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. .     

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 .     

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 .     

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.     

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.     

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      

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 the ‐action on G‐semistable locally free sheaves on

We show that the divisor of jumping lines of any , the moduli space of Gieseker‐semistable locally free sheaves of rank 2 on with , is reduced for . By a lemma of Artamkin this implies, that there are exactly ‐orbits in , the subset of those , which are trivial at a certain line .     

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.     

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.     

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 .     

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.     

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.     

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 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.     

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.     

A strict topology on Orlicz spaces

Let φ be a Young function, Ω be a locally compact space, and μ be a positive Radon measure on Ω. We consider a strict topology (in the sense of Sentilles‐Taylor) on the Orlicz function space and investigate various properties of this locally convex topology. We also study the Orlicz space of a locally compact group G with a left Haar measure under the strict topology and certain other natural locally convex topologies. Finally we present some results on various continuity properties of convolution operators on under the topology and other natural ones.