Abstract
We study the weighted boundedness of the Cauchy singular integral operator S Γ in Morrey spaces L p , λ ( Γ ) on curves satisfying the arc-chord condition, for a class of “radial type” almost monotonic weights. The non-weighted boundedness is shown to hold on an arbitrary Carleson curve. We show that the weighted boundedness is reduced to the boundedness of weighted Hardy operators in Morrey spaces L p , λ ( 0 , ℓ ) , ℓ > 0 . We find conditions for weighted Hardy operators to be bounded in Morrey spaces. To cover the case of curves we also extend the boundedness of the Hardy–Littlewood maximal operator in Morrey spaces, known in the Euclidean setting, to the case of Carleson curves.
Highlights
The well-known Morrey spaces Lp,λ introduced in [31] in relation to the study of partial differential equations, and presented in various books, see [19,27,48], were widely investigated during last decades, including the study of classical operators of harmonic analysis—maximal, singular and potential operators—in these spaces; we refer for instance to the papers [1,2,4,6,10,13,14,33,34,36,37,38,39,41,44,45,46], where Morrey spaces on metric measure spaces may be found
In this paper we deal with the one-dimensional case and study the weighted boundedness of the Cauchy singular integral operator
We make use of the known non-weighted boundedness of SΓ in this case, which enables us to reduce the boundedness of SΓ with weight (1.2) to the boundedness of weighted Hardy operators
Summary
The well-known Morrey spaces Lp,λ introduced in [31] in relation to the study of partial differential equations, and presented in various books, see [19,27,48], were widely investigated during last decades, including the study of classical operators of harmonic analysis—maximal, singular and potential operators—in these spaces; we refer for instance to the papers [1,2,4,6,10,13,14,33,34,36,37,38,39,41,44,45,46], where Morrey spaces on metric measure spaces may be found. We make use of the known non-weighted boundedness of SΓ in this case, which enables us to reduce the boundedness of SΓ with weight (1.2) to the boundedness of weighted Hardy operators We prove their boundedness in the Morrey space Lp,λ(0, ). These conditions are necessary in the case of power weights. Chiarenza–Frasca’s proof [10] of the boundedness of the maximal operator to the case of metric measure spaces X with constant dimension and prove the Fefferman–Stein inequality M f Lp,λ(X) C M# f Lp,λ(X), to derive the non-weighted boundedness of SΓ via (1.3).
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
