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

Abstract

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