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In this paper, we prove restriction theorems for the Fourier-Laguerre transform and establish Strichartz estimates for the Schr\"{o}dinger propagator $e^{-itL_\alpha}$ for the Laguerre operator $L_\alpha=-\Delta-\sum_{j=1}^{n}(\dfrac{2\alpha_j+1}{x_j}\dfrac{\partial}{\partial x_j})+\dfrac{|x|^2}{4}$, $\alpha=(\alpha_1,\alpha_2,\cdots,\alpha_n)\in{(-\frac{1}{2},\infty)^n}$ on $\mathbb{R}_+^n$ involving systems of orthonormal functions.

Poor nitrogen removal efficiency (mainly nitrate, NO3 −-N) at low temperatures strongly limits application of subsurface wastewater infiltration systems (SWISs). Seven psychrophilic strains (heterotrophic nitrifying bacteria and aerobic denitrifying bacteria) were isolated and added to SWISs to investigate the effect of embedding and direct-dosing bioaugmentation strategies on sewage treatment performance at low temperature. Both bioaugmentation strategies improved ammonium (NH4 +-N) removal efficiencies, and the embedding strategy also exhibited satisfactory NO3 −-N and total nitrogen (TN) removal efficiencies. Pyrosequencing results of the bacterial 16S rRNA gene indicated that the embedding strategy significantly decreased the indigenous soil microbial diversity (p

We consider the Grushin type operator on $\mathbb{R}^{d}_x \times \mathbb{R}^{k}_y$ with the form \begin{equation*} G_\mu=\overset{d}{\underset{i=1}{\sum}}\partial_{x_i}^2+\left(\overset{d}{\underset{i=1}{\sum}}x_i^2\right)^{2\mu}\overset{k}{\underset{j=1}{\sum}}\partial_{y_j}^2. \end{equation*} and derive weighted Hardy-Sobolev type inequalities and weighted Caffarelli-Kohn-Nirenberg type inequalities related to $G_\mu$.

We study the Grushin operators acting on $\mathbb{R}^{d_1}_x \times \mathbb{R}^{d_2}_t$ and defined by the formula \begin{equation*} L=-\overset{d_1}{\underset{j=1}{\sum}}\partial_{x_j}^2-\left(\overset{d_1}{\underset{j=1}{\sum}}|x_j|^2\right)\overset{d_2}{\underset{k=1}{\sum}}\partial_{t_k}^2. \end{equation*} We establish a restriction theorem associated with the considered operators. Our result is an analogue of the restriction theorem on the Heisenberg group obtained by D. Muller.

In this paper, we prove the dispersive estimates and Strichartz inequalities for the solution of the Schr\"{o}dinger equation related to the full Laplacian on H-type groups. This extends the results obtained by G. Furioli and A. Veneruso [Studia Math., 2004, 160: 157-178] on the Heisenberg group.

In this paper, we prove the dispersive estimates and Strichartz inequalities for the solution of the Schr��dinger equation related to the full Laplacian on H-type groups. This extends the results obtained by G. Furioli and A. Veneruso [Studia Math., 2004, 160: 157-178] on the Heisenberg group.