New local T 1 theorems on non-homogeneous spaces

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Paco Villarroya
Santa Clara University (Santa Clara, Estats Units d'Amèrica). Department of Applied Mathematics,
https://orcid.org/0009-0000-1552-5000

We develop new local T1 theorems to characterize Calder´on–Zygmund operators that extend boundedly or compactly on Lp(Rn, µ), with µ a measure of power growth. The results, whose proofs do not require random grids, have weaker hypotheses than previously known local T1 theorems since they only require a countable collection of testing functions. Moreover, a further extension of this work allows the use of testing functions supported on cubes of different dimensions. As a corollary, we describe the measures µ of the complex plane for which the Cauchy integral defines a compact operator on Lp(C, µ).

Paraules clau
Calderón–Zygmund operator, Compact operator, Non-doubling Radon measures, Cauchy integral

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Villarroya, Paco. «New local T 1 theorems on non-homogeneous spaces». Publicacions Matemàtiques, 2024, vol.VOL 68, núm. 2, p. 445-06, http://raco.cat/index.php/PublicacionsMatematiques/article/view/430120.
Referències
P. Auscher, S. Hofmann, C. Muscalu, T. Tao, and C. Thiele, Carleson measures, trees, extrapolation, and T(b) theorems, Publ. Mat. 46(2) (2002), 257–325. DOI: 10.5565/PUBLMAT_46202_01

P. Auscher and E. Routin, Local T b theorems and Hardy inequalities, J. Geom. Anal. 23(1) (2013), 303–374. DOI: 10.1007/s12220-011-9249-1

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

M. Christ, Lectures on Singular Integral Operators, CBMS Regional Conf. Ser. in Math. 77, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1990. DOI: 10.1090/CBMS/077.

G. David, Wavelets and Singular Integrals on Curves and Surfaces, Lecture Notes in Math. 1465, Springer-Verlag, Berlin, 1991. DOI: 10.1007/BFb0091544

E. B. Fabes, M. Jodeit, Jr., and N. M. Riviere ` , Potential techniques for boundary value problems on C1 -domains, Acta Math. 141(3-4) (1978), 165–186. DOI: 10.1007/BF02545747

S. Hofmann, On singular integrals of Calder´on-type in Rn, and BMO, Rev. Mat. Iberoamericana 10(3) (1994), 467–505. DOI: 10.4171/RMI/159

S. Hofmann, A proof of the local T b Theorem for standard Calder´on–Zygmund operators, Preprint (2007). arXiv:0705.0840

S. Hofmann, Local T(b) theorems and applications in PDE, in: Harmonic Analysis and Partial Differential Equations, Contemp. Math. 505, American Mathematical Society, Providence, RI, 2010, pp. 29–52. DOI: 10.1090/conm/505/09914

T. P. Hytonen ¨ , An operator-valued T b theorem, J. Funct. Anal. 234(2) (2006), 420–463. DOI: 10.1016/j.jfa.2005.11.001

T. P. Hytonen ¨ , The sharp weighted bound for general Calder´on–Zygmund operators, Ann. of Math. (2) 175(3) (2012), 1473–1506. DOI: 10.4007/annals.2012.175.3.9

T. Hytonen and H. Martikainen ¨ , On general local T b theorems, Trans. Amer. Math. Soc. 364(9) (2012), 4819–4846. DOI: 10.1090/S0002-9947-2012-05599-1

T. Hytonen and F. Nazarov ¨ , The local T b theorem with rough test functions, Adv. Math. 372 (2020), 107306, 36 pp. DOI: 10.1016/j.aim.2020.107306

T. P. Hytonen and A. V. V ¨ ah¨ akangas ¨ , The local non-homogeneous T b theorem for vectorvalued functions, Glasg. Math. J. 57(1) (2015), 17–82. DOI: 10.1017/S0017089514000123

C. E. Kenig, Harmonic Analysis Techniques for Second Order Elliptic Boundary Value Problems, CBMS Regional Conf. Ser. in Math. 83, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1994. DOI: 10.1090/cbms/083

M. T. Lacey and A. V. Vah¨ akangas ¨ , The perfect local T b theorem and twisted martingale transforms, Proc. Amer. Math. Soc. 142(5) (2014), 1689–1700. DOI: 10.1090/S0002-9939-2014-11930-7

H. Martikainen, M. Mourgoglou, and E. Vuorinen, A new approach to non-homogeneous local T b theorems, J. Anal. Math. 143(1) (2021), 95–121. DOI: 10.1007/s11854-021-0147-6

F. Nazarov, S. Treil, and A. Volberg, Cauchy integral and Calder´on–Zygmund operators on nonhomogeneous spaces, Internat. Math. Res. Notices 1997(15) (1997), 703–726. DOI: 10.1155/S1073792897000469

F. Nazarov, S. Treil, and A. Volberg, Weak type estimates and Cotlar inequalities for Calder´on–Zygmund operators on nonhomogeneous spaces, Internat. Math. Res. Notices 1998 (9) (1998), 463–487. DOI: 10.1155/S1073792898000312

F. Nazarov, S. Treil, and A. Volberg, Accretive system T b-theorems on nonhomogeneous spaces, Duke Math. J. 113(2) (2002), 259–312. DOI: 10.1215/S0012-7094-02-11323-4

F. Nazarov, S. Treil, and A. Volberg, The T b-theorem on non-homogeneous spaces, Acta Math. 190(2) (2003), 151–239. DOI: 10.1007/BF02392690

F. Nazarov, S. Treil, and A. Volberg, The T b-theorem on non-homogeneous spaces that proves a conjecture of Vitushkin, Preprint (2014). arXiv:1401.2479v2

E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, With the assistance of Timothy S. Murphy, Princeton Math. Ser. 43, Monogr. Harmon. Anal. III, Princeton University Press, Princeton, NJ, 1993. DOI: 10.1515/9781400883929

X. Tolsa, Analytic Capacity, the Cauchy Transform, and Non-homogeneous Calder´on–Zygmund Theory, Progr. Math. 307, Birkh¨auser/Springer, Cham, 2014. DOI: 10.1007/978-3-319-00596-6

P. Villarroya, A characterization of compactness for singular integrals, J. Math. Pures Appl. (9) 104(3) (2015), 485–532. DOI: 10.1016/j.matpur.2015.03.006

P. Villarroya, A global T b theorem for compactness and boundedness of Calder´on–Zygmund operators, J. Math. Anal. Appl. 480(1) (2019), 123323, 41 pp. DOI: 10.1016/j.jmaa.2019.07.013