Let L be a strictly elliptic second order operator on a bounded domain Ω ⊂ ℝⁿ. Let u be a solution to $Lu=div\overrightarrow{f}$ in Ω, u = 0 on ∂Ω. Sufficient conditions on two measures, μ and ν defined on Ω, are established which imply that the $L^q(\Omega ,d\mu )$ norm of |∇u| is dominated by the $L^p(\Omega ,dv)$ norms of $div\overrightarrow{f}$ and $|\overrightarrow{f}|$. If we replace |∇u| by a local Hölder norm of u, the conditions on μ and ν can be significantly weaker.

We establish the boundedness in $L^q$ spaces, 1 < q ≤ 2, of a “vertical” Littlewood-Paley-Stein operator associated with a reversible random walk on a graph. This result extends to certain non-reversible random walks, including centered random walks on any finitely generated discrete group.

We prove a lower bound in a law of the iterated logarithm for sums of the form ${\sum }_{k=1}^N{a}_{k}f(n_{k}x+c_k)$ where f satisfies certain conditions and the $n_k$ satisfy the Hadamard gap condition $n_{k+1}/n_k\ge q>1$.

It is proved that if $m:{ℝ}^d\to ℂ$ satisfies a suitable integral condition of Marcinkiewicz type then m is a Fourier multiplier on the $H^1$ space on the product domain ${ℝ}^{d_1}\times ...\times {ℝ}^{d_k}$. This implies an estimate of the norm $N(m,L^p\left({ℝ}^d\right)$ of the multiplier transformation of m on $L^p\left({ℝ}^d\right)$ as p→1. Precisely we get $N(m,L^p\left({ℝ}^d\right))\lesssim {(p-1)}^{-k}$. This bound is the best possible in general.

We prove unconditionality of general Franklin systems in $L^p\left(X\right)$, where X is a UMD space and where the general Franklin system corresponds to a quasi-dyadic, weakly regular sequence of knots.

We give characterizations of weighted Besov-Lipschitz and Triebel-Lizorkin spaces with $A_{\infty }$ weights via a smooth kernel which satisfies “minimal” moment and Tauberian conditions. The results are stated in terms of the mixed norm of a certain maximal function of a distribution in these weighted spaces.

A systematic method for the calculus of Bernstein's polynomial is described. It consists of reducing the problem to a homogeneous linear system of equations that may be constructed by fixed rules. Several problems about its computer implementation are discussed.

Using Bochner-Riesz means we get a multidimensional sampling theorem for band-limited functions with polynomial growth, that is, for functions which are the Fourier transform of compactly supported distributions.

We prove the central limit theorem for the multisequence ${\sum }_{1\le n₁\le N₁}\cdots {\sum }_{1\le n_d\le N_d}{a}_{n₁,...,n_d}cos(⟨2\pi m,A{₁}^{n₁}...A_d^{n_d}x⟩)$ where $m\in {\mathbb{Z}}^s$, $a_{n₁,...,n_d}$ are reals, $A₁,...,A_d$ are partially hyperbolic commuting s × s matrices, and x is a uniformly distributed random variable in ${[0,1]}^s$. The main tool is the S-unit theorem.

We define a new type of multiplier operators on $L^p{(}^N)$, where ${}^N$ is the N-dimensional torus, and use tangent sequences from probability theory to prove that the operator norms of these multipliers are independent of the dimension N. Our construction is motivated by the conjugate function operator on $L^p{(}^N)$, to which the theorem applies as a particular example.