Heat kernel estimates for the Bessel differential operator in half‐line

In the paper we consider the Bessel differential operator in half‐line , , and its Dirichlet heat kernel . For , by combining analytical and probabilistic methods, we provide sharp two‐sided estimates of the heat kernel for the whole range of the space parameters and every , which complements the recent results given in 1 , where the case was considered.     

Maximal regularity for non‐autonomous Robin boundary conditions

We consider a non‐autonomous Cauchy problem where is associated with the form , where V and H are Hilbert spaces such that V is continuously and densely embedded in H. We prove H‐maximal regularity, i.e., the weak solution u is actually in (if and ) under a new regularity condition on the form with respect to time; namely Hölder continuity with values in an interpolation space. This result is best suited to treat Robin boundary conditions. The maximal regularity allows one to use fixed point arguments to some non linear parabolic problems with Robin boundary conditions.     

Inverse bifurcation problems for diffusive Holling–Tanner population model

We consider the equation which is called Holling–Tanner population model where is a bifurcation parameter and are unknown constants. In this paper, we determine the unknown constants from the asymptotic behavior of the bifurcation curve , where .     

Sobolev estimates for (pseudo)‐differential operators and applications

In this work we show that if is a linear differential operator of order ν with smooth complex coefficients in from a complex vector space E to a complex vector space F, the Sobolev a priori estimate holds locally at any point if and only if is elliptic and the constant coefficient homogeneous operator is canceling in the sense of Van Schaftingen for every which means that Here is the homogeneous part of order ν of and is the principal symbol of . This result implies and unifies the proofs of several estimates for complexes and pseudo‐complexes of operators of order one or higher proved recently by other methods as well as it extends —in the local setup— the characterization of Van Schaftingen to operators with variable coefficients.     

Positive solutions of nonlinear Schrödinger equation with peaks on a Clifford torus

We prove the existence of large energy positive solutions for a stationary nonlinear Schrödinger equation with peaks on a Clifford type torus. Here where with for all Each is a function and is defined by the generalized notion of spherical coordinates. The solutions are obtained by a or a process.     

On ‐null sequences and their relatives

Let and , where is the conjugate index of p. We prove an omnibus theorem, which provides numerous equivalences for a sequence in a Banach space X to be a ‐null sequence. One of them is that is ‐null if and only if is null and relatively ‐compact. This equivalence is known in the “limit” case when , the case of the p‐null sequence and p‐compactness. Our approach is more direct and easier than those applied for the proof of the latter result. We apply it also to characterize the unconditional and weak versions of ‐null sequences.     

On index theory for non‐Fredholm operators: A (1 + 1)‐dimensional example

Using the general formalism of 12 , a study of index theory for non‐Fredholm operators was initiated in 9 . Natural examples arise from (1 + 1)‐dimensional differential operators using the model operator in of the type , where , and the family of self‐adjoint operators in studied here is explicitly given by Here has to be integrable on and tends to zero as and to 1 as (both functions are subject to additional hypotheses). In particular, , , has asymptotes (in the norm resolvent sense) as , respectively. The interesting feature is that violates the relative trace class condition introduced in 9 , Hypothesis 2.1 ]. A new approach adapted to differential operators of this kind is given here using an approximation technique. The approximants do fit the framework of 9 enabling the following results to be obtained. Introducing , , we recall that the resolvent regularized Witten index of , denoted by , is defined by whenever this limit exists. In the concrete example at hand, we prove Here denotes the spectral shift operator for the pair of self‐adjoint operators , and we employ the normalization, , .     

‐bounded ‐calculus for sectorial operators via generalized Gaussian estimates

We show that, for negative generators of analytic semigroups, a bounded ‐calculus self‐improves to an ‐bounded ‐calculus in an appropriate scale of ‐spaces if the semigroup satisfies suitable generalized Gaussian estimates. As application of our result we obtain that large classes of differential operators have an ‐bounded ‐calculus.     

On some structural sets and a quaternionic ‐hyperholomorphic function theory

Quaternionic analysis is regarded as a broadly accepted branch of classical analysis referring to many different types of extensions of the Cauchy‐Riemann equations to the quaternion skew field . It relies heavily on results on functions defined on domains in or with values in . This theory is centred around the concept of ψ‐hyperholomorphic functions related to a so‐called structural set ψ of or respectively. The main goal of this paper is to develop the nucleus of the ‐hyperholomorphic function theory, i.e., simultaneous null solutions of two Cauchy‐Riemann operators associated to a pair of structural sets of . Following a matrix approach, a generalized Borel‐Pompeiu formula and the corresponding Plemelj‐Sokhotzki formulae are established.     

Exponential convergence in ‐Wasserstein distance for diffusion processes without uniformly dissipative drift

By adopting the coupling by reflection and choosing an auxiliary function which is convex near infinity, we establish the exponential convergence of diffusion semigroups with respect to the standard ‐Wasserstein distance for all . In particular, we show that for the Itô stochastic differential equation if the drift term b is such that for any , holds with some positive constants K₁, K₂ and , then there is a constant such that for all , and , where is a positive constant. This improves the main result in 14 where the exponential convergence is only proved for the L¹‐Wasserstein distance.     

