A compactness criterion and application to the commutators associated with Schrödinger operators

Let T be an integral operator. In this paper, we introduce a ‐compactness criterion of , where . As an application, we apply this criterion to deal with ‐compactness of commutators associated to Schrödinger operators with potentials in the reverse Hölder's class.     

On the order of the automorphism group of foliations

Let be a holomorphic foliation with ample canonical bundle on a smooth projective surface X. We obtain an upper bound on the order of its automorphism group which depends only on and provided this group is finite. Here, and are the canonical bundles of and X, respectively.     

Anti‐periodic solutions for a gradient system with resonance via a variational approach

In this paper, we investigate a second‐order resonance anti‐periodic boundary value problem where is the m‐th eigenvalue of the corresponding eigenvalue problem. By using the dual least action principle, we obtain an existence result. In addition, we obtain the existence of 2T‐periodic solutions for .     

Extremal solutions to fourth order discontinuous functional boundary value problems

In this paper, given a L¹‐Carathéodory function, it is considered the functional fourth order equation together with the nonlinear functional boundary conditions Here , , satisfy some adequate monotonicity assumptions and are not necessarily continuous functions. It will be proved an existence and location result in presence of non ordered lower and upper solutions.     

Operators of Hadamard type on spaces of smooth functions

For an open set we study the algebra of continuous linear operators on admitting the monomials as eigenvectors. We give a concrete representation of these operators and evaluate it explicitly for the unit ball and the whole of . We also study the topology of and the algebra of eigenvalue sequences.     

Regularity theory for a class of quasilinear equations

This paper addresses the analysis of the weak solution of in a bounded domain Ω subject to the boundary condition on , when the data f belongs to and . We prove existence and uniqueness of solution for this problem in the Nikolskii space . Moreover, we obtain energy estimates regarding the Nikolskii norm of ω in terms of the norm of f.     

Volume estimates for sections of certain convex bodies

This paper deals with volume estimates for hyperplane sections of the simplex and for m‐codimensional sections of powers of m‐dimensional Euclidean balls. In the first part we consider sections through the centroid of the n‐dimensional regular simplex. We state a volume formula and give a lower bound for the volume of sections through the centroid. In the second part we study the extremal volumes of m‐codimensional sections “perpendicular” to of unit balls in the space for all . We give volume formulas and use them to show that the normal vector (1, 0, …, 0) yields the minimal volume. Furthermore we give an upper bound for the ‐dimensional volumes for natural numbers . This bound is asymptotically attained for the normal vector as .     

Determinants of classical SG‐pseudodifferential operators

We introduce a generalized trace functional TR in the spirit of Kontsevich and Vishik's canonical trace for classical SG‐pseudodifferential operators on and suitable manifolds, using a finite‐part integral regularization technique. This allows us to define a zeta‐regularized determinant for parameter‐elliptic operators , , . For , the asymptotics of as and of as are derived. For suitable pairs we show that coincides with the so‐called relative determinant .     

On unique continuation for Schrödinger operators of fractional and higher orders

In this note we study the property of unique continuation for solutions of , where V is in a function class of potentials including for . In particular, when , our result gives a unique continuation theorem for the fractional Schrödinger operator in the full range of α values.     

Fractional integration operators of variable order: continuity and compactness properties

Let be a Lebesgue‐almost everywhere positive function. We consider the Riemann‐Liouville operator of variable order defined by as an operator from to . Our first aim is to study its continuity properties. For example, we show that is always bounded (continuous) in provided that . Surprisingly, this becomes false for . In order to be bounded in L₁[0, 1], the function has to satisfy some additional assumptions. In the second, central part of this paper we investigate compactness properties of . We characterize functions for which is a compact operator and for certain classes of functions we provide order‐optimal bounds for the dyadic entropy numbers .     

Oscillation criteria for higher order nonlinear dynamic equations

In this paper, we will consider the higher‐order functional dynamic equations of the form on an above‐unbounded time scale , where and , . The function is a rd‐continuous function such that . The results extend and improve some known results in the literature on higher order nonlinear dynamic equations.     

Smoothness of the Orlicz norm in Musielak–Orlicz function spaces

In this paper, we present a characterization of support functionals and smooth points in , the Musielak–Orlicz space equipped with the Orlicz norm. As a result, criterion for the smoothness of is also obtained. Some expressions involving the norms of functionals in , the topological dual of , are proved for arbitrary Musielak–Orlicz functions.     

Some properties of a class of symmetric functions and its applications

For , the symmetric functions are defined by where , and are non‐negative integers. In this paper, the Schur convexity, geometric Schur convexity and harmonic Schur convexity of are investigated. As applications, Schur convexity for the other symmetric functions is obtained by a bijective transformation of independent variable for a Schur convex function, some analytic and geometric inequalities are established by using the theory of majorization, in particular, we derive from our results a generalization of Sharpiro's inequality, and give a new generalization of Safta's conjecture in the n‐dimensional space and others.     

Lorentz spaces with variable exponents

We introduce Lorentz spaces and with variable exponents. We prove several basic properties of these spaces including embeddings and the identity . We also show that these spaces arise through real interpolation between and . Furthermore, we answer in a negative way the question posed in 12 whether the Marcinkiewicz interpolation theorem holds in the frame of Lebesgue spaces with variable integrability.     

On an elementary operator with M‐hyponormal operator entries

A Hilbert space operator is M‐hyponormal if there exists a positive real number M such that for all . Let be M‐hyponormal and let denote either the generalized derivation or the elementary operator . We prove that if are M‐hyponormal, then satisfies the generalized Weyl's theorem and satisfies the generalized a‐Weyl's theorem for every f that is analytic on a neighborhood of .     

Bilinear decompositions for the product space

In this paper, we improve a recent result by Li and Peng on products of functions in and , where is a Schrödinger operator with V satisfying an appropriate reverse Hölder inequality. More precisely, we prove that such products may be written as the sum of two continuous bilinear operators, one from into , the other one from into , where the space is the set of distributions f whose grand maximal function satisfies      

Riemann boundary value problem for harmonic functions in Clifford analysis

In this article, we mainly deal with the boundary value problem for harmonic function with values in Clifford algebra: where is a Liapunov surface in , the Dirac operator , are unknown functions with values in a universal Clifford algebra Under some assumptions, we show that the boundary value problem is solvable.     

Subgroups and homology of extensions of centralizers of pro‐p groups

We study the growth of , where U is an open subgroup of and is a special class of pro‐p groups defined in 7 . Furthermore for non‐abelian we prove the core property: for pro‐p subgroups such that H is finitely generated and N is non‐trivial normal in G the index is always finite.     

Agmon‐type estimates for a class of jump processes

In the limit we analyse the generators of families of reversible jump processes in associated with a class of symmetric non‐local Dirichlet‐forms and show exponential decay of the eigenfunctions. The exponential rate function is a Finsler distance, given as solution of a certain eikonal equation. Fine results are sensitive to the rate function being or just Lipschitz. Our estimates are analogous to the semiclassical Agmon estimates for differential operators of second order. They generalize and strengthen previous results on the lattice . Although our final interest is in the (sub)stochastic jump process, technically this is a pure analysis paper, inspired by PDE techniques.