SHARP INEQUALITIES FOR THE CONDITIONAL SQUARE FUNCTION OF A MARTINGALE

Let f be a real martingale and s(f) its conditional square function. Then the following inequalities are sharp: parallel-to f parallel-to p less-than-or-equal-to square-root 2/p parallel-to s(f) parallel-to p, 0 < p less-than-or-equal-to 2, square-root 2/p parallel-to s(f) parallel-to p less-than-or-equal-to parallel-to f parallel-to p, p greater-than-or-equal-to 2. The second inequality is still sharp if f is replaced by the maximal function f*. Let S(f) denote the square function of f. Then the following inequalities are also sharp: parallel-to S(f) parallel-to p less-than-or-equal-to square-root 2/p parallel-to s(f) parallel-to p, 0 < p less-than-or-equal-to 2, square-root 2/p parallel-to s(f) parallel-to p less-than-or-equal-to parallel-to S(f) parallel-to p, p greater-than-or-equal-to 2. These inequalities hold for Hilbert-space-valued martingales and are strict inequalities in all of the nontrivial cases.