相关论文: Characterization of $n$-rectifiability in terms of…

Let $\mu$ be a Radon measure on $\mathbb R^{d}$ which may be non-doubling and only satisfies $\mu(Q(x,l))\le C_{0}l^{n}$} for all $x\in \mathbb R^{d}$, $l(Q)>0$, with some fixed constants $C_{0}>0$ and $n\in (0,d]$. We introduce a new type…

In this paper, we study the set of absolute continuity of p-harmonic measure, $\mu$, and $(n-1)-$dimensional Hausdorff measure, $\mathcal{H}^{n-1}$, on locally flat domains in $\mathbb{R}^{n}$, $n\geq 2$. We prove that for fixed $p$ with…

Let $f$ be a function in $\mathbb R^2$, which has a jump across a smooth curve $\mathcal S$ with nonzero curvature. We consider a family of functions $f_\epsilon$ with jumps across a family of curves $\mathcal S_\epsilon$. Each $\mathcal…

It is well-known that if a real valued function acting on a convex set satisfies the $n$-variable Jensen inequality, for some natural number $n\geq 2$, then, for all $k\in\{1,\dots, n\}$, it fulfills the $k$-variable Jensen inequality as…

We consider the dynamics of semi-hyperbolic semigroups generated by finitely many rational maps on the Riemann sphere. Assuming that the nice open set condition holds it is proved that there exists a geometric measure on the Julia set with…

In this paper we study a metric generalization of the sine function which can be extended to arbitrary normed planes. We derive its main properties and give also some characterizations of Radon planes. Furthermore, we prove that the…

We give a condition for absolute continuity of self-similar measures in arbitrary dimensions. This allows us to construct the first explicit absolutely continuous examples of inhomogeneous self-similar measures in dimension one and two. In…

We define a discrete Menger-type curvature of d+2 points in a real separable Hilbert space H by an appropriate scaling of the squared volume of the corresponding (d+1)-simplex. We then form a continuous curvature of an Ahlfors d-regular…

Let $ T\colon[0,1]^d\to [0,1]^d $ be a piecewise expanding map with an absolutely continuous invariant measure $ \mu $. Let $ \{H_n\} $ be a sequence of hyperrectangles or hyperboloids centered at the origin. Denote by $ \mathcal R(\{H_n\})…

We investigate the influence that $s$-dimensional lower and upper Hausdorff densities have on the geometry of a Radon measure in $\mathbb{R}^n$ when $s$ is a real number between $0$ and $n$. This topic in geometric measure theory has been…

In this paper, we prove that if a metric measure space satisfies the volume doubling condition and the Gagliardo-Nirenberg inequality with the same exponent $n$ $(n\geq 2)$, then it has exactly the $n$-dimensional volume growth. Besides,…

Stratified measures on Euclidean space are defined here as convex combinations of rectifiable measures. They are possibly singular with respect to the Lebesgue measure and generalize continuous-discrete mixtures. A stratified measure $\rho$…

We consider a $\varphi$-rigidity property for divergence-free vector fields in the Euclidean $n$-space, where $\varphi(t)$ is a non-negative convex function vanishing only at $t=0$. We show that this property is always satisfied in…

This article studies an integral representation of functionals of linear growth on metric measure spaces with a doubling measure and a Poincar\'e inequality. Such a functional is defined through relaxation, and it defines a Radon measure on…

Let $U\not\equiv \pm\infty$ be a $\delta$-subharmonic function on a closed disc of radius $R$ centered at zero. In the previous two parts of our paper, we obtained general and explicit estimates of the integral of the positive part of the…

In the following paper, we prove a dimension bound on the singular set of a Radon measure assuming its doubling ratio converges uniformly on compact sets. More precisely, we prove that if a Radon measure is $n$-Uniformly Asymptotically…

A locally finite multiset $(\Lambda,c),$ $\Lambda \subset \mathbb R^n, c : \Lambda \rightarrow \{1,...,b\}$ defines a Radon measure $\mu := \sum_{\lambda \in \Lambda} c(\lambda)\, \delta_\lambda$ that is Bohr almost periodic in the sense of…

Rademacher theorem states that every Lipschitz function on the Euclidean space is differentiable almost everywhere, where "almost everywhere" refers to the Lebesgue measure. In this paper we prove a differentiability result of similar type,…

Let $\mu$ be a compactly supported absolutely continuous probability measure on ${\Bbb R}^n$, we show that $\mu$ admits Fourier frames if and only if its Radon-Nikodym derivative is upper and lower bounded almost everywhere on its support.…

In this paper we study the Hausdorff volume in a non equiregular sub-Riemannian manifold and we compare it with a smooth volume. We first give the Lebesgue decomposition of the Hausdorff volume. Then we study the regular part, show that it…