Let $μ,ν$ be Radon measures on $\mathbb{R}$, with $μ$ non-atomic and $ν$ doubling, and write $μ= μ_{a} + μ_{s}$ for the Lebesgue decomposition of $μ$ relative to $ν$. For an interval $I \subset \mathbb{R}$, define $α_{μ,ν}(I) := \mathbb{W}_{1}(μ_{I},ν_{I})$, the Wasserstein distance of normalised blow-ups of $μ$ and $ν$ restricted to $I$. Let $\mathcal{S}_ν$ be the square function $$\mathcal{S}^{2}_ν(μ) = \sum_{I \in \mathcal{D}} α_{μ,ν}^{2}(I)χ_{I},$$ where $\mathcal{D}$ is the family of dyadic intervals of side-length at most one. I prove that $\mathcal{S}_ν(μ)$ is finite $μ_{a}$ almost everywhere, and infinite $μ_{s}$ almost everywhere. I also prove a version of the result for a non-dyadic variant of the square function $\mathcal{S}_ν(μ)$. The results answer the simplest "$n = d = 1"$ case of a problem of J. Azzam, G. David and T. Toro.
Absolute continuity implies uniform continuity, but generally not vice versa. In this short note, we present one sufficient condition for a uniformly continuous function to be absolutely continuous, which is the following theorem: For a uniformly continuous function f defined on an interval of the real line, if it is piecewise convex, then it is also absolutely continuous.
In this paper we develop the method of double operator integrals to prove trace formulae for functions of contractions, dissipative operators, unitary operators and self-adjoint operators. To establish the absolute continuity of spectral shift, we use the Sz.-Nagy theorem on the absolute continuity of the spectrum of the minimal unitary dilation of a completely nonunitary contraction. We also give a construction of an intermediate contraction for a pair of contractions with trace class difference.
We show that, for disjoint domains in the Euclidean space whose boundaries satisfy a non-degeneracy condition, mutual absolute continuity of their harmonic measures implies absolute continuity with respect to surface measure and rectifiability in the intersection of their boundaries.
We prove that complex Bernoulli convolutions are absolutely continuous in the supercritical parameter region, outside of an exceptional set of parameters of zero Hausdorff dimension. Similar results are also obtained in the biased case, and for other parametrized families of self-similar sets and measures in the complex plane, extending earlier results.
We consider a class of measures called autophage which was introduced and studied by Szekely for measures on the real line. We show that the autophage measures on finite-dimensional vector spaces over real or p-adic field are infinitely divisible without idempotent factors and are absolutely continuous with bounded continuous density. We also show that certain semistable measures on such vector spaces are absolutely continuous.
In this paper we consider the representing measures $μ_{x},x\in \QTR{Bbb}{R}^{d}$, and $ν_{y},y\in \QTR{Bbb}{R}^{d}$, of the Dunkl intertwining operator and of its dual. When the multiplicity function is positive, we prove that for all $x\in \QTR{Bbb}{R}_{\QTO{mbox}{reg}}^{d}$ we have $dμ_{x}(y)=\QTR{cal}{K}(x,y)dy$ and for almost all $y\in \QTR{Bbb}{R}^{d}$ we have $dν_{y}(x)=\QTR{cal}{K}(x,y)ω_{k}(x)dx,$ where $\QTR{cal}{K}(x,.)$ is a positive integrable function on $\QTR{Bbb}{R}^{d}$ with support in $\{y\in \QTR{Bbb}{R}^{d}/\Vert y\Vert \leq \Vert x\Vert \}$ and the function $\QTR{cal}{K}(.…
We prove that a self similar measure is absolutely continuous providing that it satisfies a condition depending on its Garsia entropy, contraction ratio, and the separation between different points in approximations of the self similar measure. In the special case of Bernoulli convolutions this paper gives a sufficient condition for the Bernoulli convolution with parameter lambda to be absolutely continuous in terms of lambda and its Mahler measure.
We prove that Bernoulli convolutions are absolutely continuous provided the parameter lambda is an algebraic number sufficiently close to 1 depending on the Mahler measure of lambda.
We prove absolute continuity of Gaussian measures associated to complex Brownian bridges under certain gauge transformations. As an application we prove that the invariant measure for the periodic derivative nonlinear Schrödinger equation obtained by Nahmod, Oh, Rey-Bellet and Staffilani in [20], and with respect to which they proved almost surely global well-posedness, coincides with the weighted Wiener measure constructed by Thomann and Tzvetkov [24]. Thus, in particular we prove the invariance of the measure constructed in [24].
We show that for uniform domains $Ω\subseteq \mathbb{R}^{d+1}$ whose boundaries satisfy a certain nondegeneracy condition that harmonic measure cannot be mutually absolutely continuous with respect to $α$-dimensional Hausdorff measure unless $α\leq d$. We employ a lemma that shows that at almost every nondegenerate point, we may find a tangent measure of harmonic measure whose support is the boundary of yet another uniform domain whose harmonic measure resembles the tangent measure.
