Displaying similar documents to “ω-Calderón-Zygmund operators”

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 ( d ) , and a weight w ∈ A₂, the norm of T on L²(w) is dominated by C T | | w | | A . 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...

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 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.

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 ( , w ) and F p , q s ( , 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 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 1 . Given an operator T from L c ( X ) to L¹(X), we consider the vector-valued extension T̃ of T given by T ̃ ( j f j e j ) = 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...

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 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 c | | f | | p , w for some c independent of f. Some results of similar type are discussed for the case of small...

Calderón-Zygmund operators acting on generalized Carleson measure spaces

Chin-Cheng Lin, Kunchuan Wang (2012)

Studia Mathematica

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We study Calderón-Zygmund operators acting on generalized Carleson measure spaces C M O r α , q and show a necessary and sufficient condition for their boundedness. The spaces C M O r α , q are a generalization of BMO, and can be regarded as the duals of homogeneous Triebel-Lizorkin spaces as well.

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.

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 .

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.

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 = 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.

Anisotropic classes of homogeneous pseudodifferential symbols

Árpád Bényi, Marcin Bownik (2010)

Studia Mathematica

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We define homogeneous classes of x-dependent anisotropic symbols γ , δ m ( A ) in the framework determined by an expansive dilation A, thus extending the existing theory for diagonal dilations. We revisit anisotropic analogues of Hörmander-Mikhlin multipliers introduced by Rivière [Ark. Mat. 9 (1971)] and provide direct proofs of their boundedness on Lebesgue and Hardy spaces by making use of the well-established Calderón-Zygmund theory on spaces of homogeneous type. We then show that x-dependent...

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...

Boundedness of the Hausdorff operators in H p spaces, 0 < p < 1

Elijah Liflyand, Akihiko Miyachi (2009)

Studia Mathematica

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Sufficient conditions for the boundedness of the Hausdorff operators in the Hardy spaces H p , 0 < p < 1, on the real line are proved. Two related negative results are also given.

Multilinear Calderón-Zygmund operators on weighted Hardy spaces

Wenjuan Li, Qingying Xue, Kôzô Yabuta (2010)

Studia Mathematica

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Grafakos-Kalton [Collect. Math. 52 (2001)] discussed the boundedness of multilinear Calderón-Zygmund operators on the product of Hardy spaces. Then Lerner et al. [Adv. Math. 220 (2009)] defined A p weights and built a theory of weights adapted to multilinear Calderón-Zygmund operators. In this paper, we combine the above results and obtain some estimates for multilinear Calderón-Zygmund operators on weighted Hardy spaces and also obtain a weighted multilinear version of an inequality for...

The Hardy-Lorentz spaces H p , q ( )

Wael Abu-Shammala, Alberto Torchinsky (2007)

Studia Mathematica

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We deal with the Hardy-Lorentz spaces H p , q ( ) where 0 < p ≤ 1, 0 < q ≤ ∞. We discuss the atomic decomposition of the elements in these spaces, their interpolation properties, and the behavior of singular integrals and other operators acting on them.

One-sided discrete square function

A. de la Torre, J. L. Torrea (2003)

Studia Mathematica

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Let f be a measurable function defined on ℝ. For each n ∈ ℤ we consider the average A f ( x ) = 2 - n x x + 2 f . The square function is defined as S f ( x ) = ( n = - | A f ( x ) - A n - 1 f ( x ) | ² ) 1 / 2 . The local version of this operator, namely the operator S f ( x ) = ( n = - 0 | A f ( x ) - A n - 1 f ( x ) | ² ) 1 / 2 , is of interest in ergodic theory and it has been extensively studied. In particular it has been proved [3] that it is of weak type (1,1), maps L p into itself (p > 1) and L into BMO. We prove that the operator S not only maps L into BMO but it also maps BMO into BMO. We also prove that the L p boundedness...

