Weighted inequalities for rough square functions through extrapolation

Javier Duoandikoetxea; Edurne Seijo

Displaying similar documents to “Weighted inequalities for rough square functions through extrapolation”

Characterization of associate spaces of weighted Lorentz spaces with applications

Amiran Gogatishvili, Luboš Pick, Filip Soudský (2014)

Studia Mathematica

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We characterize associate spaces of weighted Lorentz spaces GΓ(p,m,w) and present some applications of this result including necessary and sufficient conditions for a Sobolev-type embedding into $L^{\infty}$ .

Two-weighted criteria for integral transforms with multiple kernels

Vakhtang Kokilashvili, Alexander Meskhi (2006)

Banach Center Publications

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Necessary and sufficient conditions governing two-weight $L^{p}$ norm estimates for multiple Hardy and potential operators are presented. Two-weight inequalities for potentials defined on nonhomogeneous spaces are also discussed. Sketches of the proofs for most of the results are given.

Convergence a.e. of spherical partial Fourier integrals on weighted spaces for radial functions: endpoint estimates

María J. Carro, Elena Prestini (2009)

Studia Mathematica

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We prove some extrapolation results for operators bounded on radial $L^{p}$ functions with p ∈ (p₀,p₁) and deduce some endpoint estimates. We apply our results to prove the almost everywhere convergence of the spherical partial Fourier integrals and to obtain estimates on maximal Bochner-Riesz type operators acting on radial functions in several weighted spaces.

Monotonicity of generalized weighted mean values

Alfred Witkowski (2004)

Colloquium Mathematicae

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The author gives a new simple proof of monotonicity of the generalized extended mean values $M (r, s) = \leq {((\int f^{s} d μ) / (\int f^{r} d μ))}^{1 / (s - r)}$ introduced by F. Qi.

ω-Calderón-Zygmund operators

Sijue Wu (1995)

Studia Mathematica

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We prove a T1 theorem and develop a version of Calderón-Zygmund theory for ω-CZO when $ω \in A_{\infty}$ .

The minimal operator and the John--Nirenberg theorem for weighted grand Lebesgue spaces

Lihua Peng, Yong Jiao (2015)

Studia Mathematica

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We introduce the minimal operator on weighted grand Lebesgue spaces, discuss some weighted norm inequalities and characterize the conditions under which the inequalities hold. We also prove that the John-Nirenberg inequalities in the framework of weighted grand Lebesgue spaces are valid provided that the weight function belongs to the Muckenhoupt $A_{p}$ class.

Weighted Hardy inequalities and Hardy transforms of weights

Joan Cerdà, Joaquim Martín (2000)

Studia Mathematica

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Many problems in analysis are described as weighted norm inequalities that have given rise to different classes of weights, such as $A_{p}$ -weights of Muckenhoupt and $B_{p}$ -weights of Ariño and Muckenhoupt. Our purpose is to show that different classes of weights are related by means of composition with classical transforms. A typical example is the family $M_{p}$ of weights w for which the Hardy transform is $L_{p} (w)$ -bounded. A $B_{p}$ -weight is precisely one for which its Hardy transform is in $M_{p}$ , and also a weight...

Weighted bounds for variational Fourier series

Yen Do, Michael Lacey (2012)

Studia Mathematica

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For 1 < p < ∞ and for weight w in $A_{p}$ , we show that the r-variation of the Fourier sums of any function f in $L^{p} (w)$ is finite a.e. for r larger than a finite constant depending on w and p. The fact that the variation exponent depends on w is necessary. This strengthens previous work of Hunt-Young and is a weighted extension of a variational Carleson theorem of Oberlin-Seeger-Tao-Thiele-Wright. The proof uses weighted adaptation of phase plane analysis and a weighted extension of a variational...

Centered weighted composition operators via measure theory

Mohammad Reza Jabbarzadeh, Mehri Jafari Bakhshkandi (2018)

Mathematica Bohemica

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We describe the centered weighted composition operators on $L^{2} (Σ)$ in terms of their defining symbols. Our characterizations extend Embry-Wardrop-Lambert’s theorem on centered composition operators.

