Weighted Hardy and singular operators in Morrey spaces

By Natasha Samko. 2008

151 citations recorded by the source corpus.

We study the weighted boundedness of the Cauchy singular integral operator $S_\Gm$ in Morrey spaces $L^{p,\lambda}(\Gm)$ 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 $L^{p,\lambda}(0,\ell), \ell>0$. 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. Key words and phrases: Morrey space, singular operator, Hardy operator, Hardy-Littlewood maximal operator, weighted estimate.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/51c010e43e7bb32e5874d5fa391137bb7dc9a604

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In Applied Mathematics

Citations in the corpus

Maximal functions and potentials in variable exponent Morrey spaces with non-doubling measure (2010, background): “Some weighted inequalities for potentials in Morrey spaces defined on quasimetric measure spaces were derived in [5] (see also [6] for the boundedness of singular Cauchy operators along Carleson curves in weighted Morrey spaces with constant exponents).”
The singular integral operator and its commutator on weighted Morrey spaces (2016, methodology): “tion (2) for p,q and wα is equivalent to α > −q pn; see [30], then the suﬃcient condition in Theorem 1.14 turns into (20) − q p n < α < n(q −1) in the case of power weights wα. Meanwhile, in [33], Samko obtained the characterization of the boundedness of the Hilbert transform in the case of”
Commutators of Hardy operators in vanishing Morrey spaces (2012, methodology): “The approach we used in the papers [26], [27] for the study of maximal, singular and potential operators in weighted Morrey spaces, allows us to reduce the problem of the boundedness of commutators of such operators to the study of commutators of Hardy type operators.”
Weighted Hardy operators in the local generalized vanishing Morrey spaces (2012, unknown): “Most of them may be found dispersed in various papers, for instance, in [8,15,25,27,28], but for reader’s convenience we gathered them in Appendix.”
Weighted Hardy and potential operators in the generalized Morrey spaces (2011, unknown): “Just recently, in [38] we proved weighted p → p-estimates in Morrey spaces L p,λ for Hardy operators on R(1)+ and onedimensional singular operators (on R1 or on Carleson curves in the complex plane).”
Calderón–Zygmund operators in Morrey spaces (2014, unknown): “We refer in particular to [8,10,11,13–15,17,19,20,28,30–32,36].”