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      

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.     

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.     

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.     

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 .     

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.     

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.     

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 .     

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 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 .     

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.     

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.     

Random attractors for stochastic reaction‐diffusion equations with multiplicative noise in

In this paper, we study the random dynamical system generated by a stochastic reaction‐diffusion equation with multiplicative noise and prove the existence of an ‐random attractor for such a random dynamical system. The nonlinearity f is supposed to satisfy some growth of arbitrary order .     

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      

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.     

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.     

Positive solutions for an elliptic equation in an annulus with a superlinear nonlinearity with zeros

We study existence, multiplicity, and the behavior, with respect to λ, of positive radially symmetric solutions of in annular domains in . The nonlinear term has a superlinear local growth at infinity, is nonnegative, and satisfies for a suitable positive and concave function a. For this, we combine several methods such as the sub and supersolutions method, a priori estimates and degree theory.     

A necessary and sufficient condition for Hardy's operator in

The equivalent conditions are proved for the Hardy inequality to be fulfilled in the norms of a Lebesgue space with variable exponent . It is assumed that the function is monotone and positive on (0, 1).     

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.     

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.