Sharp multiplier theorem for multidimensional Bessel operators

TL;DR

This paper establishes sharp spectral multiplier theorems for multidimensional Bessel operators, extending harmonic analysis tools to these operators and analyzing their imaginary powers with precise bounds.

Contribution

It proves new multiplier theorems for Bessel operators on $L^{1, obreak ext{,} obreak} ext{ and Hardy spaces, and investigates their imaginary powers with lower bounds.

Findings

Multiplier theorems hold under Hörmander condition with $eta > d/2$.

Spectral multipliers are bounded on $L^{1, obreak ext{,} obreak} ext{ and } H^1$.

Sharpness of the multiplier theorem is demonstrated through lower estimates.

Abstract

Consider the multidimensional Bessel operator $$B f(x) = -\sum_{j=1}^N \left(\partial_j^2 f(x) +\frac{\alpha_j}{x_j} \partial_j f(x)\right), \quad x\in(0,\infty)^N. $$ Let $d = \sum_{j=1}^N \max(1,\alpha_j+1)$ be the homogeneous dimension of the space $(0,\infty)^N$ equipped with the measure $x_1^{\alpha_1}... x_N^{\alpha_N} dx_1...dx_N$. In the general case $\alpha_1,...,\alpha_N >-1$ we prove multiplier theorems for spectral multipliers $m(B)$ on $L^{1,\infty}$ and the Hardy space $H^1$. We assume that $m$ satisfies the classical H\"ormander condition $$\sup_{t>0} \left||\eta(\cdot) m(t\cdot)\right||_{W^{2,\beta}(\mathbb{R})}<\infty$$ with $\beta > d/2$. Furthermore, we investigate imaginary powers $B^{ib}$, $b\in \mathbb{R}$, and prove some lower estimates on $L^{1,\infty}$ and $L^p$, $1<p<2$. As a consequence, we deduce that our multiplier theorem is sharp.