The weak type inequality for the Walsh system
Ushangi Goginava (2008)
Studia Mathematica
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The main aim of this paper is to prove that the maximal operator is bounded from the Hardy space to weak- and is not bounded from to .
Ushangi Goginava (2008)
Studia Mathematica
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The main aim of this paper is to prove that the maximal operator is bounded from the Hardy space to weak- and is not bounded from to .
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 .
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 . ...
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 , 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 , 2/(2α+1) < p ≤ 1. Our proof is based on the atomic decomposition and molecular characterization of .
Zhixin Liu, Shanzhen Lu (1993)
Studia Mathematica
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The aim of this paper is to establish transference and restriction theorems for maximal operators defined by multipliers on the Hardy spaces and , 0 < p ≤ 1, which generalize the results of Kenig-Tomas for the case p > 1. We prove that under a mild regulation condition, an function m is a maximal multiplier on if and only if it is a maximal multiplier on . As an application, the restriction of maximal multipliers to lower dimensional Hardy spaces is considered. ...
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-...
J. J. Guadalupe, V. I. Kolyada (2001)
Studia Mathematica
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We investigate the behaviour of Fourier coefficients with respect to the system of ultraspherical polynomials. This leads us to the study of the “boundary” Lorentz space corresponding to the left endpoint of the mean convergence interval. The ultraspherical coefficients of -functions turn out to behave like the Fourier coefficients of functions in the real Hardy space ReH¹. Namely, we prove that for any the series is the Fourier series of some function φ ∈ ReH¹ with . ...
Paul Alton Hagelstein (2003)
Colloquium Mathematicae
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A necessary and sufficient condition is given on the basis of a rare maximal function such that implies f ∈ L log L([0,1]).
Muhammad Arshad, Eskandar Ameer, Aftab Hussain (2015)
Archivum Mathematicum
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The aim of this paper is to introduce some new fixed point results of Hardy-Rogers-type for ---contraction in a complete metric space. We extend the concept of -contraction into an ---contraction of Hardy-Rogers-type. An example has been constructed to demonstrate the novelty of our results.
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...
István Blahota, Lars-Erik Persson, Giorgi Tephnadze (2015)
Czechoslovak Mathematical Journal
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We prove and discuss some new -type inequalities of weighted maximal operators of Vilenkin-Nörlund means with non-increasing coefficients . These results are the best possible in a special sense. As applications, some well-known as well as new results are pointed out in the theory of strong convergence of such Vilenkin-Nörlund means. To fulfil our main aims we also prove some new estimates of independent interest for the kernels of these summability results. In the special cases of...
Yoshihiro Mizuta, Aleš Nekvinda, Tetsu Shimomura (2015)
Studia Mathematica
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Let be the n-dimensional fractional Hardy operator, where 0 < α ≤ n. It is well-known that is bounded from to with when n(1-1/p) < α ≤ n. We improve this result within the framework of Banach function spaces, for instance, weighted Lebesgue spaces and Lorentz spaces. We in fact find a ’source’ space , which is strictly larger than X, and a ’target’ space , which is strictly smaller than Y, under the assumption that is bounded from X into Y and the Hardy-Littlewood...
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...
Xuefang Yan (2015)
Czechoslovak Mathematical Journal
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Let be a metric measure space endowed with a distance and a nonnegative Borel doubling measure . Let be a non-negative self-adjoint operator of order on . Assume that the semigroup generated by satisfies the Davies-Gaffney estimate of order and satisfies the Plancherel type estimate. Let be the Hardy space associated with We show the boundedness of Stein’s square function arising from Bochner-Riesz means associated to from Hardy spaces to , and also study...
D. Cruz-Uribe, L. Diening, A. Fiorenza (2009)
Bollettino dell'Unione Matematica Italiana
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We give a new proof using the classic Calderón-Zygmund decomposition that the Hardy-Littlewood maximal operator is bounded on the variable Lebesgue space whenever the exponent function satisfies log-Hölder continuity conditions. We include the case where assumes the value infinity. The same proof also shows that the fractional maximal operator , , maps into , where .
Krzysztof Smela (2008)
Studia Mathematica
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We prove three theorems on linear operators induced by rearrangement of a subsequence of a Haar system. We find a sufficient and necessary condition for to be continuous for 0 < p < ∞.
Martin Smith (2007)
Studia Mathematica
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Given 0 < p,q < ∞ and any sequence z = zₙ in the unit disc , we define an operator from functions on to sequences by . Necessary and sufficient conditions on zₙ are given such that maps the Hardy space boundedly into the sequence space . A corresponding result for Bergman spaces is also stated.
Sunanda Naik (2010)
Annales Polonici Mathematici
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Motivated by some recent results by Li and Stević, in this paper we prove that a two-parameter family of Cesàro averaging operators is bounded on the Dirichlet spaces . We also give a short and direct proof of boundedness of on the Hardy space for 1 < p < ∞.
Feng Liu, Suzhen Mao (2017)
Czechoslovak Mathematical Journal
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In this paper we study the regularity properties of the one-dimensional one-sided Hardy-Littlewood maximal operators and . More precisely, we prove that and map with , boundedly and continuously. In addition, we show that the discrete versions and map boundedly and map continuously. Specially, we obtain the sharp variation inequalities of and , that is, if , where is the total variation of on and is the set of all functions satisfying .
Yu Liu, Jing Zhang, Jie-Lai Sheng, Li-Juan Wang (2016)
Czechoslovak Mathematical Journal
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Let be a Schrödinger operator and let be a Schrödinger type operator on , where is a nonnegative potential belonging to certain reverse Hölder class for . The Hardy type space is defined in terms of the maximal function with respect to the semigroup and it is identical to the Hardy space established by Dziubański and Zienkiewicz. In this article, we prove the -boundedness of the commutator generated by the Riesz transform , where , which is larger...
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 , .
Michalak Artur
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AbstractThe paper contains studies of relationships between properties of the “translation” mappings and the topological and geometric structure of spaces X and Hardy classes of X-valued harmonic functions on the open unit disk in ℂ (X is a Banach space). The mapping transforming the unit circle of ℂ into is associated with a function by the formula , where ϕₜ is the rotation of through t.AcknowledgmentsThis work is based in part on the author’s doctoral thesis written at...
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.