Maharam-type kernel representation for operators with a trigonometric domination

Abstract

Consider a linear and continuous operator T between Banach function spaces. We prove that under certain requirements an integral inequality for T is equivalent to a factorization of T through a specific kernel operator: in other words, the operator T has what we call a Maharam-type kernel representation. In the case that the inequality provides a domination involving trigonometric functions, a special factorization through the Fourier operator is given. We apply this result to study the problem that motivates the paper: the approximation of functions in L 2 [0, 1] by means of trigonometric series whose Fourier coefficients are given by weighted trigonometric integrals.