A note on rare maximal functions

Paul Alton Hagelstein

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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 $\int_{Q} M_{y} M_{x} h < \infty$ but $\int_{Q} M_{x} M_{y} h = \infty$ . It is shown that if f is a function supported on Q such that $\int_{Q} M_{y} M_{x} f < \infty$ but $\int_{Q} M_{x} M_{y} f = \infty$ , then there exists a set A of finite measure in ℝ² such that $\int_{A} M_{S} f = \infty$ .

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 ${| | M f | |}_{p), w} {\leq c | | f | |}_{p), w}$ holds with some c independent of f iff w belongs to the well known Muckenhoupt class $A_{p}$ , and therefore iff ${| | M f | |}_{p, w} {\leq c | | f | |}_{p, w}$ for some c independent of f. Some results of similar type are discussed for the case of small...

Transference and restriction of maximal multiplier operators on Hardy spaces

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 $H^{p} (ℝ^{n})$ and $H^{p} (^{n})$ , 0 < p ≤ 1, which generalize the results of Kenig-Tomas for the case p > 1. We prove that under a mild regulation condition, an $L^{\infty} (ℝ^{n})$ function m is a maximal multiplier on $H^{p} (ℝ^{n})$ if and only if it is a maximal multiplier on $H^{p} (^{n})$ . As an application, the restriction of maximal multipliers to lower dimensional Hardy spaces is considered. ...

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 $p_{0} \in (n / (n + 1), 1)$ 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 $p \in (p_{0}, \infty]$ . This result yields a radial maximal function characterization for Hardy spaces on $𝒳$ . ...

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 $σ ₀ * : = s u p ₙ | σ_{n, n} |$ of the Fejér means of the double Vilenkin-Fourier series is not bounded from the Hardy space $H_{1 / 2}$ to the space weak- $L_{1 / 2}$ .

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 $L_{1 / 2}$ .

Weak- and strong-type inequality for the cone-like maximal operator in variable Lebesgue spaces

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 $L_{p} (ℝ^{d})$ (in the case $p > 1$ ), but (in the case when $1 / p (\cdot)$ is log-Hölder continuous and $p_{-} = inf {p (x) : x \in ℝ^{d}} > 1$ ) on the variable Lebesgue spaces $L_{p (\cdot)} (ℝ^{d})$ , too. Furthermore, the classical Hardy-Littlewood maximal operator is of weak-type $(1, 1)$ . In the present note we generalize Besicovitch’s covering theorem for the so-called $γ$ -rectangles. We introduce a general maximal operator $M_{s}^{γ, δ}$ and with the help of generalized $Φ$ -functions, the strong-...

A remark on the $H^{1}$ -BMO duality in product domains

J. Alvarez (1989)

Colloquium Mathematicae

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Weak-type inequalities for maximal operators acting on Lorentz spaces

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 $L^{p, q}$ , 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 $C_{p, q, r}$ such that for any $ϕ \in L^{p, q}$ , ${| | ℳ ϕ | |}_{r, \infty} \leq C_{p, q, r} {| | ϕ | |}_{p, q}$ .

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 $\int_{} | M_{ω} {f (z) |}^{q} d μ (z) \leq C (\int_{} {| f (z) |}^{p} {ω (z) d V (z))}^{q / p}$ and $μ (z \in : M f (z) > λ) \leq C / (λ^{q}) (\int_{} | f (z) {|^{p} ω (z) d V (z))}^{q / p}$ , where $M_{ω}$ is the weighted Hardy-Littlewood maximal function on the upper half-plane and 1 ≤ p,q <; ∞.

A New Proof of the Boundedness of Maximal Operators on Variable Lebesgue Spaces

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 $L^{p(\cdot)}$ whenever the exponent function $p(\cdot)$ satisfies log-Hölder continuity conditions. We include the case where $p(\cdot)$ assumes the value infinity. The same proof also shows that the fractional maximal operator $M_{a}$ , $0<a<n$ , maps $L^{p(\cdot)}$ into $L^{q(\cdot)}$ , where $1/p(\cdot)-1/q(\cdot)=a/n$ .

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 $A ₙ f (x) = 2^{- n} \int_{x}^{x + 2 ⁿ} f$ . The square function is defined as $S f (x) = (\sum_{n = - \infty}^{\infty} | A ₙ f (x) - A_{n - 1} f (x) {| ²)}^{1 / 2}$ . The local version of this operator, namely the operator $S ₁ f (x) = (\sum_{n = - \infty}^{0} | A ₙ f (x) - A_{n - 1} f (x) {| ²)}^{1 / 2}$ , 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 $L^{\infty}$ into BMO. We prove that the operator S not only maps $L^{\infty}$ into BMO but it also maps BMO into BMO. We also prove that the $L^{p}$ boundedness...

