Calderon weights as Muckenhoupt weights

Abstract

Abstract. The Calderón operator S is the sum of the the Hardy averaging operator and its adjoint. The weights w for which S is bounded on L p (w) are the Calderón weights of the class C p . We give a new characterization of the weights in C p by a single condition which allows us to see that C p is the class of Muckenhoupt weights associated to a maximal operator defined through a basis in (0, ∞). The same condition characterizes the weighted weaktype inequalities for 1 < p < ∞, but that the weights for the strong type and the weak type differ for p = 1. We also prove that the weights in C p do not behave like the usual A p weights with respect to some properties and, in particular, we answer an open question on extrapolation for Muckenhoupt bases without the openness property.

The Calderón operator and the Stieltjes transform on variable Lebesgue spaces with weights

“…Our results extend those in [18] for the constant exponent L p spaces with weights. We also give two applications: the first is a weighted version of Hilbert's inequality on variable Lebesgue spaces, and the second generalizes the results in [42] for integral operators to the variable exponent setting.…”

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We characterize the weights for the Stieltjes transform and the Calderón operator to be bounded on the weighted variable Lebesgue spaces L p(·) w (0, ∞), assuming that the exponent function p(·) is log-Hölder continuous at the origin and at infinity. We obtain a single Muckenhoupt-type condition by means of a maximal operator defined with respect to the basis of intervalsOur results extend those in [18] for the constant exponent L p spaces with weights. We also give two applications: the first is a weighted version of Hilbert's inequality on variable Lebesgue spaces, and the second generalizes the results in [42] for integral operators to the variable exponent setting.2010 Mathematics Subject Classification.

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Weighted estimates for integral operators on local type spaces

“…Using the characterization of weighted boundedness for S and proved in and , we obtain. Theorem If and satisfies for almost all , then the operator is of weighted strong type for , and of weighted weak type (1, 1) if .…”

“…In , the authors consider the classical Hardy operator P and its adjoint Q given by and the Calderón operator S defined as . They prove a characterization of the weights for which S is of weighted type , , and of weighted weak type (1, 1), by a single condition.…”

“…Therefore, this class of weights is denoted by and . Remark In both articles and , some known properties of were also analyzed for and . Also, some counterexamples are provided.…”