Boundedness of Riesz transforms on weighted Carleson measure spaces

Ming-Yi Lee

Displaying similar documents to “Boundedness of Riesz transforms on weighted Carleson measure spaces”

Essential norms of weighted composition operators between Hardy spaces $H^{p}$ and $H^{q}$ for 1 ≤ p,q ≤ ∞

R. Demazeux (2011)

Studia Mathematica

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We complete the different cases remaining in the estimation of the essential norm of a weighted composition operator acting between the Hardy spaces $H^{p}$ and $H^{q}$ for 1 ≤ p,q ≤ ∞. In particular we give some estimates for the cases 1 = p ≤ q ≤ ∞ and 1 ≤ q < p ≤ ∞.

Lipschitz continuity in Muckenhoupt 𝓐₁ weighted function spaces

Dorothee D. Haroske (2011)

Banach Center Publications

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We study continuity envelopes of function spaces $B_{p, q}^{s} (ℝ ⁿ, w)$ and $F_{p, q}^{s} (ℝ ⁿ, w)$ where the weight belongs to the Muckenhoupt class ₁. This essentially extends partial forerunners in [13, 14]. We also indicate some applications of these results.

Weighted variable $L^{p}$ integral inequalities for the maximal operator on non-increasing functions

C. J. Neugebauer (2009)

Studia Mathematica

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Let $B_{p}$ be the Ariõ-Muckenhoupt weight class which controls the weighted $L^{p}$ -norm inequalities for the Hardy operator on non-increasing functions. We replace the constant p by a function p(x) and examine the associated $L^{p (x)}$ -norm inequalities of the Hardy operator.

On weighted Hardy spaces on the unit disk

Evgeny A. Poletsky, Khim R. Shrestha (2015)

Banach Center Publications

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In this paper we completely characterize those weighted Hardy spaces that are Poletsky-Stessin Hardy spaces $H_{u}^{p}$ . We also provide a reduction of $H^{\infty}$ problems to $H_{u}^{p}$ problems and demonstrate how such a reduction can be used to make shortcuts in the proofs of the interpolation theorem and corona problem.

The representation of multi-hypergraphs by set intersections

Stanisław Bylka, Jan Komar (2007)

Discussiones Mathematicae Graph Theory

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This paper deals with weighted set systems (V,,q), where V is a set of indices, $\subset 2^{V}$ and the weight q is a nonnegative integer function on . The basic idea of the paper is to apply weighted set systems to formulate restrictions on intersections. It is of interest to know whether a weighted set system can be represented by set intersections. An intersection representation of (V,,q) is defined to be an indexed family $R = {(R_{v})}_{v \in V}$ of subsets of a set S such that $| ⋂_{v \in E} R_{v} | = q (E)$ for each E ∈ . A necessary condition...

Solutions to the equation div u = f in weighted Sobolev spaces

Katrin Schumacher (2008)

Banach Center Publications

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We consider the problem div u = f in a bounded Lipschitz domain Ω, where f with $\int_{Ω} f = 0$ is given. It is shown that the solution u, constructed as in Bogovski’s approach in [1], fulfills estimates in the weighted Sobolev spaces $W_{w}^{k, q} (Ω)$ , where the weight function w is in the class of Muckenhoupt weights $A_{q}$ .

Weighted norm inequalities for maximal singular integrals with nondoubling measures

Guoen Hu, Dachun Yang (2008)

Studia Mathematica

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Let μ be a nonnegative Radon measure on $ℝ^{d}$ which satisfies μ(B(x,r)) ≤ Crⁿ for any $x \in ℝ^{d}$ and r > 0 and some positive constants C and n ∈ (0,d]. In this paper, some weighted norm inequalities with $A_{p}^{ϱ} (μ)$ weights of Muckenhoupt type are obtained for maximal singular integral operators with such a measure μ, via certain weighted estimates with $A_{\infty}^{ϱ} (μ)$ weights of Muckenhoupt type involving the John-Strömberg maximal operator and the John-Strömberg sharp maximal operator, where ϱ,p ∈ [1,∞).

