Behaviour of -Dini singular integrals in weighted spaces
Osvaldo Capri, Carlos Segovia (1989)
Studia Mathematica
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Osvaldo Capri, Carlos Segovia (1989)
Studia Mathematica
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Xiaosha Zhou, Lanzhe Liu (2013)
Colloquium Mathematicae
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Some weighted sharp maximal function inequalities for the Toeplitz type operator are established, where are a fixed singular integral operator with non-smooth kernel or ±I (the identity operator), are linear operators defined on the space of locally integrable functions, k = 1,..., m, and . The weighted boundedness of on Morrey spaces is obtained by using sharp maximal function inequalities.
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 and , where is the weighted Hardy-Littlewood maximal function on the upper half-plane and 1 ≤ p,q <; ∞.
C. J. Neugebauer (2009)
Studia Mathematica
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Let be the Ariõ-Muckenhoupt weight class which controls the weighted -norm inequalities for the Hardy operator on non-increasing functions. We replace the constant p by a function p(x) and examine the associated -norm inequalities of the Hardy operator.
Qingying Xue (2013)
Studia Mathematica
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The following iterated commutators of the maximal operator for multilinear singular integral operators and of the multilinear fractional integral operator are introduced and studied: , , where are the smooth truncations of the multilinear singular integral operators and is the multilinear fractional integral operator, for i = 1,…,m and f⃗ = (f1,…,fm). Weighted strong and L(logL) type end-point estimates for the above iterated commutators associated with two classes of multiple...
L. de Rosa, C. Segovia (2006)
Studia Mathematica
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We study the boundedness of the one-sided operator between the weighted spaces and for every weight w. If λ = 2/p whenever 1 < p < 2, and in the case p = 1 for λ > 2, we prove the weak type of . For every λ > 1 and p = 2, or λ > 2/p and 1 < p < 2, the boundedness of this operator is obtained. For p > 2 and λ > 1, we obtain the boundedness of from to , where denotes the operator M¯ iterated k times.
Paul Alton Hagelstein (2001)
Studia Mathematica
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Let denote the strong maximal operator. Let and denote the one-dimensional Hardy-Littlewood maximal operators in the horizontal and vertical directions in ℝ². A function h supported on the unit square Q = [0,1]×[0,1] is exhibited such that but . It is shown that if f is a function supported on Q such that but , then there exists a set A of finite measure in ℝ² such that .
Dorothee D. Haroske (2011)
Banach Center Publications
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We study continuity envelopes of function spaces and 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.
José García-Cuerva
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CONTENTSIntroduction.......................................................................................................................................................... 5Chapter I. Some preliminary lemmas............................................................................................................ 8Chapter II. Weighted spaces of analytic functions.......................................................................... 13 1. Behaviour at the boundary..........................................................................................................................
Keiji Izuchi (2004)
Studia Mathematica
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Let μ and λ be bounded positive singular measures on the unit circle such that μ ⊥ λ. It is proved that there exist positive measures μ₀ and λ₀ such that μ₀ ∼ μ, λ₀ ∼ λ, and , where is the associated singular inner function of μ. Let be the common zeros of equivalent singular inner functions of . Then (μ) ≠ ∅ and (μ) ∩ (λ) = ∅. It follows that μ ≪ λ if and only if (μ) ⊂ (λ). Hence (μ) is the set in the maximal ideal space of which relates naturally to the set of measures equivalent...
Kristóf Szarvas, Ferenc Weisz (2016)
Czechoslovak Mathematical Journal
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The classical Hardy-Littlewood maximal operator is bounded not only on the classical Lebesgue spaces (in the case ), but (in the case when is log-Hölder continuous and ) on the variable Lebesgue spaces , too. Furthermore, the classical Hardy-Littlewood maximal operator is of weak-type . In the present note we generalize Besicovitch’s covering theorem for the so-called -rectangles. We introduce a general maximal operator and with the help of generalized -functions, the strong-...
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 and for 1 ≤ p,q ≤ ∞. In particular we give some estimates for the cases 1 = p ≤ q ≤ ∞ and 1 ≤ q < p ≤ ∞.
Adam Osękowski (2014)
Banach Center Publications
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We prove sharp a priori estimates for the distribution function of the dyadic maximal function ℳ ϕ, when ϕ belongs to the Lorentz space , 1 < p < ∞, 1 ≤ q < ∞. The approach rests on a precise evaluation of the Bellman function corresponding to the problem. As an application, we establish refined weak-type estimates for the dyadic maximal operator: for p,q as above and r ∈ [1,p], we determine the best constant such that for any , .
Xin-Cui Guo, Ze-Hua Zhou (2015)
Czechoslovak Mathematical Journal
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Let be a holomorphic function and a holomorphic self-map of the open unit disk in the complex plane. We provide new characterizations for the boundedness of the weighted composition operators from Zygmund type spaces to Bloch type spaces in in terms of , , their derivatives, and , the -th power of . Moreover, we obtain some similar estimates for the essential norms of the operators , from which sufficient and necessary conditions of compactness of follows immediately. ...
David Cruz-Uribe, SFO (2001)
Studia Mathematica
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We prove two-weight norm inequalities in ℝⁿ for the minimal operator , 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 , 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...
Pavla Hofmanová (2016)
Commentationes Mathematicae Universitatis Carolinae
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Let be a weight on . Assume that is continuous on . Let the operator be given at measurable non-negative function on by We characterize weights on for which there exists a positive constant such that the inequality holds for every . Such inequalities have been used in the study of optimal Sobolev embeddings and boundedness of certain operators on classical Lorenz spaces.
