Sharp multiplier theorem for multidimensional Bessel operators

Abstract

Consider the multidimensional Bessel operator $$Bf(x)=-\sum _{j=1}^N\left({\partial }_j^{2}f(x)+\frac{\hat{I}{\pm }_j}{x_j}{\partial }_{j}f(x)\right),x\in (0,\infty {)}^N.$$ Let $d={\sum }_{j=1}^N\max (1,\hat{I}{\pm }_j+1)$ be the homogeneous dimension of the space $(0,\infty {)}^N$ equipped with the measure $x_1^{\hat{I}{\pm }_1}...x_N^{\hat{I}{\pm }_N}d{x}_1...d{x}_N$. In the general case $\hat{I}{\pm }_1,...,\hat{I}{\pm }_N>-1$ 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 Hörmander condition $$\underset{t>0}{\sup }\left||\hat{I}\cdot (\cdot )m(t\cdot )\right|{|}_{W^{2,{\hat{I}}^2}(\mathbb{R})}<\infty$$ with ${\hat{I}}^2>d/2$. 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.