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The Riesz Transform of Codimension Smaller Than One and the Wolff Energy
About this Title
Benjamin Jaye, Department of Mathematical Sciences, Kent State University, Kent, Ohio 44240, Fedor Nazarov, Department of Mathematical Sciences, Kent State University, Kent, Ohio 44240, Maria Carmen Reguera, School of Mathematics, University of Birmingham, Birmingham, United Kingdom and Xavier Tolsa, Institució Catalana de Recerca i Estudis Avançats (ICREA), Passeig de Lluís Companys, 23, 08010 Barcelona, Catalonia—and—Departament de Matemàtiques and BGSMath, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia
Publication: Memoirs of the American Mathematical Society
Publication Year:
2020; Volume 266, Number 1293
ISBNs: 978-1-4704-4213-2 (print); 978-1-4704-6249-9 (online)
DOI: https://doi.org/10.1090/memo/1293
Published electronically: July 21, 2020
MSC: Primary 42B37, 31B15
Table of Contents
Chapters
- 1. Introduction
- 2. Preliminaries
- 3. The general scheme: Finding a large Lipschitz oscillation coefficient
- 4. Upward and Downward Domination
Part I: The blow-up procedures
- 5. Preliminary results regarding reflectionless measures
- 6. The basic energy estimates
- 7. Blow up I: The density drop
- 8. The choice of the shell
- 9. Blow up II: Doing away with $\varepsilon$
- 10. Localization around the shell
Part II: The non-existence of an impossible object
- 11. The scheme
- 12. Suppressed kernels
- 13. Step I: Calderón-Zygmund theory (From a distribution to an $L^p$-function)
- 14. Step II: The smoothing operation
- 15. Step III: The variational argument
- 16. Contradiction
Appendices
- A. The maximum principle
- B. The small boundary mesh
- C. Lipschitz continuous solutions of the fractional Laplacian equation
- D. Index of Selected Symbols and Terms
Abstract
Fix $d\geq 2$, and $s\in (d-1,d)$. We characterize the non-negative locally finite non-atomic Borel measures $\mu$ in $\mathbb {R}^d$ for which the associated $s$-Riesz transform is bounded in $L^2(\mu )$ in terms of the Wolff energy. This extends the range of $s$ in which the Mateu-Prat-Verdera characterization of measures with bounded $s$-Riesz transform is known.
As an application, we give a metric characterization of the removable sets for locally Lipschitz continuous solutions of the fractional Laplacian operator $(-\Delta )^{\alpha /2}$, $\alpha \in (1,2)$, in terms of a well-known capacity from non-linear potential theory. This result contrasts sharply with removability results for Lipschitz harmonic functions.
- David R. Adams and Lars Inge Hedberg, Function spaces and potential theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 314, Springer-Verlag, Berlin, 1996. MR 1411441
- Michael Christ, Lectures on singular integral operators, CBMS Regional Conference Series in Mathematics, vol. 77, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1990. MR 1104656
- Guy David, Unrectifiable $1$-sets have vanishing analytic capacity, Rev. Mat. Iberoamericana 14 (1998), no. 2, 369–479 (English, with English and French summaries). MR 1654535, DOI 10.4171/RMI/242
- Guy David, Wavelets and singular integrals on curves and surfaces, Lecture Notes in Mathematics, vol. 1465, Springer-Verlag, Berlin, 1991. MR 1123480
- Guy David and Pertti Mattila, Removable sets for Lipschitz harmonic functions in the plane, Rev. Mat. Iberoamericana 16 (2000), no. 1, 137–215. MR 1768535, DOI 10.4171/RMI/272
- G. David and S. Semmes, Singular integrals and rectifiable sets in $\mathbb {R}^d$: au-delá des graphes lipschitziens. Astérique 193 (1991).
