Sharp multiplier theorem for multidimensional Bessel operators

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.