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<p>It is well known that the Riesz transform of a Lebesgue integrable function is generally not Lebesgue integrable. In this paper, using the notions of the <i>A</i>-integral and the <i>Q</i>-integral, introduced by Edward Titchmarsh, we prove analogues of Riesz’s equation and Titchmarsh’s equation for the Riesz transform of a Lebesgue integrable function. In particular, it follows from these equations that if the boundary values of a harmonic function on the upper half-space are Lebesgue integrable, then the boundary values of harmonically conjugate functions are integrable in the sense of the <i>Q</i>-integral and the harmonically conjugate functions are the Poisson <i>A</i>-integrals of their boundary values.</p>

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On the properties of the Riesz transform of Lebesgue integrable functions

  • 期刊论文

  • 发表日期：

    2026年7月7日

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  • Rashid A. Aliev,
  • Aynur F. Huseynli

摘要

It is well known that the Riesz transform of a Lebesgue integrable function is generally not Lebesgue integrable. In this paper, using the notions of the A-integral and the Q-integral, introduced by Edward Titchmarsh, we prove analogues of Riesz’s equation and Titchmarsh’s equation for the Riesz transform of a Lebesgue integrable function. In particular, it follows from these equations that if the boundary values of a harmonic function on the upper half-space are Lebesgue integrable, then the boundary values of harmonically conjugate functions are integrable in the sense of the Q-integral and the harmonically conjugate functions are the Poisson A-integrals of their boundary values.