Two-weight inequalities for the Hilbert transform of monotone functions

Two-weight inequalities for the Hilbert transform of monotone functions are characterized. The characterized weight inequality are restricted to the cones of odd or even monotone functions. The right-hand sides of inequalities is assumed to be infinite. An important problem of the theory of weight inequalities is finding conditions on nonnegative measurable functions. Weight inequalities for monotone functions are found and the discrete Hilbert transform are defined. The boundedness of the operators is also studied in the case of equal weights.