Tensor products of regularly varying sequences

We investigate how asymptotic properties of real sequences and are passed on to their tensor product . The main result concerns sequences of the form This problem is closely related to tensor stability of operator ideals on the Hilbert space.     

Hardy spaces for Bessel–Schrödinger operators

Consider the Bessel operator with a potential on , namely We assume that and is a nonnegative function. By definition, a function belongs to the Hardy space if Under certain assumptions on V we characterize the space in terms of atomic decompositions of local type. In the second part we prove that this characterization can be applied to for with no additional assumptions on the potential V.     

A unified method for boundedness in fully parabolic chemotaxis systems with signal-dependent sensitivity

This paper deals with the Keller–Segel system where Ω is a bounded domain in with smooth boundary , ; χ is a nonnegative function satisfying for some and . In the case that and , Fujie 2 established global existence of bounded solutions under the condition . On the other hand, when , Winkler 14 asserted global existence of bounded solutions for arbitrary . However, there is a gap in the proof. Recently, Fujie tried modifying the proof; nevertheless it also has a gap. It seems to be difficult to show global existence of bounded solutions for arbitrary . Moreover, the condition for K when cannot connect with the condition when . The purpose of the present paper is to obtain global existence and boundedness under more natural and proper condition for χ and to build a mathematical bridge between the cases and .     

Jakšić–Last theorem for higher rank perturbations

We consider the generalized Anderson model , where is a countable set, are i.i.d. random variables and the are rank projections. For these models we prove theorem analogous to that of Jak?i?–Last on the equivalence of the trace measure for a.e. ω. Our model covers the dimer and polymer models.     

Korovkin‐type results for multivariate functions which are periodic with respect to one variable

Let K be a compact metric space and let denote the real Banach space of all continuous functions which are 2π‐periodic with respect to the second variable. We prove the following Korovkin‐type result: Let be a continuous algebraic separating function such that for all , and let be a sequence of positive linear operators. If uniformly with respect to and uniformly on for all , then uniformly on for every . As a corollary we deduce: If , then uniformly on for every if and only if uniformly on for every , where and .     

On Fejér's inequalities for the Legendre polynomials

We present various inequalities for the sum where denotes the Legendre polynomial of degree k . Among others we prove that the inequalities hold for all and . The constant factors 2/5 and are sharp. This refines a classical result of Fejér, who proved in 1908 that is nonnegative for all and .     

The Brezis–Nirenberg problem for fractional elliptic operators

Let be a uniformly elliptic operator in divergence form in a bounded open subset Ω of . We study the effect of the operator on the existence and nonexistence of positive solutions of the nonlocal Brezis–Nirenberg problem where denotes the fractional power of with zero Dirichlet boundary values on , , and λ is a real parameter. By assuming for all and near some point , we prove existence theorems for any , where denotes the first Dirichlet eigenvalue of . Our existence result holds true for and in the interior case () and for and in the boundary case (). Nonexistence for star‐shaped domains is obtained for any .     

On the Lê–Milnor fibration for real analytic maps

In this paper, we study the topology of real analytic map‐germs with isolated critical value , with . We compare the topology of f with the topology of the compositions , where are the projections , for . As a main result, we give necessary and sufficient conditions for f to have a Lê–Milnor fibration in the tube.     

Concave univalent functions and Dirichlet finite integral

The article deals with the class consisting of non‐vanishing functions f that are analytic and univalent in such that the complement is a convex set, and the angle at ∞ is less than or equal to for some . Related to this class is the class of concave univalent mappings in , but this differs from with the standard normalization A number of properties of these classes are discussed which includes an easy proof of the coefficient conjecture for settled by Avkhadiev et al. 3 . Moreover, another interesting result connected with the Yamashita conjecture on Dirichlet finite integral for is also presented.     

