RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
General information
Latest issue
Archive
Impact factor
Subscription

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Algebra i Analiz:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


Algebra i Analiz, 2012, Volume 24, Issue 6, Pages 77–123 (Mi aa1310)  

This article is cited in 7 scientific papers (total in 7 papers)

Research Papers

The fractional Riesz transform and an exponential potential

B. Jayea, F. Nazarova, A. Volbergb

a Kent State University, Department of Mathematics, Kent, OH
b Michigan State University, Department of Mathematics, East Lansing, MI

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 $\mathcal W_{\Phi,s}(\mu)(x)=\int_0^\infty\Phi(\frac{\mu(B(x,r))}{r^s})\frac{dr}r$, 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 non-linear capacity associated to the gauge $\Phi$. It appears to be the first result of this type for $s>1$.

Keywords: Riesz transform, Calderón–Zygmund capacity, nonlinear capacity, Wolff potential, totally lower irregular measure.

Full text: PDF file (469 kB)
References: PDF file   HTML file

English version:
St. Petersburg Mathematical Journal, 2013, 24:6, 903–938

Bibliographic databases:

Received: 11.07.2012
Language:

Citation: B. Jaye, F. Nazarov, A. Volberg, “The fractional Riesz transform and an exponential potential”, Algebra i Analiz, 24:6 (2012), 77–123; St. Petersburg Math. J., 24:6 (2013), 903–938

Citation in format AMSBIB
\Bibitem{JayNazVol12}
\by B.~Jaye, F.~Nazarov, A.~Volberg
\paper The fractional Riesz transform and an exponential potential
\jour Algebra i Analiz
\yr 2012
\vol 24
\issue 6
\pages 77--123
\mathnet{http://mi.mathnet.ru/aa1310}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3097554}
\zmath{https://zbmath.org/?q=an:1281.42011}
\elib{http://elibrary.ru/item.asp?id=20730184}
\transl
\jour St. Petersburg Math. J.
\yr 2013
\vol 24
\issue 6
\pages 903--938
\crossref{https://doi.org/10.1090/S1061-0022-2013-01272-6}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000331545300004}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84888126456}


Linking options:
  • http://mi.mathnet.ru/eng/aa1310
  • http://mi.mathnet.ru/eng/aa/v24/i6/p77

    SHARE: VKontakte.ru FaceBook Twitter Mail.ru Livejournal Memori.ru


    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. A. L. Volberg, V. Ya. Èiderman, “Non-homogeneous harmonic analysis: 16 years of development”, Russian Math. Surveys, 68:6 (2013), 973–1026  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    2. F. Nazarov, X. Tolsa, A. Volberg, “On the uniform rectifiability of AD-regular measures with bounded Riesz transform operator: the case of codimension 1”, Acta Math., 213:2 (2014), 237–321  crossref  mathscinet  zmath  isi  scopus
    3. Prat L., Tolsa X., “Non-Existence of Reflectionless Measures For the S-Riesz Transform When 0 < S < 1”, Ann. Acad. Sci. Fenn. Ser. A1-Math., 40:2 (2015), 957–968  crossref  mathscinet  zmath  isi
    4. Azzam J., David G., Toro T., “Wasserstein distance and the rectifiability of doubling measures: part I”, Math. Ann., 364:1-2 (2016), 151–224  crossref  mathscinet  zmath  isi  scopus
    5. Chousionis V., Prat L., Tolsa X., “Square Functions of Fractional Homogeneity and Wolff Potentials”, Int. Math. Res. Notices, 2016, no. 8, 2295–2319  crossref  mathscinet  isi  scopus
    6. Carmen Reguera M., Tolsa X., “Riesz transforms of non-integer homogeneity on uniformly disconnected sets”, Trans. Am. Math. Soc., 368:10 (2016), 7045–7095  crossref  mathscinet  zmath  isi  scopus
    7. Jaye B., Nazarov F., “Reflectionless Measures For Calderon-Zygmund Operators II: Wolff Potentials and Rectifiability”, J. Eur. Math. Soc., 21:2 (2019), 549–583  crossref  mathscinet  zmath  isi  scopus
  • Алгебра и анализ St. Petersburg Mathematical Journal
    Number of views:
    This page:239
    Full text:43
    References:38
    First page:20

     
    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2019