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

Abstract

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. Primary 42B25; Secondary 26D15, 42B35.

Calderón Operator on Local Morrey Spaces with Variable Exponents

“…The boundedness of the Calderón operator on Lebesgue spaces is a well known result [2]. Recently, the boundedness property has been extended to the weighted Lebesgue spaces [3] and the weighted Lebesgue spaces with variable exponents [4]. In this paper, we further extend the boundedness of the Calderón operator to local Morrey spaces with variable exponents.…”

“…In this paper, we use another maximal function from [3] which is defined via the basis {(0, r) : r > 0}. Similar to the results in [4], by using this maximal function, the exponent functions for the local Morrey spaces with variable exponents is not required to be globally log-Hölder continuous function. The exponent function is just required to be log-Hölder continuous at origin and infinity.…”