For all $1\leq m\leq n-1$, we investigate the interaction of locally finite measures in $\mathbb{R}^n$ with the family of $m$-dimensional Lipschitz graphs. For instance, we characterize Radon measures $μ$, which are carried by Lipschitz graphs in the sense that there exist graphs $Γ_1,Γ_2,\dots$ such that $μ(\mathbb{R}^n\setminus\bigcup_1^\inftyΓ_i)=0$, using only countably many evaluations of the measure. This problem in geometric measure theory was classically studied within smaller classes of measures, e.g.~for the restrictions of $m$-dimensional Hausdorff measure $\mathcal{H}^m$ to $E\subseteq \mathbb{R}^n$ with $0<\mathcal{H}^m(E)<\infty$. However, an example of Csörnyei, Käenmäki, Rajala, and Suomala shows that classical methods are insufficient to detect when a general measure charges a Lipschitz graph. To develop a characterization of Lipschitz graph rectifiability for arbitrary Radon measures, we look at the behavior of coarse doubling ratios of the measure on dyadic cubes that intersect conical annuli. This extends a characterization of graph rectifiability for pointwise doubling measures by Naples by mimicking the approach used in the characterization of Radon measures carried by rectifiable curves by Badger and Schul.
A set in the Euclidean plane is constructed whose image under the classical Radon transform is Lipschitz in every direction. It is also shown that, under mild hypotheses, for any such set the function which maps a direction to the corresponding Lipschitz constant cannot be bounded.
We add to the literature the following observation. If $μ$ is a singular measure on $\mathbb{R}^n$ which assigns measure zero to every porous set and $f:\mathbb{R}^n\rightarrow\mathbb{R}$ is a Lipschitz function which is non-differentiable $μ$-a.e. then for every $C^1$ function $g:\mathbb{R}^n\rightarrow\mathbb{R}$ it holds $$μ\{x\in\mathbb{R}^n: f(x)=g(x)\}=0.$$ 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.
We construct a (Lipschitz) differentiability space which has at generic points a disconnected tangent and thus does not contain positive measure subsets isometric to positive measure subsets of spaces admitting a Poincaré inequality. We also prove that $l^2$-valued Lipschitz maps are differentiable a.e., but there are also Lipschitz maps taking values in some other Banach spaces having the Radon-Nikodym property which fail to be differentiable on sets of positive measure.
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.
Let 0<n<d be integers and let H denote the n-dimensional Hausdorff measure restricted to an n-dimensional Lipschitz graph in R^d with slope strictly less than 1. For r>2, we prove that the r-variation and oscillation for Calderón-Zygmund singular integrals with odd kernel are bounded operators in L^p(H) for 1<p finite, from L^1(H) to weak-L^1(H), and from the space of bounded H-measurable functions to BMO(H). Concerning the first endpoint estimate, we actually show that such operators are bounded from the space of finite complex Radon measures in R^d to weak-L^1(H).
We study differentiability properties of a potential of the type $K\star μ$, where $μ$ is a finite Radon measure in $\mathbb{R}^N$ and the kernel $K$ satisfies $|\nabla^j K(x)| \le C\, |x|^{-(N-1+j)}, \quad j=0,1,2.$ We introduce a notion of differentiability in the capacity sense, where capacity is classical capacity in the de la Vallée Poussin sense associated with the kernel $|x|^{-(N-1)}.$ We require that the first order remainder at a point is small when measured by means of a normalized weak capacity "norm" in balls of small radii centered at the point. This implies weak $L^{N/(N-1)}$ di…
In this paper we prove generalizations of Lusin-type theorems for gradients due to Giovanni Alberti, where we replace the Lebesgue measure with any Radon measure $μ$. We apply this to go beyond the known result on the existence of Lipschitz functions which are non-differentiable at $μ$-almost every point $x$ in any direction which is not contained in the decomposability bundle $V(μ,x)$, recently introduced by Alberti and the first named author. More precisely, we prove that it is possible to construct a Lipschitz function which attains any prescribed admissible blowup at every point except for…
We demonstrate the necessity of a Poincaré type inequality for those metric measure spaces that satisfy Cheeger's generalization of Rademacher's theorem for all Lipschitz functions taking values in a Banach space with the Radon-Nikodym property. This is done by showing the existence of a rich structure of curve fragments that connect near by points, similar in nature to Semmes's pencil of curves for the standard Poincaré inequality. Using techniques similar to Cheeger-Kleiner, we show that our conditions are also sufficient. We also develop another characterization of "RNP Lipschitz differenti…
In this paper we prove self-improvement properties of strong Muckenhoupt and Reverse Hölder 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.
