On Entropy Bumps for Calder\'on-Zygmund Operators

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