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. ...
Shuichi Sato (2019)
Czechoslovak Mathematical Journal
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We consider Littlewood-Paley functions associated with a non-isotropic dilation group on . We prove that certain Littlewood-Paley functions defined by kernels with no regularity concerning smoothness are bounded on weighted spaces, , with weights of the Muckenhoupt class. This, in particular, generalizes a result of N. Rivière (1971).
Fabio Berra (2022)
Czechoslovak Mathematical Journal
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We give a quantitative characterization of the pairs of weights for which the dyadic version of the one-sided Hardy-Littlewood maximal operator satisfies a restricted weak type inequality for . More precisely, given any measurable set , the estimate holds if and only if the pair belongs to , that is, for every dyadic cube and every measurable set . The proof follows some ideas appearing in S. Ombrosi (2005). We also obtain a similar quantitative characterization for the...
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...
Nayna Govindbhai Kalsariya, Bhikha Lila Ghodadra (2024)
Mathematica Bohemica
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We consider the Vilenkin orthonormal system on a Vilenkin group and the Vilenkin-Fourier coefficients , , of functions for some . We obtain certain sufficient conditions for the finiteness of the series , where is a given sequence of positive real numbers satisfying a mild assumption and . We also find analogous conditions for the double Vilenkin-Fourier series. These sufficient conditions are in terms of (either global or local) moduli of continuity of and give multiplicative...
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...