Multipliers for weighted Lp-spaces, transference, and the q-variation of functions

If 1 < p < infinity and omega is a weight function satisfying the A(p) condition on R (written omega is an element of A(p) (R), let M-p,M-omega (R) denote the algebra of multipliers for L-p(omega). By definition, M-p,M-omega (R) consists of the functions phi is an element of (R) such that the mapping f bar right arrow (phi (f) over cap)(V), defined initially on the Schwartz class, extends to a bounded linear transformation of L-p(omega) into itself. For 1 less than or equal to q < infinity, let M-q (R) denote the algebra of functions f is an element of L-infinity (R) such that the q-variation of f over the dyadic intervals is uniformly bounded. We show that if 2 less than or equal to p < infinity and omega is an element of A(p/2) (R), then there is a real number s > 2 such that M-q (R) subset of or equal to M-p,M-omega, (R) for 1 less than or equal to q < s. We also establish the counterpart of this result for sequence spaces l(p)(omega), omega is an element of A(p/2)(Z), by developing some machinery for transferring multipliers from one weighted setting to another. Our result on the inclusion M-q(R) subset of or equal to M-p,M-omega(R) states that, under suitable circumstances, Kurtz's weighted Marcinkiewicz Multiplier Theorem extends from M-1(R) to M-q(R), and also furnishes a counterpart for the (unweighted) generalization of the classical Marcinkiewicz Multiplier Theorem due to COIFMAN, de FRANCIA, and SEMMES. (C) Elsevier, Paris