The maximal theorem for weighted grand Lebesgue spaces

Alberto Fiorenza, Babita Gupta, Pankaj Jain (2008)

Studia Mathematica

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We study the Hardy inequality and derive the maximal theorem of Hardy and Littlewood in the context of grand Lebesgue spaces, considered when the underlying measure space is the interval (0,1) ⊂ ℝ, and the maximal function is localized in (0,1). Moreover, we prove that the inequality ${| | M f | |}_{p), w} {\leq c | | f | |}_{p), w}$ holds with some c independent of f iff w belongs to the well known Muckenhoupt class $A_{p}$ , and therefore iff ${| | M f | |}_{p, w} {\leq c | | f | |}_{p, w}$ for some c independent of f. Some results of similar type are discussed for the case of small...

The linear bound in A₂ for Calderón-Zygmund operators: a survey

Michael Lacey (2011)

Banach Center Publications

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For an L²-bounded Calderón-Zygmund Operator T acting on $L ² (ℝ^{d})$ , and a weight w ∈ A₂, the norm of T on L²(w) is dominated by $C_{T} {| | w | |}_{A ₂}$ . The recent theorem completes a line of investigation initiated by Hunt-Muckenhoupt-Wheeden in 1973 (MR0312139), has been established in different levels of generality by a number of authors over the last few years. It has a subtle proof, whose full implications will unfold over the next few years. This sharp estimate requires that the A₂ character of the weight can...

Maximal function and Carleson measures in the theory of Békollé-Bonami weights

Carnot D. Kenfack, Benoît F. Sehba (2016)

Colloquium Mathematicae

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Let ω be a Békollé-Bonami weight. We give a complete characterization of the positive measures μ such that $\int_{} | M_{ω} {f (z) |}^{q} d μ (z) \leq C (\int_{} {| f (z) |}^{p} {ω (z) d V (z))}^{q / p}$ and $μ (z \in : M f (z) > λ) \leq C / (λ^{q}) (\int_{} | f (z) {|^{p} ω (z) d V (z))}^{q / p}$ , where $M_{ω}$ is the weighted Hardy-Littlewood maximal function on the upper half-plane and 1 ≤ p,q <; ∞.

On the Fejér-F. Riesz inequality in $L^{p}$

Yoram Sagher (1977)

Studia Mathematica

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Existence of solutions to the (rot,div)-system in $L_{p}$ -weighted spaces

Wojciech M. Zajączkowski (2010)

Applicationes Mathematicae

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The existence of solutions to the elliptic problem rot v = w, div v = 0 in a bounded domain Ω ⊂ ℝ³, ${v \cdot n ̅ |}_{S} = 0$ , S = ∂Ω in weighted $L_{p}$ -Sobolev spaces is proved. It is assumed that an axis L crosses Ω and the weight is a negative power function of the distance to the axis. The main part of the proof is devoted to examining solutions of the problem in a neighbourhood of L. The existence in Ω follows from the technique of regularization.

The continuity of pseudo-differential operators on weighted local Hardy spaces

Ming-Yi Lee, Chin-Cheng Lin, Ying-Chieh Lin (2010)

Studia Mathematica

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We first show that a linear operator which is bounded on $L ²_{w}$ with w ∈ A₁ can be extended to a bounded operator on the weighted local Hardy space $h ¹_{w}$ if and only if this operator is uniformly bounded on all $h ¹_{w}$ -atoms. As an application, we show that every pseudo-differential operator of order zero has a bounded extension to $h ¹_{w}$ .

Jacobi decomposition of weighted Triebel-Lizorkin and Besov spaces

George Kyriazis, Pencho Petrushev, Yuan Xu (2008)

Studia Mathematica

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The Littlewood-Paley theory is extended to weighted spaces of distributions on [-1,1] with Jacobi weights $w (t) = {(1 - t)}^{α} {(1 + t)}^{β}$ . Almost exponentially localized polynomial elements (needlets) $φ_{ξ}$ , $ψ_{ξ}$ are constructed and, in complete analogy with the classical case on ℝⁿ, it is shown that weighted Triebel-Lizorkin and Besov spaces can be characterized by the size of the needlet coefficients $⟨ f, φ_{ξ} ⟩$ in respective sequence spaces.

Weighted sub-Bergman Hilbert spaces

Maria Nowak, Renata Rososzczuk (2014)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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We consider Hilbert spaces which are counterparts of the de Branges-Rovnyak spaces in the context of the weighted Bergman spaces $A_{α}^{2}$ , $- 1 < α < \infty$ . These spaces have already been studied in [8], [7], [5] and [1]. We extend some results from these papers.

