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

Denise Szecsei

Displaying similar documents to “A convolution property of some measures with self-similar fractal support”

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

David L. Ritter (1987)

Colloquium Mathematicae

Similarity:

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

Kathryn E. Hare (2015)

Colloquium Mathematicae

Similarity:

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.

$L^{p}$ -improving properties of measures of positive energy dimension

Kathryn E. Hare, Maria Roginskaya (2005)

Colloquium Mathematicae

Similarity:

A measure is called $L^{p}$ -improving if it acts by convolution as a bounded operator from $L^{p}$ to $L^{q}$ for some q > p. Positive measures which are $L^{p}$ -improving are known to have positive Hausdorff dimension. We extend this result to complex $L^{p}$ -improving measures and show that even their energy dimension is positive. Measures of positive energy dimension are seen to be the Lipschitz measures and are characterized in terms of their improving behaviour on a subset of $L^{p}$ -functions.

$L^{p} - L^{q}$ estimates for some convolution operators with singular measures on the Heisenberg group

T. Godoy, P. Rocha (2013)

Colloquium Mathematicae

Similarity:

We consider the Heisenberg group ℍⁿ = ℂⁿ × ℝ. Let ν be the Borel measure on ℍⁿ defined by $ν (E) = \int_{ℂ ⁿ} χ_{E} (w, φ (w)) η (w) d w$ , where $φ (w) = \sum_{j = 1}^{n} a_{j} | w_{j} | ²$ , w = (w₁,...,wₙ) ∈ ℂⁿ, $a_{j} \in ℝ$ , and η(w) = η₀(|w|²) with $η ₀ \in C_{c}^{\infty} (ℝ)$ . We characterize the set of pairs (p,q) such that the convolution operator with ν is $L^{p} (ℍ ⁿ) - L^{q} (ℍ ⁿ)$ bounded. We also obtain $L^{p}$ -improving properties of measures supported on the graph of the function $φ (w) = {| w |}^{2 m}$ .

On the isotropic constant of marginals

Grigoris Paouris (2012)

Studia Mathematica

Similarity:

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.

On a theorem of Saeki concerning convolution squares of singular measures

Thomas Körner (2008)

Bulletin de la Société Mathématique de France

Similarity:

If $1 > α > 1 / 2$ , then there exists a probability measure $μ$ such that the Hausdorff dimension of the support of $μ$ is $α$ and $μ * μ$ is a Lipschitz function of class $α - \frac{1}{2}$ .

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

Gogi Pantsulaia (2009)

Bulletin of the Polish Academy of Sciences. Mathematics

Similarity:

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,...).

The type set for homogeneous singular measures on ℝ ³ of polynomial type

E. Ferreyra, T. Godoy (2006)

Colloquium Mathematicae

Similarity:

Let φ:ℝ ² → ℝ be a homogeneous polynomial function of degree m ≥ 2, let μ be the Borel measure on ℝ ³ defined by $μ (E) = \int_{D} χ_{E} (x, φ (x)) d x$ with D = x ∈ ℝ ²:|x| ≤ 1 and let $T_{μ}$ be the convolution operator with the measure μ. Let $φ = φ ₁^{e ₁} \dots φ ₙ^{e ₙ}$ be the decomposition of φ into irreducible factors. We show that if $e_{i} \neq m / 2$ for each $φ_{i}$ of degree 1, then the type set $E_{μ} : = (1 / p, 1 / q) \in [0, 1] \times [0, 1] : | | T_{μ} {| |}_{p, q} < \infty$ can be explicitly described as a closed polygonal region.

Osgood type conditions for an m th-order differential equation

Stanisaw Szufla (1998)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

Similarity:

We present a new theorem on the differential inequality $u^{(m)} \leq w (u)$ . Next, we apply this result to obtain existence theorems for the equation $x^{(m)} = f (t, x)$ .

