Boundedness vs. blow-up in a chemotaxis system

We determine the critical blow-up exponent for a Keller-Segel-type chemotaxis model, where the chemotactic sensitivity equals some nonlinear function of the particle density. Assuming some growth conditions for the chemotactic sensitivity function we establish an a priori estimate for the solution of the problem considered and conclude the global existence and boundedness of the solution. Furthermore, we prove the existence of solutions that become unbounded in finite or infinite time in that situation where this a priori estimate fails.     

The Helgason Fourier transform for homogeneous vector bundles over compact Riemannian symmetric spaces—the local theory

The Helgason Fourier transform on noncompact Riemannian symmetric spaces G/K is generalized to the homogeneous vector bundles over the compact dual spaces U/K. The scalar theory on U/K was considered by Sherman (the local theory for U/K of arbitrary rank, and the global theory for U/K of rank one). In this paper we extend the local theory of Sherman to arbitrary homogeneous vector bundles on U/K. For U/K of rank one we also obtain a generalization of the Cartan-Helgason theorem valid for any K-type.     

Regular Carathéodory-type selectors under no convexity assumptions

We prove the existence of Carathéodory-type selectors (that is, measurable in the first variable and having certain regularity properties like Lipschitz continuity, absolute continuity or bounded variation in the second variable) for multifunctions mapping the product of a measurable space and an interval into compact subsets of a metric space or metric semigroup.     

Groups with a positive law on commutators

Let p be a prime number and let G be a finitely generated group that is residually a finite p-group. We prove that if G satisfies a positive law on all elements of the form [a,b][c,d]ⁱ, a,b,c,d∈G and i?0, then the entire derived subgroup G^′ satisfies a positive law. In fact, G^′ is an extension of a nilpotent group by a locally finite group of finite exponent.     

Mappings of finite distortion: the degree of regularity

This paper investigates the self-improving integrability properties of the so-called mappings of finite distortion. Let K(x)1 be a measurable function defined on a domain ΩRⁿ, n2, and such that exp(βK(x))L_loc¹(Ω), β>0. We show that there exist two universal constants c₁(n),c₂(n) with the following property: Let f be a mapping in W_loc^1,1(Ω,Rⁿ) with |Df(x)|ⁿK(x)J(x,f) for a.e. xΩ and such that the Jacobian determinant J(x,f) is locally in L¹ log^−c₁(n)βL. Then automatically J(x,f) is locally in L¹ log^c₂(n)βL(Ω). This result constitutes the appropriate analog for the self-improving regularity of quasiregular mappings and clarifies many other interesting properties of mappings of finite distortion. Namely, we obtain novel results on the size of removable singularities for bounded mappings of finite distortion, and on the area distortion under this class of mappings.     

Brownian motion characterization of some Besov-Lipschitz spaces on domains

We characterize the Besov-Lipschitz spaces with zero boundary conditions on bounded smooth domains. We prove that the appropriate first and second difference norms are equivalent to the norm given in terms of the transition kernel of the Brownian motion killed upon exit from the domain.     

Balls have the worst best Sobolev inequalities

Using transportation techniques in the spirit of Cordero-Erausquin, Nazaret and Villani [7], we establish an optimal non parametric trace Sobolev inequality, for arbitrary locally Lipschitz domains in ℝⁿ. We deduce a sharp variant of the Brézis-Lieb trace Sobolev inequality [4], containing both the isoperimetric inequality and the sharp Euclidean Sobolev embedding as particular cases. This inequality is optimal for a ball, and can be improved for any other bounded, Lipschitz, connected domain. We also derive a strengthening of the Brézis-Lieb inequality, suggested and left as an open problem in [4]. Many variants will be investigated in a companion article [10].     

Spectral properties of generalized Cesàro operators on Hardy and weighted Bergman spaces

We investigate spectral properties of integral operators of the form

Positive Galois coverings of self-injective algebras

In the article, we study the structure of Galois coverings of self-injective artin algebras with infinite cyclic Galois groups. In particular, we characterize all basic, connected, self-injective artin algebras having Galois coverings by the repetitive algebras of basic connected artin algebras and with the Galois groups generated by positive automorphisms of the repetitive algebras.