On the linear polarization constants of finite dimensional spaces

We study the linear polarization constants of finite dimensional Banach spaces. We obtain the correct asymptotic behaviour of these constants for the spaces : they behave as if and as if . For we get the asymptotic behaviour up to a logarithmic factor.     

Ground states for a nonhomogeneous elliptic system involving Hardy–Sobolev critical exponents

This paper studies the following nonhomogeneous elliptic system involving Hardy–Sobolev critical exponents where , Ω is a C ¹ open bounded domain in containing the origin, and . The existence result of positive ground state solution is established.     

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.     

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 .     

The essential spectrum of a singular Sturm–Liouville operator

In this paper we study the essential spectrum of the operator where is a positive absolutely continuous function on (0, 1) that resembles for some . We prove that the essential spectrum of coincides with the essential spectrum of the operator .     

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 .     

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 .     

Well‐posedness of fractional degenerate differential equations with finite delay on vector‐valued functional spaces

We study the well‐posedness of the fractional degenerate differential equations with finite delay on Lebesgue–Bochner spaces , periodic Besov spaces and periodic Triebel–Lizorkin spaces , where A and M are closed linear operators on a Banach space X satisfying , F is a bounded linear operator from (resp. and ) into X, where is given by when and . Using known operator‐valued Fourier multiplier theorems, we give necessary or sufficient conditions for the well‐posedness of in the above three function spaces.     

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 .     

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.     

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.     

Existence of solutions to a non‐variational singular elliptic system with unbounded weights

In this paper we prove an existence result for the following singular elliptic system where Ω is a bounded open set in (), is the p‐laplacian operator, and are suitable Lebesgue functions and , , are positive parameters satisfying suitable assumptions.     

Boundedness and compactness of Hardy operators on Lorentz‐type spaces

Given non‐negative measurable functions on we study the high dimensional Hardy operator between Orlicz–Lorentz spaces , where f is a measurable function of and is the ball of radius t in . We give sufficient conditions of boundedness of and . We investigate also boundedness and compactness of between weighted and classical Lorentz spaces. The function spaces considered here do not need to be Banach spaces. Specifying the weights and the Orlicz functions we recover the existing results as well as we obtain new results in the new and old settings.     

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.     

Upper bound for the ratios of eigenvalues of Schrödinger operators with nonnegative single‐barrier potentials

In this paper we prove the optimal upper bound for one‐dimensional Schrödinger operators with a nonnegative differentiable and single‐barrier potential , such that , where . In particular, if satisfies the additional condition , then for . For this result, we develop a new approach to study the monotonicity of the modified Prüfer angle function.     

The canonical central exact sequence for locally compact quantum groups

For a locally compact quantum group we define its center, , and its quantum group of inner automorphisms, . We show that one obtains a natural isomorphism between and , we characterize normal quantum subgroups of a compact quantum group as those left invariant by the action of the quantum group of inner automorphisms and discuss several examples.     

Isomorphic classification of mixed sequence spaces and of Besov spaces over [0, 1]d

The aim of this paper is to establish the isomorphic classification of Besov spaces over [0, 1]^d . Using the identification of the Besov space with the ‐infinite direct sum of finite‐dimensional spaces (which holds independently of the dimension and of the smoothness degree of the space ) we show that , , is a family of mutually non‐isomorphic spaces. The only exception is the isomorphism between the spaces and , which follows from Pełczyński's isomorphism between and . We also tell apart the isomorphic classes of spaces from the isomorphic classes of Besov spaces over the Euclidean space .     

Boundedness in a fully parabolic chemotaxis‐consumption system with nonlinear diffusion and sensitivity,and logistic source

In this paper we study the zero‐flux chemotaxis‐system Ω being a convex smooth and bounded domain of , , and where , and . For any the chemotactic sensitivity function is assumed to behave as the prototype , with and . We prove that for nonnegative and sufficiently regular initial data and , the corresponding initial‐boundary value problem admits a unique globally bounded classical solution provided μ is large enough.     

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.