Let $u$ be a $\delta$-subharmonic function with associated measure $\mu$, and let $v$ be a superharmonic function with associated measure $\nu$, 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 $\left(ℳ\right(u,x,s)-ℳ(u,x,t\left)\right)/\left(ℳ\right(v,x,s)-ℳ(v,x,t\left)\right)$, lie between the upper and lower limits as $r\to 0+$ of the quotient $\mu \left(B\right(x,r\left)\right)/\nu \left(B\right(x,r\left)\right)$. This enables us to use some well-known measure-theoretic results to prove new variants...

The classical Whitney extension theorem states that every function in Lip$(\beta ,F)$, $F\subset {\mathbf{R}}^n$, $F$ closed, $k\&lt;\beta \le k+1$, $k$ a non-negative integer, can be extended to a function in Lip$(\beta ,{\mathbf{R}}^n)$. Her Lip$(\beta ,F)$ 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 ${\mathbf{R}}^d$, $d\&lt;n$, of the Bessel potential spaces in ${\mathbf{R}}^n$ and the Besov or generalized Lipschitz...

Given a positive measure μ in ${ℝ}^n$, there is a natural variant of the noncentered Hardy-Littlewood maximal operator $M_{\mu }f\left(x\right)=su{p}_{x\in B}1/\mu \left(B\right){ʃ}_B\left|f\right|d\mu$, where the supremum is taken over all balls containing the point x. In this paper we restrict our attention to rotation invariant, strictly positive measures μ in ${ℝ}^n$. We give some necessary and sufficient conditions for $M_{\mu }$ to be bounded from $L^1\left(d\mu \right)$ to $L^{1,\infty }\left(d\mu \right)$.

Let $\mu$ be a nonnegative Radon measure on ${ℝ}^d$ which only satisfies $\mu \left(B(x,r)\right)\le C_0{r}^n$ for all $x\in {ℝ}^d$, $r>0$, with some fixed constants $C_0>0$ and $n\in (0,d].$ In this paper, a new characterization for the space $\mathrm{RBMO}\left(\mu \right)$ of Tolsa in terms of the John-Strömberg sharp maximal function is established.

Let $u(x,y)$ be a harmonic function in the half-plane ${\mathbf{R}}_{+}^{n+1}$, $n\ge 2$. We define a family of functionals $D(u;r),-\infty \&gt;r\&gt;\infty$, 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 ${\mathbf{R}}_{+}^{n+1}$. We show that $D\left(u\right)={sup}_{r}D(u;r)$ 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.