The type set for some measures on $ℝ^{2 n}$ with $n$ -dimensional support

E. Ferreyra, T. Godoy, Marta Urciuolo (2002)

Czechoslovak Mathematical Journal

Similarity:

Let $ϕ_{1}, \dots, ϕ_{n}$ be real homogeneous functions in $C^{\infty} (ℝ^{n} - {0})$ of degree $k \geq 2$ , let $ϕ (x) = (ϕ_{1} (x), \dots, ϕ_{n} (x))$ and let $μ$ be the Borel measure on $ℝ^{2 n}$ given by $μ (E) = \int_{ℝ^{n}} χ_{E} (x, ϕ (x)) {| x |}^{γ - n} d x$ where $d x$ denotes the Lebesgue measure on $ℝ^{n}$ and $γ > 0$ . Let $T_{μ}$ be the convolution operator $T_{μ} f (x) = (μ * f) (x)$ and let $E_{μ} = {(1 / p, 1 / q) ∥ T_{μ} ∥_{p, q} < \infty, 1 \leq p, q \leq \infty} .$ Assume that, for $x \neq 0$ , the following two conditions hold: $det (d^{2} ϕ (x) h)$ vanishes only at $h = 0$ and $det (d ϕ (x)) \neq 0$ . In this paper we show that if $γ > n (k + 1) / 3$ then $E_{μ}$ is the empty set and if $γ \leq n (k + 1) / 3$ then $E_{μ}$ is the closed segment with endpoints $D = (1 - \frac{γ}{n (k + 1)}, 1 - \frac{2 γ}{n (k + 1)})$ and $D^{'} = (\frac{2 γ}{n (1 + k)}, \frac{γ}{n (1 + k)})$ . Also, we give some examples.

Sets of $β$ -expansions and the Hausdorff measure of slices through fractals

Tom Kempton (2016)

Journal of the European Mathematical Society

Similarity:

We study natural measures on sets of $β$ -expansions and on slices through self similar sets. In the setting of $β$ -expansions, these allow us to better understand the measure of maximal entropy for the random $β$ -transformation and to reinterpret a result of Lindenstrauss, Peres and Schlag in terms of equidistribution. Each of these applications is relevant to the study of Bernoulli convolutions. In the fractal setting this allows us to understand how to disintegrate Hausdorff measure by slicing,...

Integral representation and relaxation for functionals defined on measures

Ennio De Giorgi, Luigi Ambrosio, Giuseppe Buttazzo (1987)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

Similarity:

Given a separable metric locally compact space $\Omega$ , and a positive finite non-atomic measure $\lambda$ on $\Omega$ , we study the integral representation on the space of measures with bounded variation $\Omega$ of the lower semicontinuous envelope of the functional $F(u)=\int_{\Omega}f(x,u)d\lambda\qquad u\in L^{1}(\Omega,\lambda,\mathbb{R}^{n})$ with respect to the weak convergence of measures.

On inhomogeneous self-similar measures and their $L^{q}$ spectra

Przemysław Liszka (2013)

Annales Polonici Mathematici

Similarity:

Let $S_{i} : ℝ^{d} \to ℝ^{d}$ for i = 1,..., N be contracting similarities, let $(p ₁, . . ., p_{N}, p)$ be a probability vector and let ν be a probability measure on $ℝ^{d}$ with compact support. It is well known that there exists a unique inhomogeneous self-similar probability measure μ on $ℝ^{d}$ such that $μ = \sum_{i = 1}^{N} p_{i} μ \circ S_{i}^{- 1} + p ν$ . We give satisfactory estimates for the lower and upper bounds of the $L^{q}$ spectra of inhomogeneous self-similar measures. The case in which there are a countable number of contracting similarities and probabilities is considered. In particular,...

Level by level equivalence and the number of normal measures over $P_{κ} (λ)$

Arthur W. Apter (2007)

Fundamenta Mathematicae

Similarity:

We construct two models for the level by level equivalence between strong compactness and supercompactness in which if κ is λ supercompact and λ ≥ κ is regular, we are able to determine exactly the number of normal measures $P_{κ} (λ)$ carries. In the first of these models, $P_{κ} (λ)$ carries $2^{2^{{[λ]}^{< κ}}}$ many normal measures, the maximal number. In the second of these models, $P_{κ} (λ)$ carries $2^{2^{{[λ]}^{< κ}}}$ many normal measures, except if κ is a measurable cardinal which is not a limit of measurable cardinals. In this case, κ (and...

