Letting $A \subset \mathbb{R}^n$ be Borel measurable and $W_0 : A \to \mathbb{G}(n,m)$ Lipschitzian, we establish that \begin{equation*} \limsup_{r \to 0^+} \frac{\mathcal{H}^m \left[ A \cap B(x,r) \cap (x+ W_0(x))\right]}{α(m)r^m} \geq \frac{1}{2^n}, \end{equation*} for $\mathcal{L}^n$-almost every $x \in A$. In particular, it follows that $A$ is $\mathcal{L}^n$-negligible if and only if $\mathcal{H}^m(A \cap (x+W_0(x))=0$, for $\mathcal{L}^n$-almost every $x \in A$.
In the setting of $\R^d$ with an $n-$dimensional measure $μ,$ we give several characterizations of Lipschitz spaces in terms of mean oscillations involving $μ.$ We also show that Lipschitz spaces are preserved by those Calderon-Zygmund operators $T$ associated to the measure $μ$ for which T(1) is the Lipschitz class $0.$
In this note we will show a Calderón--Zygmund decomposition associated with a function $f\in L^1(\mathbb{T}^ω)$. The idea relies on an adaptation of a more general result by J. L. Rubio de Francia in the setting of locally compact groups. Some related results about differentiation of integrals on the infinite-dimensional torus are also discussed.
Consider $v$ a Lipschitz unit vector field on $R^n$ and $K$ its Lipschitz constant. We show that the maps $S_s:S_s(X) = X + sv(X)$ are invertible for $0\leq |s|<1/K$ and define nonsingular point transformations. We use these properties to prove first the differentiation in L^p norm for $1\le p<\infty.$ Then we show the existence of a universal set of values $s\in [-1/2K,1/2K]$ of measure 1/K for which the Lipschitz unit vector fields $v\circ S_s^{-1}$ satisfy Zygmund's conjecture for all functions in $L^p(\R^n)$ and for each p, $1\leq p< \infty.$
Let $T$ be a Calderón-Zygmund singular integral operator. In this paper, we will show some weighted boundedness properties of commutator $[b,T]$ on the weighted Morrey spaces $L^{p,κ}(w)$ under appropriate conditions on the weight $w$, where the symbol $b$ belongs to weighted $BMO$ or Lipschitz space or weighted Lipschitz space.
We extend the recently much-studied Hardy factorization theorems to the weight case. The key point of this paper is to establish the factorization theorems without individual condition on the weight functions. As a direct application, we obtain the characterizations of $BMO(\mathbb{R}^n)$ space and Lipschitz spaces via the weighted boundedness of commutators of multilinear Calderón-Zygmund operators with the genuinely multilinear weights.
We prove that a function in several variables is in the local Zygmund class $\mathcal Z^{m,1}$ if and only if its composite with every smooth curve is of class $\mathcal Z^{m,1}$. This complements the well-known analogous result for local Hölder--Lipschitz classes $\mathcal C^{m,α}$ which we reprove along the way. We demonstrate that these results generalize to mappings between Banach spaces and use them to study the regularity of the superposition operator $f_* : g \mapsto f \circ g$ acting on the global Zygmund space $Λ_{m+1}(\mathbb R^d)$. We prove that, for all integers $m,k\ge 1$, the map…
We consider the Calderón-Zygmund kernels $K_ {α,n}(x)=(x_i^{2n-1}/|x|^{2n-1+α})_{i=1}^d$ in $\mathbb{R}^n$ for $0<α\leq 1$ and $n\in\mathbb{N}$. We show that, on the plane, for $0<α<1$, the capacity associated to the kernels $K_{α,n}$ is comparable to the Riesz capacity $C_{\frac23(2-α),\frac 3 2}$ of non-linear potential theory. As consequences we deduce the semiadditivity and bi-Lipschitz invariance of this capacity. Furthermore we show that for any Borel set $E\subset\mathbb{R}^n$ with finite length the $L^2(\mathcal{H}^1 \lfloor E)$-boundedness of the singular integral associated to $K_{1,…
The aim of this article is to give a complete solution to the problem of the bilinear decompositions of the products of some Hardy spaces $H^p(\mathbb{R}^n)$ and their duals in the case when $p<1$ and near to $1$, via wavelets, paraproducts and the theory of bilinear Calderón-Zygmund operators. Precisely, the authors establish the bilinear decompositions of the product spaces $H^p(\mathbb{R}^n)\times\dotΛ_α (\mathbb{R}^n)$ and $H^p(\mathbb{R}^n)\timesΛ_α(\mathbb{R}^n)$, where, for all $p\in(\frac{n}{n+1},\,1)$ and $α:=n(\frac{1}{p}-1)$, $H^p(\mathbb{R}^n)$ denotes the classical real Hardy spac…
The purpose of this note is to point out a simple consequence of some earlier work of the authors, "Hard Sard: Quantitative implicit function and extension theorems for Lipschitz maps". For $f$, a Lipschitz function from a Euclidean space into a metric space, we give quantitative estimates for how often the pullback of the metric under $f$ is approximately a seminorm. This is a quantitative version of Kirchheim's metric differentiation result from 1994. Our result is in the form of a Carleson-type estimate.
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).
Suppose that $(X,d,μ)$ is a metric measure space of finite Hausdorff dimension and that, for every Lipschitz $f \colon X \to \mathbb R$, $\operatorname{Lip}(f,\cdot)$ is dominated by every upper gradient of $f$. We show that $X$ is a Lipschitz differentiability space, and the differentiable structure of $X$ has dimension at most $\dim_{\mathrm{H}} X$. Since our assumptions are satisfied whenever $X$ is doubling and satisfies a Poincaré inequality, we thus obtain a new proof of Cheeger's generalisation of Rademacher's theorem. Our approach uses Guth's multilinear Kakeya inequality for neighbour…
Let $μ$ be a non-negative Radon measure on ${\mathbb R}^d$ which only satisfies the polynomial growth condition. Let ${\mathcal Y}$ be a Banach space and $H^1(μ)$ the Hardy space of Tolsa. In this paper, the authors prove that a linear operator $T$ is bounded from $H^1(μ)$ to ${\mathcal Y}$ if and only if $T$ maps all $(p, γ)$-atomic blocks into uniformly bounded elements of ${\mathcal Y}$; moreover, the authors prove that for a sublinear operator $T$ bounded from $L^1(μ)$ to $L^{1, \infty}(μ)$, if $T$ maps all $(p, γ)$-atomic blocks with $p\in(1, \infty)$ and $γ\in{\mathbb N}$ into uniformly …