- Guy David and Stephen Semmes, Analysis of and on uniformly rectifiable sets, Mathematical Surveys and Monographs, vol. 38, American Mathematical Society, Providence, RI, 1993. MR 1251061
- V. Eiderman, F. Nazarov, and A. Volberg, Vector-valued Riesz potentials: Cartan-type estimates and related capacities, Proc. Lond. Math. Soc. (3) 101 (2010), no. 3, 727–758. MR 2734959, DOI 10.1112/plms/pdq003
- Vladimir Eiderman, Fedor Nazarov, and Alexander Volberg, The $s$-Riesz transform of an $s$-dimensional measure in $\Bbb {R}^2$ is unbounded for $1<s<2$, J. Anal. Math. 122 (2014), 1–23. MR 3183521, DOI 10.1007/s11854-014-0001-1
- Vladimir Eiderman and Alexander Volberg, $L^2$-norm and estimates from below for Riesz transforms on Cantor sets, Indiana Univ. Math. J. 60 (2011), no. 4, 1077–1112. MR 2975336, DOI 10.1512/iumj.2011.60.4304
- Hany M. Farag, The Riesz kernels do not give rise to higher-dimensional analogues of the Menger-Melnikov curvature, Publ. Mat. 43 (1999), no. 1, 251–260. MR 1697524, DOI 10.5565/PUBLMAT_{4}3199_{1}1
- Steve Hofmann and José María Martell, Uniform rectifiability and harmonic measure I: Uniform rectifiability implies Poisson kernels in $L^p$, Ann. Sci. Éc. Norm. Supér. (4) 47 (2014), no. 3, 577–654 (English, with English and French summaries). MR 3239100, DOI 10.24033/asens.2223
- Steve Hofmann, José María Martell, and Svitlana Mayboroda, Uniform rectifiability and harmonic measure III: Riesz transform bounds imply uniform rectifiability of boundaries of 1-sided NTA domains, Int. Math. Res. Not. IMRN 10 (2014), 2702–2729. MR 3214282, DOI 10.1093/imrn/rnt002
- Steve Hofmann, José María Martell, and Ignacio Uriarte-Tuero, Uniform rectifiability and harmonic measure, II: Poisson kernels in $L^p$ imply uniform rectifiability, Duke Math. J. 163 (2014), no. 8, 1601–1654. MR 3210969, DOI 10.1215/00127094-2713809
- Benjamin Jaye and Fedor Nazarov, Reflectionless measures for Calderón-Zygmund operators I: general theory, J. Anal. Math. 135 (2018), no. 2, 599–638. MR 3829611, DOI 10.1007/s11854-018-0047-6
- Benjamin Jaye and Fedor Nazarov, Reflectionless measures for Calderón-Zygmund operators II: Wolff potentials and rectifiability, J. Eur. Math. Soc. (JEMS) 21 (2019), no. 2, 549–583. MR 3896210, DOI 10.4171/JEMS/844
- B. Jaye, F. Nazarov, and A. Volberg, The fractional Riesz transform and an exponential potential, Algebra i Analiz 24 (2012), no. 6, 77–123; English transl., St. Petersburg Math. J. 24 (2013), no. 6, 903–938. MR 3097554, DOI 10.1090/S1061-0022-2013-01272-6
- Peter W. Jones, Square functions, Cauchy integrals, analytic capacity, and harmonic measure, Harmonic analysis and partial differential equations (El Escorial, 1987) Lecture Notes in Math., vol. 1384, Springer, Berlin, 1989, pp. 24–68. MR 1013815, DOI 10.1007/BFb0086793
- N. S. Landkof, Foundations of modern potential theory, Springer-Verlag, New York-Heidelberg, 1972. Translated from the Russian by A. P. Doohovskoy; Die Grundlehren der mathematischen Wissenschaften, Band 180. MR 0350027
- Joan Mateu, Laura Prat, and Joan Verdera, The capacity associated to signed Riesz kernels, and Wolff potentials, J. Reine Angew. Math. 578 (2005), 201–223. MR 2113895, DOI 10.1515/crll.2005.2005.578.201
- Joan Mateu and Xavier Tolsa, Riesz transforms and harmonic $\textrm {Lip}_1$-capacity in Cantor sets, Proc. London Math. Soc. (3) 89 (2004), no. 3, 676–696. MR 2107011, DOI 10.1112/S0024611504014790
