04056cam  22005178i 450000100090000000300050000900500170001400600190003100700150005000800410006502000270010604000350013305000210016808200170018908400220020610000360022824501700026426300090043426400610044330000340050433600260053833700280056433800270059249000740061950000440069350400410073750509190077850600500169752007760174753300950252353800360261858800470265465000230270165000220272465000240274665000320277065000350280265001790283765001500301670000470316670000430321370000280325677601610328485600450344585600480349021684820RPAM20210409153632.0m      b    000 0 cr/|||||||||||210409s2020    riu     ob    000 0 eng    a9781470462499 (online)  aLBSOR/DLCbengerdacDLCdRPAM00aQA403b.J38 202000a515/.785223  a42B37a31B152msc1 aJaye, Benjamin,d1984-eauthor.14aThe Riesz transform of codimension smaller than one and the Wolff energy /h[electronic resource] cBenjamin Jaye, Fedor Nazorov, Maria Carmen Reguera, Xavier Tolsa.  a2010 1aProvidence, RI :bAmerican Mathematical Society,c[2020]  a1 online resource (pages cm.)  atextbtxt2rdacontent  aunmediatedbn2rdamedia  avolumebnc2rdacarrier0 aMemoirs of the American Mathematical Society, x1947-6221 ; vv. 1293  a"Forthcoming, volume 266, number 1293."  aIncludes bibliographical references.00tChapter 1. IntroductiontChapter 2. PreliminariestChapter 3. The general scheme: Finding a large Lipschitz oscillation coefficienttChapter 4. Upward and Downward DominationtChapter 5. Preliminary results regarding reflectionless measurestChapter 6. The basic energy estimatestChapter 7. Blow up I: The density droptChapter 8. The choice of the shelltChapter 9. Blow up II: Doing away with $\eps $tChapter 10. Localization around the shelltChapter 11. The schemetChapter 12. Suppressed kernelstChapter 13. Step I: Calderâon-Zygmund theory (From a distribution to an $L^p$-function)tChapter 14. Step II: The smoothing operationtChapter 15. Step III: The variational argumenttChapter 16. ContradictiontAppendix A. The maximum principletAppendix B. The small boundary meshtAppendix C. Lipschitz continuous solutions of the fractional Laplacian equationtAppendix D. Index of Selected Symbols and Terms1 aAccess is restricted to licensed institutions  a"Fix d [greater than or equal to] 2, and s [epsilon] (d - 1, d). We characterize the non-negative locally finite non-atomic Borel measures [mu] in Rd for which the associated s-Riesz transform is bounded in Lp2s([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])[infinity]/2, [infinity] [epsilon] (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"--cProvided by publisher.  aElectronic reproduction.bProvidence, Rhode Island :cAmerican Mathematical Society.d2020  aMode of access : World Wide Web  aDescription based on print version record. 0aHarmonic analysis. 0aLipschitz spaces. 0aLaplacian operator. 0aCalderâon-Zygmund operator. 0aPotential theory (Mathematics) 7aHarmonic analysis on Euclidean spaces -- Harmonic analysis in several variables {For automorphic theory, see mainly 11F30} -- Harmonic analysis and PDE [See also 35-XX].2msc 7aPotential theory {For probabilistic potential theory, see 60J45} -- Higher-dimensional theory -- Potentials and capacities, extremal length.2msc1 aNazorov, Fedorq(Fedya L'vovich),eauthor.1 aReguera, Maria Carmen,d1981-eauthor.1 aTolsa, Xavier,eauthor.0 iPrint version: aJaye, Benjamin, 1984-tRiesz transform of codimension smaller than one and the Wolff energy /w(DLC)   2020032234x0065-9266z97814704421324 3Contentsuhttps://www.ams.org/memo/1293/4 3Contentsuhttps://doi.org/10.1090/memo/1293