An α-number characterization of Lp spaces on uniformly rectifiable sets
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Jonas Azzam
University of Edinburgh. School of Mathematics
Damian Dąbrowski
Universitat Autònoma de Barcelona. Departament de Matemàtiques
We give a characterization of Lp(σ) for uniformly rectifiable measures σ using Tolsa’s α-numbers, by showing, for 1 < p < ∞ and f ∈ Lp(σ), that kfkLp(σ) ∼Z ∞0(αfσ(x, r) + |f|x,rασ(x, r))2drr12Lp(σ).
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quantitative rectifiability, uniformly rectifiable sets, α-numbers, Lp spaces
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Azzam, Jonas; and Dąbrowski, Damian. “An α-number characterization of Lp spaces on uniformly rectifiable sets”. Publicacions Matemàtiques, vol.VOL 67, no. 2, pp. 819-50, https://raco.cat/index.php/PublicacionsMatematiques/article/view/418513.
Referències
J. Azzam, G. David, and T. Toro, Wasserstein distance and the rectifiability of doubling measures: part I, Math. Ann. 364(1-2) (2016), 151–224.
DOI: 10.1007/s00208-015-1206-z
J. Azzam and X. Tolsa, Characterization of n-rectifiability in terms of Jones’ square function: Part II, Geom. Funct. Anal. 25(5) (2015), 1371–1412. DOI: 10.1007/s00039-015-0334-7
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DOI: 10.1090/tran/8170
M. Christ, A T(b) theorem with remarks on analytic capacity and the Cauchy integral, Colloq. Math. 60/61(2) (1990), 601–628.
DOI: 10.4064/cm-60-61-2-601-628
D. Dąbrowski, Necessary condition for rectifiability involving Wasserstein distance W2, Int. Math. Res. Not. IMRN 2020(22) (2020), 8936–8972.
DOI: 10.1093/imrn/rnaa012
D. Dąbrowski, Sufficient condition for rectifiability involving Wasserstein distance W2, J. Geom. Anal. 31(8) (2021), 8539–8606.
DOI: 10.1007/s12220-020-00603-y
G. David, Morceaux de graphes lipschitziens et intégrales singulières sur une surface, Rev. Mat. Iberoamericana 4(1) (1988), 73–114. DOI: 10.4171/RMI/64
G. David, M. Engelstein, and S. Mayboroda, Square functions, nontangential limits, and harmonic measure in codimension larger than 1, Duke Math.
J. 170(3) (2021), 455–501. DOI: 10.1215/00127094-2020-0048
G. David, J.-L. Journé, and S. Semmes, Opérateurs de Calderón–Zygmund, fonctions para-accrétives et interpolation, Rev. Mat. Iberoamericana 1(4)
(1985), 1–56. DOI: 10.4171/RMI/17
G. David and S. Mayboroda, Harmonic measure is absolutely continuous with respect to the Hausdorff measure on all low-dimensional uniformly rectifiable sets, Int. Math. Res. Not. IMRN (2022), rnac109. DOI: 10.1093/imrn/rnac109
G. David and S. Semmes, Singular integrals and rectifiable sets in Rn. Au-delà des graphes lipschitziens, Astérisque 193 (1991), 147 pp.
J. Feneuil, Absolute continuity of the harmonic measure on low dimensional rectifiable sets, J. Geom. Anal. 32(10) (2022), Paper No. 247, 36 pp. DOI: 10.1007/s12220-022-00978-0
L. Grafakos, “Classical Fourier Analysis”, Third edition, Graduate Texts in Mathematics 249, Springer, New York, 2014. DOI: 10.1007/978-1-4939-1194-3
T. Hytönen and O. Tapiola, Almost Lipschitz-continuous wavelets in metric spaces via a new randomization of dyadic cubes, J. Approx. Theory 185 (2014), 12–30. DOI: 10.1016/j.jat.2014.05.017
P. Mattila, “Geometry of Sets and Measures in Euclidean Spaces. Fractals and Rectifiability”, Cambridge Studies in Advanced Mathematics 44, Cambridge University Press, Cambridge, 1995. DOI: 10.1017/CBO9780511623813
T. Orponen, Absolute continuity and α-numbers on the real line, Anal. PDE 12(4) (2019), 969–996. DOI: 10.2140/apde.2019.12.969
X. Tolsa, Littlewood–Paley theory and the T(1) theorem with non-doubling measures, Adv. Math. 164(1) (2001), 57–116. DOI: 10.1006/aima.2001.2011
X. Tolsa, Principal values for Riesz transforms and rectifiability, J. Funct. Anal. 254(7) (2008), 1811–1863. DOI: 10.1016/j.jfa.2007.07.020
X. Tolsa, Uniform rectifiability, Calderón–Zygmund operators with odd kernel, and quasiorthogonality, Proc. Lond. Math. Soc. (3) 98(2) (2009), 393–426. DOI: 10.1112/plms/pdn035
X. Tolsa, Mass transport and uniform rectifiability, Geom. Funct. Anal. 22(2) (2012), 478–527. DOI: 10.1007/s00039-012-0160-0
X. Tolsa, “Analytic Capacity, the Cauchy Transform, and Non-homogeneous Calderón–Zygmund Theory”, Progress in Mathematics 307, Birkhäuser/
Springer, Cham, 2014. DOI: 10.1007/978-3-319-00596-6
X. Tolsa, Rectifiable measures, square functions involving densities, and the Cauchy transform, Mem. Amer. Math. Soc. 245(1158) (2017), 130 pp. DOI: 10.1090/memo/1158
DOI: 10.1007/s00208-015-1206-z
J. Azzam and X. Tolsa, Characterization of n-rectifiability in terms of Jones’ square function: Part II, Geom. Funct. Anal. 25(5) (2015), 1371–1412. DOI: 10.1007/s00039-015-0334-7
J. Azzam, X. Tolsa, and T. Toro, Characterization of rectifiable measures in terms of α-numbers, Trans. Amer. Math. Soc. 373(11) (2020), 7991–8037.
