A new proof of the rectifiable slices theorem

Robert L. Jerrard

Displaying similar documents to “A new proof of the rectifiable slices theorem”

A Whitney extension theorem in $L^{p}$ and Besov spaces

Alf Jonsson, Hans Wallin (1978)

Annales de l'institut Fourier

Similarity:

The classical Whitney extension theorem states that every function in Lip $(β, F)$ , $F \subset R^{n}$ , $F$ closed, $k < β \leq k + 1$ , $k$ a non-negative integer, can be extended to a function in Lip $(β, R^{n})$ . Her Lip $(β, F)$ stands for the class of functions which on $F$ have continuous partial derivatives up to order $k$ satisfying certain Lipschitz conditions in the supremum norm. We formulate and prove a similar theorem in the $L^{p}$ -norm. The restrictions to $R^{d}$ , $d < n$ , of the Bessel potential spaces in $R^{n}$ and the Besov or generalized Lipschitz...

Radon-Nikodym property for vector-valued integrable functions

Surjit Singh Khurana (1978)

Annales de l'institut Fourier

Similarity:

It is proved that if a Frechet space $E$ has $R - N$ property, then $L_{p} (E, ν)$ also has $R - N$ property, for $1 < p < \infty$ .

Metric transitivity and integer valued functions

Solomon Schwartzman (1960)

Annales de l'institut Fourier

Similarity:

Soit $X$ un espace mesurable de mesure $μ$ finie ; $φ : X \to X$ une application vérifiant $μ (φ^{- 1} (S)) = μ (S)$ pour chaque ensemble mesurable $S \subseteq X$ . On donne des conditions nécessaires et suffisantes pour que $X$ soit un ensemble ergodique.

Mean values and associated measures of $δ$ -subharmonic functions

Neil A. Watson (2002)

Mathematica Bohemica

Similarity:

Let $u$ be a $δ$ -subharmonic function with associated measure $μ$ , and let $v$ be a superharmonic function with associated measure $ν$ , on an open set $E$ . For any closed ball $B (x, r)$ , of centre $x$ and radius $r$ , contained in $E$ , let $ℳ (u, x, r)$ denote the mean value of $u$ over the surface of the ball. We prove that the upper and lower limits as $s, t \to 0$ with $0 < s < t$ of the quotient $(ℳ (u, x, s) - ℳ (u, x, t)) / (ℳ (v, x, s) - ℳ (v, x, t))$ , lie between the upper and lower limits as $r \to 0 +$ of the quotient $μ (B (x, r)) / ν (B (x, r))$ . This enables us to use some well-known measure-theoretic results to prove new variants...

Algebraic genericity of strict-order integrability

Luis Bernal-González (2010)

Studia Mathematica

Similarity:

We provide sharp conditions on a measure μ defined on a measurable space X guaranteeing that the family of functions in the Lebesgue space $L^{p} (μ, X)$ (p ≥ 1) which are not q-integrable for any q > p (or any q < p) contains large subspaces of $L^{p} (μ, X)$ (without zero). This improves recent results due to Aron, García, Muñoz, Palmberg, Pérez, Puglisi and Seoane. It is also shown that many non-q-integrable functions can even be obtained on any nonempty open subset of X, assuming that X is a topological...

The max norm in $R^{n}$ -isometries and measure

Regina Cohen, James W. Fickett (1982)

Colloquium Mathematicae

Similarity:

The density of the area integral in $ℝ_{+}^{n + 1}$

Richard F. Gundy, Martin L. Silverstein (1985)

Annales de l'institut Fourier

Similarity:

Let $u (x, y)$ be a harmonic function in the half-plane $R_{+}^{n + 1}$ , $n \geq 2$ . We define a family of functionals $D (u; r), - \infty > r > \infty$ , that are analogs of the family of local times associated to the process $u (x_{t}, y_{t})$ where $(x_{t}, y_{t})$ is Brownian motion in $R_{+}^{n + 1}$ . We show that $D (u) = {sup}_{r} D (u; r)$ is bounded in $L^{p}$ if and only if $u (x, y)$ belongs to $H^{p}$ , an equivalence already proved by Barlow and Yor for the supremum of the local times. Our proof relies on the theory of singular integrals due to Caldéron and Zygmund, rather than the stochastic calculus.

Metric spaces admitting only trivial weak contractions

Richárd Balka (2013)

Fundamenta Mathematicae

Similarity:

If (X,d) is a metric space then a map f: X → X is defined to be a weak contraction if d(f(x),f(y)) < d(x,y) for all x,y ∈ X, x ≠ y. We determine the simplest non-closed sets X ⊆ ℝⁿ in the sense of descriptive set-theoretic complexity such that every weak contraction f: X → X is constant. In order to do so, we prove that there exists a non-closed $F_{σ}$ set F ⊆ ℝ such that every weak contraction f: F → F is constant. Similarly, there exists a non-closed $G_{δ}$ set G ⊆ ℝ such that every weak...