Hardy and Cowling–Price theorems associated with the Jacobi–Cherednik operator

This paper is devoted to the proof of Hardy and Cowling–Price type theorems for the Fourier transform tied to the Jacobi–Cherednik operator.     

Hilbert transform and related topics associated with the differential Jacobi operator on (0, +∞)

We define Hilbert transform and conjugate Poisson integrals associated with the Jacobi differential operator on (0, +∞). We prove that these operators are bounded in the appropriate Lebesgue spaces L ^p , 1 < p < +∞. In this study, the tools used are the Littlewood–Paley g-functions associated with the Poisson semigroup and the supplementary Poisson semigroup which we introduce in this paper.     

The Schatten–von Neumann class associated with the Gabor–Riemann–Liouville operator

Periodica Mathematica Hungarica - In this paper, we define the localization operator associated with the Riemann–Liouville operator, and show that it is not only bounded, but it is also in...     

Second order Hamilton–Jacobi–Bellman equations with an unbounded operator

This work is devoted to the study of a class of Hamilton–Jacobi–Bellman equations associated to an optimal control problem where the state equation is a stochastic differential inclusion with a maximal monotone operator. We show that the value function minimizing a Bolza-type cost functional is a viscosity solution of the HJB equation. The proof is based on the perturbation of the initial problem by approximating the unbounded operator. Finally, by providing a comparison principle we are able to show that the solution of the equation is unique.     

q–differ-integral operator on p–valent functions associated with operator on Hilbert space

Making use of multivalent functions with negative coefficients of the type f (z)=z^p-^∑∞_k=p+1a_kz^k,which are analytic in the open unit disk and applying the q-derivative a q–differintegral operator is considered.Furthermore by using the familiar Riesz-Dunford integral,a linear operator on Hilbert space H is introduced.A new subclass of p-valent functions related to an operator on H is defined.Coefficient estimate,distortion bound and extreme po...     

Lipschitz conditions for the generalized discrete Fourier transform associated with the Jacobi operator on [0,π]

Our aim in this paper is to prove an analog of the classical Titchmarsh theorem on the image under the discrete Fourier–Jacobi transform of a set of functions satisfying a generalized Lipschitz condition in the space ${\mathbb{L}}_2^{(\alpha ,\beta )}$.     

Spectral asymptotics for the Sturm—Liouville operator with point interaction

For the Sturm-Liouville operator with point interaction, weak asymptotics of the discrete spectrum are found. A class of operators for which zero is the unique spectrum accumulation point is specified.     

Spectral structure of the Neumann–Poincaré operator on tori

We address the question whether there is a three-dimensional bounded domain such that the Neumann–Poincaré operator defined on its boundary has infinitely many negative eigenvalues. It is proved in this paper that tori have such a property. It is done by decomposing the Neumann–Poincaré operator on tori into infinitely many self-adjoint compact operators on a Hilbert space defined on the circle using the toroidal coordinate system and the Fourier basis, and then by proving that the numerical range of infinitely many operators in the decomposition has both positive and negative values.     

Potential operators associated with Jacobi and Fourier–Bessel expansions

We study potential operators (Riesz and Bessel potentials) associated with classical Jacobi and Fourier–Bessel expansions. We prove sharp estimates for the corresponding potential kernels. Then we characterize those 1≤p,q≤∞$1\le p,q\le \infty$, for which the potential operators are of strong type (p,q)$(p,q)$, of weak type (p,q)$(p,q)$ and of restricted weak type (p,q)$(p,q)$. These results may be thought of as analogues of the celebrated Hardy–Littlewood–Sobolev fractional integration theorem in the Jacobi and Fourier–Bessel settings. As an ingredient of our line of reasoning, we also obtain sharp estimates of the Poisson kernel related to Fourier–Bessel expansions.     

Spectral bounds for the Neumann-Poincaré operator on planar domains with corners

The boundary double layer potential, or the Neumann-Poincaré operator, is studied on the Sobolev space of order 1/2 along the boundary, coinciding with the space of charges giving rise to double layer potentials with finite energy in the whole space. Poincaré’s program of studying the spectrum of the boundary double layer potential is developed in complete generality on closed Lipschitz hypersurfaces in euclidean space. Furthermore, the Neumann-Poincaré operator is realized as a singular integral transform bearing similarities to the Beurling-Ahlfors transform in 2 dimensions. As an application, in the case of planar curves with corners, bounds for the spectrum of the Neumann-Poincaré operator are derived from recent results in quasi-conformal mapping theory.     

Qualitative properties of solutions of an integral equation associated with the Benjamin–Ono–Zakharov–Kuznetsov operator

We study qualitative properties of solutions of an integral equation associated the Benjamin–Ono–Zakharov–Kuznetsov operator. We establish the regularity of the positive solutions without the assumption of being in fractional Sobolev–Liouville spaces. Moreover we show that the solutions are axially symmetric. Furthermore we establish Lipschitz continuity and the decay rate of the solutions.