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 .     

Pre‐Hilbert spaces with anomalous splitting and orthogonally‐closed subspace structures

Four classes of closed subspaces of an inner product space S that can naturally replace the lattice of projections in a Hilbert space are: the complete/cocomplete subspaces , the splitting subspaces , the quasi‐splitting subspaces and the orthogonally‐closed subspaces . It is well‐known that in general the algebraic structure of these families differ remarkably and they coalesce if and only if S is a Hilbert space. It is also known that when S is a hyperplane in its completion i.e. then and . On the other extreme, when i.e. then and . Motivated by this and in contrast to it, we show that in general the codimension of S in bears very little relation to the properties of these families. In particular, we show that the equalities and can hold for inner product spaces with arbitrary codimension in . At the end we also contribute to the study of the algebraic structure of by testing it for the Riesz interpolation property. We show that may fail to enjoy the Riesz interpolation property in both extreme situations when S is “very small” (i.e. and when S is ‘very big’ (i.e. .     

Dimension of images of subspaces under mappings in Triebel‐Lizorkin spaces

Let and let be a ‐quasicontinuous representative of a mapping in the Triebel‐Lizorkin space . We find an optimal value of such that for a.e. the Hausdorff dimension of is at most α. We construct examples to show that the value of β is optimal and we show that it does not increase once p goes below the critical value α.     

On convolution in weighted ‐spaces

A distributional generalization of Young's inequality was stated by L. Schwartz. It asserts that convolution yields a continuous bilinear map if A generalization to the weighted ‐spaces in the form was given by N. Ortner and the author in 1989. By means of interpolation theory, we improve this result with respect to the image space under certain restrictions on μ and ν. This implies limit relations in for the Poisson kernel and yields a solution of the Dirichlet problem for the half‐space with boundary values in the space By this we generalize a former result of J. Alvarez, M. Guzmán‐Partida and S. Pérez‐Esteva referring to the special case of      

Riesz minimal energy problems on ‐manifolds

In , , we study the constructive and numerical solution of minimizing the energy relative to the Riesz kernel , where , for the Gauss variational problem, considered for finitely many compact, mutually disjoint, boundaryless ‐dimensional ‐manifolds , , where , each being charged with Borel measures with the sign prescribed. We show that the Gauss variational problem over a convex set of Borel measures can alternatively be formulated as a minimum problem over the corresponding set of surface distributions belonging to the Sobolev–Slobodetski space , where and . An equivalent formulation leads in the case of two manifolds to a nonlinear system of boundary integral equations involving simple layer potential operators on Γ. A corresponding numerical method is based on the Galerkin–Bubnov discretization with piecewise constant boundary elements. Wavelet matrix compression is applied to sparsify the system matrix. Numerical results are presented to illustrate the approach.     

Metrics of Kaluza–Klein type on the anti‐de Sitter space

We introduce and study a new family of pseudo‐Riemannian metrics on the anti‐de Sitter three‐space . These metrics will be called “of Kaluza‐Klein type” , as they are induced in a natural way by the corresponding metrics defined on the tangent sphere bundle . For any choice of three real parameters , the pseudo‐Riemannian manifold is homogeneous. Moreover, we shall introduce and study some natural almost contact and paracontact structures , compatible with , such that is a homogeneous almost contact (respectively, paracontact) metric structure. These structures will be then used to show the existence of a three‐parameter family of homogeneous metric mixed 3‐structures on the anti‐de Sitter three‐space.     

Semi‐classical resolvent estimates and regions free of resonances

We prove resolvent estimates for self‐adjoint operators of the form on , , where is a semi‐classical parameter and , , is a real‐valued potential. The potential is supposed to have very little regularity with respect to the radial variable, only. As a consequence, we obtain a region free of resonances in the case when V is of compact support.     

Precise rates of the first moment convergence in the LIL for NA sequences

Let be a strictly stationary sequence of negatively associated random variables with zero mean and finite variance. We set and , . If , then for any , we show the precise rates of the first moment convergence in the law of the iterated logarithm for a kind of weighted infinite series of and as , and as .     

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.     

Uniqueness results for semilinear elliptic systems on

In this paper we establish uniqueness criteria for positive radially symmetric finite energy solutions of semilinear elliptic systems of the form As an application we consider the nonlinear Schrödinger system for and exponents q which satisfy in case and in case . Generalizing the results of Wei and Yao for we find new sufficient conditions and necessary conditions on such that precisely one positive solution exists. Our results dealing with the special case are optimal. Finally, an application to a multi‐component nonlinear Schrödinger system is given.     

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 .     

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.     

Hyponormality of bounded‐type Toeplitz operators

In this paper we deal with the hyponormality of Toeplitz operators with matrix‐valued symbols. The aim of this paper is to provide a tractable criterion for the hyponormality of bounded‐type Toeplitz operators (i.e., the symbol is a matrix‐valued function such that Φ and are of bounded type). In particular, we get a much simpler criterion for the hyponormality of when the co‐analytic part of the symbol Φ is a left divisor of the analytic part.     

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.     

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      

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.     

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 .     

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.