On Herz-Bochkarev limiting problem

TL;DR

This paper investigates optimal bounds for Hausdorff-Young-type inequalities in Lorentz spaces, focusing on the limiting case as the integrability parameter approaches 2, and introduces new grand Lorentz space techniques.

Contribution

It provides refined bounds for the Herz-Bochkarev problem using innovative grand Lorentz space methods, advancing previous estimates.

Findings

Derived optimal bounds for constants as p approaches 2

Refined estimates improve upon previous results

Introduced new grand Lorentz space techniques

Abstract

This paper studies Hausdorff-Young-type inequalities within the framework of Lorentz spaces $L_{p,q}$. Focusing on the dependence of the associated constants on the integrability parameter $p$, we derive optimal bounds in the limiting case $p\rightarrow 2$, addressing the Herz-Bochkarev problem. The results obtained refine the pioneering estimates in [3] and are comparable to recent advances in [16]. The main ingredients of our approach are new grand Lorentz space techniques.