Let $\mathcal{M} =\{m_{j}\}_{j=1}^{\infty}$ be a family of Marcinkiewicz multipliers of sufficient uniform smoothness in $\mathbb{R}^{n}$ . We show that the$L$^{$p$}norm, 1<$p$<∞, of the related maximal operator $$M_Nf(x)= \sup_{1\leq j \leq N} |\mathcal{F}^{-1} ( m_j \mathcal{F} f)|(x) $$ is at most$C$(log($N$+2))^{$n$/2}. We show that this bound is sharp.

Let Ω be a Greenian domain in ℝ^{$d$},$d$≥2, or—more generally—let Ω be a connected $\mathcal{P}$ -Brelot space satisfying axiom D, and let$u$be a numerical function on Ω, $u\not\equiv\infty$ , which is locally bounded from below. A short proof yields the following result: The function$u$is the infimum of its superharmonic majorants if and only if each set {$x$:$u$($x$)>$t$},$t$∈ℝ, differs from an analytic set only by a polar set and $\int u\,d\mu_{x}^{V}\le u(x)$ , whenever$V$is a relatively compact open set in Ω and$x$∈$V$.

Let$f$be a germ of an analytic function at infinity that can be analytically continued along any path in the complex plane deprived of a finite set of points, $${f \in \mathcal{A}(\bar{\mathbb{C}} \setminus A)}$$ , $${\# A< \infty}$$ . J. Nuttall has put forward the important relation between the$maximal domain$of$f$where the function has a single-valued branch and the$domain of convergence$of the diagonal Padé approximants for$f$. The Padé approximants, which are rational functions and thus single-valued, approximate a holomorphic branch of$f$in the domain of their convergence. At the same time most of their poles tend to the boundary of the domain of convergence and the support of their limiting distribution models the system of cuts that makes the function$f$single-valued. Nuttall has conjectured (and proved for many important special cases) that this system of cuts has$minimal logarithmic capacity$among all other systems converting the function$f$to a single-valued branch. Thus the domain of convergence corresponds to the$maximal$(in the sense of$minimal$boundary) domain of single-valued holomorphy for the analytic function $${f\in\mathcal{A}(\bar{\mathbb{C}} \setminus A)}$$ . The complete proof of Nuttall’s conjecture (even in a more general setting where the set$A$has logarithmic capacity 0) was obtained by H. Stahl. In this work, we derive$strong asymptotics$for the denominators of the diagonal Padé approximants for this problem in a rather general setting. We assume that$A$is a finite set of branch points of$f$which have the$algebro-logarithmic character$and which are placed in a$generic position$. The last restriction means that we exclude from our consideration some degenerated “constellations” of the branch points.

Let$D$($μ$) be the Dirichlet space weighted by the Poisson integral of the positive measure$μ$. We give a characterization of the measures$μ$equal to a countable sum of atoms for which the Blaschke condition is a necessary and sufficient condition for a sequence to be a zero set for$D$($μ$).

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