The John-Nirenberg type inequality for non-doubling measures

Yoshihiro Sawano; Hitoshi Tanaka

Displaying similar documents to “The John-Nirenberg type inequality for non-doubling measures”

Non-compact Littlewood-Paley theory for non-doubling measures

Michael Wilson (2007)

Studia Mathematica

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We prove weighted Littlewood-Paley inequalities for linear sums of functions satisfying mild decay, smoothness, and cancelation conditions. We prove these for general “regular” measure spaces, in which the underlying measure is not assumed to satisfy any doubling condition. Our result generalizes an earlier result of the author, proved on $ℝ^{d}$ with Lebesgue measure. Our proof makes essential use of the technique of random dyadic grids, due to Nazarov, Treil, and Volberg.

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 <; ∞.

Some remarks on the dyadic Rademacher maximal function

Mikko Kemppainen (2013)

Colloquium Mathematicae

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Properties of a maximal function for vector-valued martingales were studied by the author in an earlier paper. Restricting here to the dyadic setting, we prove the equivalence between (weighted) $L^{p}$ inequalities and weak type estimates, and discuss an extension to the case of locally finite Borel measures on ℝⁿ. In addition, to compensate for the lack of an $L^{\infty}$ inequality, we derive a suitable BMO estimate. Different dyadic systems in different dimensions are also considered.

Weighted norm inequalities for maximal singular integrals with nondoubling measures

Guoen Hu, Dachun Yang (2008)

Studia Mathematica

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Let μ be a nonnegative Radon measure on $ℝ^{d}$ which satisfies μ(B(x,r)) ≤ Crⁿ for any $x \in ℝ^{d}$ and r > 0 and some positive constants C and n ∈ (0,d]. In this paper, some weighted norm inequalities with $A_{p}^{ϱ} (μ)$ weights of Muckenhoupt type are obtained for maximal singular integral operators with such a measure μ, via certain weighted estimates with $A_{\infty}^{ϱ} (μ)$ weights of Muckenhoupt type involving the John-Strömberg maximal operator and the John-Strömberg sharp maximal operator, where ϱ,p ∈ [1,∞).

Lineability and spaceability on vector-measure spaces

Giuseppina Barbieri, Francisco J. García-Pacheco, Daniele Puglisi (2013)

Studia Mathematica

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It is proved that if X is infinite-dimensional, then there exists an infinite-dimensional space of X-valued measures which have infinite variation on sets of positive Lebesgue measure. In term of spaceability, it is also shown that $c a (ℬ, λ, X) ∖ M_{σ}$ , the measures with non-σ-finite variation, contains a closed subspace. Other considerations concern the space of vector measures whose range is neither closed nor convex. All of those results extend in some sense theorems of Muñoz Fernández et al. [Linear...

Simple fractions and linear decomposition of some convolutions of measures

Jolanta K. Misiewicz, Roger Cooke (2001)

Discussiones Mathematicae Probability and Statistics

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Every characteristic function φ can be written in the following way: φ(ξ) = 1/(h(ξ) + 1), where h(ξ) = ⎧ 1/φ(ξ) - 1 if φ(ξ) ≠ 0 ⎨ ⎩ ∞ if φ(ξ) = 0 This simple remark implies that every characteristic function can be treated as a simple fraction of the function h(ξ). In the paper, we consider a class C(φ) of all characteristic functions of the form $φ_{a} (ξ) = [a / (h (ξ) + a)]$ , where φ(ξ) is a fixed characteristic function. Using the well known theorem on simple fraction decomposition of rational functions we obtain...

A generalization of the Aleksandrov operator and adjoints of weighted composition operators

Eva A. Gallardo-Gutiérrez, Jonathan R. Partington (2013)

Annales de l’institut Fourier

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A generalization of the Aleksandrov operator is provided, in order to represent the adjoint of a weighted composition operator on $ℋ^{2}$ by means of an integral with respect to a measure. In particular, we show the existence of a family of measures which represents the adjoint of a weighted composition operator under fairly mild assumptions, and we discuss not only uniqueness but also the generalization of Aleksandrov–Clark measures which corresponds to the unweighted case, that is, to the...

