N^{m\left(p-1\right)}.$$

Similar results hold for more general regions. These results are various special cases of the inequalities

$$\gamma_{1}N^{m\left(p-1\right) }\leq\int_{\mathbb{T}^{m}}\left\ve... ...NB}e\left(k\cdot x\right) \right\vert ^{p}dx\leq\gamma_{2} \phi\left(N\right),$$

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.