General Radon Measure Research Articles

On the Converse of Pansu’s Theorem

We provide a suitable generalisation of Pansu’s differentiability theorem to general Radon measures on Carnot groups and we show that if Lipschitz maps between Carnot groups are Pansu-differentiable almost everywhere for some Radon measures μ, then μ must be absolutely continuous with respect to the Haar measure of the group.

Gradient of the single layer potential and quantitative rectifiability for general Radon measures

We identify a set of sufficient local conditions under which a significant portion of a Radon measure μ on Rn+1 with compact support can be covered by an uniformly n-rectifiable set, at the level of a ball B⊂Rn+1 such that μ(B)≈r(B)n. This result involves a flatness condition, formulated in terms of the so-called β1-number of B, and the L2(μ|B)-boundedness, as well as a control on the mean oscillation on the ball, of the operatorTμf(x)=∫∇xE(x,y)f(y)dμ(y). Here E(⋅,⋅) is the fundamental solution for a uniformly elliptic operator in divergence form associated with an (n+1)×(n+1) matrix with Hölder continuous coefficients. This generalizes a work by Girela-Sarrión and Tolsa for the n-Riesz transform. The motivation for our result stems from a two-phase problem for the elliptic harmonic measure.

Renormalized solutions of elliptic equations with variable exponents and general measure data

A class of second-order elliptic equations with variable nonlinearity exponents and the right-hand side in the form of the general Radon measure with finite total variation is considered. The existence of a renormalized solution of the Dirichlet problem is proved as a consequence of stability with respect to the convergence of the right-hand side of the equation. Bibliography: 37 titles.

A Weak Type Vector-Valued Inequality for the Modified Hardy–Littlewood Maximal Operator for General Radon Measure on ℝ n

Abstract The aim of this paper is to prove the weak type vector-valued inequality for the modified Hardy– Littlewood maximal operator for general Radon measure on ℝ n . Earlier, the strong type vector-valued inequality for the same operator and the weak type vector-valued inequality for the dyadic maximal operator were obtained. This paper will supplement these existing results by proving a weak type counterpart.

A two-weight inequality for essentially well localized operators with general measures

We develop a new formulation of well localized operators as well as a new proof for the necessary and sufficient conditions to characterize their boundedness between L2(Rn,u) and L2(Rn,v) for general Radon measures u and v.

Dimension-Free Properties of Strong Muckenhoupt and Reverse Hölder Weights for Radon Measures

In this paper, we prove self-improvement properties of strong Muckenhoupt and Reverse Holder weights with respect to a general Radon measure on $$\mathbb {R}^n$$ . We derive our result via a Bellman function argument. An important feature of our proof is that it uses only the Bellman function for the one-dimensional problem for Lebesgue measure; with this function in hand, we derive dimension-free results for general measures and dimensions.

The Riesz transform and quantitative rectifiability for general Radon measures

In this paper we show that if $$\mu $$ is a Borel measure in $${{\mathbb {R}}}^{n+1}$$ with growth of order n, such that the n-dimensional Riesz transform $${{\mathcal {R}}}_\mu $$ is bounded in $$L^2(\mu )$$ , and $$B\subset {{\mathbb {R}}}^{n+1}$$ is a ball with $$\mu (B)\approx r(B)^n$$ such that: then there exists a uniformly n-rectifiable set $$\Gamma $$ , with $$\mu (\Gamma \cap B)\gtrsim \mu (B)$$ , and such that $$\mu |_\Gamma $$ is absolutely continuous with respect to $${{\mathcal {H}}}^n|_\Gamma $$ . This result is an essential tool to solve an old question on a two phase problem for harmonic measure in subsequent papers by Azzam, Mourgoglou, Tolsa, and Volberg.

Lusin type theorems for Radon measures

We add to the literature the following observation. If \mu is a singular measure on \mathbb{R}^n which assigns measure zero to every porous set and f\colon \mathbb{R}^n\rightarrow\mathbb{R} is a Lipschitz function which is non-differentiable \mu -a.e., then for every C^1 function g\colon \mathbb{R}^n\rightarrow\mathbb{R} it holds \mu\{x\in\mathbb{R}^n\colon f(x)=g(x)\}=0. In other words the Lusin type approximation property of Lipschitz functions with C^1 functions does not hold with respect to a general Radon measure.

Existence of weak solutions to refraction problems in Negative Refractive Index Materials

We prove existence of weak solutions to certain refraction problems in the setting of materials with negative refractive index, both in the near and far field. Solutions are obtained when the target measure is discrete, as well as when it is given by a general Radon measure. The far field problem is treated using techniques from optimal mass transport. As such a fully nonlinear PDE of Monge–Ampère type arises in this setting, and we calculate the coefficients explicitly. Finally, we produce an example in which the A3w condition holds but the A3s condition does not.

