MathSciDoc: An Archive for Mathematician ∫

Let σ be the geodesic inversion on a Heisenberg type group$N$with homogeneous dimension$Q$, and denote by$S$the jacobian of σ. We prove that, for $$ - \frac{1}{2}Q< \alpha< \frac{1}{2}Q$$ , the operators $$T_\alpha :f \mapsto S^{1/2 - \alpha /Q} (f \circ \sigma )$$ are bounded on certain homogeneous Sobolev spaces $$\mathcal{H}^\alpha (N)$$ if and only if$N$is an Iwasawa$N$-group.

Let$f$be meromorphic in the open unit disc$D$and strongly normal; that is, $$(1 - |z|^2 )f^\# (z) \to 0as|z| \to 1,$$

First we prove a new inequality comparing uniformly the relative volume of a Borel subset with respect to any given complex euclidean ball$B$⊂$C$^{$n$}with its relative logarithmic capacity in$C$^{$n$}with respect to the same ball$B$. An analogous comparison inequality for Borel subsets of euclidean balls of any generic real subspace of$C$^{$n$}is also proved.

In this paper we prove a well-posedness result for the Cauchy problem. We study a class of first order hyperbolic differential [2] operators of rank zero on an involutive submanifold of$T$^{*}$R$^{$n$+1}-{0} and prove that under suitable assumptions on the symmetrizability of the lifting of the principal symbol to a natural blow up of the “singular part” of the characteristic set, the operator is strongly hyperbolic.

Mukai and Sakai proved that given a vector bundle$E$of rank$n$on a connected smooth projective curve of genus$g$and any$r$∈[1,$n$], there is subbundle$S$of rank$r$such that deg Hom($S, E/S$)≤$r$($n−r$)$g$. We prove a generalization of this bound for equivariant principal bundles. Our proof even simplifies the one given by Holla and Narasimhan for usual principal bundles.

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Let ($A$_{$0, A$}_{$1$}) be a compatible pair of quasi-Banach spaces and 1et$A$be a corresponding space of real interpolation type such that$A$_{$0$}∩$A$_{$1$}is not dense in$A$. Upper and lower estimates are obtained for the distance of any element$f$of$A$from$A$_{$0$}∩$A$_{$1$}. These lead to formulae for the distance in a large number of concrete situations, such as when$A$_{$0$}∩$A$_{$1$}=$L$^{$∞$}and$A$is either weak-$L$^{q}, a ‘grand’ Lebesgue space or an Orlicz space of exponential type.

A similar formula to the one established by Ansemil and Floret for symmetric tensor products of direct sums is proved for alternating and Jacobian tensor products. It is then applied to stable spaces where a number of isomorphisms between spaces of tensors or multilinear forms are unveiled. A connection between these problems and irreducible group representations is made.

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Let$Z$be the zero set of a holomorphic section$f$of a Hermitian vector bundle. It is proved that the current of integration over the irreducible components of$Z$of top degree, counted with multiplicities, is a product of a residue factor$R$^{$f$}and a “Jacobian factor”. There is also a relation to the Monge-Ampère expressions ($dd$^{$c$}log|$f$|)^{$k$}, which we define for all positive powers$k$.