We characterize Radon measures $μ$ in $\mathbb{R}^{n}$ that are $d$-rectifiable in the sense that their supports are covered up to $μ$-measure zero by countably many $d$-dimensional Lipschitz graphs and $μ\ll \mathcal{H}^{d}$. The characterization is in terms of a Jones function involving the so-called $α$-numbers. This answers a question left open in a former work by Azzam, David, and Toro.
We show that the Hilger derivative on time scales is a special case of the Radon--Nikodym derivative with respect to the natural measure associated with every time scale. Moreover, we show that the concept of delta absolute continuity agrees with the one from measure theory in this context.
Given positive measures $ν,μ$ on an arbitrary measurable space $(Ω, \mathcal F)$, we construct a sequence of finite partitions $(π_n)_n$ of $(Ω, \mathcal F)$ s.t. $$ \sum_{A\in π_n: μ(A)>0} 1_{A} \frac{ν(A)}{μ(A)} \longrightarrow \frac{dν^a}{dμ} \quad μ\text{ a.e. as } n\to \infty . $$ As an application, we modify the probabilistic proof of the Radon-Nikodym Theorem so that it uses convergence along a properly chosen sequence (instead of along a net), and so that it does not rely on the martingale convergence theorem (nor any probability theory), obtaining a completely elementary proof.
Araci et al. introduced a $p$-adic $(ρ,q)$-analogue of the Haar distribution. By means of the distribution, they constructed the $p$-adic $(ρ,q)$-Volkenborn integral. In this paper, by virtue of the Mahler expansion of continuous functions, the author gives the Radon-Nikodym theorem with respect to the $p$-adic $(ρ,q)$-distribution on $\Bbb Z_p$.
This paper is concerned with establishing uniform weighted $L^p$-$L^q$ estimates for a class of operators generalizing both Radon-like operators and sublevel set operators. Such estimates are shown to hold under general circumstances whenever a scalar inequality holds for certain associated measures (the inequality is of the sort studied by Oberlin, relating measures of parallelepipeds to powers of their Euclidean volumes). These ideas lead to previously unknown, weighted affine-invariant estimates for Radon-like operators as well as new $L^p$-improving estimates for degenerate Radon-like oper…
Convolution with an appropriate surface measure on a paraboloid in R^d defines a bounded operator T from L^p(R^d) to L^q(R^d) for certain exponents p,q. In this article it is proved that there exist functions which extremize the associated inequality.
We prove several variations on the results of Ricci and Travaglini concerning bounds for convolution with all rotations of a measure supported by a fixed convex curve in the plane. Estimates are obtained for averages over higher-dimensional convex hypersurfaces, smooth k-dimensional surfaces and non-translation invariant families of surfaces.
In the late `90s there was a flurry of activity relating $1$-rectifiable sets, boundedness of singular integral operators, the analytic capacity of a set, and the integral Menger curvature in the plane. In `99 Leger extended the results for Menger curvature to $1$-rectifiable sets in higher dimension, as well as to the codimension one case. A decade later, Lerman and Whitehouse, and later Meurer, found higher-dimensional geometrically motivated generalizations of Menger curvature that yield results about the uniform rectifiability of measures and the rectifiability of sets respectively. In thi…
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 extensively studied when $s$ is an integer. In this paper, we focus on the non-integer case, building upon a series of papers on $s$-sets by Martín and Mattila from 1988 to 2000. When $0<s<1$, we prove that measures with almost everywhere positive lower density and finite upper density are carried by countably many bi-Lipschitz curves. When $1\leq s<n$, we id…
This paper establishes a necessary and sufficient condition for $L^p$-boundedness of a class of multilinear functionals which includes both the Brascamp-Lieb inequalities and generalized Radon transforms associated to algebraic incidence relations. The testing condition involves bounding the average of an inverse power of certain Jacobian-type quantities along fibers of associated projections and covers many widely-studied special cases, including convolution with measures on nondegenerate hypersurfaces or on nondegenerate curves. The heart of the proof is based on Guth's visibility lemma in o…
We identify a set of sufficient local conditions under which a significant portion of a Radon measure $μ$ on $\mathbb{R}^{n+1}$ with compact support can be covered by an $n$-uniformly rectifiable set at the level of a ball $B\subset \mathbb{R}^{n+1}$ such that $μ(B)\approx r(B)^n$. This result involves a flatness condition, formulated in terms of the so-called $β_1$-number of $B$, and the $L^2(μ|_B)$-boundedness, as well as a control on the mean oscillation on the ball, of the operator \begin{equation} T_μf(x)=\int \nabla_x\mathcal{E}(x,y)f(y)\,dμ(y). \end{equation} Here $\mathcal{E}(\cdot,\cd…