Denseness and Borel complexity of some sets of vector measures

Zbigniew Lipecki (2004)

Studia Mathematica

Similarity:

Let ν be a positive measure on a σ-algebra Σ of subsets of some set and let X be a Banach space. Denote by ca(Σ,X) the Banach space of X-valued measures on Σ, equipped with the uniform norm, and by ca(Σ,ν,X) its closed subspace consisting of those measures which vanish at every ν-null set. We are concerned with the subsets $_{ν} (X)$ and $_{ν} (X)$ of ca(Σ,X) defined by the conditions |φ| = ν and |φ| ≥ ν, respectively, where |φ| stands for the variation of φ ∈ ca(Σ,X). We establish necessary and sufficient...

Common zero sets of equivalent singular inner functions

Keiji Izuchi (2004)

Studia Mathematica

Similarity:

Let μ and λ be bounded positive singular measures on the unit circle such that μ ⊥ λ. It is proved that there exist positive measures μ₀ and λ₀ such that μ₀ ∼ μ, λ₀ ∼ λ, and $| ψ_{μ ₀} | < 1 \cap | ψ_{λ ₀} | < 1 = \emptyset$ , where $ψ_{μ}$ is the associated singular inner function of μ. Let $(μ) = ⋂_{ν; ν \sim μ} Z (ψ_{ν})$ be the common zeros of equivalent singular inner functions of $ψ_{μ}$ . Then (μ) ≠ ∅ and (μ) ∩ (λ) = ∅. It follows that μ ≪ λ if and only if (μ) ⊂ (λ). Hence (μ) is the set in the maximal ideal space of $H^{\infty}$ which relates naturally to the set of measures equivalent...

Simple fractions and linear decomposition of some convolutions of measures

Jolanta K. Misiewicz, Roger Cooke (2001)

Discussiones Mathematicae Probability and Statistics

Similarity:

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...

Limit theorems for random fields

Nguyen van Thu

Similarity:

CONTENTSIntroduction............................................................................................................................................................................ 51. Notation and preliminaries............................................................................................................................................ 52. Statement of the problem..................................................................................................................................................

Erratum to “Geometric rigidity of $\times m$ invariant measures” (J. Eur. Math. Soc. 14, 1539–1563 (2012))

Michael Hochman (2013)

Journal of the European Mathematical Society

Similarity:

Wasserstein metric and subordination

Philippe Clément, Wolfgang Desch (2008)

Studia Mathematica

Similarity:

Let $(X, d_{X})$ , $(Ω, d_{Ω})$ be complete separable metric spaces. Denote by (X) the space of probability measures on X, by $W_{p}$ the p-Wasserstein metric with some p ∈ [1,∞), and by $_{p} (X)$ the space of probability measures on X with finite Wasserstein distance from any point measure. Let $f : Ω \to_{p} (X)$ , $ω \mapsto f_{ω}$ , be a Borel map such that f is a contraction from $(Ω, d_{Ω})$ into $(_{p} (X), W_{p})$ . Let ν₁,ν₂ be probability measures on Ω with $W_{p} (ν ₁, ν ₂)$ finite. On X we consider the subordinated measures $μ_{i} = \int_{Ω} f_{ω} d ν_{i} (ω)$ . Then $W_{p} (μ ₁, μ ₂) \leq W_{p} (ν ₁, ν ₂)$ . As an application we show that the solution measures $ϱ_{α} (t)$ ...

A convolution property of the Cantor-Lebesgue measure, II

Daniel M. Oberlin (2003)

Colloquium Mathematicae

Similarity:

For 1 ≤ p,q ≤ ∞, we prove that the convolution operator generated by the Cantor-Lebesgue measure on the circle is a contraction whenever it is bounded from $L^{p} ()$ to $L^{q} ()$ . We also give a condition on p which is necessary if this operator maps $L^{p} ()$ into L²().

