The problem of trigonometric Fourier series multipliers of classes in λp,q spaces

In this article, we consider weighted spaces of numerical sequences λp,q, which are defined as sets of sequences a = {ak}^∞_k=1, for which the norm ||a||λp,q :=\sum^∞_k=1|ak|^q k^(q/p −1)^1/q<∞ is finite. In the case of non-increasing sequences, the norm of the space λp,q coincides with the norm of the classical Lorentz space lp,q. Necessary and sufficient conditions are obtained for embeddings of the space λp,q into the space λp1,q1. The interpolation properties of these spaces with respect to the real interpolation method are studied. It is shown that the scale of spaces λp,q is closed in the relative real interpolation method, as well as in relative to the complex interpolation method. A description of the dual space to the weighted space λp,q is obtained. Specifically, it is shown that the space is reflective, where p', q' are conjugate to the parameters p and q. The paper also studies the properties of the convolution operator in these spaces. The main result of this work is an O’Neil type inequality. The resulting inequality generalizes the classical Young-O’Neil inequality. The research methods are based on the interpolation theorems proved in this paper for the spaces λp,q.