Related papers: Sampling measures, Muckenhoupt Hamiltonians, and t…

Let $\mu$ and $\nu$ be two Borel probability measures on two separable metric spaces $\X$ and $\Y$ respectively. For $h, g$ be two Hausdorff functions and $q\in \R$, we introduce and investigate the generalized pseudo-packing measure…

For a compactly supported absolutely continuous measure $\mu$ on ${\mathbb{R}}^2$ having a density function equal to a finite linear combination of indicator functions of rectangles $\left[a_{i}, b_{i}\right]\times \left[c_{i},…

Let $\Omega\subsetneq\mathbb R^{n+1}$ be open and let $\mu$ be some measure supported on $\partial\Omega$ such that $\mu(B(x,r))\leq C\,r^n$ for all $x\in\mathbb R^{n+1}$, $r>0$. We show that if the harmonic measure in $\Omega$ satisfies…

Using the invariance of Quadratic Gravity in FLRW metric under the group of diffeomorphisms of the time coordinate, we rewrite the action $A$ of the theory in terms of the invariant dynamical variable $g(\tau)\,.$ We propose to consider the…

Let $({\mathcal X},\,d,\,\mu)$ be a metric measure space satisfying the upper doubling condition and the geometrically doubling condition in the sense of T. Hyt\"onen. In this paper, the authors prove that the $L^p(\mu)$ boundedness with…

We continue the study on Kurzweil--Stieltjes integration on compact lines initiated in [doi:10.1007/s11117-025-01161-9]. Given a real valued function $G$ on a compact line, the presented integral is called the Kurzweil--Stieltjes integral…

Let $\mu$ be a general stochastic measure, where we assume for $\mu$ only $\sigma$-additivity in probability and continuity of paths. We prove that the symmetric integral $\int_{[0,T]}f(\mu_t, t)\circ\,{\rm d}\mu_t$ is well defined. For…

We prove that if $\mu$ is a self-affine measure in the plane whose defining IFS acts totally irreducibly on $\mathbb{RP}^1$ and satisfies an exponential separation condition, then its dimension is equal to its Lyapunov dimension. We also…

We consider a Hamiltonian system on the symplectic space $({\mathbb{R}}^{2n}, dy\wedge dx)$ with a real-analytic Hamiltonian $H : {\mathbb{R}}^{2n}\to {\mathbb{R}}$. We assume that the system has a non-degenerate equilibrium position at the…

This paper relates to the Fourier decay properties of images of self-similar measures $\mu$ on $\mathbb{R}^k$ under nonlinear smooth maps $f \colon \mathbb{R}^k \to \mathbb{R}$. For example, we prove that if the linear parts of the…

Let $\omega$ be a B\'ekoll\'e-Bonami weight. We give a complete characterization of the positive measures $\mu$ such that $$\int_{\mathcal H}|M_\omega f(z)|^qd\mu(z)\le C\left(\int_{\mathcal H}|f(z)|^p\omega(z)dV(z)\right)^{q/p}$$ and…

Let $\mu$ be a positive measure on $R^d$. It is known that if the space $L^2(\mu)$ has a frame of exponentials then the measure $\mu$ must be of "pure type": it is either discrete, absolutely continuous or singular continuous. It has been…

Regular normalized W-valued spectral measures on a compact Hausdorff space X are in one-to-one correspondence with unital *-representations \rho:C(X)\to W, where W stands for a von Neumann algebra. In this paper we show that for every…

In this paper, we consider the non-spectral problem for the planar self-affine measures $\mu_{M,D}$ generated by an expanding integer matrix $M\in M_2(\mathbb{Z})$ and a finite digit set $D\subset\mathbb{Z}^2$. Let $p\geq2$ be a positive…

Let $(E,\mathcal E,\mu)$ be a measure space and $G\colon E\times E\to [0,\infty]$ be measurable. Moreover, let $\mathcal F\!_{ui}$ denote the set of all $q\in\mathcal E^+$ (measurable numerical functions $q\ge 0$ on $E$) such that…

We study the asymptotic quantization error of order $r$ for Markov-type measures $\mu$ on a class of ratio-specified graph directed fractals. We show that the quantization dimension of $\mu$ exists and determine its exact value $s_{r}$ in…

Given a $\mathbb Z^r$-action $\alpha$ on a nilmanifold $X$ by automorphisms and an ergodic $\alpha$-invariant probability measure $\mu$, we show that $\mu$ is the uniform measure on $X$, unless modulo finite index modification, one of the…

We consider an ergodic invariant measure $\mu$ for a smooth action of $Z^k$, $k \ge 2$, on a $(k+1)$-dimensional manifold or for a locally free smooth action of $R^k$, $k \ge 2$ on a $(2k+1)$-dimensional manifold. We prove that if $\mu$ is…

If $\mu $ is a positive Borel measure on the interval $[0, 1)$ we let $\mathcal H_\mu $ be the Hankel matrix $\mathcal H_\mu =(\mu_{n, k})_{n,k\ge 0}$ with entries $\mu_{n, k}=\mu_{n+k}$, where, for $n\,=\,0, 1, 2, \dots $, $\mu_n$ denotes…

Let H be a regular element of an irreducible Lie Algebra g, and let mu be the orbital measure supported on the Adjoint orbit of H. We show that the k-th power of the Fourier transform of mu is in L^2(g) if and only if k > dim g/(dim g-rank…