Bounds on moments of weighted sums of finite Riesz products

TL;DR

This paper establishes bounds on the $L^p$ norms of linear combinations of finite Riesz products with lacunary sequences, showing their equivalence to the $ell^p$ norm of coefficients for large lacunarity.

Contribution

It proves that for large enough lacunarity, the $L^p$ norm of linear combinations of normalized Riesz products is equivalent to the $ell^p$ norm of the coefficients, extending to vector-valued spaces.

Findings

$L^p$ norm of linear combinations is equivalent to coefficient norms for large lacunarity.

Results generalize to vector-valued $L^p$ spaces.

Provides a characterization of embeddings of $ell^p$ into $L^p$ via Fourier coefficients.

Abstract

Let $n_j$ be a lacunary sequence of integers, such that $n_{j+1}/n_j\geq r$. We are interested in linear combinations of the sequence of finite Riesz products $\prod_{j=1}^N(1+\cos(n_j t))$. We prove that, whenever the Riesz products are normalized in $L^p$ norm ($p\geq 1$) and when $r$ is large enough, the $L^p$ norm of such a linear combination is equivalent to the $\ell^p$ norm of the sequence of coefficients. In other words, one can describe many ways of embedding $\ell^p$ into $L^p$ based on Fourier coefficients. This generalizes to vector valued $L^p$ spaces.