Discrete Ahlfors--Beurling transform and its properties

Rashid A. Aliev ; Aynur N. Ahmadova

Problemy analiza, Tome 9 (2020) no. 2, pp. 3-15.

Résumé

The Ahlfors–Beurling transform has been well studied on classical Lebesgue, Morrey, Sobolev, Besov, Campanato, etc. spaces. However, its discrete version is still not studied well. In this paper, we study the properties of the discrete Ahlfors–Beurling transform on discrete Lebesgue spaces.

Détail

Keywords: discrete Ahlfors–Beurling transform, boundedness.
Mots-clés : Lebesgue spaces

Bibliographie

[1] Ahlfors L. V., Lectures on Quasiconformal Mappings, University Lecture Series, 38, 2nd ed., AMS, Providence, RI, 2006 | DOI | MR | Zbl

[2] Aliev R. A., Amrahova A. F., “Properties of the discrete Hilbert transform”, Complex Analysis and Operator Theory, 13 (2019), 3883–3897 | DOI | MR | Zbl

[3] Aliev R. A., Amrahova A. F., “On the summability of the discrete Hilbert transform”, Ural Math. J., 4:2 (2018), 6–12 | DOI | MR | Zbl

[4] Aliev R. A., Nabiyeva Kh. I., “The A-integral and restricted Ahlfors-Beurling transform”, Integral Transforms and Special Functions, 29:10 (2018), 820–830 | DOI | MR | Zbl

[5] Astala K., Iwaniec T., Martin G., Elliptic partial differential equations and quasiconformal mappings in the plane, University Press, Princeton, 2009 | MR | Zbl

[6] Banuelos R., Janakiraman P., “$L^{p} $-bounds for the Beurling-Ahlfors transform”, Trans. Amer. Math. Soc., 360:7 (2008), 3603–3612 | DOI | MR | Zbl

[7] Calderon A. P., Zygmund A., “On the existence of certain singular integrals”, Acta Mathematica, 88 (1952), 85–139 | DOI | MR | Zbl

[8] Cruz V., Mateu J., Orobitg J., “Beltrami equation with coefficient in Sobolev and Besov spaces”, Canadian J. Math., 65:6 (2013), 1217–1235 | DOI | MR | Zbl

[9] Cruz V., Tolsa X., “Smoothness of the Beurling transform in Lipschitz domains”, J. Func. Anal., 262:10 (2012), 4423–4457 | DOI | MR | Zbl

[10] Doubtsov E., Vasin A. V., “Restricted Beurling transforms on Campanato spaces”, Complex Variables and Elliptic Eq., 62:3 (2017), 333–346 | DOI | MR | Zbl

[11] Dragicevic O., “Weighted estimates for powers of the Ahlfors-Beurling operators”, Proc. Amer. Math. Soc., 139:6 (2011), 2113–2120 | DOI | MR | Zbl

[12] Kwok-Pun H., “The Ahlfors–Beurling transform on Morrey spaces with variable exponents”, Integral Transforms and Special Functions, 29:3 (2018), 207–220 | DOI | MR | Zbl

[13] Mateu J., Orobitg J., Verdera J., “Extra cancellation of even Calderon-Zygmund operators and quasiconformal mappings”, J. Math. Pures et Appl., 91:4 (2009), 402–431 | DOI | MR | Zbl

[14] Prats M., “$L^{p} $-bounds for the Beurling-Ahlfors transform”, Publicacions Mat., 61:2 (2017), 291–336 | DOI | MR | Zbl

[15] Riesz M., “Sur les fonctions conjuguees”, Mathematische Zeitschrift, 27 (1928), 218–244 | DOI | MR

[16] Stein E.M., Singular Integrals and Differentiability Properties of Functions, University Press, Princeton, 1970 | MR | Zbl

[17] Titchmarsh E. C., “Reciprocal formulae involving series and integrals”, Math. Z., 25 (1926), 321–347 | DOI | MR | Zbl

[18] Tolsa X., “Regularity of $C^{1} $ and Lipschitz domains in terms of the Beurling transform”, J. Math. Pures et Appl., 100 (2013), 137–165 | DOI | MR | Zbl

[19] Vasin A.V., “Regularity of the Beurling Transform in Smooth Domains”, J. Math. Sciences, 215:5 (2016), 577–584 | DOI | MR | Zbl

[20] Vekua I.N., Generalized analytic functions, Pergamon Press, 1962 | MR | Zbl