Let f# be the sharp
function introduced by Fefferman and Stein. Suppose that T and U are
operators acting on the space of Schwartz functions which satisfy the pointwise
estimate (Tf)#(x) ≤ A|Uf(x)|. Then, on the Lp spaces, the operator norm of
T divided by the operator norm of U is bounded by a constant times p.
This result allows us to obtain the best possible rate of growth estimate, as
p →∞, on the norms of singular integrals, multipliers, and pseudo-differential
operators. These estimates remain valid on weighted Lp spaces defined by an A∞
weight.