We show that in many parametrized families of self-similar measures, their projections, and their convolutions, the set of parameters for which the measure fails to be absolutely continuous is very small - of co-dimension at least one in parameter space. This complements an active line of research concerning similar questions for dimension. Moreover, we establish some regularity of the density outside this small exceptional set, which applies in particular to Bernoulli convolutions; along the way, we prove some new results about the dimensions of self-similar measures and the absolute continui…
We characterize the absolute continuity of convolution products of orbital measures on the classical, irreducible Riemannian symmetric spaces $G/K$ of Cartan type $III$, where $G$ is a non-compact, connected Lie group and $K$ is a compact, connected subgroup. By the orbital measures, we mean the uniform measures supported on the double cosets, $KzK,$ in $G$. The characterization can be expressed in terms of dimensions of eigenspaces or combinatorial properties of the annihilating roots of the elements $z$. A consequence of our work is to show that the convolution product of any rank% $G/K,$ co…
A finite Borel measure $μ$ in ${\mathbb R}^d$ is called a frame-spectral measure if it admits an exponential frame (or Fourier frame) for $L^2(μ)$. It has been conjectured that a frame-spectral measure must be translationally absolutely continuous, which is a criterion describing the local uniformity of a measure on its support. In this paper, we show that if any measures $ν$ and $λ$ without atoms whose supports form a packing pair, then $ν\ast λ+δ_t\astν$ is translationally singular and it does not admit any Fourier frame. In particular, we show that the sum of one-fourth and one-sixteenth Ca…
Let $d \geq 2$ and $d - 1 < s < d$. Let $μ$ be a compactly supported Radon measure in $\mathbb{R}^{d}$ with finite $s$-energy. I prove that the radial projections $π_{x\sharp}μ$ of $μ$ are absolutely continuous with respect to $\mathcal{H}^{d - 1}$ for every centre $x \in \mathbb{R}^{d} \setminus \operatorname{spt} μ$, outside an exceptional set of dimension at most $2(d - 1) - s$. This is sharp. In fact, for $x$ outside an exceptional set as above, $π_{x\sharp}μ\in L^{p}(S^{d - 1})$ for some $p > 1$.
We consider the spectral function $ρ(μ)$ $(μ\geq 0)$ for the Sturm-Liouville equation $y^{''}+(λ-q)y =0$ on $[0,\infty)$ with the boundary condition $y(0)=0$ and where $q$ has slow decay $O(x^{-α})$ $(a>0)$ as $x\to \infty$. We develop our previous methods of locating spectral concentration for $q$ with rapid exponential decay (JCAM 81 (1997) 333-348) to deal with the new theoretical and computational complexities which arise for slow decay.
We study absolute continuity of harmonic measure with respect to surface measure on domains $Ω$ that have large complements. We show that if $Γ\subset \mathbb{R}^{d+1}$ is $d$-Ahlfors regular and splits $ \mathbb{R}^{d+1}$ into two NTA domains then $ω_Ω\ll \mathscr{H}^{d}$ on $Γ\cap \partialΩ$. This result is a natural generalisation of a result of Wu in [Wu86]. We also prove that almost every point in $Γ\cap\partialΩ$ is a cone point if $Γ$ is a Lipschitz graph. Combining these results and a result from [AHMMMTV], we characterize sets of absolute continuity with finite $\mathscr{H}^{d}$-measu…
We show how to turn the question of the absolute continuity of Bernoulli convolutions into one of counting the growth of the number of overlaps in the system. When the contraction parameter is a hyperbolic algebraic integer, we turn this question of absolute continuity into a question involving the ergodic theory of cocycles over domain exchange transformations.
We prove that the set of exceptional $λ\in (1/2,1)$ such that the associated Bernoulli convolution is singular has zero Hausdorff dimension, and likewise for biased Bernoulli convolutions, with the exceptional set independent of the bias. This improves previous results by Erdös, Kahane, Solomyak, Peres and Schlag, and Hochman. A theorem of this kind is also obtained for convolutions of homogeneous self-similar measures. The proofs are very short, and rely on old and new results on the dimensions of self-similar measures and their convolutions, and the decay of their Fourier transform.
Let $μ$ be a planar self-similar measure with similarity dimension exceeding $1$, satisfying a mild separation condition, and such that the fixed points of the associated similitudes do not share a common line. Then, we prove that the orthogonal projections $π_{e\sharp}(μ)$ are absolutely continuous for all $e \in S^{1} \setminus E$, where the exceptional set $E$ has zero Hausdorff dimension. The result is obtained from a more general framework which applies to certain parametrised families of self-similar measures on the real line. Our results extend previous work of Shmerkin and Solomyak fro…
We study measures $μ$ on the plane with two independent Alberti representations. It is known, due to Alberti, Csörnyei, and Preiss, that such measures are absolutely continuous with respect to Lebesgue measure. The purpose of this paper is to quantify the result of A-C-P. Assuming that the representations of $μ$ are bounded from above, in a natural way to be defined in the introduction, we prove that $μ\in L^{2}$. If the representations are also bounded from below, we show that $μ$ satisfies a reverse Hölder inequality with exponent $2$, and is consequently in $L^{2 + ε}$ by Gehring's lemma. A…
We show that if $Ω$ is an NTA domain with harmonic measure $w$ and $E\subseteq \partialΩ$ is contained in an Ahlfors regular set, then $w|_{E}\ll \mathscr{H}^{d}|_{E}$. Moreover, this holds quantitatively in the sense that for all $τ>0$ $w$ obeys an $A_{\infty}$-type condition with respect to $\mathscr{H}^{d}|_{E'}$, where $E'\subseteq E$ is so that $w(E\backslash E')<τw(E)$, even though $\partialΩ$ may not even be locally $\mathscr{H}^{d}$-finite. We also show that, for uniform domains with uniform complements, if $E\subseteq\partialΩ$ is the Lipschitz image of a subset of $\mathbb{R}^{d}$, t…
We show that, for disjoint domains in the Euclidean space, mutual absolute continuity of their harmonic measures implies absolute continuity with respect to surface measure and rectifiability in the intersection of their boundaries. This improves on our previous result which assumed that the boundaries satisfied the capacity density condition.
Let $μ$ denote a Borel probability measure and let $\{ μ_{t} \}_{t\geq 1}$ denote the free additive convolution semigroup of Nica and Speicher. We show that the support of these measures varies continuously in the Hausdorff metric for $t >1$. We utilize complex analytic methods and, in particular, a characterization of the absolutely continuous portion of these supports due to Huang.
