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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 < p ≠ < ∞, with new short proofs. A result proved is as follows.Let ε be a monotonic increasing function on ( , ∞) which satisfy ∞ dt ε(t)t = . Let σ and w be two weights onthen any Calderón-Zygmund operator T satis es the bound ||Tσ f || L p (w) ||f || L p (σ) .

Abstract. In this short note, we give a very efficient proof of a recent result of TreilVolberg and Lacey-Spencer giving sufficient conditions for the two-weight boundedness of a sparse operator. We also give a new sufficient condition for the two-weight boundedness of a sparse operator. We make critical use of a formula of Hytönen in [6].

Abstract:We investigate weighted inequalities for fractional maximal operators and fractional integral operators. We work within the innovative framework of "entropy bounds" introduced by Treil-Volberg. Using techniques developed by Lacey and the second author, we are able to efficiently prove the weighted inequalities.