Problems on averages and lacunary maximal functions

Andreas Seeger, James Wright (2011)

Banach Center Publications

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We prove three results concerning convolution operators and lacunary maximal functions associated to dilates of measures. First we obtain an H¹ to $L^{1, \infty}$ bound for lacunary maximal operators under a dimensional assumption on the underlying measure and an assumption on an $L^{p}$ regularity bound for some p > 1. Secondly, we obtain a necessary and sufficient condition for L² boundedness of lacunary maximal operator associated to averages over convex curves in the plane. Finally we prove an $L^{p}$ ...

On the Rademacher maximal function

Mikko Kemppainen (2011)

Studia Mathematica

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This paper studies a new maximal operator introduced by Hytönen, McIntosh and Portal in 2008 for functions taking values in a Banach space. The $L^{p}$ -boundedness of this operator depends on the range space; certain requirements on type and cotype are present for instance. The original Euclidean definition of the maximal function is generalized to σ-finite measure spaces with filtrations and the $L^{p}$ -boundedness is shown not to depend on the underlying measure space or the filtration. Martingale...

Sums of commuting operators with maximal regularity

Christian Le Merdy, Arnaud Simard (2001)

Studia Mathematica

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Let Y be a Banach space and let $S \subset L_{p}$ be a subspace of an $L_{p}$ space, for some p ∈ (1,∞). We consider two operators B and C acting on S and Y respectively and satisfying the so-called maximal regularity property. Let ℬ and be their natural extensions to $S (Y) \subset L_{p} (Y)$ . We investigate conditions that imply that ℬ + is closed and has the maximal regularity property. Extending theorems of Lamberton and Weis, we show in particular that this holds if Y is a UMD Banach lattice and $e^{- t B}$ is a positive contraction...

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.

Weighted variable $L^{p}$ integral inequalities for the maximal operator on non-increasing functions

C. J. Neugebauer (2009)

Studia Mathematica

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Let $B_{p}$ be the Ariõ-Muckenhoupt weight class which controls the weighted $L^{p}$ -norm inequalities for the Hardy operator on non-increasing functions. We replace the constant p by a function p(x) and examine the associated $L^{p (x)}$ -norm inequalities of the Hardy operator.

$L^{p}$ Maximal Regularity for Second Order Cauchy Problems is Independent of $p$

Ralph Chill, Sachi Srivastava (2008)

Bollettino dell'Unione Matematica Italiana

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If the second order problem $\ddot{u}+\dot{u}+Au=f$ has $L^{p}$ maximal regularity for some $p\in(1,\infty)$ , then it has $L^{p}$ maximal regularity for every $p\in(1,\infty)$ .

Maximal regularity of discrete and continuous time evolution equations

Sönke Blunck (2001)

Studia Mathematica

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We consider the maximal regularity problem for the discrete time evolution equation $u_{n + 1} - T u ₙ = f ₙ$ for all n ∈ ℕ₀, u₀ = 0, where T is a bounded operator on a UMD space X. We characterize the discrete maximal regularity of T by two types of conditions: firstly by R-boundedness properties of the discrete time semigroup ${(T ⁿ)}_{n \in ℕ ₀}$ and of the resolvent R(λ,T), secondly by the maximal regularity of the continuous time evolution equation u’(t) - Au(t) = f(t) for all t > 0, u(0) = 0, where A:= T - I. By recent...

Optimal estimates for the fractional Hardy operator

Yoshihiro Mizuta, Aleš Nekvinda, Tetsu Shimomura (2015)

Studia Mathematica

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Let $A_{α} f (x) = {| B (0, | x |) |}^{- α / n} \int_{B (0, | x |)} f (t) d t$ 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...

On the regularity of the one-sided Hardy-Littlewood maximal functions

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 $ℳ^{+}$ and $ℳ^{-}$ . More precisely, we prove that $ℳ^{+}$ and $ℳ^{-}$ map $W^{1, p} (ℝ) \to W^{1, p} (ℝ)$ with $1 < p < \infty$ , boundedly and continuously. In addition, we show that the discrete versions $M^{+}$ and $M^{-}$ map $BV (ℤ) \to BV (ℤ)$ boundedly and map $l^{1} (ℤ) \to BV (ℤ)$ continuously. Specially, we obtain the sharp variation inequalities of $M^{+}$ and $M^{-}$ , that is, $Var (M^{+} (f)) \leq Var (f) and Var (M^{-} (f)) \leq Var (f)$ if $f \in BV (ℤ)$ , where $Var (f)$ is the total variation of $f$ on $ℤ$ and $BV (ℤ)$ is the set of all functions $f : ℤ \to ℝ$ satisfying $Var (f) < \infty$ .