Boundedness in chemotaxis systems with rotational flux terms

We consider the chemotaxis system with rotation under no‐flux boundary conditions in the bounded domain , . Here the matrix‐valued function fulfills () for all with some nondecreasing function S₀ and is a nonnegative function with for all . Moreover, f satisfies for all with nondecreasing function f₀. It is shown that for the nonnegative initial data and with , if at least one of the following assumptions holds:

• ,
• , and ,
• ,

then the corresponding initial‐boundary value problem possesses a unique global classical solution that is uniformly bounded.     

Spectrality of certain self‐affine measures on the generalized spatial Sierpinski gasket

The self‐affine measure corresponding to a upper or lower triangle expanding matrix M and the digit set in the space is supported on the generalized spatial Sierpinski gasket, where are the standard basis of unit column vectors in . We consider in this paper the existence of orthogonal exponentials on the Hilbert space , i.e., the spectrality of . Such a property is directly connected with the entries of M and is not completely determined. For this generalized spatial Sierpinski gasket, we present a method to deal with the spectrality or non‐spectrality of . As an application, the spectral property of a class of such self‐affine measures are clarified. The results here generalize the corresponding results in a simple manner.     

Infinitely many homoclinic solutions for a second‐order Hamiltonian system

In this paper, we study the homoclinic solutions of the following second‐order Hamiltonian system where , and . Applying the symmetric Mountain Pass Theorem, we establish a couple of sufficient conditions on the existence of infinitely many homoclinic solutions. Our results significantly generalize and improve related ones in the literature. For example, is not necessary to be uniformly positive definite or coercive; through is still assumed to be superquadratic near , it is not assumed to be superquadratic near .     

‐maximal regularity for fractional difference equations on UMD spaces

Let T be a bounded linear operator defined on a Banach space X. We introduce an operator theoretical method for linear fractional difference equations based on the notion of α‐resolvent sequence of bounded and linear operators. Then, we define and characterize the ‐ maximal regularity of solutions for the problem solely in terms of the R‐boundedness of the set      

Circular symmetrization,subordination and arclength problems on convex functions

We study the class of univalent analytic functions f in the unit disk of the form satisfying where Ω will be a proper subdomain of which is starlike with respect to . Let be the unique conformal mapping of onto Ω with and and . Let denote the arclength of the image of the circle , . The first result in this paper is an inequality for , which solves the general extremal problem , and contains many other well‐known results of the previous authors as special cases. Other results of this article cover another set of related problems about integral means in the general setting of the class .     

On a ‐Kirchhoff equation with critical exponent and an additional nonlocal term via truncation argument

We study existence and multiplicity of solutions of the following nonlocal ‐Kirchhoff equation with critical exponent, via truncation argument on the Sobolev space with variable exponent, where Ω is a bounded smooth domain of , , M, f are continuous functions, , and are real parameter.     

Weak Musielak–Orlicz Hardy spaces and applications

Let satisfy that , for any given , is an Orlicz function and is a Muckenhoupt weight uniformly in . In this article, the authors introduce the weak Musielak–Orlicz Hardy space via the grand maximal function and then obtain its vertical or its non–tangential maximal function characterizations. The authors also establish other real‐variable characterizations of , respectively, in terms of the atom, the molecule, the Lusin area function, the Littlewood–Paley g‐function or ‐function. All these characterizations for weighted weak Hardy spaces (namely, and with and ) are new and part of these characterizations even for weak Hardy spaces (namely, and with ) are also new. As an application, the boundedness of Calderón–Zygmund operators from to in the critical case is presented.     

On the geometry of complex ‐spaces

It is known that applying an ‐homothetic deformation to a complex contact manifold whose vertical space is annihilated by the curvature yields a condition which is invariant under ‐homothetic deformations. A complex contact manifold satisfying this condition is said to be a complex ‐space. In this paper, we deal with the questions of Bochner, conformal and conharmonic flatness of complex ‐spaces when , and prove that such kind of spaces cannot be Bochner flat, conformally flat or conharmonically flat.     

A Mehta–Ramanathan theorem for linear systems with basepoints

Let be a normal complex projective polarized variety and an H‐semistable sheaf on X. We prove that the restriction to a sufficiently positive general complete intersection curve passing through a prescribed finite set of points remains semistable, provided that at each , the variety X is smooth and the factors of a Jordan–Hölder filtration of are locally free. As an application, we obtain a generalization of Miyaoka's generic semipositivity theorem.     

Eigenvalues of the k‐th power of a graph

The k‐th power of a graph G, denoted by , is a graph with the same set of vertices as G such that two vertices are adjacent in if and only if their distance in G is at most k. In this paper, we give the bounds on the spectral radius of and . The Nordhaus–Gaddum‐type inequality for the spectral radius of the graph is also presented. Moreover, we obtain an upper bound on the energy of the second power of graphs.