No abstract

Part of the intrinsic structure of singular integrals in the Bessel setting is captured by Muckenhoupt-type weights. Anderson-Kerman showed that the Bessel Riesz transform is bounded on weighted L p w if and only if w is in the class A p,λ . We introduce a new class of Muckenhoupt-type weights A p,λ in the Bessel setting, which is different from A p,λ but characterizes the weighted boundedness for the Hardy-Littlewood maximal operators. We also establish the weighted L p boundedness and compactness, as well as the endpoint weak type boundedness of Riesz commutators. The quantitative weighted bound is also established.

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.

Let A p α (B n ; C d ) be the weighted Bergman space on the unit ball B n of C n of functions taking values in C d . For 1 < p < ∞ let T p,α be the algebra generated by finite sums of finite products of Toeplitz operators with bounded matrix-valued symbols (this is called the Toeplitz algebra in the case d = 1). We show that every S ∈ T p,α can be approximated by localized operators. This will be used to obtain several equivalent expressions for the essential norm of operators in T p,α . We then use this to characterize compact operators in A p α (B n ; C d ). The main result generalizes previous results and states that an operator in A p α (B n ; C d ) is compact if only if it is in T p,α and its Berezin transform vanishes on the boundary.2000 Mathematics Subject Classification. 32A36, 32A, 47B05, 47B35.