Bochner-Riesz means with respect to a rough distance function

Abstract:

The generalized Bochner-Riesz operator $S^{R,\lambda }$ may be defined as \begin{equation*} S^{R,\lambda }f(x) = \mathcal {F}^{-1}\left [\left (1-\frac {\rho }{R}\right )^{\lambda }_+ \widehat {f}\right ](x) \end{equation*} where $\rho$ is an appropriate distance function and $\mathcal F^{-1}$ is the inverse Fourier transform. The behavior of $S^{R,\lambda }$ on $L^p(\mathbf {R}^d\times \mathbf {R})$ is described for $\rho (\xi ’,\xi _{d+1})=\max \{|\xi ’|,|\xi _{d+1}|\}$, a rough distance function. We conjecture that this operator is bounded on $\mathbf {R}^{d}\times \mathbf {R}$ when $\lambda >\max \{d(\frac {1}{2}-\frac {1}{p})-\frac {1}{2},0\}$ and $p<2+\frac {6}{d-3}$, and unbounded when $p \!\geq \!2\!+\!\frac {6}{d-3}$. This conjecture is verified for large ranges of $p$.