Optimal transport, Cheeger energies and contractivity of dynamic transport distances in extended spaces

We introduce the setting of extended metric–topological measure spaces as a general “Wiener like” framework for optimal transport problems and nonsmooth metric analysis in infinite dimension.After a brief review of optimal transport tools for general Radon measures, we discuss the notions of the Cheeger energy, of the Radon measures concentrated on absolutely continuous curves, and of the induced “dynamic transport distances”. We study their main properties and their links with the theory of Dirichlet forms and the Bakry–Émery curvature condition, in particular concerning the contractivity properties and the EVI formulation of the induced Heat semigroup.

Erratum: On transport equations driven by a non‐divergence free force field Math. Meth. Appl. Sci. 2007 (DOI: 10.1002/mma.817)

AbstractDue to a prolonged but thorough refereeing process of 1 and updates subsequently made to 1, some of the references made at the beginning of Section 4 of 2 to the text of 1 are no longer correct. Precisely speaking, as stated, the results of Section 4 are valid if the measure µ is the Lebesgue measure on Ω as this yields the density of the set 𝒞1(Ω)∩𝒟(Tmax) in 𝒟(Tmax) endowed with the graph norm ∥ · ∥𝒟. This result is then used to show the existence of traces, as in 3. In the new version of 1, we managed to deal with a general Radon measure that required, however, a new definition of the maximal transport operator Tmax to ensure the existence of traces. It follows that the non‐divergence case can be dealt in a similar way to Proposition 3.2 of 1 but using a divergence‐free version of 2, Equation (16) as the definition of Tmax.

A Marstrand theorem for measures with polytope density

Given \({s\in (0,2]}\) and any centrally symmetric convex polytope \({\Theta\subset\mathbb{R}^{n}}\) , define \({\Theta_r(x):=r\Theta+x}\) we prove that if a Radon measure μ has the property \({\label{df1} 0 < \mathop {\lim }\limits_{r \to 0} \frac{\mu\big(\Theta_r(x)\big)}{r^s} < \infty\quad {\rm for}\;\mu\textrm{ a.e. }x}\) then s is an integer. For the case Θ is the Euclidean ball, this result was first proved by Marstrand in 1955 for Hausdorff measure in the plane (Marstrand in Proc Lond Math Soc 3(4):257–302, 1954) and later for general Radon measures in \({\mathbb {R}^n}\) (Marstrand in Trans Am Math Soc 205:369–392, 1964).

RECTIFIABILITY OF MEASURES WITH LOCALLY UNIFORM CUBE DENSITY

The conjecture that Radon measures in Euclidean space with positive finite density are rectifiable was a central problem in Geometric Measure Theory for fifty years. This conjecture was positively resolved by Preiss in 1986, using methods entirely dependent on the symmetry of the Euclidean unit ball. Since then, due to reasons of isometric immersion of metric spaces into $l_{\infty}$ and the uncommon nature of the sup norm even in finite dimensions, a popular model problem for generalising this result to non-Euclidean spaces has been the study of 2-uniform measures in $l^{3}_{\infty}$. The rectifiability or otherwise of these measures has been a well-known question.In this paper the stronger result that locally 2-uniform measures in $l^{3}_{\infty}$ are rectifiable is proved. This is the first result that proves rectifiability, from an initial condition about densities, for general Radon measures of dimension greater than 1 outside Euclidean space.2000 Mathematical Subject Classification: 28A75.

Singular perturbations and homogenization in stratified media

We consider a heterogeneous structure which is stratified in some direction (say x1). The strips are assumed to have very high conductivity which means that the conductivity coefficient a{x1) ranges over |0.+ ∞| oscillating without the usual boundedness assumption (with respect to the L∞-norm) of a and a-1 This problem has been already considered by C.Picard J. Mossino and B.Heron who studied in particular the asymptotic behaviour as ∊-0 of the diffusion equation: boundary condition under the assumption that the sequences (a∊) and (a∊)-1 have limits for the weak topology of L1. In this paper (see also preprint [19]). we propose a new method (convex functionals on measures ranging into L2 which allow us to pass to the limit in (*) when (a∊) and (a∊)1converges ∗-weakly to general Radon measures, say m and tr. Due to the loss of coercivity, the limit of the sequence of solutions (u∊) may have discontinuities whose energy (associated to the singular part of ∊) is concentrated on hyperplanes orthogonal to the On the same way , diffusions surface energies will appear associated with the singular part of m. Extensions to nonlinear problems are also considered.