arXiv subjects
Publications and source records attributed to Chaabane Rejeb.
Consider the Dunkl Laplacian $\Delta_k$ associated with a root system $\Phi$ in $\R^d$ and a nonnegative multiplicity function $k$ on $\Phi$. In this paper, we establish a Caffarelli-Silvestre characterization for the fractional Dunkl Laplacian through an extension problem. We also express the corresponding fundamental solution in terms of the $\Delta_k$-Riesz kernel and prove a fractional Nash-type inequality.
Consider the genus one Hurwitz space $\mathcal{H}_1(n_0,\dots,n_m)$ of ramified covering of fixed degree with $m+1$ prescribed poles of order $n_0+1,\dots,n_m+1$, respectively. Based on a recent formula proved in \cite{Rejeb23}, we derive an explicit solution to the WDVV equations associated with the genus one Dubrovin-Hurwitz-Frobenius manifold structure induced by the normalized holomorphic differential. The obtained solution is written in terms of Bell polynomials, Eisenstein series as well as the Weierstrass functions.
We consider the Hurwitz spaces of ramified coverings of $\mathbb{P}^1$ with prescribed ramification profile over the point at infinity. By means of a particular symmetric bidifferential on a compact Riemann surface, we introduce quasi-homogeneous differentials. By following Dubrovin, we construct on Hurwitz spaces a family of Frobenius manifold structures associated with the quasi-homogeneous differentials. We explicitly derive new generating formulas for the corresponding prepotentials. This produces quasi-homogeneous solutions to the following generalized WDVV associativity equations: $F_i\eta^{-1}F_j=F_j\eta^{-1}F_i$, where the invertible constant matrix $\eta$ is a linear combination of the matrices $F_j$. In particular, our approach provides another look at Dubrovin's construction of semi-simple Hurwitz-Frobenius manifolds and establishes an alternative practical method to calculate their primary free energy functions. As applications, we use our formalism to obtain various explicit quasi-homogeneous solutions to the WDVV equations in genus zero and one and give a new proof of Ramanujan's differential equations for Eisenstein series.
Let $\Delta_k$ be the Dunkl Laplacian relative to a fixed root system $\mathcal{R}$ in $\mathbb{R}^d$, $d\geq2$, and to a nonnegative multiplicity function $k$ on $\mathcal{R}$. Our first purpose in this paper is to solve the $\Delta_k$-Dirichlet problem for annular regions. Secondly, we introduce and study the $\Delta_k$-Green function of the annulus and we prove that it can be expressed by means of $\Delta_k$-spherical harmonics. As applications, we obtain a Poisson-Jensen formula for $\Delta_k$-subharmonic functions and we study positive continuous solutions for a $\Delta_k$-semilinear problem.