When is a Riesz distribution a complex measure?

Alan D. Sokal

Displaying similar documents to “When is a Riesz distribution a complex measure?”

Variation for the Riesz transform and uniform rectifiability

Albert Mas, Xavier Tolsa (2014)

Journal of the European Mathematical Society

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For $1 \leq n < d$ integers and $ρ > 2$ , we prove that an $n$ -dimensional Ahlfors-David regular measure $μ$ in $ℝ^{d}$ is uniformly $n$ -rectifiable if and only if the $ρ$ -variation for the Riesz transform with respect to $μ$ is a bounded operator in $L^{2} (μ)$ . This result can be considered as a partial solution to a well known open problem posed by G. David and S. Semmes which relates the $L^{2} (μ)$ boundedness of the Riesz transform to the uniform rectifiability of $μ$ .

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...

On the Fejér-F. Riesz inequality in $L^{p}$

Yoram Sagher (1977)

Studia Mathematica

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The max norm in $R^{n}$ -isometries and measure

Regina Cohen, James W. Fickett (1982)

Colloquium Mathematicae

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Unique Bernoulli $g$ -measures

Anders Johansson, Anders Öberg, Mark Pollicott (2012)

Journal of the European Mathematical Society

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We improve and subsume the conditions of Johansson and Öberg and Berbee for uniqueness of a $g$ -measure, i.e., a stationary distribution for chains with complete connections. In addition, we prove that these unique $g$ -measures have Bernoulli natural extensions. We also conclude that we have convergence in the Wasserstein metric of the iterates of the adjoint transfer operator to the $g$ -measure.

Sharp inequalities for Riesz transforms

Adam Osękowski (2014)

Studia Mathematica

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We establish the following sharp local estimate for the family ${R_{j}}_{j = 1}^{d}$ of Riesz transforms on $ℝ^{d}$ . For any Borel subset A of $ℝ^{d}$ and any function $f : ℝ^{d} \to ℝ$ , $\int_{A} | R_{j} f (x) | d x \leq C_{p} {| | f | |}_{L^{p} (ℝ^{d})} {| A |}^{1 / q}$ , 1 < p < ∞. Here q = p/(p-1) is the harmonic conjugate to p, $C_{p} = {[2^{q + 2} Γ (q + 1) / π^{q + 1} \sum_{k = 0}^{\infty} {(- 1)}^{k} / {(2 k + 1)}^{q + 1}]}^{1 / q}$ , 1 < p < 2, and $C_{p} = {[4 Γ (q + 1) / π^{q} \sum_{k = 0}^{\infty} 1 / {(2 k + 1)}^{q}]}^{1 / q}$ , 2 ≤ p < ∞. This enables us to determine the precise values of the weak-type constants for Riesz transforms for 1 < p < ∞. The proof rests on appropriate martingale inequalities, which are of independent interest.

Dichotomy of global density of Riesz capacity

Hiroaki Aikawa (2016)

Studia Mathematica

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Let $C_{α}$ be the Riesz capacity of order α, 0 < α < n, in ℝⁿ. We consider the Riesz capacity density $̲ (C_{α}, E, r) = i n f_{x \in ℝ ⁿ} C_{α} (E \cap B (x, r)) / C_{α} (B (x, r))$ for a Borel set E ⊂ ℝⁿ, where B(x,r) stands for the open ball with center at x and radius r. In case 0 < α ≤ 2, we show that $l i m_{r \to \infty} ̲ (C_{α}, E, r)$ is either 0 or 1; the first case occurs if and only if $̲ (C_{α}, E, r)$ is identically zero for all r > 0. Moreover, it is shown that the densities with respect to more general open sets enjoy the same dichotomy. A decay estimate for α-capacitary potentials is also...

An inversion formula and a note on the Riesz kernel

Andrejs Dunkels (1976)

Annales de l'institut Fourier

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For potentials $U_{K}^{T} = K * T$ , where $K$ and $T$ are certain Schwartz distributions, an inversion formula for $T$ is derived. Convolutions and Fourier transforms of distributions in $(D_{L}^{'} p)$ -spaces are used. It is shown that the equilibrium distribution with respect to the Riesz kernel of order $α$ , $0 < α < m$ , of a compact subset $E$ of $R^{m}$ has the following property: its restriction to the interior of $E$ is an absolutely continuous measure with analytic density which is expressed by an explicit formula.

Generalized Riesz products produced from orthonormal transforms

Nikolaos Atreas, Antonis Bisbas (2012)

Colloquium Mathematicae

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Let $ℳ_{p} = {m_{k}}_{k = 0}^{p - 1}$ be a finite set of step functions or real valued trigonometric polynomials on = [0,1) satisfying a certain orthonormality condition. We study multiscale generalized Riesz product measures μ defined as weak-* limits of elements $μ_{N} \in V_{N} (N \in ℕ)$ , where $V_{N}$ are $p^{N}$ -dimensional subspaces of L₂() spanned by an orthonormal set which is produced from dilations and multiplications of elements of $ℳ_{p}$ and $\bar{⋃_{N \in ℕ} V_{N}} = L ₂ ()$ . The results involve mutual absolute continuity or singularity of such Riesz products extending previous...

