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The fractional Riesz transform and an exponential potential

Authors: B. Jaye, F. Nazarov and A. Volberg
Original publication: Algebra i Analiz, tom 24 (2012), nomer 6.
Journal: St. Petersburg Math. J. 24 (2013), 903-938
MSC (2010): Primary 42B20
Published electronically: September 23, 2013
MathSciNet review: 3097554
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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$.

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Additional Information

B. Jaye
Affiliation: Kent State University, Department of Mathematics, Kent, Ohio 44240

F. Nazarov
Affiliation: Kent State University, Department of Mathematics, Kent, Ohio 44240

A. Volberg
Affiliation: Michigan State University, Department of Mathematics, East Lansing, Michigan 48824

Keywords: Riesz transform, Calder\'on--Zygmund capacity, nonlinear capacity, Wolff potential, totally lower irregular measure
Received by editor(s): July 11, 2012
Published electronically: September 23, 2013
Article copyright: © Copyright 2013 American Mathematical Society

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