A Best Constant for Zygmund's Conjugate Function Inequality

When the space L log+L is given the Hardy-Littlewood norm the best constant in the corresponding version of Zygmund's conjugate function inequality is shown to be r2 3~2 + 5-2 7-2 + • ■ ■ K = I-2 + 3"2 + 5"2 + 7" This complements the recent result of Burgess Davis that the best constant in Kolmogorov's inequality is K"1. The symbol K will be used throughout for the constant p2 _ 3-2 + 5-2 _ 7-2 + . . . K = —,--.--.--.-= 0.7424537 • • •. r2 + 3~2 + 5~2 + i~2 + ■ ■ ■ Let /be a real-valued 27r-periodic function of class L'(0,2-n) = Lx, and let/ be its conjugate function [10, Chapter IV]. It is well known that/is finite a.e. on (0, 2vt) but need not itself be integrable. There is however the following weak-type estimate for/due to Kolmogorov [10, p. 134], the best constant for which was determined recently by Davis [5] (cf. also [4]). Theorem 1 (Kolmogorov, Davis). When f G Lx the conjugate function f satisfies the inequality (1) ym{x: \f(x)\ > y) < K"1 fQ " \f(x)\dx, y > 0,2 and the constant K~ ' is best-possible. Complementing Kolmogorov's inequality is the result of Zygmund [10, p. 254] that the conjugate function/is integrable whenever/is of class L log+L. The admissible constants in Zygmund's inequality have been determined by Pichorides [8] as follows. Theorem 2 (Zygmund, Pichorides). There are absolute constants A and B = B(A) such that for every f G L log+L the conjugate function f satisfies the inequality (2) ¡^ \f(x)\ dx i C ^g(2m/t)dt=l. ¿m JO ¿m JO We claim that the inequality (7 ) fS log cot (// 8 ) dt < a f ' log (2m/1) dt holds for every í with 0 < s < 27r; equivalently, if we set License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use