Hausdorff–Young-type inequalities for vector-valued Dirichlet series

CARANDO, DANIEL; SEVILLA-PERIS, PABLO; MARCECA, FELIPE

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY

AMER MATHEMATICAL SOC

Año: 2020 vol. 373 p. 5627 - 5652

0002-9947

We study Hausdorff-Young type inequalities for vector-valued Dirichlet series which allow tocompare the norm of a Dirichlet series in the Hardy space Hp(X) with the q-norm of its coefficients.In order to obtain inequalities completely analogous to the scalar case, a Banach space must satisfythe restrictive notion of Fourier type/cotype. We show that variants of these inequalities hold for themuch broader range of spaces enjoying type/cotype. We also consider Hausdorff-Young type inequal-ities for functions defined on the infinite torus T∞ or the boolean cube {−1, 1}∞ . As a fundamentaltool we show that type and cotype are equivalent to hypercontractive homogeneous polynomial typeand cotype, a result of independent interest.