Continuous rearrangements of the Haar system in $H_{p}$ for 0 < p < ∞

Krzysztof Smela (2008)

Studia Mathematica

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We prove three theorems on linear operators $T_{τ, p} : H_{p} (ℬ) \to H_{p}$ induced by rearrangement of a subsequence of a Haar system. We find a sufficient and necessary condition for $T_{τ, p}$ to be continuous for 0 < p < ∞.

A radial estimate for the maximal operator associated with the free Schrödinger equation

Sichun Wang (2006)

Studia Mathematica

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Let d > 0 be a positive real number and n ≥ 1 a positive integer and define the operator $S_{d}$ and its associated global maximal operator $S * *_{d}$ by $(S_{d} f) (x, t) = 1 / (2 π) ⁿ \int_{ℝ ⁿ} e^{i x \cdot ξ} e^{{i t | ξ |}^{d}} f ̂ (ξ) d ξ$ , f ∈ (ℝⁿ), x ∈ ℝⁿ, t ∈ ℝ, $(S * *_{d} f) (x) = s u p_{t \in ℝ} | 1 / (2 π) ⁿ \int_{ℝ ⁿ} e^{i x \cdot ξ} e^{{i t | ξ |}^{d}} f ̂ (ξ) d ξ |$ , f ∈ (ℝⁿ), x ∈ ℝⁿ, where f̂ is the Fourier transform of f and (ℝⁿ) is the Schwartz class of rapidly decreasing functions. If d = 2, $S_{d} f$ is the solution to the initial value problem for the free Schrödinger equation (cf. (1.3) in this paper). We prove that for radial functions f ∈ (ℝⁿ), if n ≥ 3, 0 < d ≤ 2, and p ≥...

Some weighted norm inequalities for a one-sided version of $g *_{λ}$

L. de Rosa, C. Segovia (2006)

Studia Mathematica

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We study the boundedness of the one-sided operator $g ⁺_{λ, φ}$ between the weighted spaces $L^{p} (M ¯ w)$ and $L^{p} (w)$ 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 $g ⁺_{λ, φ}$ . 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 $g ⁺_{λ, φ}$ from $L^{p} ({(M ¯)}^{[p / 2] + 1} w)$ to $L^{p} (w)$ , where ${(M ¯)}^{k}$ denotes the operator M¯ iterated k times.

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...

Translations of functions iv vector Hardy classes on the unit disk

Michalak Artur

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AbstractThe paper contains studies of relationships between properties of the “translation” mappings $T_{F}$ and the topological and geometric structure of spaces X and Hardy classes $h^{p} (, X)$ of X-valued harmonic functions on the open unit disk in ℂ (X is a Banach space). The mapping $T_{F}$ transforming the unit circle of ℂ into $h^{p} (, X)$ is associated with a function $F \in h^{p} (, X)$ by the formula $T_{F} (t) = F \circ ϕ ₜ$ , where ϕₜ is the rotation of through t.AcknowledgmentsThis work is based in part on the author’s doctoral thesis written at...

On contractive projections in Hardy spaces

Florence Lancien, Beata Randrianantoanina, Eric Ricard (2005)

Studia Mathematica

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We prove a conjecture of Wojtaszczyk that for 1 ≤ p < ∞, p ≠ 2, $H_{p} ()$ does not admit any norm one projections with dimension of the range finite and greater than 1. This implies in particular that for 1 ≤ p < ∞, p ≠ 2, $H_{p}$ does not admit a Schauder basis with constant one.

A complete characterization of R-sets in the theory of differentiation of integrals

G. A. Karagulyan (2007)

Studia Mathematica

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Let $_{s}$ 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 $_{s}$ differentiates the integral of f if s ∉ S, and $D ̅_{s} f (x) = l i m s u p_{d i a m (R) \to 0, x \in R \in_{s}} {| R |}^{- 1} \int_{R} f = \infty$ almost everywhere if s ∈ S. If the condition $D ̅_{s} f (x) = \infty$ 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 $G_{δ}$ (resp. a $G_{δ σ}$ ).