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.

No abstract uploaded!

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≤∞$

No abstract uploaded!