arXiv subjects

Mohamed Gaidi

Publications and source records attributed to Mohamed Gaidi.

3 recordsLinked to original sources

arXiv2023

On the Study of the Klein-Gordon Equation in the Dunkl Setting

In Dunkl theory on $\mathbb{R}^{n}$ which generalizes classical Fourier analysis, we study the solution of the Klein-Gordon-equation defined by: \begin{eqnarray} \nonumber \partial_{t}^{2}u-Δ_{k}u=-m^{2}u \ , \ \ \ u (x,0)=g(x) \ , \ \ \ \partial_{t}u(x,0)=f(x) \end{eqnarray} with \ $m > 0$ \ and \ $\partial_{t}^{2}u$ \ is the second derivative of the solution $u$ with respect to $t$ and $Δ_{k}u$ is the Dunkl Laplacian with respect to $x$ where $f$ and $g$ the two functions in $\mathcal{S}(\mathbb{R}^{n})$ which surround the initial conditions. We obtain an integral representation for its solution which we gives some properties. As a specific result, we studied the associated energies to the Dunkl-Klein-Gordon equation.

math.AP↗

arXiv2017

$L^p-L^q$ estimates for the solution of the Dunkl wave equation

In this paper, our main aim is to derive $L^p-L^q$ estimates of the solution $u_k(x,t)$ ( t fixed) of the Cauchy problem for the homogeneous linear wave equation associated to the Dunkl Laplacian $Δ_k$, $$Δ_ku_k(x,t)= \partial_t^2u_k (x,t),\quad \partial_tu_k(x,0)= f(x),\quad u_k(x,0)= g(x).$$ We extend to Dunkl setting the estimates given by Srichartz in \cite{Sti} for the ordinary wave equation .

math.CA↗

arXiv2017

$L^p$ estimates for an oscillating Dunkl multiplier

In this paper, we study the $L^p$ boundedness of a class of oscillating multiplier operator for the Dunkl transform, $T_{m_α}=\mathcal{F}_k^{-1}(m_α\mathcal{F}_k(f))$ with $m(ξ)=|ξ|^{-α}e^{\pm i|ξ|}ϕ(ξ)$. We obtain an $L^p$-bound result for the corresponding maximal functions. As a specific applications, we give an extension of the $L^p$ estimate for the wave equation and of Stein's theorem for the analytic family of maximal spherical means \cite{Stein}

math.CA↗