Triebel-Lizorkin spaces with non-doubling measures

Yongsheng Han, Dachun Yang (2004)

Studia Mathematica

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Suppose that μ is a Radon measure on $ℝ^{d}$ , which may be non-doubling. The only condition assumed on μ is a growth condition, namely, there is a constant C₀ > 0 such that for all x ∈ supp(μ) and r > 0, μ(B(x,r)) ≤ C₀rⁿ, where 0 < n ≤ d. The authors provide a theory of Triebel-Lizorkin spaces $Ḟ_{p q}^{s} (μ)$ for 1 < p < ∞, 1 ≤ q ≤ ∞ and |s| < θ, where θ > 0 is a real number which depends on the non-doubling measure μ, C₀, n and d. The method does not use the vector-valued maximal function...

ω-Calderón-Zygmund operators

Sijue Wu (1995)

Studia Mathematica

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We prove a T1 theorem and develop a version of Calderón-Zygmund theory for ω-CZO when $ω \in A_{\infty}$ .

The multifractal box dimensions of typical measures

Frédéric Bayart (2012)

Fundamenta Mathematicae

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We compute the typical (in the sense of Baire’s category theorem) multifractal box dimensions of measures on a compact subset of $ℝ^{d}$ . Our results are new even in the context of box dimensions of measures.

A convolution property of some measures with self-similar fractal support

Denise Szecsei (2007)

Colloquium Mathematicae

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We define a class of measures having the following properties: (1) the measures are supported on self-similar fractal subsets of the unit cube $I^{M} = {[0, 1)}^{M}$ , with 0 and 1 identified as necessary; (2) the measures are singular with respect to normalized Lebesgue measure m on $I^{M}$ ; (3) the measures have the convolution property that $μ * L^{p} \subseteq L^{p + ε}$ for some ε = ε(p) > 0 and all p ∈ (1,∞). We will show that if (1/p,1/q) lies in the triangle with vertices (0,0), (1,1) and (1/2,1/3), then $μ * L^{p} \subseteq L^{q}$ for any measure μ in our...

Lipschitz continuity in Muckenhoupt 𝓐₁ weighted function spaces

Dorothee D. Haroske (2011)

Banach Center Publications

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We study continuity envelopes of function spaces $B_{p, q}^{s} (ℝ ⁿ, w)$ and $F_{p, q}^{s} (ℝ ⁿ, w)$ 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.

Some singular measures on the circle which improve $L^{p}$ spaces

David L. Ritter (1987)

Colloquium Mathematicae

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A unified Lorenz-type approach to divergence and dependence

Teresa Kowalczyk

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AbstractThe paper deals with function-valued and numerical measures of absolute and directed divergence of one probability measure from another. In case of absolute divergence, some new results are added to the known ones to form a unified structure. In case of directed divergence, new concepts are introduced and investigated. It is shown that the notions of absolute and directed divergences complement each other and provide a good insight into the extent and the type of discrepancy...

Convex Corson compacta and Radon measures

Grzegorz Plebanek (2002)

Fundamenta Mathematicae

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Assuming the continuum hypothesis, we show that (i) there is a compact convex subset L of $Σ (ℝ^{ω ₁})$ , and a probability Radon measure on L which has no separable support; (ii) there is a Corson compact space K, and a convex weak*-compact set M of Radon probability measures on K which has no $G_{δ}$ -points.

Monotonicity of generalized weighted mean values

Alfred Witkowski (2004)

Colloquium Mathematicae

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The author gives a new simple proof of monotonicity of the generalized extended mean values $M (r, s) = \leq {((\int f^{s} d μ) / (\int f^{r} d μ))}^{1 / (s - r)}$ introduced by F. Qi.

