On Entropy Bumps for Calder\'on-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 \neq 2 < \infty $, with new short proofs. A result proved is as follows. Let $ \varepsilon $ be a monotonic increasing function on $ (1, \infty)$ which satisfy $ \int _{1} ^{\infty} \frac {dt} {\varepsilon (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 $, $$ \sup _{Q} \biggl[ \frac {\sigma (Q)} {\lvert Q\rvert} \biggr]^{p-1} \frac {\int _{Q} M (\sigma \chi_{Q})} {\sigma (Q)} \cdot \frac {w (Q)} {\lvert Q\rvert}\biggl[ \frac {\int _{Q} M (w \chi_{Q})} {w (Q)}\biggr]^{p-1} < \infty, $$ then any Calder\'on-Zygmund operator $ T$ satisfies the bound $ \lVert T _{\sigma} f \rVert _{L ^{p} (w)} \lesssim \lVert f\rVert _{L ^{p} (\sigma)} $.