Bohr radius for locally univalent harmonic mappings

We consider the class of all sense‐preserving harmonic mappings of the unit disk , where h and g are analytic with , and determine the Bohr radius if any one of the following conditions holds:

• 1. h is bounded in .
• 2. h satisfies the condition in with .
• 3. both h and g are bounded in .
• 4. h is bounded and .

We also consider the problem of determining the Bohr radius when the supremum of the modulus of the dilatation of f in is strictly less than 1. In addition, we determine the Bohr radius for the space of analytic Bloch functions and the space of harmonic Bloch functions. The paper concludes with two conjectures.     

On the structure of sets with positive reach

We give a complete characterization of compact sets with positive reach (proximally C ¹ sets) in the plane and of one‐dimensional sets with positive reach in . Further, we prove that if is a set of positive reach of topological dimension , then A has its “k‐dimensional regular part” which is a k‐dimensional “uniform” C ^{1, 1} manifold open in A and can be locally covered by finitely many ‐dimensional DC surfaces. We also show that if has positive reach, then can be locally covered by finitely many semiconcave hypersurfaces.     

Some characterizations of BLO space

For , a real‐valued function belongs to space if In this paper, we establish a version of John–Nirenberg inequality suitable for the space with . As a corollary, it is proved that spaces are independent of the scale in sense of norm. Also, we characterize the space through weighted Lebesgue spaces and variable Lebesgue spaces, respectively.     

Strict topologies on measure spaces

Let X be a measurable space, let be a family of measurable subsets of it, and let be a subspace of complex measures on X that is also closed under restrictions of measures. In this paper we introduce the ‐convergence topology and the ‐strict topology on . Among other results, we find necessary and sufficient conditions for Hausdorff‐ness and coincide‐ness of these topologies. Applications to Lebesgue spaces, and also examples in Hausdorff topological spaces and locally compact groups are given.     

Littlewood–Paley characterizations of higher‐order Sobolev spaces via averages on balls

In this article, the authors characterize higher‐order Sobolev spaces , with , and , or with , and , via the Lusin area function and the Littlewood–Paley ‐function in terms of ball averages, where denotes the maximal integer not greater than . Moreover, the authors also complement the above results in the endpoint cases of p via establishing some weak type estimates. These improve and develop the corresponding known results for Sobolev spaces with smoothness order .     

On a generalized nonlinear heat equation in Besov and Triebel–Lizorkin spaces

The paper deals with solutions of the Cauchy problem of a nonlinear generalized heat equation in the context of Besov and Triebel–Lizorkin spaces and where and with initial data belonging to some spaces where and .     

The automorphism group of a rigid affine variety

An irreducible algebraic variety X is rigid if it admits no nontrivial action of the additive group of the ground field. We prove that the automorphism group of a rigid affine variety contains a unique maximal torus . If the grading on the algebra of regular functions defined by the action of  is pointed, the group  is a finite extension of . As an application, we describe the automorphism group of a rigid trinomial affine hypersurface and find all isomorphisms between such hypersurfaces.     

Low dimensional instability for quasilinear problems of weighted exponential nonlinearity

We prove a sharp Liouville type theorem for stable solutions of the equation on the entire Euclidean space , where and f is a continuous and nonnegative function in such that as , where and . Our theorem holds true for and is sharp in the case .     

Eigenvalues of Robin Laplacians in infinite sectors

For , let denote the infinite planar sector of opening 2α, and be the Laplacian in , , with the Robin boundary condition , where stands for the outer normal derivative and . The essential spectrum of does not depend on the angle α and equals , and the discrete spectrum is non‐empty if and only if . In this case we show that the discrete spectrum is always finite and that each individual eigenvalue is a continous strictly increasing function of the angle α. In particular, there is just one discrete eigenvalue for . As α approaches 0, the number of discrete eigenvalues becomes arbitrary large and is minorated by with a suitable , and the nth eigenvalue of behaves as and admits a full asymptotic expansion in powers of α². The eigenfunctions are exponentially localized near the origin. The results are also applied to δ‐interactions on star graphs.