On Ordinary and Standard Lebesgue Measures on $ℝ^{\infty}$

Gogi Pantsulaia (2009)

Bulletin of the Polish Academy of Sciences. Mathematics

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New concepts of Lebesgue measure on $ℝ^{\infty}$ are proposed and some of their realizations in the ZFC theory are given. Also, it is shown that Baker’s both measures [1], [2], Mankiewicz and Preiss-Tišer generators [6] and the measure of [4] are not α-standard Lebesgue measures on $ℝ^{\infty}$ for α = (1,1,...).

A new characterization of $RBMO (μ)$ by John-Strömberg sharp maximal functions

Guoen Hu, Dachun Yang, Dongyong Yang (2009)

Czechoslovak Mathematical Journal

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Let $μ$ be a nonnegative Radon measure on $ℝ^{d}$ which only satisfies $μ (B (x, r)) \leq C_{0} r^{n}$ for all $x \in ℝ^{d}$ , $r > 0$ , with some fixed constants $C_{0} > 0$ and $n \in (0, d] .$ In this paper, a new characterization for the space $RBMO (μ)$ of Tolsa in terms of the John-Strömberg sharp maximal function is established.

Continuous linear functionals on the space of Borel vector measures

Pola Siwek (2008)

Annales Polonici Mathematici

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We study properties of the space ℳ of Borel vector measures on a compact metric space X, taking values in a Banach space E. The space ℳ is equipped with the Fortet-Mourier norm ${| | \cdot | |}_{ℱ}$ and the semivariation norm ||·||(X). The integral introduced by K. Baron and A. Lasota plays the most important role in the paper. Investigating its properties one can prove that in most cases the space $(ℳ, | | \cdot | |_{ℱ}) *$ is contained in but not equal to the space (ℳ,||·||(X))*. We obtain a representation of the continuous functionals...

Boundedness of Riesz transforms on weighted Carleson measure spaces

Ming-Yi Lee (2012)

Studia Mathematica

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Let w be in the Muckenhoupt $A_{\infty}$ weight class. We show that the Riesz transforms are bounded on the weighted Carleson measure space $C M O_{w}^{p}$ , the dual of the weighted Hardy space $H_{w}^{p}$ , 0 < p ≤ 1.

On the maximal function for rotation invariant measures in $ℝ^{n}$

Ana Vargas (1994)

Studia Mathematica

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Given a positive measure μ in $ℝ^{n}$ , there is a natural variant of the noncentered Hardy-Littlewood maximal operator $M_{μ} f (x) = s u p_{x \in B} 1 / μ (B) ʃ_{B} | f | d μ$ , where the supremum is taken over all balls containing the point x. In this paper we restrict our attention to rotation invariant, strictly positive measures μ in $ℝ^{n}$ . We give some necessary and sufficient conditions for $M_{μ}$ to be bounded from $L^{1} (d μ)$ to $L^{1, \infty} (d μ)$ .

Weighted inequalities for gradients on non-smooth domains

Caroline Sweezy, J. Michael Wilson

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We prove sufficiency of conditions on pairs of measures μ and ν, defined respectively on the interior and the boundary of a bounded Lipschitz domain Ω in d-dimensional Euclidean space, which ensure that, if u is the solution of the Dirichlet problem. Δu = 0 in Ω, ${u |}_{\partial Ω} = f$ , with f belonging to a reasonable test class, then $(\int_{Ω} {| \nabla u |}^{q} {d μ)}^{1 / q} \leq (\int_{\partial Ω} {| f |}^{p} {d ν)}^{1 / p}$ , where 1 < p ≤ q < ∞ and q ≥ 2. Our sufficiency conditions resemble those found by Wheeden and Wilson for the Dirichlet problem on $ℝ ₊^{d + 1}$ . As in that case we attack the...

