Let $\Phi \left(t\right)={ʃ}_0^{t}a\left(s\right)ds$ and $\Psi \left(t\right)={ʃ}_0^{t}b\left(s\right)ds$, where a(s) is a positive continuous function such that ${ʃ}_1^{\infty }a\left(s\right)/sds=\infty$ and b(s) is quasi-increasing and $li{m}_{s\to \infty }b\left(s\right)=\infty$. Then the following statements for the Hardy-Littlewood maximal function Mf(x) are equivalent: (j) there exist positive constants $c_1$ and $s_0$ such that ${ʃ}_1^{s}a\left(t\right)/tdt\ge c_{1}b\left(c_{1}s\right)$ for all $s\ge s_0$; (jj) there exist positive constants $c_2$ and $c_3$ such that ${ʃ}_0^{2\pi }\Psi (\left(c_2\right)/{\left(\right|⨍|}_{}\left)\right|⨍\left(x\right)\left|\right)dx\le c_3+c_3{ʃ}_0^{2\pi }{\Phi (1/(\left|⨍\right|}_{}\left)\right)Mf\left(x\right)dx$ for all $⨍\in L^1\left(\right)$.

We establish a decomposition of non-negative Radon measures on ${ℝ}^d$ which extends that obtained by Strichartz [6] in the setting of $\alpha$-dimensional measures. As consequences, we deduce some well-known properties concerning the density of non-negative Radon measures. Furthermore, some properties of non-negative Radon measures having their Riesz potential in a Lebesgue space are obtained.

The Hardy inequality ${\int }_{\Omega }{\left|u\left(x\right)\right|}^{p}d{\left(x\right)}^{-p}\ddot{x}\le c{\int }_{\Omega }{\left|\nabla u\left(x\right)\right|}^p\ddot{x}$ with $d\left(x\right)=dist(x,\partial \Omega )$ holds for $u\in C_0^{\infty }\left(\Omega \right)$ if $\Omega \subset {ℝ}^n$ is an open set with a sufficiently smooth boundary and if $1<p<\infty$. P. Hajlasz proved the pointwise counterpart to this inequality involving a maximal function of Hardy-Littlewood type on the right hand side and, as a consequence, obtained the integral Hardy inequality. We extend these results for gradients of higher order and also for $p=1$.

In their celebrated paper [3], Burkholder, Gundy, and Silverstein used Brownian motion to derive a maximal function characterization of $H^p$ spaces for 0 < p < ∞. In the present paper, we show that the methods in [3] extend to higher dimensions and yield a dimension-free weak type (1,1) estimate for a conjugate function on the N-dimensional torus.

We give a proof of an integral formula of Berndtsson which is related to the inversion of Fourier-Laplace transforms of $\overline{\partial }$-closed $(n,n-1)$-forms in the complement of a compact convex set in ${ℂ}^n$.

Let ũ denote the conjugate Poisson integral of a function $f\in L^p\left(ℝ\right)$. We give conditions on a region Ω so that $li{m}_{(v,\epsilon )\to (0,0)}(v,\epsilon )\in \Omega ũ(x+v,\epsilon )=Hf\left(x\right)$, the Hilbert transform of f at x, for a.e. x. We also consider more general Calderón-Zygmund singular integrals and give conditions on a set Ω so that $su{p}_{(v,r)\in \Omega }\left|{ʃ}_{\left|t\right|>r}k(x+v-t)f\left(t\right)dt\right|$ is a bounded operator on $L^p$, 1 < p < ∞, and is weak (1,1).