Paley's theorem for Hankel matrices via the Schur test

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 ${\mathrm{\ell }}^2$ 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 $H^2$.