Displaying similar documents to “Multi-dimensional Fejér summability and local Hardy spaces”

On the Nörlund means of Vilenkin-Fourier series

István Blahota, Lars-Erik Persson, Giorgi Tephnadze (2015)

Czechoslovak Mathematical Journal

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We prove and discuss some new ( H p , L p ) -type inequalities of weighted maximal operators of Vilenkin-Nörlund means with non-increasing coefficients { q k : k 0 } . 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...

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 ( ) .

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 σ is bounded from the Hardy space H 1 / 2 to weak- L 1 / 2 and is not bounded from H 1 / 2 to L 1 / 2 .

Hardy-Rogers-type fixed point theorems for α - G F -contractions

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 α - η - G F -contraction in a complete metric space. We extend the concept of F -contraction into an α - η - G F -contraction of Hardy-Rogers-type. An example has been constructed to demonstrate the novelty of our results.

Maximal operators of Fejér means of double Vilenkin-Fourier series

István Blahota, György Gát, Ushangi Goginava (2007)

Colloquium Mathematicae

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The main aim of this paper is to prove that the maximal operator σ * : = s u p | σ n , n | of the Fejér means of the double Vilenkin-Fourier series is not bounded from the Hardy space H 1 / 2 to the space weak- L 1 / 2 .

A note on a critical problem with natural growth in the gradient

Boumediene Abdellaoui, Ireneo Peral (2006)

Journal of the European Mathematical Society

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The paper analyzes the influence on the meaning of natural growth in the gradient of a perturbation by a Hardy potential in some elliptic equations. Indeed, in the case of the Laplacian the natural problem becomes Δ u Λ N u | x | 2 = u + N 2 2 u | x | 2 x 2 | x | ( N 2 ) / 2 + λ f ( x ) in Ω , u = 0 on Ω , Λ N = ( ( N 2 ) / 2 ) 2 . This problem is a particular case of problem (2). Notice that ( N 2 ) / 2 is optimal as coefficient and exponent on the right hand side.

Local integrability of strong and iterated maximal functions

Paul Alton Hagelstein (2001)

Studia Mathematica

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Let M S denote the strong maximal operator. Let M x and M y 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 Q M y M x h < but Q M x M y h = . It is shown that if f is a function supported on Q such that Q M y M x f < but Q M x M y f = , then there exists a set A of finite measure in ℝ² such that A M S f = .

Boundedness of Stein's square functions and Bochner-Riesz means associated to operators on Hardy spaces

Xuefang Yan (2015)

Czechoslovak Mathematical Journal

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Let ( X , d , μ ) be a metric measure space endowed with a distance d and a nonnegative Borel doubling measure μ . Let L be a non-negative self-adjoint operator of order m on L 2 ( X ) . Assume that the semigroup e - t L generated by L satisfies the Davies-Gaffney estimate of order m and L satisfies the Plancherel type estimate. Let H L p ( X ) be the Hardy space associated with L . We show the boundedness of Stein’s square function 𝒢 δ ( L ) arising from Bochner-Riesz means associated to L from Hardy spaces H L p ( X ) to L p ( X ) , and also study...

Restricted weak type inequalities for the one-sided Hardy-Littlewood maximal operator in higher dimensions

Fabio Berra (2022)

Czechoslovak Mathematical Journal

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We give a quantitative characterization of the pairs of weights ( w , v ) for which the dyadic version of the one-sided Hardy-Littlewood maximal operator satisfies a restricted weak ( p , p ) type inequality for 1 p < . More precisely, given any measurable set E 0 , the estimate w ( { x n : M + , d ( 𝒳 E 0 ) ( x ) > t } ) C [ ( w , v ) ] A p + , d ( ) p t p v ( E 0 ) holds if and only if the pair ( w , v ) belongs to A p + , d ( ) , that is, | E | | Q | [ ( w , v ) ] A p + , d ( ) v ( E ) w ( Q ) 1 / p for every dyadic cube Q and every measurable set E Q + . The proof follows some ideas appearing in S. Ombrosi (2005). We also obtain a similar quantitative characterization for the...

Optimal estimates for the fractional Hardy operator

Yoshihiro Mizuta, Aleš Nekvinda, Tetsu Shimomura (2015)

Studia Mathematica

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Let A α f ( x ) = | B ( 0 , | x | ) | - α / n B ( 0 , | x | ) f ( t ) d t be the n-dimensional fractional Hardy operator, where 0 < α ≤ n. It is well-known that A α is bounded from L p to L p α with p α = n p / ( α p - n p + n ) 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 S α , Y , which is strictly larger than X, and a ’target’ space T Y , which is strictly smaller than Y, under the assumption that A α is bounded from X into Y and the Hardy-Littlewood...

Bounded evaluation operators from H p into q

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 T z , p ( f ) = ( 1 - | z | ² ) 1 / p f ( z ) . Necessary and sufficient conditions on zₙ are given such that T z , p maps the Hardy space H p boundedly into the sequence space q . A corresponding result for Bergman spaces is also stated.

Weighted local Orlicz-Hardy spaces with applications to pseudo-differential operators

Dachun Yang, Sibei Yang

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Let Φ be a concave function on (0,∞) of strictly critical lower type index p Φ ( 0 , 1 ] and ω A l o c ( ) (the class of local weights introduced by V. S. Rychkov). We introduce the weighted local Orlicz-Hardy space h ω Φ ( ) via the local grand maximal function. Let ρ ( t ) t - 1 / Φ - 1 ( t - 1 ) for all t ∈ (0,∞). We also introduce the BMO-type space b m o ρ , ω ( ) and establish the duality between h ω Φ ( ) and b m o ρ , ω ( ) . Characterizations of h ω Φ ( ) , including the atomic characterization, the local vertical and the local nontangential maximal function characterizations, are...

Boundedness of Littlewood-Paley operators relative to non-isotropic dilations

Shuichi Sato (2019)

Czechoslovak Mathematical Journal

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We consider Littlewood-Paley functions associated with a non-isotropic dilation group on n . We prove that certain Littlewood-Paley functions defined by kernels with no regularity concerning smoothness are bounded on weighted L p spaces, 1 < p < , with weights of the Muckenhoupt class. This, in particular, generalizes a result of N. Rivière (1971).

Radial maximal function characterizations for Hardy spaces on RD-spaces

Loukas Grafakos, Liguang Liu, Dachun Yang (2009)

Bulletin de la Société Mathématique de France

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An RD-space 𝒳 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 𝒳 having “dimension” n , there exists a p 0 ( n / ( n + 1 ) , 1 ) such that for certain classes of distributions, the L p ( 𝒳 ) quasi-norms of their radial maximal functions and grand maximal functions are equivalent when p ( p 0 , ] . This result yields a radial maximal function characterization for Hardy spaces on 𝒳 . ...