A Convolution Operator Related to the Generalized Mehler–Fock and Kontorovich–Lebedev Transforms

In this paper we study a generalization of an index integral involving the product of modified Bessel functions and associated Legendre functions. It is applied to a convolution construction associated with this integral, which is related to the classical Kontorovich–Lebedev and generalized Mehler–Fock transforms. Mapping properties and norm estimates in weighted L _p -spaces, 1 ≤ p ≤ 2, are investigated. An application to a class of convolution integral equations is considered. Necessary and sufficient conditions are found for the solvability of these equations in L ₂.     

Lévy–Khintchine Formula and Dual Convolution Semigroups Associated with Laguerre and Bessel Functions

In this work we consider a system of partial differential operators D ₁,D ₂ on K=[0,+[×R, whose eigenfunctions are the functions (x,t), (x,t)K, =((R0)×N)(0×[0,+[), which are related to the Laguerre functions for ((R 0)×N)(0,0) and which are the Bessel functions for (0×[0,+[). We provide K and with a convolution structure. We prove a Lévy–Khintchine formula on K, which permits us to characterize dual convolution semigroups on .     

On Potential Functions Associated with Eigenfunctions of the Discrete Sturm–Liouville Operator

Mathematical Notes - Potential functions associated with eigenfunctions of the discrete Sturm–Liouville operator are studied in a loaded space. On the basis of representations of kernels with...     

Asymptotic Solutions of Viscous Hamilton–Jacobi Equations with Ornstein–Uhlenbeck Operator

We study the long time behavior of solutions of the Cauchy problem for semilinear parabolic equations with the Ornstein–Uhlenbeck operator in ? ^N . The long time behavior in the main results is stated with help of the corresponding to ergodic problem, which complements, in the case of unbounded domains, the recent developments on long time behaviors of solutions of (viscous) Hamilton–Jacobi equations due to Namah (1996 Namah , G. ( 1996 ). Asymptotic solution of a Hamilton–Jacobi equation . Asymptotic Anal. 12 ( 4 ): 355 – 370 . [CSA] [Web of Science ®] , [Google Scholar]), Namah and Roquejoffre (1999 Namah , G. , Roquejoffre , J.-M . ( 1999 ). Remarks on the long-time behavior of the solutions of Hamilton–Jacobi equations . Comm. PDE 24 ( 5–6 ): 883 – 893 . [CSA] [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), Roquejoffre (1998 Roquejoffre , J.-M . ( 1998 ). Comportement asymptotique des solutions d’équations de Hamilton–Jacobi monodimensionnelles . C. R. Acad. Sci. Paris Sér. I Math. 326 ( 2 ): 185 – 189 . [CSA] [Crossref] , [Google Scholar]), Fathi (1998 Fathi , A. ( 1998 ). Sur la convergence du semi-groupe de Lax–Oleinik semigroup . C. R. Acad. Sci. Paris Sér. I Math. 327 ( 3 ): 267 – 270 . [CSA] [Crossref] , [Google Scholar]), Barles and Souganidis (2000 Barles , G. , Souganidis , P. E. ( 2000 ). On the large time behavior of solutions of Hamilton–Jacobi equaitons . SIAM J. Math. Anal. 31 ( 4 ): 925 – 939 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar] 2001 Barles , G. , Souganidis , P. E. ( 2001 ). Space-time periodic solutions and long-time behavior of solutions to quasi-periodic parabolic equations . SIAM J. Math. Anal. 32 ( 6 ): 1311 – 1323 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]). We also establish existence and uniqueness results for solutions of the Cauchy problem and ergodic problem for semilinear parabolic equations with the Ornstein–Uhlenbeck operator.     

Besov and Hardy Spaces Associated with the Schrödinger Operator on the Heisenberg Group

We introduce the Besov space $\dot{B}^{0,L}_{1,1}$ associated with the Schrödinger operator L with a nonnegative potential satisfying a reverse Hölder inequality on the Heisenberg group, and obtain the molecular decomposition. We also develop the Hardy space $H_{L}^{1}$ associated with the Schrödinger operator via the Littlewood–Paley area function and give equivalent characterizations via atoms, molecules, and the maximal function. Moreover, using the molecular decomposition, we prove that $\dot{B}^{0,L}_{1,1}$ is a subspace of $H_{L}^{1}$ .     

$$L^p$$-Analysis of the Hodge–Dirac Operator Associated with Witten Laplacians on Complete Riemannian Manifolds

We prove R-bisectoriality and boundedness of the \(H^\infty \)-functional calculus in \(L^p\) for all \(1<p<\infty \) for the Hodge–Dirac operator associated with Witten Laplacians on complete Riemannian manifolds with non-negative Bakry–Emery Ricci curvature on k-forms.     

An $$L^1$$-Estimate for Certain Spectral Multipliers Associated with the Ornstein–Uhlenbeck Operator

We study a class of spectral multipliers \(\phi (L)\) for the Ornstein–Uhlenbeck operator L arising from the Gaussian measure on \(\mathbb {R}^n\) and find a sufficient condition for integrability of \(\phi (L)f\) in terms of the admissible conical square function and a maximal function.     

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.     

Hahn Difference Operator and Associated Jackson–Nörlund Integrals

This paper is devoted for a rigorous investigation of Hahn’s difference operator and the associated calculus. Hahn’s difference operator generalizes both the difference operator and Jackson’s q-difference operator. Unlike these two operators, the calculus associated with Hahn’s difference operator receives no attention. In particular, its right inverse has not been constructed before. We aim to establish a calculus of differences based on Hahn’s difference operator. We construct a right inverse of Hahn’s operator and study some of its properties. This inverse also generalizes both Nörlund sums and the Jackson q-integrals. We also define families of corresponding exponential and trigonometric functions which satisfy first and second order difference equations, respectively.     

Lagrangian submanifolds and the Hamilton–Jacobi equation

Lagrangian submanifolds are becoming a very essential tool to generalize and geometrically understand results and procedures in the area of mathematical physics. The geometric version of the Hamilton–Jacobi equation in terms of Lagrangian submanifolds enables here some novel interesting applications of the Hamilton–Jacobi equation in holonomic, nonholonomic and time-dependent dynamics from a geometrical point of view.     

Calderon’s Reproducing Formula Related to the Dunkl Operator on the Real Line

 We consider a generalized convolution , , on the real line generated by the Dunkl operator

Calderon’s Reproducing Formula Related to the Dunkl Operator on the Real Line

 We consider a generalized convolution , , on the real line generated by the Dunkl operator

Burchnall–Chaundy polynomials and Dunkl–Darboux operators

Mathematical Notes - Certain properties of Burchnall–Chaundy polynomials are studied. The first two nonzero coefficients following the leading coefficient are calculated in explicit form. The...     

Paley–Wiener Theorems of Generalized Fourier Transform Associated with a Cherednik Type Operator on the Real Line

We consider a singular differential-difference operator \(\Lambda \) on the real line which generalizes the Cherednik operator associated with the reflection group \(\mathbb {Z}_2\) on \(\mathbb {R}\). We establish the Paley–Wiener theorems for the generalized Fourier transform on \(\mathbb {R}\) tied to \(\Lambda \).     

On the Convolution of a Finite Number of Analytic Functions Involving a Generalized Srivastava–Attiya Operator

The present paper gives several subordination results involving a generalized Srivastava–Attiya operator (defined below). Among the results presented in this paper include also a sufficiency condition for the convexity of the convolution of certain functions and a sharp result relating to the convolution structure. We also mention various useful special cases of the main results including those which are related to the Zeta function.