Spectrum of a non-self-adjoint operator associated with the periodic heat equation

We study the spectrum of the linear operator L=−θ_∂−?θ_∂(sinθθ_∂) subject to the periodic boundary conditions on θ∈[−π,π]. We prove that the operator is closed in with the domain in for |?|<2, its spectrum consists of an infinite sequence of isolated eigenvalues and the set of corresponding eigenfunctions is complete. By using numerical approximations of eigenvalues and eigenfunctions, we show that all eigenvalues are simple, located on the imaginary axis and the angle between two subsequent eigenfunctions tends to zero for larger eigenvalues. As a result, the complete set of linearly independent eigenfunctions does not form a basis in .     

Resolvents of operators and partial functional differential equations with nonautonomous past

To a backward evolution family on a Banach space X we associate an abstract differential operator G through the integral equation on a Banach space of X-valued functions on . We compute the resolvent of the restriction of this operator to a smaller domain to obtain a generator. We then apply the results to prove existence, exponential stability and exponential dichotomy of solutions to partial functional equations with nonautonomous past as discussed in [S. Brendle, R. Nagel, Dist. Contin. Dynam. Systems 8 (2002) 953-966]. Our main tools are spectral mapping theorems for evolution semigroups and hyperbolicity criteria.     

The exact solution of a linear integral equation with weakly singular kernel

A space , which is proved to be a reproducing kernel space with simple reproducing kernel, is defined. The expression of its reproducing kernel function is given. Subsequently, a class of linear Volterra integral equation (VIE) with weakly singular kernel is discussed in the new reproducing kernel space. The reproducing kernel method of linear operator equation Au=f, which request the image space of operator A is and operator A is bounded, is improved. Namely, the request for the image space is weakened to be L²[a,b], and the boundedness of operator A is also not required. As a result, the exact solution of the equation is obtained. The numerical experiments show the efficiency of our method.     

On uniqueness for nonlinear elliptic equation involving the Pucci's extremal operator

In this article we study uniqueness of positive solutions for the nonlinear uniformly elliptic equation in RN, limr→∞u(r)=0, where denotes the Pucci's extremal operator with parameters 0<λ?Λ and p>1. It is known that all positive solutions of this equation are radially symmetric with respect to a point in RN, so the problem reduces to the study of a radial version of this equation. However, this is still a nontrivial question even in the case of the Laplacian (λ=Λ). The Pucci's operator is a prototype of a nonlinear operator in no-divergence form. This feature makes the uniqueness question specially challenging, since two standard tools like Pohozaev identity and global integration by parts are no longer available. The corresponding equation involving is also considered.     

The Cauchy process and the Steklov problem

Let X_t be a Cauchy process in . We investigate some of the fine spectral theoretic properties of the semigroup of this process killed upon leaving a domain D. We establish a connection between the semigroup of this process and a mixed boundary value problem for the Laplacian in one dimension higher, known as the “Mixed Steklov Problem.” Using this we derive a variational characterization for the eigenvalues of the Cauchy process in D. This characterization leads to many detailed properties of the eigenvalues and eigenfunctions for the Cauchy process inspired by those for Brownian motion. Our results are new even in the simplest geometric setting of the interval (−1,1) where we obtain more precise information on the size of the second and third eigenvalues and on the geometry of their corresponding eigenfunctions. Such results, although trivial for the Laplacian, take considerable work to prove for the Cauchy processes and remain open for general symmetric α-stable processes. Along the way we present other general properties of the eigenfunctions, such as real analyticity, which even though well known in the case of the Laplacian, are not available for more general symmetric α-stable processes.     

Normalized system of functions with respect to the Laplace operator and its applications

A system of functions 0-normalized with respect to the operator Δ in some domain is constructed. Application of this system to boundary value problems for the polyharmonic equation is considered. Connection between harmonic functions and solutions of the Helmholtz equation is investigated.     

