Sharp estimates for imaginary powers of Bessel operators

TL;DR

This paper establishes sharp weak type (1,1) bounds for the imaginary powers of the Bessel operator, advancing understanding of its harmonic analysis properties.

Contribution

It provides the first sharp weak type (1,1) estimates for the imaginary powers of the Bessel operator, a key step in harmonic analysis.

Findings

Proved sharp weak type (1,1) bounds for $L^{ieta}$

Extended the theory of imaginary powers to Bessel operators

Enhanced understanding of harmonic analysis on weighted spaces

Abstract

Let $L f(x):=-\frac{d^2}{dx^2}f(x)-\frac{ r}{x}\frac{d}{dx}f(x),\quad x>0, r>0$ be the Bessel operator on $((0,\infty), |\cdot|, x^rdx)$. In this paper, we prove the sharp weak type $(1,1)$ estimate for the imaginary power $L^{i\alpha}, \alpha\in \mathbb R$, of the Bessel operator.