Paley's theorem for Hankel matrices via the Schur test

TL;DR

This paper provides a Schur test-based proof of Paley's theorem for Hankel matrices, revealing interesting patterns and recovering the optimal constant, and also reestablishes Paley multipliers characterization.

Contribution

It introduces Schur test proofs for Paley's theorem in Hankel matrices and characterizes Paley multipliers from $H^1$ to $H^2$, with optimal constants.

Findings

Schur test proofs for Paley's theorem

Identification of patterns in vectors for the Schur test

Recovery of the best constant in the main case

Abstract

Paley's theorem about lacunary coefficients of functions in the classical space $H^1$ on the unit circle is equivalent to the statement that certain Hankel matrices define bounded operators on $\ell^2$ of the nonnegative integers. Since that statement reduces easily to the case where the entries in the matrix are all nonnegative, it must be provable by the Schur test. We give such proofs with interesting patterns in the vectors used in the test, and we recover the best constant in the main case. We use related ideas to reprove the characterization of Paley multipliers from $H^1$ to $H^2$.