The fractional Riesz transform and an exponential potential
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- by B. Jaye, F. Nazarov and A. Volberg
- St. Petersburg Math. J. 24 (2013), 903-938
- DOI: https://doi.org/10.1090/S1061-0022-2013-01272-6
- Published electronically: September 23, 2013
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Abstract:
In this paper we study the $s$-dimensional Riesz transform of a finite measure $\mu$ in $\mathbf {R}^d$, with $s\in (d-1,d)$. We show that the boundedness of the Riesz transform of $\mu$ yields a weak type estimate for the Wolff potential $\mathcal {W}_{\Phi ,s}(\mu )(x) = \int _0^{\infty }\Phi \bigl (\frac {\mu (B(x,r))}{r^s}\bigr ) \frac {dr}{r},$ where $\Phi (t) = e^{-1/t^{\beta }}$ with $\beta >0$ depending on $s$ and $d$. In particular, this weak type estimate implies that $\mathcal {W}_{\Phi ,s}(\mu )$ is finite $\mu$-almost everywhere. As an application, we obtain an upper bound for the Calderón–Zygmund capacity $\gamma _s$ in terms of the nonlinear capacity associated to the gauge $\Phi$. It appears to be the first result of this type for $s>1$.References
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Bibliographic Information
- B. Jaye
- Affiliation: Kent State University, Department of Mathematics, Kent, Ohio 44240
- MR Author ID: 975566
- Email: bjaye@kent.edu
- F. Nazarov
- Affiliation: Kent State University, Department of Mathematics, Kent, Ohio 44240
- MR Author ID: 233855
- Email: nazarov@math.kent.edu
- A. Volberg
- Affiliation: Michigan State University, Department of Mathematics, East Lansing, Michigan 48824
- Email: volberg@math.msu.edu
- Received by editor(s): July 11, 2012
- Published electronically: September 23, 2013
- © Copyright 2013 American Mathematical Society
- Journal: St. Petersburg Math. J. 24 (2013), 903-938
- MSC (2010): Primary 42B20
- DOI: https://doi.org/10.1090/S1061-0022-2013-01272-6
- MathSciNet review: 3097554