AN INEQUALITY FOR THE HILBERT TRANSFORM
Ray Redheffer · Proceedings of the National Academy of Sciences · 1968
Throughout this note f and g denote complex-valued functions of the real variable u with domain (-0 ,0) U (0, co), m > 1 is an integer, and p and q are finite complementary Lebesgue exponents.We define K(r) = r 1 for Ir<, K(r) + K =1 and, if the integral exists as a Cauchy principal value at 0, x, and A= co, f*(x) = Pf [x u -KX ]Xf(u)du.For example, when m = 1, the expression is