Mohammad Reza Jabbarzadeh, Rana Hajipouri (2017)
Mathematica Bohemica
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First, some classic properties of a weighted Frobenius-Perron operator on as a predual of weighted Koopman operator on will be investigated using the language of the conditional expectation operator. Also, we determine the spectrum of under certain conditions.
Loukas Grafakos, Liguang Liu, Dachun Yang (2009)
Bulletin de la Société Mathématique de France
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An RD-space is a space of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds. The authors prove that for a space of homogeneous type having “dimension” , there exists a such that for certain classes of distributions, the quasi-norms of their radial maximal functions and grand maximal functions are equivalent when . This result yields a radial maximal function characterization for Hardy spaces on . ...
Armin Rainer, Gerhard Schindl (2014)
Studia Mathematica
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We characterize stability under composition of ultradifferentiable classes defined by weight sequences M, by weight functions ω, and, more generally, by weight matrices , and investigate continuity of composition (g,f) ↦ f ∘ g. In addition, we represent the Beurling space and the Roumieu space as intersection and union of spaces and for associated weight sequences, respectively.
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 . Given an operator T from to L¹(X), we consider the vector-valued extension T̃ of T given by . 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...
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 holds with some c independent of f iff w belongs to the well known Muckenhoupt class , and therefore iff for some c independent of f. Some results of similar type are discussed for the case of small...
Krzysztof Zajkowski (2010)
Studia Mathematica
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We prove that for the spectral radius of a weighted composition operator , acting in the space , the following variational principle holds: , where X is a Hausdorff compact space, α: X → X is a continuous mapping preserving a Borel measure μ with suppμ = X, is the set of all α-invariant ergodic probability measures on X, and a: X → ℝ is a continuous and -measurable function, where . This considerably extends the range of validity of the above formula, which was previously known...
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 is given. It is shown that the solution u, constructed as in Bogovski’s approach in [1], fulfills estimates in the weighted Sobolev spaces , where the weight function w is in the class of Muckenhoupt weights .
G. A. Karagulyan (2007)
Studia Mathematica
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Let be the family of open rectangles in the plane ℝ² with a side of angle s to the x-axis. We say that a set S of directions is an R-set if there exists a function f ∈ L¹(ℝ²) such that the basis differentiates the integral of f if s ∉ S, and almost everywhere if s ∈ S. If the condition holds on a set of positive measure (instead of a.e.) we say that S is a WR-set. It is proved that S is an R-set (resp. a WR-set) if and only if it is a (resp. a ).
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 . The square function is defined as . The local version of this operator, namely the operator , 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 into itself (p > 1) and into BMO. We prove that the operator S not only maps into BMO but it also maps BMO into BMO. We also prove that the boundedness...
Anna Kamińska, Lech Maligranda (2004)
Studia Mathematica
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We study order convexity and concavity of quasi-Banach Lorentz spaces , where 0 < p < ∞ and w is a locally integrable positive weight function. We show first that contains an order isomorphic copy of . We then present complete criteria for lattice convexity and concavity as well as for upper and lower estimates for . We conclude with a characterization of the type and cotype of in the case when is a normable space.
Dongyang Chen, William B. Johnson, Bentuo Zheng (2014)
Studia Mathematica
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We give a corrected proof of Theorem 2.10 in our paper “Commutators on ” [Studia Math. 206 (2011), 175-190] for the case 1 < q < p < ∞. The case when 1 = q < p < ∞ remains open. As a consequence, the Main Theorem and Corollary 2.17 in that paper are only valid for 1 < p,q < ∞.
Sergei V. Astashkin, Lech Maligranda (2010)
Studia Mathematica
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The Rademacher sums are investigated in the Cesàro spaces (1 ≤ p ≤ ∞) and in the weighted Korenblyum-Kreĭn-Levin spaces on [0,1]. They span l₂ space in for any 1 ≤ p < ∞ and in if and only if the weight w is larger than on (0,1). Moreover, the span of the Rademachers is not complemented in for any 1 ≤ p < ∞ or in for any quasi-concave weight w. In the case when p > 1 and when w is such that the span of the Rademacher functions is isomorphic to l₂, this span is...
Karma Dajani, Martijn de Vries (2007)
Journal of the European Mathematical Society
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Let be a non-integer. We consider expansions of the form , where the digits are generated by means of a Borel map defined on . We show existence and uniqueness of a -invariant probability measure, absolutely continuous with respect to , where is the Bernoulli measure on with parameter () and is the normalized Lebesgue measure on . Furthermore, this measure is of the form , where is equivalent to . We prove that the measure of maximal entropy and are mutually...
Wojciech M. Zajączkowski
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CONTENTS1. Introduction.......................................................................52. Notation and auxiliary results............................................93. Statement of the problem (1.1)-(1.3)..............................204. The problem (3.14).........................................................225. Auxiliary results in ...............................................346. Existence of solutions of (3.14) in ............417. Green function................................................................528....
Songqing Chen, Huoxiong Wu, Qingying Xue (2014)
Studia Mathematica
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This paper is devoted to investigating the properties of multilinear conditions and conditions, which are suitable for the study of multilinear operators on Lebesgue spaces. Some monotonicity properties of and classes with respect to P⃗ and q are given, although these classes are not in general monotone with respect to the natural partial order. Equivalent characterizations of multilinear classes in terms of the linear classes are established. These results essentially improve...
Dongyang Chen, William B. Johnson, Bentuo Zheng (2011)
Studia Mathematica
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Let T be a bounded linear operator on with 1 ≤ q < ∞ and 1 < p < ∞. Then T is a commutator if and only if for all non-zero λ ∈ ℂ, the operator T - λI is not X-strictly singular.