On integrals of conjugate functions

This paper is devoted to problems of integrability of conjugate functions. We study the problem of when the validity of the Riesz equation $\int_0^{2\pi}\varphi\overline f\,dx=-\int_0^{2\pi}f\overline\varphi\,dx$ for a single function $\varphi$ implies its validity for other functions, and also the problems of the integrability in the Denjoy–Hinchin sense of functions conjugate to summable functions, of the integrability in the Lebesgue sense of functions conjugate to Denjoy–Hinchin integrable functions, and of the $A$-, $A^*$- and $U$-integrability of functions conjugate to functions that have improper Lebesgue integrals.
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