in this paper, using a generalized dunkl translation operator, we obtain an analog of titchmarsh's theorem for the dunkl transform for functions satisfying the lipschitz-dunkl condition in $mathrm{l}_{2,alpha}=mathrm{l}_{alpha}^{2}(mathbb{r})=mathrm{l}^{2}(mathbb{r}, |x|^{2alpha+1}dx), alpha>frac{-1}{2}$.

in this paper, we study the convexity of the integral operator

Abstract In mathematics, a common approach to solving new problem is break down the into sub-problems that you know how solve and then try glue pieces together yield solution of problem. This also very useful in software development; well-tested can be glued order obtain solutions wider class problems. Operator splitting technique illustrates this principle well. Rather complex equations broken...

We prove some sharp Hardy type inequalities related to the Dirac operator by elementary, direct methods. Some of these inequalities have been obtained previously using spectral information about the Dirac-Coulomb operator. Our results are stated under optimal conditions on the asymptotics of the potentials near zero and near infinity.