Bochner-Riesz means with respect to a rough distance function
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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 $p$.References
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Additional Information
- Paul Taylor
- Affiliation: Department of Mathematics, University of Wisconsin–Madison, 480 Lincoln Drive, Madison, Wisconsin 53706
- Address at time of publication: Department of Mathematics, Shippensburg University, 1871 Old Main Drive, Shippensburg, Pennsylvania 17257-2299
- Email: pttaylor@ship.edu
- Received by editor(s): July 26, 2004
- Received by editor(s) in revised form: November 16, 2004
- Published electronically: November 17, 2006
- Additional Notes: The author thanks Andreas Seeger for his guidance
- © Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 359 (2007), 1403-1432
- MSC (2000): Primary 42B15; Secondary 42B25
- DOI: https://doi.org/10.1090/S0002-9947-06-03918-3
- MathSciNet review: 2272131