SHARP SQUARE-FUNCTION INEQUALITIES FOR CONDITIONALLY SYMMETRICAL MARTINGALES

Let f be a conditionally symmetric martingale taking values in a Hilbert space H and let S(f) be its square function. If nu-p is the smallest positive zero of the confluent hypergeometric function and mu-p is the largest positive zero of the parabolic cylinder function of parameter p, then the following inequalities are sharp: parallel-to f parallel-to p less-than-or-equal-to nu-p parallel-to S(f) parallel-to p if 0 < p less-than-or-equal-to 2, parallel-to f parallel-to p less-than-or-equal-to mu-p parallel-to S(f) parallel-to p if p greater-than-or-equal-to 3, nu-p parallel-to S(f) parallel-to p less-than-or-equal-to parallel-to f parallel-to p if p greater-than-or-equal-to 2. Moreover, the constants nu-p and mu-p for the cases mentioned above are also best possible for the Marcinkiewicz-Paley inequalities for Haar functions.