DOI: 10.1090/tran/8170
M. Christ, A T(b) theorem with remarks on analytic capacity and the Cauchy integral, Colloq. Math. 60/61(2) (1990), 601–628.
DOI: 10.4064/cm-60-61-2-601-628
D. Dąbrowski, Necessary condition for rectifiability involving Wasserstein distance W2, Int. Math. Res. Not. IMRN 2020(22) (2020), 8936–8972.
DOI: 10.1093/imrn/rnaa012
D. Dąbrowski, Sufficient condition for rectifiability involving Wasserstein distance W2, J. Geom. Anal. 31(8) (2021), 8539–8606.
DOI: 10.1007/s12220-020-00603-y
G. David, Morceaux de graphes lipschitziens et intégrales singulières sur une surface, Rev. Mat. Iberoamericana 4(1) (1988), 73–114. DOI: 10.4171/RMI/64
G. David, M. Engelstein, and S. Mayboroda, Square functions, nontangential limits, and harmonic measure in codimension larger than 1, Duke Math.
J. 170(3) (2021), 455–501. DOI: 10.1215/00127094-2020-0048
G. David, J.-L. Journé, and S. Semmes, Opérateurs de Calderón–Zygmund, fonctions para-accrétives et interpolation, Rev. Mat. Iberoamericana 1(4)
(1985), 1–56. DOI: 10.4171/RMI/17
G. David and S. Mayboroda, Harmonic measure is absolutely continuous with respect to the Hausdorff measure on all low-dimensional uniformly rectifiable sets, Int. Math. Res. Not. IMRN (2022), rnac109. DOI: 10.1093/imrn/rnac109
G. David and S. Semmes, Singular integrals and rectifiable sets in Rn. Au-delà des graphes lipschitziens, Astérisque 193 (1991), 147 pp.
J. Feneuil, Absolute continuity of the harmonic measure on low dimensional rectifiable sets, J. Geom. Anal. 32(10) (2022), Paper No. 247, 36 pp. DOI: 10.1007/s12220-022-00978-0
L. Grafakos, “Classical Fourier Analysis”, Third edition, Graduate Texts in Mathematics 249, Springer, New York, 2014. DOI: 10.1007/978-1-4939-1194-3
T. Hytönen and O. Tapiola, Almost Lipschitz-continuous wavelets in metric spaces via a new randomization of dyadic cubes, J. Approx. Theory 185 (2014), 12–30. DOI: 10.1016/j.jat.2014.05.017
P. Mattila, “Geometry of Sets and Measures in Euclidean Spaces. Fractals and Rectifiability”, Cambridge Studies in Advanced Mathematics 44, Cambridge University Press, Cambridge, 1995. DOI: 10.1017/CBO9780511623813
T. Orponen, Absolute continuity and α-numbers on the real line, Anal. PDE 12(4) (2019), 969–996. DOI: 10.2140/apde.2019.12.969
X. Tolsa, Littlewood–Paley theory and the T(1) theorem with non-doubling measures, Adv. Math. 164(1) (2001), 57–116. DOI: 10.1006/aima.2001.2011
X. Tolsa, Principal values for Riesz transforms and rectifiability, J. Funct. Anal. 254(7) (2008), 1811–1863. DOI: 10.1016/j.jfa.2007.07.020
X. Tolsa, Uniform rectifiability, Calderón–Zygmund operators with odd kernel, and quasiorthogonality, Proc. Lond. Math. Soc. (3) 98(2) (2009), 393–426. DOI: 10.1112/plms/pdn035
X. Tolsa, Mass transport and uniform rectifiability, Geom. Funct. Anal. 22(2) (2012), 478–527. DOI: 10.1007/s00039-012-0160-0
X. Tolsa, “Analytic Capacity, the Cauchy Transform, and Non-homogeneous Calderón–Zygmund Theory”, Progress in Mathematics 307, Birkhäuser/
Springer, Cham, 2014. DOI: 10.1007/978-3-319-00596-6
X. Tolsa, Rectifiable measures, square functions involving densities, and the Cauchy transform, Mem. Amer. Math. Soc. 245(1158) (2017), 130 pp. DOI: 10.1090/memo/1158
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