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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

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≤∞$

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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