On Extrapolation of Carleson Measures
We give an alternate proof of three versions of the theorem on extrapolation of Carleson measures.
math.CA↗
arXiv subjects
John Garnett
Publications and source records attributed to John Garnett.
Carleson measure estimates and $ε$-approximation of bounded harmonic functions, without Ahlfors regularity assumptions
Let $Ω$ be a domain in $\mathbb{R}^{d+1}$, $d \geq 1$. In the paper's references [HMM2] and [GMT] it was proved that if $Ω$ satisfies a corkscrew condition and if $\partial Ω$ is $d$-Ahlfors regular, i.e. Hausdorff measure $\mathcal{H}^d(B(x,r) \cap \partial Ω) \sim r^d$ for all $x \in \partial Ω$ and $0 < r < {\rm diam}(\partial Ω)$, then $\partial Ω$ is uniformly rectifiable if and only if (a) a square function Carleson measure estimate holds for every bounded harmonic function on $Ω$ or (b) an $\varepsilon$-approximation property for all $0 < \varepsilon <1$ for every such function. Here we explore (a) and (b) when $\partial Ω$ is not required to be Ahlfors regular. We first prove that (a) and (b) hold for any domain $Ω$ for which there exists a domain $\widetilde Ω\subset Ω$ such that $\partial Ω\subset \partial \widetilde Ω$ and $\partial \widetilde Ω$ is uniformly rectifiable. We next assume $Ω$ satisfies a corkscrew condition and $\partial Ω$ satisfies a capacity density condition. Under these assumptions we prove conversely that the existence of such $\widetilde Ω$ implies (a) and (b) hold on $Ω$ and give further characterizations of domains for which (a) or (b) holds. One is that harmonic measure satisfies a Carleson packing condition for diameters similar to the corona decompositionm proved equivalent to uniform rectifiability in [GMT]. The second characterization is reminiscent of the Carleson measure description of $H^{\infty}$ interpolating sequences in the unit disc.
math.CA↗
Uniform rectifiability, elliptic measure, square functions, and $\varepsilon$-approximability via an ACF monotonicity formula
Let $Ω\subset\mathbb{R}^{n+1}$, $n\geq2$, be an open set with Ahlfors-David regular boundary that satisfies the corkscrew condition. We consider a uniformly elliptic operator $L$ in divergence form associated with a matrix $A$ with real, merely bounded and possibly non-symmetric coefficients, which are also locally Lipschitz and satisfy suitable Carleson type estimates. In this paper we show that if $L^*$ is the operator in divergence form associated with the transpose matrix of $A$, then $\partialΩ$ is uniformly $n$-rectifiable if and only if every bounded solution of $Lu=0$ and every bounded solution of $L^*v=0$ in $Ω$ is $\varepsilon$-approximmable if and only if every bounded solution of $Lu=0$ and every bounded solution of $L^*v=0$ in $Ω$ satisfies a suitable square-function Carleson measure estimate. Moreover, we obtain two additional criteria for uniform rectifiability. One is given in terms of the so-called $S<N$ estimates, and another in terms of a suitable corona decomposition involving $L$-harmonic and $L^*$-harmonic measures. We also prove that if $L$-harmonic measure and $L^*$-harmonic measure satisfy a weak $A_\infty$-type condition, then $\partial Ω$ is $n$-uniformly rectifiable. In the process we obtain a version of Alt-Caffarelli-Friedman monotonicity formula for a fairly wide class of elliptic operators which is of independent interest and plays a fundamental role in our arguments.
math.CA↗
Uniform rectifiability from Carleson measure estimates and $\varepsilon$-approximability of bounded harmonic functions
Let $Ω\subset\mathbb R^{n+1}$, $n\geq1$, be a corkscrew domain with Ahlfors-David regular boundary. In this paper we prove that $\partialΩ$ is uniformly $n$-rectifiable if every bounded harmonic function on $Ω$ is $\varepsilon$-approximable or if every bounded harmonic function on $Ω$ satisfies a suitable square-function Carleson measure estimate. In particular, this applies to the case when $Ω=\mathbb R^{n+1}\setminus E$ and $E$ is Ahlfors-David regular. Our results solve a conjecture posed by Hofmann, Martell, and Mayboroda in a recent work where they proved the converse statements. Here we also obtain two additional criteria for uniform rectifiability. One is given in terms of the so-called "$S<N$" estimates, and another in terms of a suitable corona decomposition involving harmonic measure.
math.CA↗
The regularity of the boundary of a multidimensional aggregation patch
Let $d \geq 2$ and let $N(y)$ be the fundamental solution of the Laplace equation in $R^d$ We consider the aggregation equation $$ \frac{\partial ρ}{\partial t} + \operatorname{div}(ρv) =0, v = -\nabla N * ρ$$ with initial data $ρ(x,0) = χ_{D_0}$, where $χ_{D_0}$ is the indicator function of a bounded domain $D_0 \subset R^d.$ We now fix $0 < γ< 1$ and take $D_0$ to be a bounded $C^{1+γ}$ domain (a domain with smooth boundary of class $C^{1+γ}$). Then we have Theorem: If $D_0$ is a $C^{1 + γ}$ domain, then the initial value problem above has a solution given by $$ρ(x,t) = \frac{1}{1 -t} χ_{D_t}(x), \quad x \in R^d, \quad 0 \le t < 1$$ where $D_t$ is a $C^{1 + γ}$ domain for all $0 \leq t < 1$.
math.AP↗
Square functions and uniform rectifiability
In this paper it is shown that an Ahlfors-David $n$-dimensional measure $μ$ on $\mathbb{R}^d$ is uniformly $n$-rectifiable if and only if for any ball $B(x_0,R)$ centered at $\operatorname{supp}(μ)$, $$ \int_0^R \int_{x\in B(x_0,R)} \left|\frac{μ(B(x,r))}{r^n} - \frac{μ(B(x,2r))}{(2r)^n} \right|^2\,dμ(x)\,\frac{dr}r \leq c\, R^n.$$ Other characterizations of uniform $n$-rectifiability in terms of smoother square functions are also obtained.
math.CA↗
A doubling measure on $\R^d$ can charge a rectifiable curve
For $d\geq 2$, we construct a doubling measure $ν$ on $\R^d$ and a rectifiable curve $Γ$ such that $ν(Γ)>0$.
math.MG↗