The A-Integral and Restricted Riesz Transform

Year 2020, Volume: 3 Issue: 3, 104 - 112, 14.09.2020

Rashid Aliev Khanim Nebiyeva

https://doi.org/10.33205/cma.728156

Abstract

It is known that the restricted Riesz transform of a Lebesgue integrable function is not Lebesgue integrable. In this paper we prove that the restricted Riesz transform of a Lebesgue integrable function is A-integrable and the analogue of Riesz's equality holds.

ABSTRACT.It is known that the restricted Riesz transform of a Lebesgue integrable function is not Lebesgue inte-grable. In this paper, we prove that the restricted Riesz transform of a Lebesgue integrable function isA-integrableand the analogue of Riesz’s equality holds

Keywords

Riesz transform, A-integral, Riesz's inequality, covering theorem

References

• A. B. Aleksandrov: A-integrability of the boundary values of harmonic functions. Math. Notes 30(1) (1981), 515–523.
• R. A. Aliev: N ± -integrals and boundary values of Cauchy-type integrals of finite measures. Sbornik: Mathematics 205(7) (2014), 913–935.
• R. A. Aliev: On properties of Hilbert transform of finite complex measures. Complex Analysis and Operator Theory 10(1) (2016), 171–185.
• R. A. Aliev: Riesz’s equality for the Hilbert transform of the finite complex measures. Azerb. J. Math. 6(1) (2016), 126–135.
• R. A. Aliev: Representability of Cauchy-type integrals of finite complex measures on the real axis in terms of their boundary values. Complex Variables and Elliptic Equations 62(4) (2017), 536–553.
• R. A. Aliev, K. I. Nebiyeva: The A-integral and Restricted Complex Riesz Transform. Azerbaijan Journal of Mathe- matics 10(1) (2020), 209–221.
• A. S. Besicovitch: On a general metric property of summable functions. J. London Math. Soc. 1(2) (1926), 120–128.
• M. P. Efimova: On the properties of the Q-integral. Math. Notes 90(3-4) (2011), 322–332.
• M. P. Efimova: The sufficient condition for integrability of a generalized Q-integral and points of integrability. Moscow Univ. Math. Bull. 70(4) (2015), 181–184.
• L. C. Evans, R. F. Gariepy: Measure theory and fine properties of functions. CRC Press, Boca Raton (1992).
• M. A. Ragusa: Elliptic boundary value problem in Vanishing Mean Oscillation hypothesis. Comment. Math. Univ. Carolin 40(4) (1999), 651–663.
• M. A. Ragusa: Necessary and sufficient condition for a VMO function. Applied Mathematics and Computation 218(24) (2012), 11952–11958.
• T. S. Salimov: The A-integral and boundary values of analytic functions. Math. USSR-Sbornik 64(1) (1989), 23–40.
• V. A. Skvortsov: A-integrable martingale sequences and Walsh series. Izvestia: Math. 65(3) (2001), 607–616.
• E. M. Stein: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton (1970).
• E. C. Titchmarsh: On conjugate functions. Proc. London Math. Soc. 9 (1929), 49–80.
• P. L. Ul’yanov: The A-integral and conjugate functions. Mathematics, vol.7 (1956), Uch. Zap. Mosk. Gos. Univ. 181, 139–157, (in Russian).
• P. L. Ul’yanov: Integrals of Cauchy type. Twelve Papers on Approximations and Integrals. Amer. Math. Soc. Trans. 2(44) (1965), 129-150.