On Entropy Bumps for Calderón-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 $1< p \neq 2 < \infty$, 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 $σ$ and $w$ be two weights on $\mathbb R ^{d}$. If this supremum is finite, for a choice of $1< p < \infty$, $\sup _{Q} \biggl[ \frac {σ(Q)} {\lvert Q\rvert} \biggr]^{p-1} \frac {\int _{Q} M (σχ_{Q})} {σ(Q)} \cdot \frac {w (Q)} {\lvert Q\rvert}\biggl[ \frac {\int _{Q} M (w χ_{Q})} {w (Q)}\biggr]^{p-1} < \infty,$ then any Calderón-Zygmund operator $T$ satisfies the bound $\lVert T _σ f \rVert _{L ^{p} (w)} \lesssim \lVert f\rVert _{L ^{p} (σ)}$.

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