On some subspaces of Morrey-Sobolev spaces and boundedness of Riesz integrals

Mouhamadou Dosso, Ibrahim Fofana, Moumine Sanogo (2013)

Annales Polonici Mathematici

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For 1 ≤ q ≤ α ≤ p ≤ ∞, ${(L^{q}, l^{p})}^{α}$ is a complex Banach space which is continuously included in the Wiener amalgam space $(L^{q}, l^{p})$ and contains the Lebesgue space $L^{α}$ . We study the closure ${(L^{q}, l^{p})}_{c, 0}^{α}$ in ${(L^{q}, l^{p})}^{α}$ of the space of test functions (infinitely differentiable and with compact support in $ℝ^{d}$ ) and obtain norm inequalities for Riesz potential operators and Riesz transforms in these spaces. We also introduce the Sobolev type space $W ¹ ({(L^{q}, l^{p})}^{α})$ (a subspace of a Morrey-Sobolev space, but a superspace of the classical Sobolev space...

Inequalities involving heat potentials and Green functions

Neil A. Watson (2015)

Mathematica Bohemica

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We take some well-known inequalities for Green functions relative to Laplace’s equation, and prove not only analogues of them relative to the heat equation, but generalizations of those analogues to the heat potentials of nonnegative measures on an arbitrary open set $E$ whose supports are compact polar subsets of $E$ . We then use the special case where the measure associated to the potential has point support, in the following situation. Given a nonnegative supertemperature on an open set...

Riesz potentials derived by one-mode interacting Fock space approach

Nobuhiro Asai (2007)

Colloquium Mathematicae

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The main aim of this short paper is to study Riesz potentials on one-mode interacting Fock spaces equipped with deformed annihilation, creation, and neutral operators with constants $c_{0, 0}, c_{1, 1} \in ℝ$ and $c_{0, 1} > 0$ , $c_{1, 2} \geq 0$ as in equations (1.4)-(1.6). First, to emphasize the importance of these constants, we summarize our previous results on the Hilbert space of analytic L² functions with respect to a probability measure on ℂ. Then we consider the Riesz kernels of order 2α, $α = c_{0, 1} / c_{1, 2}$ , on ℂ if $0 < c_{0, 1} < c_{1, 2}$ , which can be derived from...

Invariant subspaces for operators in a general II1-factor

Uffe Haagerup, Hanne Schultz (2009)

Publications Mathématiques de l'IHÉS

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Let ℳ be a von Neumann factor of type II1 with a normalized trace τ. In 1983 L. G. Brown showed that to every operator T∈ℳ one can in a natural way associate a spectral distribution measure μ T (now called the Brown measure of T), which is a probability measure in ℂ with support in the spectrum σ(T) of T. In this paper it is shown that for every T∈ℳ and every Borel set B in ℂ, there is a unique closed T-invariant subspace $𝒦 = 𝒦_{T} (B)$ affiliated with ℳ, such that the Brown measure of ${T |}_{𝒦}$ is concentrated...

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,...).

Characteristic points, rectifiability and perimeter measure on stratified groups

Valentino Magnani (2006)

Journal of the European Mathematical Society

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We establish an explicit connection between the perimeter measure of an open set $E$ with $C^{1}$ boundary and the spherical Hausdorff measure $S^{Q - 1}$ restricted to $\partial E$ , when the ambient space is a stratified group endowed with a left invariant sub-Riemannian metric and $Q$ denotes the Hausdorff dimension of the group. Our formula implies that the perimeter measure of $E$ is less than or equal to $S^{Q - 1} (\partial E)$ up to a dimensional factor. The validity of this estimate positively answers a conjecture raised by Danielli,...

On Peano derivatives in $L^{p} (E_{n})$

S. Lasher (1968)

Studia Mathematica

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Interpolating sequences, Carleson measures and Wirtinger inequality

Eric Amar (2008)

Annales Polonici Mathematici

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Let S be a sequence of points in the unit ball of ℂⁿ which is separated for the hyperbolic distance and contained in the zero set of a Nevanlinna function. We prove that the associated measure $μ_{S} : = \sum_{a \in S} (1 - | a | ²) ⁿ δ_{a}$ is bounded, by use of the Wirtinger inequality. Conversely, if X is an analytic subset of such that any δ -separated sequence S has its associated measure $μ_{S}$ bounded by C/δⁿ, then X is the zero set of a function in the Nevanlinna class of . As an easy consequence, we prove that if S is a dual...

L¹ representation of Riesz spaces

Bahri Turan (2006)

Studia Mathematica

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Let E be a Riesz space. By defining the spaces $L ¹_{E}$ and $L_{E}^{\infty}$ of E, we prove that the center $Z (L ¹_{E})$ of $L ¹_{E}$ is $L_{E}^{\infty}$ and show that the injectivity of the Arens homomorphism m: Z(E)” → Z(E˜) is equivalent to the equality $L ¹_{E} = Z {(E)}^{'}$ . Finally, we also give some representation of an order continuous Banach lattice E with a weak unit and of the order dual E˜ of E in $L ¹_{E}$ which are different from the representations appearing in the literature.

Relations between Shy Sets and Sets of $ν_{p}$ -Measure Zero in Solovay’s Model

G. Pantsulaia (2004)

Bulletin of the Polish Academy of Sciences. Mathematics

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An example of a non-zero non-atomic translation-invariant Borel measure $ν_{p}$ on the Banach space $ℓ_{p} (1 \leq p \leq \infty)$ is constructed in Solovay’s model. It is established that, for 1 ≤ p < ∞, the condition " $ν_{p}$ -almost every element of $ℓ_{p}$ has a property P" implies that “almost every” element of $ℓ_{p}$ (in the sense of [4]) has the property P. It is also shown that the converse is not valid.