Non-compact Littlewood-Paley theory for non-doubling measures

Michael Wilson (2007)

Studia Mathematica

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We prove weighted Littlewood-Paley inequalities for linear sums of functions satisfying mild decay, smoothness, and cancelation conditions. We prove these for general “regular” measure spaces, in which the underlying measure is not assumed to satisfy any doubling condition. Our result generalizes an earlier result of the author, proved on $ℝ^{d}$ with Lebesgue measure. Our proof makes essential use of the technique of random dyadic grids, due to Nazarov, Treil, and Volberg.

Embeddings of doubling weighted Besov spaces

Dorothee D. Haroske, Philipp Skandera (2014)

Banach Center Publications

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We study continuous embeddings of Besov spaces of type $B_{p, q}^{s} (ℝ ⁿ, w)$ , where s ∈ ℝ, 0 < p < ∞, 0 < q ≤ ∞, and the weight w is doubling. This approach generalises recent results about embeddings of Muckenhoupt weighted Besov spaces. Our main argument relies on appropriate atomic decomposition techniques of such weighted spaces; here we benefit from earlier results by Bownik. In addition, we discuss some other related weight classes briefly and compare corresponding results.

Singularities in Muckenhoupt weighted function spaces

Dorothee D. Haroske (2008)

Banach Center Publications

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We study weighted function spaces of Lebesgue, Besov and Triebel-Lizorkin type where the weight function belongs to some Muckenhoupt $_{p}$ class. The singularities of functions in these spaces are characterised by means of envelope functions.

The Properties of the Weighted Space $H_{2, α}^{k} (Ω)$ and Weighted Set $W_{2, α}^{k} (Ω, δ)$

V.A. Rukavishnikov, E.V. Matveeva, E.I. Rukavishnikova (2018)

Communications in Mathematics

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We study the properties of the weighted space $H_{2, α}^{k} (Ω)$ and weighted set $W_{2, α}^{k} (Ω, δ)$ for boundary value problem with singularity.

Boundedness of Riesz transforms on weighted Carleson measure spaces

Ming-Yi Lee (2012)

Studia Mathematica

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Let w be in the Muckenhoupt $A_{\infty}$ weight class. We show that the Riesz transforms are bounded on the weighted Carleson measure space $C M O_{w}^{p}$ , the dual of the weighted Hardy space $H_{w}^{p}$ , 0 < p ≤ 1.

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.

The minimal operator and the geometric maximal operator in ℝⁿ

David Cruz-Uribe, SFO (2001)

Studia Mathematica

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We prove two-weight norm inequalities in ℝⁿ for the minimal operator $f (x) = i n f_{Q ∋ x} 1 / | Q | \int_{Q} | f | d y$ , extending to higher dimensions results obtained by Cruz-Uribe, Neugebauer and Olesen [8] on the real line. As an application we extend to ℝⁿ weighted norm inequalities for the geometric maximal operator $M ₀ f (x) = {s u p}_{Q ∋ x} e x p (1 / | Q | \int_{Q} l o g | f | d x)$ , proved by Yin and Muckenhoupt [27]. We also give norm inequalities for the centered minimal operator, study powers of doubling weights and give sufficient conditions for the geometric maximal operator to be equal...

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...

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.

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

Earl Berkson, T. A. Gillespie (1990)

Colloquium Mathematicae

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Disjoint hypercyclic powers of weighted translations on groups

Liang Zhang, Hui-Qiang Lu, Xiao-Mei Fu, Ze-Hua Zhou (2017)

Czechoslovak Mathematical Journal

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Let $G$ be a locally compact group and let $1 \leq p < \infty .$ Recently, Chen et al. characterized hypercyclic, supercyclic and chaotic weighted translations on locally compact groups and their homogeneous spaces. There has been an increasing interest in studying the disjoint hypercyclicity acting on various spaces of holomorphic functions. In this note, we will study disjoint hypercyclic and disjoint supercyclic powers of weighted translation operators on the Lebesgue space $L^{p} (G)$ in terms of the weights. Sufficient...

