Luzin Inequality for the Complement of Complex Ellipsoids in Сⁿ

We consider a generalization of Luzin area integral inequality for functions defined by Cauchy --- Leray --- Fantappie adjoint integral in the complement of complex ellipsoids in Сⁿ. In this work we obtain estimates that are useful for characterization of the smoothness of holomorphic functions by pseudoanalytical continuations. These results are a technical part of the investigation devoted to the description of spaces of holomorphic functions by polynomial approximations. Our methods could be considered as a model example of the application of vector-valued T1-theorem to the proof of nonlinear inequality

References

[1] Stein E.M. On the functions of Littlewood — Paley, Lusin, Marcinkiewicz. Trans. Amer. Math. Soc., 1958, vol. 88, pp. 430–466.

[2] Dynkin E.M. Estimates of analytic functions in Jordan domain. Journal of Soviet Mathematics, 1986, vol. 34, iss. 6, pp. 2060–2073. DOI: 10.1007/BF01741580

[3] Ahern P., Bruna J. Maximal and area integral characterization of Hardy — Sobolev spaces in the unit ball in Cn. Rev. Mat. Iberoamericana, 1988, vol. 4, no. 1, pp. 123–153.

[4] Krantz S., Li S.Y. Area integral characterizations of functions in Hardy spaces on domains in Cn. Complex Variables, 1997, vol. 32, no. 4, pp. 373–399.

[5] Sandrine G. Complex tangential characterizations of Hardy — Sobolev spaces of holomorphic functions. Rev. Mat. Iberoamericana, 1993, vol. 9, no. 2, pp. 201–255.

[6] Rotkevich A.S. External area integral inequality for the Cauchy — Leray — Fantappie integral. Available at: https://arxiv.org/abs/1707.08181 (accessed: 15.09.2017).

[7] Rotkevich A.S. Constructive description of the Besov classes in convex domains in ℂd. Journal of Mathematical Sciences, 2014, vol. 202, iss. 4, pp. 573–600. DOI: 10.1007/s10958-014-2064-z

[8] Bonami A., Lohoue N. Projecteurs de Bergman et Szego pour une classe de domaines faiblement pseudo-convexes et estimations. Comp. Math., 1982, vol. 46, no. 2, pp. 159–226.

[9] Hansson T. On Hardy spaces in complex ellipsoids. Ann. Inst. Fourier, 1999, vol. 49, no. 5, pp. 1477–1501. DOI: 10.5802/aif.1727

[10] Shirokov N.A. Uniform polynomial approximations on convex domains in ℂn. Journal of Mathematical Sciences, 2007, vol. 141, iss. 5, pp. 1564–1572. DOI: 10.1007/s10958-007-0064-y

[11] Hytonen T., Weis L. A T1 theorem for integral transformations with operator-valued kernel. J. Reine Angew. Math., 2006, vol. 599, pp. 155–200.

[12] Leray J. Le calcul differentiel et integral sur une variete analytique complexe (Probleme de Cauchy. III). Bull. Soc. Math. Fr., 1959, vol. 87, pp. 81–180. DOI: 10.24033/bsmf.1515