Annals of Functional Analysis

Nonlinear harmonic analysis of integral operators in weighted grand Lebesgue spaces and applications

Alberto Fiorenza and Vakhtang Kokilashvili

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Abstract

In this article, we give a boundedness criterion for Cauchy singular integral operators in generalized weighted grand Lebesgue spaces. We establish a necessary and sufficient condition for the couple of weights and curves ensuring boundedness of integral operators generated by the Cauchy singular integral defined on a rectifiable curve. We characterize both weak and strong type weighted inequalities. Similar problems for Calderón–Zygmund singular integrals defined on measured quasimetric space and for maximal functions defined on curves are treated. Finally, as an application, we establish existence and uniqueness, and we exhibit the explicit solution to a boundary value problem for analytic functions in the class of Cauchy-type integrals with densities in weighted grand Lebesgue spaces.

Article information

Source
Ann. Funct. Anal., Volume 9, Number 3 (2018), 413-425.

Dates
Received: 6 June 2017
Accepted: 5 September 2017
First available in Project Euclid: 6 February 2018

Permanent link to this document
https://projecteuclid.org/euclid.afa/1517886227

Digital Object Identifier
doi:10.1215/20088752-2017-0056

Mathematical Reviews number (MathSciNet)
MR3835228

Zentralblatt MATH identifier
06946365

Subjects
Primary: 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.)
Secondary: 42B25: Maximal functions, Littlewood-Paley theory 46E30: Spaces of measurable functions (Lp-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)

Keywords
Cauchy singular integral operator Carleson curve Muckenhoupt $A_{p}$ class Calderón–Zygmund singular integrals Riemann boundary value problem

Citation

Fiorenza, Alberto; Kokilashvili, Vakhtang. Nonlinear harmonic analysis of integral operators in weighted grand Lebesgue spaces and applications. Ann. Funct. Anal. 9 (2018), no. 3, 413--425. doi:10.1215/20088752-2017-0056. https://projecteuclid.org/euclid.afa/1517886227


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References

  • [1] A. Böttcher and Y. I. Karlovich, Carleson Curves, Muckenhoupt Weights, and Toeplitz Operators, Progr. Math. 154, Birkhäuser, Basel, 1997.
  • [2] A.-P. Calderón, Cauchy integrals on Lipschitz curves and related operators, Proc. Natl. Acad. Sci. USA 74 (1977), no. 4, 1324–1327.
  • [3] C. Capone, M. R. Formica, and R. Giova, Grand Lebesgue spaces with respect to measurable functions, Nonlinear Anal. 85 (2013), 125–131.
  • [4] D’Onofrio, C. Sbordone, and R. Schiattarella, Grand Sobolev spaces and their applications in geometric function theory and PDEs, J. Fixed Point Theory Appl. 13 (2013), no. 2, 309–340.
  • [5] E. M. Dyn’kin, Methods of the theory of singular integrals (the Hilbert transform and Calderón-Zygmund theory), Itogi Nauki. Tekh. Ser. Sovrem. Mat. Prilozh. Temat. Obz. 15 (1987), 197–202.
  • [6] A. Fiorenza, M. R. Formica, and J. M. Rakotoson, Pointwise estimates for $G\Gamma$-functions and applications, Differential Integral Equations 30 (2017), no. 11-12, 809–824.
  • [7] A. Fiorenza and J. M. Rakotoson, Some estimates in $G\Gamma(p,m,w)$ spaces, J. Math. Anal. Appl. 340 (2008), no. 2, 793–805.
  • [8] N. Fusco, P.-L. Lions, and C. Sbordone, Sobolev imbedding theorems in borderline cases, Proc. Amer. Math. Soc. 124 (1996), no. 2, 561–565.
  • [9] E. Greco, T. Iwaniec, and C. Sbordone, Inverting the $p$-harmonic operator, Manuscripta Math. 92 (1997), no. 2, 249–258.
  • [10] V. P. Havin, Boundary properties of integrals of Cauchy type and of conjugate harmonic functions in regions with rectifiable boundary, Math. Sb. (N.S.) 68 (1965), no. 4, 499–517.
  • [11] T. Iwaniec and C. Sbordone, On the integrability of the Jacobian under minimal hypotheses, Arch. Ration. Mech. Anal. 119 (1992), no. 2, 129–143.
  • [12] G. Khuskivadze, V. Kokilashvili, and V. Paatashvili, Boundary value problems for analytic and harmonic functions in domain with nonsmooth boundaries: Application to conformal mappings, Mem. Differential Equations Math. Phys. 14 (1998), 1–195.
  • [13] V. Kokilashvili, A. Meskhi, H. Rafeiro, and S. Samko, Integral Operators in Non-Standard Function Spaces, II: Variable Exponent Hölder, Morrey-Campanato and grand spaces, Oper. Theory Adv. Appl. 249, Birkhäuser, Cham, 2016.
  • [14] V. Kokilashvili and V. Paatashvili, Boundary Value Problems for Analytic and Harmonic Functions in Nonstandard Banach Function Spaces, Nova Science, New York, 2012.
  • [15] I. I. Privalov, Randeigenschaften analytischer Funktionen, II, Hochschulbücher für Mathematik 25, VEB Dt. Verl. d. Wiss., Berlin, 1956.
  • [16] J.-O. Strömberg and A. Torchinsky, Weighted Hardy Spaces, Lecture Notes in Math. 1381, Springer, Berlin, 1989.