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Reflectionless measures for Calderón-Zygmund operators I: general theory

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Abstract

We study the properties of reflectionless measures for an s-dimensional Calderón-Zygmund operator T acting in Rd, where s ∈ (0, d). Roughly speaking, these are measures μ for which Tμ(1) is constant on the support of the measure. In this series of papers, we develop the basic theory of reflectionless measures, and describe the relationship between the description of reflectionless measures and certain well-known problems in harmonic analysis and geometric measure theory.

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References

  1. G. David and P. Mattila, Removable sets for Lipschitz harmonic functions in the plane, Rev. Mat. Iberoamer. 16 (2000), 137–215.

    Article  MathSciNet  MATH  Google Scholar 

  2. G. David and S. Semmes, Analysis of and on uniformly rectifiable sets, Amer. Math. Soc., Providence, RI, 1993.

    Book  MATH  Google Scholar 

  3. V. Eiderman, F. Nazarov, and A. Volberg Vector-valued Riesz potentials: Cartan-type estimates and related capacities, Proc. London Math. Soc. (3) 101 (2010), 727–758.

    Article  MathSciNet  MATH  Google Scholar 

  4. V. Eiderman, F. Nazarov, and A. Volberg The s-Riesz transform of an s-dimensional measure in R2 is unbounded for 1 < s < 2, J. Anal. Math. 122 (2014), 1–23.

    Article  MathSciNet  MATH  Google Scholar 

  5. L. Grafakos, Classical Fourier Analysis, second edition, Springer, New York, 2008.

    MATH  Google Scholar 

  6. B. Jaye and F. Nazarov, Reflectionless measures and the Mattila-Melnikov-Verdera uniform rectifiability theorem, Geometric Aspects of Functional Analysis, Springer, Cham, 2014, pp. 199–229.

    MATH  Google Scholar 

  7. B. Jaye and F. Nazarov, Three revolutions in the kernel are worse than one, Int. Math. Res. Not. (IMRN) rnx101, 2017.

    Google Scholar 

  8. B. Jaye and F. Nazarov Reflectionless measures for Calderón-Zygmund operators II: Wolff potentials and rectifiability, J. Eur. Math. Soc. (JEMS), to appear.

  9. B. Jaye, F. Nazarov, M. C. Reguera, and X. Tolsa, The Riesz transform of codimension smaller than one and the Wolff energy, Mem. Amer. Math. Soc., to appear. arXiv:1602.02821 [math.AP].

  10. J. Korevaar, A Distribution Proof of Wiener’s Tauberian Theorem, Proc. Amer. Math. Soc. 16 (1965), 353–355.

    MathSciNet  MATH  Google Scholar 

  11. J. Mateu, L. Prat, and J. Verdera, The capacity associated to signed Riesz kernels, and Wolff potentials, J. Reine Angew. Math. 578 (2005), 201–223.

    MathSciNet  MATH  Google Scholar 

  12. P. Mattila, Geometry of Sets and Measures in Euclidean Spaces. Fractals and Rectifiability, Cambridge University Press, Cambridge, 1995.

    Book  MATH  Google Scholar 

  13. P. Mattila, Cauchy singular integrals and rectifiability in measures of the plane, Adv. Math. 115 (1995), 1–34.

    Article  MathSciNet  MATH  Google Scholar 

  14. P. Mattila, M. Melnikov, and J. Verdera, The Cauchy integral, analytic capacity, and uniform rectifiability, Ann. of Math. (2) 144 (1996), 127–136.

    Article  MathSciNet  MATH  Google Scholar 

  15. P. Mattila and D. Preiss, Rectifiable measures in Rn and existence of principal values for singular integrals, J. London Math. Soc. (2) 52 (1995), 482–496.

    Article  MathSciNet  MATH  Google Scholar 

  16. P. Mattila and J. Verdera, Convergence of singular integrals with general measures, J. Eur. Math. Soc. (JEMS) 11 (2009), 257–271.

    Article  MathSciNet  MATH  Google Scholar 

  17. M. Melnikov, A. Poltoratski, and A. Volberg, Uniqueness theorems for Cauchy integrals, Publ. Mat. 52 (2008), 289–314.

    Article  MathSciNet  MATH  Google Scholar 

  18. M. C. Reguera and X. Tolsa, em Riesz transforms of non-integer homogeneity on uniformly disconnected sets Trans, Amer. Math. Soc. 368 (2016), 7045–7095.

    Article  MathSciNet  MATH  Google Scholar 

  19. F. Nazarov, X. Tolsa, and A. Volberg, On the uniform rectifiability of AD regular measures with bounded Riesz transform operator: the case of codimension 1, ActaMath. 213 (2014), 237–321.

    MathSciNet  MATH  Google Scholar 

  20. F. Nazarov, S. Treil, and A. Volberg, Weak type estimates and Cotlar inequalities for Calderón-Zygmund operators on nonhomogeneous spaces, Int. Math. Res. Not. IMRN 9 (1998), 463–487.

    Article  MATH  Google Scholar 

  21. A. Poltoratski and C. Remling, Reflectionless Herglotz functions and Jacobi matrices, Comm. Math. Phys. 288 (2009), 1007–1021.

    Article  MathSciNet  MATH  Google Scholar 

  22. L. Prat, Potential theory of signed Riesz kernels: capacity and Hausdorff measure, Int. Math. Res. Not. IMRN 19 (2004), 937–981.

    Article  MathSciNet  MATH  Google Scholar 

  23. D. Preiss Geometry of measures in Rn: distribution, rectifiability, and densities, Ann. of Math. (2) 125 (1987), 537–643.

    Article  MathSciNet  MATH  Google Scholar 

  24. C. Remling, The absolutely continuous spectrum of Jacobi matrices, Ann. of Math. (2) 174 (2011), 125–171.

    Article  MathSciNet  MATH  Google Scholar 

  25. X. Tolsa and J. Verdera, May the Cauchy transform of a non-trivial finite measure vanish on the support of the measure? Ann. Acad. Sci. Fenn. Math. 31 (2006), 479–494.

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematical Sciences, Clemson University, Clemson, SC, 29631, USA

    Benjamin Jaye

  2. Department of Mathematical Sciences, Kent State University, Kent, OH, 44240, USA

    Fedor Nazarov

Authors
  1. Benjamin Jaye
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  2. Fedor Nazarov
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Correspondence to Benjamin Jaye.

Additional information

Supported in part by NSF DMS-1500881 and a Kent State Summer Research award.

Supported in part by NSF DMS-1265623.

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Jaye, B., Nazarov, F. Reflectionless measures for Calderón-Zygmund operators I: general theory. JAMA 135, 599–638 (2018). https://doi.org/10.1007/s11854-018-0047-6

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  • DOI: https://doi.org/10.1007/s11854-018-0047-6

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