AnL¹-existence theorem is proved for the nonlinear stationary Boltzmann equation with hard forces and no small velocity truncation—only the Grad angular cut-off-in a setting between two coaxial rotating cylinders when the indata are given on the cylinders.

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.

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

Wheref^# denotes the spherical derivative of f. We prove results about the existence of asymptotic values of f at points of C=∂D. For example, f has asymptotic values at an uncountably dense subset of C, and the asymptotic values of f form a set of positive linear measure.

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

Then we give several interesting applications of these inequalities. First we obtain sharp uniform estimates on the relative size of plurisubharmonic lemniscates associated to the Lelong class of plurisubharmonic functions of logarithmic singularities at infinity on Cⁿ as well as the Cegrell class of plurisubharmonic functions of bounded Monge-Ampère mass on a hyperconvex domain Ω⊂(Cⁿ.

Then we also deduce new results on the global behaviour of both the Lelong class and the Cegrell class of plurisubharmonic functions.

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^*Rⁿ⁺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.

Let (A_{0, A1}) be a compatible pair of quasi-Banach spaces and 1etA be a corresponding space of real interpolation type such that A₀∩A₁ is not dense in A. Upper and lower estimates are obtained for the distance of any element f of A from A₀∩A₁. These lead to formulae for the distance in a large number of concrete situations, such as when A₀∩A₁=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.