The linear bound in A₂ for Calderón-Zygmund operators: a survey

Michael Lacey (2011)

Banach Center Publications

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For an L²-bounded Calderón-Zygmund Operator T acting on $L ² (ℝ^{d})$ , and a weight w ∈ A₂, the norm of T on L²(w) is dominated by $C_{T} {| | w | |}_{A ₂}$ . The recent theorem completes a line of investigation initiated by Hunt-Muckenhoupt-Wheeden in 1973 (MR0312139), has been established in different levels of generality by a number of authors over the last few years. It has a subtle proof, whose full implications will unfold over the next few years. This sharp estimate requires that the A₂ character of the weight can...

Weak compactness in the space of operator valued measures $M_{b} a (Σ, (X, Y))$ and its applications

N.U. Ahmed (2011)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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In this note we present necessary and sufficient conditions characterizing conditionally weakly compact sets in the space of (bounded linear) operator valued measures $M_{b a} (Σ, (X, Y))$ . This generalizes a recent result of the author characterizing conditionally weakly compact subsets of the space of nuclear operator valued measures $M_{b a} (Σ, ₁ (X, Y))$ . This result has interesting applications in optimization and control theory as illustrated by several examples.

The minimal operator and the geometric maximal operator in ℝⁿ

David Cruz-Uribe, SFO (2001)

Studia Mathematica

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We prove two-weight norm inequalities in ℝⁿ for the minimal operator $f (x) = i n f_{Q ∋ x} 1 / | Q | \int_{Q} | f | d y$ , 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 $M ₀ f (x) = {s u p}_{Q ∋ x} e x p (1 / | Q | \int_{Q} l o g | f | d x)$ , 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...

Measures of pseudorandomness of finite binary lattices, I. The measures $Q_{k}$ , normality

Katalin Gyarmati, Christian Mauduit, András Sárközy (2010)

Acta Arithmetica

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Commutators with fractional integral operators

Irina Holmes, Robert Rahm, Scott Spencer (2016)

Studia Mathematica

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We investigate weighted norm inequalities for the commutator of a fractional integral operator and multiplication by a function. In particular, we show that, for $μ, λ \in A_{p, q}$ and α/n + 1/q = 1/p, the norm $| | [b, I_{α}] : L^{p} (μ^{p}) \to L^{q} (λ^{q}) | |$ is equivalent to the norm of b in the weighted BMO space BMO(ν), where $ν = μ λ^{- 1}$ . This work extends some of the results on this topic existing in the literature, and continues a line of investigation which was initiated by Bloom in 1985 and was recently developed further by the first author, Lacey,...

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.

Self-affine measures that are $L^{p}$ -improving

Kathryn E. Hare (2015)

Colloquium Mathematicae

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A measure is called $L^{p}$ -improving if it acts by convolution as a bounded operator from $L^{q}$ to L² for some q < 2. Interesting examples include Riesz product measures, Cantor measures and certain measures on curves. We show that equicontractive, self-similar measures are $L^{p}$ -improving if and only if they satisfy a suitable linear independence property. Certain self-affine measures are also seen to be $L^{p}$ -improving.

On the isotropic constant of marginals

Grigoris Paouris (2012)

Studia Mathematica

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We show that if μ₁, ..., μₘ are log-concave subgaussian or supergaussian probability measures in $ℝ^{n_{i}}$ , i ≤ m, then for every F in the Grassmannian $G_{N, n}$ , where N = n₁ + ⋯ + nₘ and n< N, the isotropic constant of the marginal of the product of these measures, $π_{F} (μ ₁ \otimes \dots \otimes μ ₘ)$ , is bounded. This extends known results on bounds of the isotropic constant to a larger class of measures.

Estimates of capacity of self-similar measures

Jozef Myjak, Tomasz Szarek (2002)

Annales Polonici Mathematici

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We give lower and upper estimates of the capacity of self-similar measures generated by iterated function systems $(S_{i}, p_{i}) : i = 1, . . ., N$ where $S_{i}$ are bi-lipschitzean transformations.