On the duality between $p$ -modulus and probability measures

Luigi Ambrosio, Simone Di Marino, Giuseppe Savaré (2015)

Journal of the European Mathematical Society

Similarity:

Motivated by recent developments on calculus in metric measure spaces $(X, d, m)$ , we prove a general duality principle between Fuglede’s notion [15] of $p$ -modulus for families of finite Borel measures in $(X, d)$ and probability measures with barycenter in $L^{q} (X, m)$ , with $q$ dual exponent of $p \in (1, \infty)$ . We apply this general duality principle to study null sets for families of parametric and non-parametric curves in $X$ . In the final part of the paper we provide a new proof, independent of optimal transportation, of the...

The non-uniqueness of the limit solutions of the scalar Chern-Simons equations with signed measures

Adilson Eduardo Presoto (2021)

Mathematica Bohemica

Similarity:

We investigate the effect of admitting signed measures as a datum at the scalar Chern-Simons equation $- Δ u + e^{u} (e^{u} - 1) = μ in Ω$ with the Dirichlet boundary condition. Approximating $μ$ by a sequence ${(μ_{n})}_{n \in ℕ}$ of $L^{1}$ functions or finite signed measures such that this equation has a solution $u_{n}$ for each $n \in ℕ$ , we are interested in establishing the convergence of the sequence ${(u_{n})}_{n \in ℕ}$ to a function $u^{#}$ and describing the form of the measure which appears on the right-hand side of the scalar Chern-Simons equation solved by $u^{#}$ .

Path functionals over Wasserstein spaces

Alessio Brancolini, Giuseppe Buttazzo, Filippo Santambrogio (2006)

Journal of the European Mathematical Society

Similarity:

Given a metric space $X$ we consider a general class of functionals which measure the cost of a path in $X$ joining two given points $x_{0}$ and $x_{1}$ , providing abstract existence results for optimal paths. The results are then applied to the case when $X$ is aWasserstein space of probabilities on a given set $Ω$ and the cost of a path depends on the value of classical functionals over measures. Conditions for linking arbitrary extremal measures $μ_{0}$ and $μ_{1}$ by means of finite cost paths are given. ...

On nearly radial marginals of high-dimensional probability measures

Bo'az Klartag (2010)

Journal of the European Mathematical Society

Similarity:

Suppose that $μ$ is an absolutely continuous probability measure on $^{R}$ n, for large $n$ . Then $μ$ has low-dimensional marginals that are approximately spherically-symmetric. More precisely, if $n \geq {(C / ε)}^{C d}$ , then there exist $d$ -dimensional marginals of $μ$ that are $ε$ -far from being sphericallysymmetric, in an appropriate sense. Here $C > 0$ is a universal constant.

Estimates of capacity of self-similar measures

Jozef Myjak, Tomasz Szarek (2002)

Annales Polonici Mathematici

Similarity:

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.

Self-affine measures and vector-valued representations

Qi-Rong Deng, Xing-Gang He, Ka-Sing Lau (2008)

Studia Mathematica

Similarity:

Let A be a d × d integral expanding matrix and let $S_{j} (x) = A^{- 1} (x + d_{j})$ for some $d_{j} \in ℤ^{d}$ , j = 1,...,m. The iterated function system (IFS) ${S_{j}}_{j = 1}^{m}$ generates self-affine measures and scale functions. In general this IFS has overlaps, and it is well known that in many special cases the analysis of such measures or functions is facilitated by expressing them in vector-valued forms with respect to another IFS that satisfies the open set condition. In this paper we prove a general theorem on such representation. The proof...

On NIP and invariant measures

Ehud Hrushovski, Anand Pillay (2011)

Journal of the European Mathematical Society

Similarity:

We study forking, Lascar strong types, Keisler measures and definable groups, under an assumption of NIP (not the independence property), continuing aspects of the paper [16]. Among key results are (i) if $p = tp (b / A)$ does not fork over $A$ then the Lascar strong type of $b$ over $A$ coincides with the compact strong type of $b$ over $A$ and any global nonforking extension of $p$ is Borel definable over $bdd (A)$ , (ii) analogous statements for Keisler measures and definable groups, including the fact that $G^{000} = G^{00}$ for $G$ ...