Norm estimates for Bessel-Riesz operators on generalized Morrey spaces

Mochammad Idris; Hendra Gunawan; A. Eridani

Displaying similar documents to “Norm estimates for Bessel-Riesz operators on generalized Morrey spaces”

On (n,k)-quasiparanormal operators

Jiangtao Yuan, Guoxing Ji (2012)

Studia Mathematica

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Let T be a bounded linear operator on a complex Hilbert space . For positive integers n and k, an operator T is called (n,k)-quasiparanormal if $| | T^{1 + n} (T^{k} x) {| |}^{1 / (1 + n)} | | T^{k} {x | |}^{n / (1 + n)} \geq | | T (T^{k} x) | |$ for x ∈ . The class of (n,k)-quasiparanormal operators contains the classes of n-paranormal and k-quasiparanormal operators. We consider some properties of (n,k)-quasiparanormal operators: (1) inclusion relations and examples; (2) a matrix representation and SVEP (single valued extension property); (3) ascent and Bishop’s property (β); (4)...

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))...

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 < ∞.

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.

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...

Generalized $H_{m, m}^{m, 0}$ function fractional integration operators in some classes of analytic functions

V. Kiryakova (1988)

Matematički Vesnik

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On Sobolev spaces of fractional order and ε-families of operators on spaces of homogeneous type

A. Gatto, Stephen Vági (1999)

Studia Mathematica

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We introduce Sobolev spaces $L_{α}^{p}$ for 1 < p < ∞ and small positive α on spaces of homogeneous type as the classes of functions f in $L^{p}$ with fractional derivative of order α, $D^{α} f$ , as introduced in [2], in $L^{p}$ . We show that for small α, $L_{α}^{p}$ coincides with the continuous version of the Triebel-Lizorkin space $F_{p}^{α, 2}$ as defined by Y. S. Han and E. T. Sawyer in [4]. To prove this result we give a more general definition of ε-families of operators on spaces of homogeneous type, in which the identity...

Mapping properties of fundamental operators in harmonic analysis related to Bessel operators

Jorge J. Betancor, Eleonor Harboure, Adam Nowak, Beatriz Viviani (2010)

Studia Mathematica

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We obtain sharp power-weighted $L^{p}$ , weak type and restricted weak type inequalities for the heat and Poisson integral maximal operators, Riesz transform and a Littlewood-Paley type square function, emerging naturally in the harmonic analysis related to Bessel operators.

Some remarks on Bochner-Riesz means

S. Thangavelu (2000)

Colloquium Mathematicae

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We study $L^{p}$ norm convergence of Bochner-Riesz means $S_{R}^{δ} f$ associated with certain non-negative differential operators. When the kernel $S_{R}^{m} (x, y)$ satisfies a weak estimate for large values of m we prove $L^{p}$ norm convergence of $S_{R}^{δ} f$ for δ > n|1/p-1/2|, 1 < p < ∞, where n is the dimension of the underlying manifold.

Besov spaces and 2-summing operators

M. A. Fugarolas (2004)

Colloquium Mathematicae

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Let Π₂ be the operator ideal of all absolutely 2-summing operators and let $I_{m}$ be the identity map of the m-dimensional linear space. We first establish upper estimates for some mixing norms of $I_{m}$ . Employing these estimates, we study the embedding operators between Besov function spaces as mixing operators. The result obtained is applied to give sufficient conditions under which certain kinds of integral operators, acting on a Besov function space, belong to Π₂; in this context, we also...

$H^{p}$ Sobolev spaces

Robert S. Strichartz (1990)

Colloquium Mathematicae

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Dichotomy of global density of Riesz capacity

Hiroaki Aikawa (2016)

Studia Mathematica

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Let $C_{α}$ be the Riesz capacity of order α, 0 < α < n, in ℝⁿ. We consider the Riesz capacity density $̲ (C_{α}, E, r) = i n f_{x \in ℝ ⁿ} C_{α} (E \cap B (x, r)) / C_{α} (B (x, r))$ for a Borel set E ⊂ ℝⁿ, where B(x,r) stands for the open ball with center at x and radius r. In case 0 < α ≤ 2, we show that $l i m_{r \to \infty} ̲ (C_{α}, E, r)$ is either 0 or 1; the first case occurs if and only if $̲ (C_{α}, E, r)$ is identically zero for all r > 0. Moreover, it is shown that the densities with respect to more general open sets enjoy the same dichotomy. A decay estimate for α-capacitary potentials is also...

Modulation space estimates for multilinear pseudodifferential operators

Árpád Bényi, Kasso A. Okoudjou (2006)

Studia Mathematica

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We prove that for symbols in the modulation spaces $ℳ^{p, q}$ , p ≥ q, the associated multilinear pseudodifferential operators are bounded on products of appropriate modulation spaces. In particular, the symbols we study here are defined without any reference to smoothness, but rather in terms of their time-frequency behavior.

$L^{p}$ estimates for Schrödinger operators with certain potentials

Zhongwei Shen (1995)

Annales de l'institut Fourier

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We consider the Schrödinger operators $- Δ + V (x)$ in $ℝ^{n}$ where the nonnegative potential $V (x)$ belongs to the reverse Hölder class $B_{q}$ for some $q \geq n / 2$ . We obtain the optimal $L^{p}$ estimates for the operators $(- Δ + V)^{i γ}, \nabla^{2} (- Δ + V)^{- 1}, \nabla (- Δ + V)^{- 1 / 2}$ and $\nabla (- Δ + V)^{- 1}$ where $γ \in ℝ$ . In particular we show that $(- Δ + V)^{i γ}$ is a Calderón-Zygmund operator if $V \in B_{n / 2}$ and $\nabla (- Δ + V)^{- 1 / 2}, \nabla (- Δ + V)^{- 1} \nabla$ are Calderón-Zygmund operators if $V \in B_{n}$ .

$L^{p}$ 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.

Large time behaviour of a conservation law regularised by a Riesz-Feller operator: the sub-critical case

Carlota Maria Cuesta, Xuban Diez-Izagirre (2023)

Czechoslovak Mathematical Journal

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We study the large time behaviour of the solutions of a nonlocal regularisation of a scalar conservation law. This regularisation is given by a fractional derivative of order $1 + α$ , with $α \in (0, 1)$ , which is a Riesz-Feller operator. The nonlinear flux is given by the locally Lipschitz function ${| u |}^{q - 1} u / q$ for $q > 1$ . We show that in the sub-critical case, $1 < q < 1 + α$ , the large time behaviour is governed by the unique entropy solution of the scalar conservation law. Our proof adapts the proofs of the analogous results for...