Commutators with fractional integral operators

Irina Holmes, Robert Rahm, Scott Spencer (2016)

Studia Mathematica

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We investigate weighted norm inequalities for the commutator of a fractional integral operator and multiplication by a function. In particular, we show that, for μ , λ A p , q and α/n + 1/q = 1/p, the norm | | [ b , I α ] : L p ( μ p ) L q ( λ q ) | | is equivalent to the norm of b in the weighted BMO space BMO(ν), where ν = μ λ - 1 . This work extends some of the results on this topic existing in the literature, and continues a line of investigation which was initiated by Bloom in 1985 and was recently developed further by the first author, Lacey,...

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 + weight belongs, in terms of the R H + constant. Similar problems for A p + , 1 < p < and R H r + , 1 < r < are solved using factorization.

Boundedness of commutators of strongly singular convolution operators on Herz-type spaces

Zongguang Liu (2003)

Studia Mathematica

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The author investigates the boundedness of the higher order commutator of strongly singular convolution operator, T b m , on Herz spaces K ̇ q α , p ( ) and K q α , p ( ) , and on a new class of Herz-type Hardy spaces H K ̇ q , b , m α , p , 0 ( ) and H K q , b , m α , p , 0 ( ) , where 0 < p ≤ 1 < q < ∞, α = n(1-1/q) and b ∈ BMO(ℝⁿ).

Toeplitz operators on Bergman spaces and Hardy multipliers

Wolfgang Lusky, Jari Taskinen (2011)

Studia Mathematica

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We study Toeplitz operators T a with radial symbols in weighted Bergman spaces A μ p , 1 < p < ∞, on the disc. Using a decomposition of A μ p into finite-dimensional subspaces the operator T a can be considered as a coefficient multiplier. This leads to new results on boundedness of T a and also shows a connection with Hardy space multipliers. Using another method we also prove a necessary and sufficient condition for the boundedness of T a for a satisfying an assumption on the positivity of certain...

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 | M ω f ( z ) | q d μ ( z ) C ( | f ( z ) | p ω ( z ) d V ( z ) ) q / p and μ ( z : M f ( z ) > λ ) C / ( λ q ) ( | 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 <; ∞.

Commutators of Littlewood-Paley [...] g κ ∗ g κ * -functions on non-homogeneous metric measure spaces

Guanghui Lu, Shuangping Tao (2017)

Open Mathematics

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The main purpose of this paper is to prove that the boundedness of the commutator [...] Mκ,b∗ κ , b * generated by the Littlewood-Paley operator [...] Mκ∗ κ * and RBMO (μ) function on non-homogeneous metric measure spaces satisfying the upper doubling and the geometrically doubling conditions. Under the assumption that the kernel of [...] Mκ∗ κ * satisfies a certain Hörmander-type condition, the authors prove that [...] Mκ,b∗ κ , b * is bounded on Lebesgue spaces Lp(μ) for 1 < p < ∞, bounded from...

Extending Hardy fields by non- -germs

Krzysztof Grelowski (2008)

Annales Polonici Mathematici

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For a large class of Hardy fields their extensions containing non- -germs are constructed. Hardy fields composed of only non- -germs, apart from constants, are also considered.

The Hausdorff operators on the real Hardy spaces H p ( )

Yuichi Kanjin (2001)

Studia Mathematica

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We prove that the Hausdorff operator generated by a function ϕ is bounded on the real Hardy space H p ( ) , 0 < p ≤ 1, if the Fourier transform ϕ̂ of ϕ satisfies certain smoothness conditions. As a special case, we obtain the boundedness of the Cesàro operator of order α on H p ( ) , 2/(2α+1) < p ≤ 1. Our proof is based on the atomic decomposition and molecular characterization of H p ( ) .

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 d which satisfies μ(B(x,r)) ≤ Crⁿ for any x 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 ϱ ( μ ) weights of Muckenhoupt type involving the John-Strömberg maximal operator and the John-Strömberg sharp maximal operator, where ϱ,p ∈ [1,∞).