Let $f : V → \mathbf{R}$ be a function defined on a subset $V$ of $\mathbf{R}^n × \mathbf{R}^d$, let $\varphi : x \mapsto \inf \lbrace f(x, t); t$ such that $(x, t) ∈ V \rbrace$ denote the shadow of $f$, and let $Φ = \lbrace (x, t) ∈ V; f(x, t) = \varphi(x) \rbrace$. This paper deals with the characterization of some properties of $\varphi$ in terms of the infinitesimal behavior of $f$ near points $\xi ∈ Φ$, proving in particular a conjecture of J.-M Trépreau concerning the case $d=1$. Characterizations of this type are provided for the convexity, the subharmonicity, or the $C^{1,1}$-regularity of $\varphi$ in the interior of $I = \lbrace x ∈ \mathbf{R}^n; \exists t \in \mathbf{R}^d, (x,t) ∈ V \rbrace$, and, in the $C^{1,1}$ case, an expression for $D^2 \varphi$ is given. To some extent, an answer is given to the following question: which convex function $\varphi : I → \mathbf{R}, I$ interval $\subset \mathbf{R}$ (resp. which function $\varphi : I → \mathbf{R}$ of class $C^{1,1}$) is the shadow of a $C^2$ function $f:I × \mathbf{R}→\mathbf{R}$?

Some new characterizations of the class of positive measures $\gamma$ on $\mathbf R^n$ such that $H{_{p}^{l}} \subset L_p (\gamma)$ are given, where $H{_{p}^{l}} (1\lt p \lt ∞, 0 \lt l \lt ∞)$ is the space of Bessel potentials. This imbedding, as well as the corresponding trace inequality $$||J_l u||_{L_p (\gamma )} \leq C||u||_{L_p },$$ for Bessel potentials $J_l = (1-Δ)^{-1/2}$, is shown to be equivalent to one of the following conditions.

1. $J_l (J_{l\gamma})^{p^\prime} \leq C J_{l\gamma}$ a.e.

2. $M_l (M_{l\gamma})^{p^\prime} \leq C M_{l\gamma}$ a.e.

3. For all compact subsets $E$ of $\mathbf R^n$

$$\int_E (J_{l\gamma)})^{p^\prime} dx \leq C \text{ cap} (E, H{_{p}^{l}}),$$ where $1/p+1/p'=1$, $M_l$ is the fractional maximal operator, and cap$(\cdot, H{_{p}^{l}})$ is the Bessel capacity. In particular it is shown that the trace inequality for a positive measure $\gamma$ holds if and only if it holds for the measure $(J_{l\gamma})^{p^\prime} dx$. Similar results are proved for the Riesz potentials $I_{l\gamma} = |x|^{l-n}*\gamma$.

These results are used to get a complete characterization of the positive measures on $\mathbf R^n$ giving rise to bounded pointwise multipliers $M (H{_{p}^{m}} → H {_{p}^{−l}})$. Some applications to elliptic partial differential equations are considered, including coercive estimates for solutions of the Poisson equation, and existence of positive solutions for certain linear and semi-linear equations.

Given a torsion section of a semistable elliptic surface, we prove equidistribution results for the components of singular fibers which are hit by the section, and for the root of unity (identifying the zero component with $\mathbf C$) which is hit by the section in case the section hits the zero component.

Let $\Delta$ be the Laplace-Beltrami operator on an $n$-dimensional complete $C^∞$ manifold $M$. In this paper, we establish an estimate of $e^{t\Delta} (d\mu)$ valid for all $t \gt 0$, where $d\mu$ is a locally uniformly $\alpha$-dimensional measure on $M, 0 ≤ \alpha ≤ n$. The result is used to study the mapping properties of $(I - t\Delta)^{-\beta}$ considered as an operator from $L^p (M, d\mu)$ to $L^p (M, dx)$, where $dx$ is the Riemannian measure on $M$, $\beta \gt (n−\alpha)/2p′, 1/p+1/p′=1, 1≤ p ≤ ∞$.

In the paper [B2], Baernstein constructs a simply connected domain $\Omega$ in the plane for which the conformal mapping $f$ of $\Omega$ into the unit disc $\Delta$ satisfies $$\int_{\mathbf{R} \cap \Omega} |f'(z)|^p |dz| = \infty,$$ for some $p ∈ (1,2)$, where $\mathbf R$ is the real line.

This gives a counterexample to a conjecture stating that for any simply connected domain $\Omega$ in the plane, all the above integrals are finite for any $1 \lt p \lt 2$.

In this paper, we give a conceptual proof of the basic estimate of Baenstein.