Let$f:V$→$R$be a function defined on a subset$V$of$R$^{$n$}×$R$^{$d$}let$⃜:x→$inf{$f(x t)$;$t$such that$(x t)∈V}$denote the$shadow$of$f$and let$Φ$=${(x t)∈V; f(x t)=⃜(x)}$This paper deals with the characterization of some properties of ⃜ in terms of the infinitesimal behavior of$f$near points ζ∈$Φ$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 ⃜ in the interior of$I={x∈$$R$^{n};ε$R$^{d}$(x t)∈V}$and in the$C$^{1 1}case an expression for$D$^{2}⃜ is given To some extent an answer is given to the following question: which convex function ⃜:$I$→$R$$I$interval ϒ$R$(resp which function √:$I$→$R$of class$C$^{1 1}) is the shadow of a$C$^{2}function$f:I$×$R→R$?

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Some new characterizations of the class of positive measures γ on$R$^{$n$}such that$H$_{$p$}^{$l$}∉L_{$p$}(γ) are given where$H$_{$p$}^{$l$}(1<$p$<∞ 0<$l$∞) is the space of Bessel potentials This imbed ding as well as the corresponding trace inequality $$||J_l u||_{L_p (\gamma )} \leqslant C||u||_{L_p } $$ for Bessel potentials$J$_{$l$}=(1-Δ)^{-1/2}is shown to be equivalent to one of the following conditions(a)$J$_{$l$}($J$_{$l$γ})^{$p$}≤$CJ$_{$lγ$}a e(b)$M$_{$l$}($M$_{$l$γ})^{$p’$}≤$CM$_{$lγ$}a e(c)For all compact subsets$E$of$R$^{$n$} $$\int_E {(J_{l\gamma } )^p dx} \leqslant C{\text{ }}cap (E H_p^l )$$ where 1/$p$+1/$p'$=1$M$_{$l$}is the fractional maximal operator and cap ($H$_{$p$}^{$l$}) is the Bessel capacity In particular it is shown that the trace inequality for a positive measure \gg holds if and only if it holds for the measure$(J$_{$l\gg$})^{$p'$}$dx$Similar results are proved for the Riesz potentials$I$_{$l$γ}=|$x$|^{$l-n$}* γ

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$C$) which is hit by the section in case the section hits the zero component

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

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

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Perturbations by analytic discs along generating CR-submanifolds of C^{n}are considered. In the case all partial indices of a closed path$p$in a generating CR-fibration {$M$(ξ)}_{ξ∈∂$D$}are greater or equal to −1 we can completely parametrize all small holomorphic perturbations of the path$p$along the fibration {$M$(ξ)}_{ξ∈∂$D$}. In this case we also study the geometry of perturbations by analytic discs and their relation to the conormal bundle of the fibration.

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Let$E$ç$S$^{1}be a set with Minkowski dimension$d(E)1$. We consider the Hardy-Littlewood maximal function, the Hilbert transform and the maximal Hilbert transform along the directions of$E$. The main result of this paper shows that these operators are bounded on$L$_{$rad$}^{$p$}(R^{2}) for$p>1+d(E)$and unbounded when$p<1+d(E)$. We also give some end-point results.

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Weighted weak type estimates are proved for some maximal operators on the weighted Hardy spaces$H$_{ω}^{$p$}(0 <$p$< 1, ω ∈$A$_{1}) (0<p<1, ω∞A_{1}); in particular, weighted weak type endpoint estimates are obtained for the maximal operators arising from the Bochner-Riesz means and the spherical means on$H$_{ω}^{$p$}.

We introduce and study the Jarník limit set ℐ_{σ}of a geometrically finite Kleinian group with parabolic elements. The set ℐ_{σ}is the dynamical equivalent of the classical set of well approximable limit points. By generalizing the method of Jarník in the theory of Diophantine approximations, we estimate the dimension of ℐ_{σ}with respect to the Patterson measure. In the case in which the exponent of convergence of the group does not exceed the maximal rank of the parabolic fixed points, and hence in particular for all finitely generated Fuchsian groups, it is shown that this leads to a complete description of ℐ_{σ}in terms of Hausdorff dimension. For the remaining case, we derive some estimates for the Hausdorff dimension and the packing dimension of ℐ_{σ}.