- Pertti Mattila, Geometry of sets and measures in Euclidean spaces, Cambridge Studies in Advanced Mathematics, vol. 44, Cambridge University Press, Cambridge, 1995. Fractals and rectifiability. MR 1333890
- Pertti Mattila, On the analytic capacity and curvature of some Cantor sets with non-$\sigma$-finite length, Publ. Mat. 40 (1996), no. 1, 195–204. MR 1397014, DOI 10.5565/PUBLMAT_{4}0196_{1}2
- Pertti Mattila, Mark S. Melnikov, and Joan Verdera, The Cauchy integral, analytic capacity, and uniform rectifiability, Ann. of Math. (2) 144 (1996), no. 1, 127–136. MR 1405945, DOI 10.2307/2118585
- Pertti Mattila and Joan Verdera, Convergence of singular integrals with general measures, J. Eur. Math. Soc. (JEMS) 11 (2009), no. 2, 257–271. MR 2486933, DOI 10.4171/JEMS/149
- M. S. Mel′nikov, Analytic capacity: a discrete approach and the curvature of measure, Mat. Sb. 186 (1995), no. 6, 57–76 (Russian, with Russian summary); English transl., Sb. Math. 186 (1995), no. 6, 827–846. MR 1349014, DOI 10.1070/SM1995v186n06ABEH000045
- Fedor Nazarov, Xavier Tolsa, and Alexander Volberg, On the uniform rectifiability of AD-regular measures with bounded Riesz transform operator: the case of codimension 1, Acta Math. 213 (2014), no. 2, 237–321. MR 3286036, DOI 10.1007/s11511-014-0120-7
- Fedor Nazarov, Xavier Tolsa, and Alexander Volberg, The Riesz transform, rectifiability, and removability for Lipschitz harmonic functions, Publ. Mat. 58 (2014), no. 2, 517–532. MR 3264510
- F. Nazarov, S. Treil, and A. Volberg, Weak type estimates and Cotlar inequalities for Calderón-Zygmund operators on nonhomogeneous spaces, Internat. Math. Res. Notices 9 (1998), 463–487. MR 1626935, DOI 10.1155/S1073792898000312
- F. Nazarov, S. Treil, and A. Volberg, The $Tb$-theorem on non-homogeneous spaces that proves a conjecture of Vitushkin. Preprint (1999). arXiv:1401.2479.
- P. V. Paramonov, Harmonic approximations in the $C^1$-norm, Mat. Sb. 181 (1990), no. 10, 1341–1365 (Russian); English transl., Math. USSR-Sb. 71 (1992), no. 1, 183–207. MR 1085885, DOI 10.1070/SM1992v071n01ABEH002129
- Laura Prat, On the semiadditivity of the capacities associated with signed vector valued Riesz kernels, Trans. Amer. Math. Soc. 364 (2012), no. 11, 5673–5691. MR 2946926, DOI 10.1090/S0002-9947-2012-05724-2
- Maria Carmen Reguera and Xavier Tolsa, Riesz transforms of non-integer homogeneity on uniformly disconnected sets, Trans. Amer. Math. Soc. 368 (2016), no. 10, 7045–7095. MR 3471085, DOI 10.1090/S0002-9947-2016-06587-3
- Aleix Ruiz de Villa and Xavier Tolsa, Non existence of principal values of signed Riesz transforms of non integer dimension, Indiana Univ. Math. J. 59 (2010), no. 1, 115–130. MR 2666475, DOI 10.1512/iumj.2010.59.3884
- Xavier Tolsa, Analytic capacity, the Cauchy transform, and non-homogeneous Calderón-Zygmund theory, Progress in Mathematics, vol. 307, Birkhäuser/Springer, Cham, 2014. MR 3154530
- Xavier Tolsa, Calderón-Zygmund capacities and Wolff potentials on Cantor sets, J. Geom. Anal. 21 (2011), no. 1, 195–223. MR 2755682, DOI 10.1007/s12220-010-9145-0
- Xavier Tolsa, Painlevé’s problem and the semiadditivity of analytic capacity, Acta Math. 190 (2003), no. 1, 105–149. MR 1982794, DOI 10.1007/BF02393237
- Merja Vihtilä, The boundedness of Riesz $s$-transforms of measures in $\mathbf R^n$, Proc. Amer. Math. Soc. 124 (1996), no. 12, 3797–3804. MR 1343727, DOI 10.1090/S0002-9939-96-03522-8
- Alexander Volberg, Calderón-Zygmund capacities and operators on nonhomogeneous spaces, CBMS Regional Conference Series in Mathematics, vol. 100, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2003. MR 2019058