On Entropy Bumps for Calderón-Zygmund Operators

Abstract

Abstract We study twoweight inequalities in the recent innovative language of ‘entropy’ due to Treil-Volberg. The inequalities are extended to Lp, for 1 < p ≠ 2 < ∞, with new short proofs. A result proved is as follows. Let ɛ be a monotonic increasing function on (1,∞) which satisfy Let σ and w be two weights on ℝd. If this supremum is finite, for a choice of 1 < p < ∞, then any Calderón-Zygmund operator T satisfies the bound ||Tof||Lp(w) ≲ ||f|| Lp(o).