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Boundedness and compactness of some operators on discrete Morrey spaces

Martha Guzmán-Partida (2021)

Commentationes Mathematicae Universitatis Carolinae

We consider discrete versions of Morrey spaces introduced by Gunawan et al. in papers published in 2018 and 2019. We prove continuity and compactness of multiplication operators and commutators acting on them.

Boundedness for a bilinear model sum operator on ℝⁿ

Erin Terwilleger (2007)

Studia Mathematica

The purpose of this article is to obtain a multidimensional extension of Lacey and Thiele's result on the boundedness of a model sum which plays a crucial role in the boundedness of the bilinear Hilbert transform in one dimension. This proof is a simplification of the original proof of Lacey and Thiele modeled after the presentation of Bilyk and Grafakos.

Boundedness from H 1 to L 1 of Riesz transforms on a Lie group of exponential growth

Peter Sjögren, Maria Vallarino (2008)

Annales de l’institut Fourier

Let G be the Lie group 2 + endowed with the Riemannian symmetric space structure. Let X 0 , X 1 , X 2 be a distinguished basis of left-invariant vector fields of the Lie algebra of G and define the Laplacian Δ = - ( X 0 2 + X 1 2 + X 2 2 ) . In this paper we consider the first order Riesz transforms R i = X i Δ - 1 / 2 and S i = Δ - 1 / 2 X i , for i = 0 , 1 , 2 . We prove that the operators R i , but not the S i , are bounded from the Hardy space H 1 to L 1 . We also show that the second-order Riesz transforms T i j = X i Δ - 1 X j are bounded from H 1 to L 1 , while the transforms S i j = Δ - 1 X i X j and R i j = X i X j Δ - 1 , for i , j = 0 , 1 , 2 , are not.

Boundedness of certain oscillatory singular integrals

Dashan Fan, Yibiao Pan (1995)

Studia Mathematica

We prove the L p and H 1 boundedness of oscillatory singular integral operators defined by Tf = p.v.Ω∗f, where Ω ( x ) = e i Φ ( x ) K ( x ) , K(x) is a Calderón-Zygmund kernel, and Φ satisfies certain growth conditions.

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