Some subclasses of analytic functions involving the generalized Noor integral operator

In this paper, we define the generalized Noor integral operator by using convolution. By applying this operator, we introduce some subclasses , and of analytic functions and study their subordinate relations, inclusion relations, the integral operator, the sufficient conditions for a function to be in the class and the radius problems.     

The Cauchy operator for basic hypergeometric series

We introduce the Cauchy augmentation operator for basic hypergeometric series. Heine's transformation formula and Sears' transformation formula can be easily obtained by the symmetric property of some parameters in operator identities. The Cauchy operator involves two parameters, and it can be considered as a generalization of the operator T(bD_q). Using this operator, we obtain extensions of the Askey–Wilson integral, the Askey–Roy integral, Sears' two-term summation formula, as well as the q-analogs of Barnes' lemmas. Finally, we find that the Cauchy operator is also suitable for the study of the bivariate Rogers–Szegö polynomials, or the continuous big q-Hermite polynomials.     

Approximation of the semigroup generated by the Robin Laplacian in terms of the Gaussian semigroup

We consider the Laplacian ΔR subject to Robin boundary conditions on the space , where Ω is a smooth, bounded, open subset of RN. It is known that ΔR generates an analytic contraction semigroup. We show how this semigroup can be obtained from the Gaussian semigroup on C₀(RN) via a Trotter formula. As the main ingredient, we construct a positive, contractive, linear extension operator Eβ from to C₀(RN) which maps an operator core for ΔR into the domain of the generator of the Gaussian semigroup.     

On fractional maximal function and fractional integral on the Laguerre hypergroup

Let be the Laguerre hypergroup which is the fundamental manifold of the radial function space for the Heisenberg group. In this paper we obtain necessary and sufficient conditions on the parameters for the boundedness of the fractional maximal operator and the fractional integral operator on the Laguerre hypergroup from the spaces to the spaces and from the spaces to the weak spaces .     

Wythoff polytopes and low-dimensional homology of Mathieu groups

We describe two methods for computing the low-dimensional integral homology of the Mathieu simple groups and use them to make computations such as and . One method works via Sylow subgroups. The other method uses a Wythoff polytope and perturbation techniques to produce an explicit free -resolution. Both methods apply in principle to arbitrary finite groups.     

Transmission problems and spectral theory for singular integral operators on Lipschitz domains

We prove the well-posedness of the transmission problem for the Laplacian across a Lipschitz interface, with optimal non-tangential maximal function estimates, for data in Lebesgue and Hardy spaces on the boundary. As a corollary, we show that the spectral radius of the (adjoint) harmonic double layer potential K∗ in is less than , whenever is a bounded convex domain and 1<p?2.     

Hardy space estimates for the wave equation on compact Lie groups

Let −L be the Laplacian. In this paper, we prove that on a compact Lie group G of dimension n, the multiplier operator , s∈(0,1], extends to a bounded operator on the Hardy space Hp(G), 0<p<∞, if and only if . The result is an analogue of a well-known theorem in Euclidean space.     

Positive integral operators in unbounded domains

We study positive integral operators in with continuous kernel k(x,y). We show that if the operator is compact and Hilbert-Schmidt. If in addition k(x,x)→0 as |x|→∞, k is represented by an absolutely and uniformly convergent bilinear series of uniformly continuous eigenfunctions and is trace class. Replacing the first assumption by the stronger then and the bilinear series converges also in L¹. Sharp norm bounds are obtained and Mercer's theorem is derived as a special case.     

Mixed means over balls and annuli and lower bounds for operator norms of maximal functions

In this paper we prove mixed-means inequalities for integral power means of an arbitrary real order, where one of the means is taken over the ball , centered at and of radius , δ>0. Therefrom we deduce the corresponding Hardy-type inequality, that is, the operator norm of the operator S_δ which averages over , introduced by Christ and Grafakos in Proc. Amer. Math. Soc. 123 (1995) 1687-1693. We also obtain the operator norm of the related limiting geometric mean operator, that is, Carleman or Levin-Cochran-Lee-type inequality. Moreover, we indicate analogous results for annuli and discuss estimations related to the Hardy-Littlewood and spherical maximal functions.