A sufficient condition for the boundedness of operator-weighted martingale transforms and Hilbert transform

Sandra Pot (2007)

Studia Mathematica

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Let W be an operator weight taking values almost everywhere in the bounded positive invertible linear operators on a separable Hilbert space . We show that if W and its inverse $W^{- 1}$ both satisfy a matrix reverse Hölder property introduced by Christ and Goldberg, then the weighted Hilbert transform $H : L ²_{W} (ℝ,) \to L ²_{W} (ℝ,)$ and also all weighted dyadic martingale transforms $T_{σ} : L ²_{W} (ℝ,) \to L ²_{W} (ℝ,)$ are bounded. We also show that this condition is not necessary for the boundedness of the weighted Hilbert transform.

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 John-Nirenberg inequality for functions of bounded mean oscillation with bounded negative part

Min Hu, Dinghuai Wang (2022)

Czechoslovak Mathematical Journal

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A version of the John-Nirenberg inequality suitable for the functions $b \in BMO$ with $b^{-} \in L^{\infty}$ is established. Then, equivalent definitions of this space via the norm of weighted Lebesgue space are given. As an application, some characterizations of this function space are given by the weighted boundedness of the commutator with the Hardy-Littlewood maximal operator.

On the best ranges for $A_{p}^{+}$ and $R H_{r}^{+}$

María Silvina Riveros, A. de la Torre (2001)

Czechoslovak Mathematical Journal

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In this paper we study the relationship between one-sided reverse Hölder classes $R H_{r}^{+}$ and the $A_{p}^{+}$ classes. We find the best possible range of $R H_{r}^{+}$ to which an $A_{1}^{+}$ weight belongs, in terms of the $A_{1}^{+}$ constant. Conversely, we also find the best range of $A_{p}^{+}$ to which a $R H_{\infty}^{+}$ weight belongs, in terms of the $R H_{\infty}^{+}$ constant. Similar problems for $A_{p}^{+}$ , $1 < p < \infty$ and $R H_{r}^{+}$ , $1 < r < \infty$ are solved using factorization.

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.

Existence of solutions to the (rot,div)-system in L₂-weighted spaces

Wojciech M. Zajączkowski (2009)

Applicationes Mathematicae

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The existence of solutions to the elliptic problem rot v = w, div v = 0 in Ω ⊂ ℝ³, ${v \cdot n ̅ |}_{S} = 0$ , S = ∂Ω, in weighted Hilbert spaces is proved. It is assumed that Ω contains an axis L and the weight is a negative power of the distance to the axis. The main part of the proof is devoted to examining solutions in a neighbourhood of L. Their existence in Ω follows by regularization.

Weighted norm inequalities for vector-valued singular integrals on homogeneous spaces

Sergio Antonio Tozoni (2004)

Studia Mathematica

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Let X be a homogeneous space and let E be a UMD Banach space with a normalized unconditional basis ${(e_{j})}_{j \geq 1}$ . Given an operator T from $L_{c}^{\infty} (X)$ to L¹(X), we consider the vector-valued extension T̃ of T given by $T ̃ (\sum_{j} f_{j} e_{j}) = \sum_{j} T (f_{j}) e_{j}$ . We prove a weighted integral inequality for the vector-valued extension of the Hardy-Littlewood maximal operator and a weighted Fefferman-Stein inequality between the vector-valued extensions of the Hardy-Littlewood and the sharp maximal operators, in the context of Orlicz spaces. We give...

Generalized Hörmander conditions and weighted endpoint estimates

María Lorente, José María Martell, Carlos Pérez, María Silvina Riveros (2009)

Studia Mathematica

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We consider two-weight estimates for singular integral operators and their commutators with bounded mean oscillation functions. Hörmander type conditions in the scale of Orlicz spaces are assumed on the kernels. We prove weighted weak-type estimates for pairs of weights (u,Su) where u is an arbitrary nonnegative function and S is a maximal operator depending on the smoothness of the kernel. We also obtain sufficient conditions on a pair of weights (u,v) for the operators to be bounded...

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.

The John-Nirenberg type inequality for non-doubling measures

Yoshihiro Sawano, Hitoshi Tanaka (2007)

Studia Mathematica

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X. Tolsa defined a space of BMO type for positive Radon measures satisfying some growth condition on $ℝ^{d}$ . This new BMO space is very suitable for the Calderón-Zygmund theory with non-doubling measures. Especially, the John-Nirenberg type inequality can be recovered. In the present paper we introduce a localized and weighted version of this inequality and, as applications, we obtain some vector-valued inequalities and weighted inequalities for Morrey spaces.