On Entropy Bumps for Calderón-Zygmund Operators

Abstract

We study two weight inequalities in the recent innovative language of `entropy' due to Treil-Volberg. The inequalities are extended to $L^p$, for $1<p\ne 2<\infty$, with new short proofs. A result proved is as follows. Let $\epsilon$ be a monotonic increasing function on $(1,\infty )$ which satisfy ${\int }_1^{\infty }\frac{dt}{\epsilon (t)t}=1$. Let $\sigma$ and $w$ be two weights on ${\mathbb{R}}^d$. If this supremum is finite, for a choice of $1<p<\infty$, $$\underset{Q}{\sup }[\frac{\sigma (Q)}{|Q|}{]}^{p-1}\frac{{\int }_{Q}M(\sigma {\chi }_Q)}{\sigma (Q)}\cdot \frac{w(Q)}{|Q|}[\frac{{\int }_{Q}M(w{\chi }_Q)}{w(Q)}{]}^{p-1}<\infty ,$$ then any Calderón-Zygmund operator $T$ satisfies the bound $\Vert T_{\sigma }f{\Vert }_{L^p(w)}\lesssim \Vert f{\Vert }_{L^p(\sigma )}$.

8 pages. V2 typos corrected. To appear in Concrete Operators