Riesz transforms of the Hodge–de Rham Laplacian on Riemannian manifolds

Let M be a complete non‐compact Riemannian manifold satisfying the volume doubling property. Let be the Hodge–de Rham Laplacian acting on 1‐differential forms. According to the Bochner formula, where and are respectively the positive and negative part of the Ricci curvature and ? is the Levi–Civita connection. We study the boundedness of the Riesz transform from to and of the Riesz transform from to . We prove that, if the heat kernel on functions satisfies a Gaussian upper bound and if the negative part of the Ricci curvature is ε‐sub‐critical for some , then is bounded from to and is bounded from to for where depends on ε and on a constant appearing in the volume doubling property. A duality argument gives the boundedness of the Riesz transform from to for where Δ is the non‐negative Laplace–Beltrami operator. We also give a condition on to be ε‐sub‐critical under both analytic and geometric assumptions.     

Weylness of 2 × 2 operator matrices

Let and be complex separable infinite‐dimensional Hilbert spaces. Given the operators and , we define where is an unknown element. In this paper, a necessary and sufficient condition is given for to be a right Weyl (left Weyl, or Weyl) operator for some . Moreover, some relevant properties and illustrating examples are also given.     

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 .     

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.     

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 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.     

New gap results on the 4‐dimensional sphere

A result showed by M. Gursky in 4 ensures that any metric g on the 4‐dimensional sphere satisfying and is isometric to the round metric. In this note, we prove that there exists a universal number i ₀ such that any metric g on the 4‐dimensional sphere satisfying and is isometric to the round metric. Moreover, there exists a universal such that any metric g on the 4‐dimensional sphere with nonnegative sectional curvature, and is isometric to the round metric. This last result slightly improves a rigidity theorem also proved in 4 .     

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 .     

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.     

A symmetry property for polyharmonic functions vanishing on equidistant hyperplanes

Let be a polyharmonic function of order N defined on the strip satisfying the growth condition (0.1) for and any compact subinterval K of , and suppose that vanishes on equidistant hyperplanes of the form for and Then it is shown that is odd at t ₀, i.e. that for . The second main result states that u is identically zero provided that u satisfies 0.1 and vanishes on 2N equidistant hyperplanes with distance c .     

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.     

Optimizers for the singular Trudinger–Moser inequalities in the critical case in

The main purpose of this paper is to study the existence of extremal functions for the singular Trudinger–Moser inequalities in the critical case in . More precisely, let and denote then we will prove in this article that there exists such that can be achieved for all , .     

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.     

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 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 closed Lie ideals of certain tensor products of ‐algebras

For a simple ‐algebra A and any other ‐algebra B, it is proved that every closed ideal of is a product ideal if either A is exact or B is nuclear. Closed commutator of a closed ideal in a Banach algebra whose every closed ideal possesses a quasi‐central approximate identity is described in terms of the commutator of the Banach algebra. If α is either the Haagerup norm, the operator space projective norm or the ‐minimal norm, then this allows us to identify all closed Lie ideals of , where A and B are simple, unital ‐algebras with one of them admitting no tracial functionals, and to deduce that every non‐central closed Lie ideal of contains the product ideal . Closed Lie ideals of are also determined, A being any simple unital ‐algebra with at most one tracial state and X any compact Hausdorff space. And, it is shown that closed Lie ideals of are precisely the product ideals, where A is any unital ‐algebra and α any completely positive uniform tensor norm.     

Bifurcation properties for a class of fractional Laplacian equations in

This paper concerns with the study of some bifurcation properties for the following class of nonlocal problems (P) where , , , is a positive continuous function, is a bounded continuous function and is the fractional Laplacian. The main tools used are the Leray–Shauder degree theory and the global bifurcation result due to Rabinowitz.     

Characterisation of zero trace functions in variable exponent Sobolev spaces

It is well known that if u belongs to the Sobolev space , where Ω is an open subset of and , then if belongs to weak , where dist . Results of this type are given here for Sobolev spaces with a variable exponent p , under the conditions that Ω is bounded and satisfies a mild regularity condition, and p is a bounded, log‐Hölder continuous function that is bounded away from 1. The outcome includes theorems that are new even when p is constant. In particular it is shown that if and only if and .