Weighted $H^{p}$ spaces

José García-Cuerva

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CONTENTSIntroduction.......................................................................................................................................................... 5Chapter I. Some preliminary lemmas............................................................................................................ 8Chapter II. Weighted $H^{p}$ spaces of analytic functions.......................................................................... 13 1. Behaviour at the boundary..........................................................................................................................

Weighted composition operators from Zygmund spaces to Bloch spaces on the unit ball

Yu-Xia Liang, Chang-Jin Wang, Ze-Hua Zhou (2015)

Annales Polonici Mathematici

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Let H() denote the space of all holomorphic functions on the unit ball ⊂ ℂⁿ. Let φ be a holomorphic self-map of and u∈ H(). The weighted composition operator $u C_{φ}$ on H() is defined by $u C_{φ} f (z) = u (z) f (φ (z))$ . We investigate the boundedness and compactness of $u C_{φ}$ induced by u and φ acting from Zygmund spaces to Bloch (or little Bloch) spaces in the unit ball.

On some subspaces of Morrey-Sobolev spaces and boundedness of Riesz integrals

Mouhamadou Dosso, Ibrahim Fofana, Moumine Sanogo (2013)

Annales Polonici Mathematici

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For 1 ≤ q ≤ α ≤ p ≤ ∞, ${(L^{q}, l^{p})}^{α}$ is a complex Banach space which is continuously included in the Wiener amalgam space $(L^{q}, l^{p})$ and contains the Lebesgue space $L^{α}$ . We study the closure ${(L^{q}, l^{p})}_{c, 0}^{α}$ in ${(L^{q}, l^{p})}^{α}$ of the space of test functions (infinitely differentiable and with compact support in $ℝ^{d}$ ) and obtain norm inequalities for Riesz potential operators and Riesz transforms in these spaces. We also introduce the Sobolev type space $W ¹ ({(L^{q}, l^{p})}^{α})$ (a subspace of a Morrey-Sobolev space, but a superspace of the classical Sobolev space...

Generalized Riesz products produced from orthonormal transforms

Nikolaos Atreas, Antonis Bisbas (2012)

Colloquium Mathematicae

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Let $ℳ_{p} = {m_{k}}_{k = 0}^{p - 1}$ be a finite set of step functions or real valued trigonometric polynomials on = [0,1) satisfying a certain orthonormality condition. We study multiscale generalized Riesz product measures μ defined as weak-* limits of elements $μ_{N} \in V_{N} (N \in ℕ)$ , where $V_{N}$ are $p^{N}$ -dimensional subspaces of L₂() spanned by an orthonormal set which is produced from dilations and multiplications of elements of $ℳ_{p}$ and $\bar{⋃_{N \in ℕ} V_{N}} = L ₂ ()$ . The results involve mutual absolute continuity or singularity of such Riesz products extending previous...

The spectral decomposition of weighted shifts and the $A^{p}$ condition

Earl Berkson, T. A. Gillespie (1990)

Colloquium Mathematicae

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Regularity properties of commutators and $B M O$ -Triebel-Lizorkin spaces

Abdellah Youssfi (1995)

Annales de l'institut Fourier

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In this paper we consider the regularity problem for the commutators $([b, R_{k}])_{1 \leq k \leq n}$ where $b$ is a locally integrable function and $(R_{j})_{1 \leq j \leq n}$ are the Riesz transforms in the $n$ -dimensional euclidean space $ℝ^{n}$ . More precisely, we prove that these commutators $([b, R_{k}])_{1 \leq k \leq n}$ are bounded from $L^{p}$ into the Besov space ${\dot{B}}_{p}^{s, p}$ for $1 < p < + \infty$ and $0 < s < 1$ if and only if $b$ is in the $B M O$ -Triebel-Lizorkin space ${\dot{F}}_{\infty}^{s, p}$ . The reduction of our result to the case $p = 2$ gives in particular that the commutators $([b, R_{k}])_{1 \leq k \leq n}$ are bounded form $L^{2}$ into the Sobolev space ${\dot{H}}^{s}$ if and only if $b$ ...

Variation for the Riesz transform and uniform rectifiability

Albert Mas, Xavier Tolsa (2014)

Journal of the European Mathematical Society

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For $1 \leq n < d$ integers and $ρ > 2$ , we prove that an $n$ -dimensional Ahlfors-David regular measure $μ$ in $ℝ^{d}$ is uniformly $n$ -rectifiable if and only if the $ρ$ -variation for the Riesz transform with respect to $μ$ is a bounded operator in $L^{2} (μ)$ . This result can be considered as a partial solution to a well known open problem posed by G. David and S. Semmes which relates the $L^{2} (μ)$ boundedness of the Riesz transform to the uniform rectifiability of $μ$ .

