Bounds of Riesz Transforms on $L^p$ Spaces for Second Order Elliptic Operators

Zhongwei Shen

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Riesz transforms for the Dunkl Ornstein-Uhlenbeck operator

Adam Nowak, Luz Roncal, Krzysztof Stempak (2010)

Colloquium Mathematicae

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We propose a definition of Riesz transforms associated to the Ornstein-Uhlenbeck operator based on the Dunkl Laplacian. In the case related to the group ℤ ₂ it is proved that the Riesz transform is bounded on the corresponding $L^{p}$ spaces, 1 < p < ∞.

Riesz potentials and amalgams

Michael Cowling, Stefano Meda, Roberta Pasquale (1999)

Annales de l'institut Fourier

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Let $(M, d)$ be a metric space, equipped with a Borel measure $μ$ satisfying suitable compatibility conditions. An amalgam $A_{p}^{q} (M)$ is a space which looks locally like $L^{p} (M)$ but globally like $L^{q} (M)$ . We consider the case where the measure $μ (B (x, ρ)$ of the ball $B (x, ρ)$ with centre $x$ and radius $ρ$ behaves like a polynomial in $ρ$ , and consider the mapping properties between amalgams of kernel operators where the kernel $ker K (x, y)$ behaves like $d (x, y)^{- a}$ when $d (x, y) \leq 1$ and like $d (x, y)^{- b}$ when $d (x, y) \geq 1$ . As an application, we describe Hardy–Littlewood–Sobolev type regularity...

Characterization of Globally Lipschitz Nemytskiĭ Operators Between Spaces of Set-Valued Functions of Bounded φ-Variation in the Sense of Riesz

N. Merentes, J. L. Sánchez Hernández (2004)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let (X,∥·∥) and (Y,∥·∥) be two normed spaces and K be a convex cone in X. Let CC(Y) be the family of all non-empty convex compact subsets of Y. We consider the Nemytskiĭ operators, i.e. the composition operators defined by (Nu)(t) = H(t,u(t)), where H is a given set-valued function. It is shown that if the operator N maps the space $R V_{φ ₁} ([a, b]; K)$ into $R W_{φ ₂} ([a, b]; C C (Y))$ (both are spaces of functions of bounded φ-variation in the sense of Riesz), and if it is globally Lipschitz, then it has to be of the form H(t,u(t))...

Remark on the inequality of F. Riesz

W. Łenski (2005)

Banach Center Publications

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We prove F. Riesz’ inequality assuming the boundedness of the norm of the first arithmetic mean of the functions ${| φ ₙ |}^{p}$ with p ≥ 2 instead of boundedness of the functions φₙ of an orthonormal system.

On definitions of superharmonic functions

Seizô Itô (1975)

Annales de l'institut Fourier

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Let $A$ be an elliptic differential operator of second order with variable coefficients. In this paper it is proved that any $A$ -superharmonic function in the Riesz-Brelot sense is locally summable and satisfies the $A$ -superharmonicity in the sense of Schwartz distribution.

$L^{p}$ boundedness of Riesz transforms for orthogonal polynomials in a general context

Liliana Forzani, Emanuela Sasso, Roberto Scotto (2015)

Studia Mathematica

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Nowak and Stempak (2006) proposed a unified approach to the theory of Riesz transforms and conjugacy in the setting of multi-dimensional orthogonal expansions, and proved their boundedness on L². Following them, we give easy to check sufficient conditions for their boundedness on $L^{p}$ , 1 < p < ∞. We also discuss the symmetrized version of these transforms.

Riesz spaces of order bounded disjointness preserving operators

Fethi Ben Amor (2007)

Commentationes Mathematicae Universitatis Carolinae

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Let $L$ , $M$ be Archimedean Riesz spaces and $ℒ_{b} (L, M)$ be the ordered vector space of all order bounded operators from $L$ into $M$ . We define a Lamperti Riesz subspace of $ℒ_{b} (L, M)$ to be an ordered vector subspace $ℒ$ of $ℒ_{b} (L, M)$ such that the elements of $ℒ$ preserve disjointness and any pair of operators in $ℒ$ has a supremum in $ℒ_{b} (L, M)$ that belongs to $ℒ$ . It turns out that the lattice operations in any Lamperti Riesz subspace $ℒ$ of $ℒ_{b} (L, M)$ are given pointwise, which leads to a generalization of the classic Radon-Nikod’ym theorem...

Triebel-Lizorkin spaces with non-doubling measures

Yongsheng Han, Dachun Yang (2004)

Studia Mathematica

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Suppose that μ is a Radon measure on $ℝ^{d}$ , which may be non-doubling. The only condition assumed on μ is a growth condition, namely, there is a constant C₀ > 0 such that for all x ∈ supp(μ) and r > 0, μ(B(x,r)) ≤ C₀rⁿ, where 0 < n ≤ d. The authors provide a theory of Triebel-Lizorkin spaces $Ḟ_{p q}^{s} (μ)$ for 1 < p < ∞, 1 ≤ q ≤ ∞ and |s| < θ, where θ > 0 is a real number which depends on the non-doubling measure μ, C₀, n and d. The method does not use the vector-valued maximal function...

Norm estimates for Bessel-Riesz operators on generalized Morrey spaces

Mochammad Idris, Hendra Gunawan, A. Eridani (2018)

Mathematica Bohemica

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We revisit the properties of Bessel-Riesz operators and present a different proof of the boundedness of these operators on generalized Morrey spaces. We also obtain an estimate for the norm of these operators on generalized Morrey spaces in terms of the norm of their kernels on an associated Morrey space. As a consequence of our results, we reprove the boundedness of fractional integral operators on generalized Morrey spaces, especially of exponent $1$ , and obtain a new estimate for their...

Logarithmic inequalities for second-order Riesz transforms and related Fourier multipliers

Adam Osękowski (2013)

Colloquium Mathematicae

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We study logarithmic estimates for a class of Fourier multipliers which arise from a nonsymmetric modulation of jumps of Lévy processes. In particular, this leads to corresponding tight bounds for second-order Riesz transforms on $ℝ^{d}$ .

Displaying similar documents to “Bounds of Riesz Transforms on L p Spaces for Second Order Elliptic Operators”

Displaying similar documents to “Bounds of Riesz Transforms on $L^{p}$ Spaces for Second Order Elliptic Operators”