Sharp inequalities for Riesz transforms

Adam Osękowski (2014)

Studia Mathematica

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We establish the following sharp local estimate for the family ${R_{j}}_{j = 1}^{d}$ of Riesz transforms on $ℝ^{d}$ . For any Borel subset A of $ℝ^{d}$ and any function $f : ℝ^{d} \to ℝ$ , $\int_{A} | R_{j} f (x) | d x \leq C_{p} {| | f | |}_{L^{p} (ℝ^{d})} {| A |}^{1 / q}$ , 1 < p < ∞. Here q = p/(p-1) is the harmonic conjugate to p, $C_{p} = {[2^{q + 2} Γ (q + 1) / π^{q + 1} \sum_{k = 0}^{\infty} {(- 1)}^{k} / {(2 k + 1)}^{q + 1}]}^{1 / q}$ , 1 < p < 2, and $C_{p} = {[4 Γ (q + 1) / π^{q} \sum_{k = 0}^{\infty} 1 / {(2 k + 1)}^{q}]}^{1 / q}$ , 2 ≤ p < ∞. This enables us to determine the precise values of the weak-type constants for Riesz transforms for 1 < p < ∞. The proof rests on appropriate martingale inequalities, which are of independent interest.

Boundedness of Stein's square functions and Bochner-Riesz means associated to operators on Hardy spaces

Xuefang Yan (2015)

Czechoslovak Mathematical Journal

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Let $(X, d, μ)$ be a metric measure space endowed with a distance $d$ and a nonnegative Borel doubling measure $μ$ . Let $L$ be a non-negative self-adjoint operator of order $m$ on $L^{2} (X)$ . Assume that the semigroup $e^{- t L}$ generated by $L$ satisfies the Davies-Gaffney estimate of order $m$ and $L$ satisfies the Plancherel type estimate. Let $H_{L}^{p} (X)$ be the Hardy space associated with $L .$ We show the boundedness of Stein’s square function $𝒢_{δ} (L)$ arising from Bochner-Riesz means associated to $L$ from Hardy spaces $H_{L}^{p} (X)$ to $L^{p} (X)$ , and also study...

Compactness of composition operators acting on weighted Bergman-Orlicz spaces

Ajay K. Sharma, S. Ueki (2012)

Annales Polonici Mathematici

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We characterize compact composition operators acting on weighted Bergman-Orlicz spaces $_{α}^{ψ} = f \in H () : \int_{} ψ (| f (z) |) d A_{α} (z) < \infty$ , where α > -1 and ψ is a strictly increasing, subadditive convex function defined on [0,∞) and satisfying ψ(0) = 0, the growth condition $l i m_{t \to \infty} ψ (t) / t = \infty$ and the Δ₂-condition. In fact, we prove that $C_{φ}$ is compact on $_{α}^{ψ}$ if and only if it is compact on the weighted Bergman space $²_{α}$ .

Weighted boundedness of Toeplitz type operators related to singular integral operators with non-smooth kernel

Xiaosha Zhou, Lanzhe Liu (2013)

Colloquium Mathematicae

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Some weighted sharp maximal function inequalities for the Toeplitz type operator $T_{b} = \sum_{k = 1}^{m} T^{k, 1} M_{b} T^{k, 2}$ are established, where $T^{k, 1}$ are a fixed singular integral operator with non-smooth kernel or ±I (the identity operator), $T^{k, 2}$ are linear operators defined on the space of locally integrable functions, k = 1,..., m, and $M_{b} (f) = b f$ . The weighted boundedness of $T_{b}$ on Morrey spaces is obtained by using sharp maximal function inequalities.

Weighted Frobenius-Perron operators and their spectra

Mohammad Reza Jabbarzadeh, Rana Hajipouri (2017)

Mathematica Bohemica

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First, some classic properties of a weighted Frobenius-Perron operator $𝒫_{ϕ}^{u}$ on $L^{1} (Σ)$ as a predual of weighted Koopman operator $W = u U_{ϕ}$ on $L^{\infty} (Σ)$ will be investigated using the language of the conditional expectation operator. Also, we determine the spectrum of $𝒫_{ϕ}^{u}$ under certain conditions.