Let L be the full laplacian on the Heisenberg group ${ℍ}^n$ of arbitrary dimension n. Then for $f\in L^2\left({ℍ}^n\right)$ such that ${(I-L)}^{s/2}f\in L^2\left({ℍ}^n\right)$, s > 3/4, for a $\varphi \in C_c\left({ℍ}^n\right)$ we have ${ʃ}_{{ℍ}^n}\left|\varphi \left(x\right)\right|su{p}_{0<t\le 1}{\left|e^{(\surd -1)tL}f\left(x\right)\right|}^{2}dx\le C_{\varphi }\parallel f{\parallel }_{W^s}^2$. On the other hand, the above maximal estimate fails for s < 1/4. If Δ is the sublaplacian on the Heisenberg group ${ℍ}^n$, then for every s < 1 there exists a sequence $f_n\in L^2\left({ℍ}^n\right)$ and $C_n>0$ such that ${(I-L)}^{s/2}{f}_n\in L^2\left({ℍ}^n\right)$ and for a $\varphi \in C_c\left({ℍ}^n\right)$ we have ${ʃ}_{{ℍ}^n}\left|\varphi \left(x\right)\right|su{p}_{0<t\le 1}{\left|e^{(\surd -1)t\Delta }{f}_n\left(x\right)\right|}^{2}dx\ge C_n\parallel f_n{\parallel }_{W^s}^2,li{m}_{n\to \infty }{C}_n=+\infty$.

Let A be a pseudodifferential operator on ${ℝ}^N$ whose Weyl symbol a is a strictly positive smooth function on $W={ℝ}^N\times {ℝ}^N$ such that $|{\partial }^{\alpha }a|\le C_{\alpha }{a}^{1-\varrho }$ for some ϱ>0 and all |α|>0, ${\partial }^{\alpha }a$ is bounded for large |α|, and $li{m}_{w\to \infty }a\left(w\right)=\infty$. Such an operator A is essentially selfadjoint, bounded from below, and its spectrum is discrete. The remainder term in the Weyl asymptotic formula for the distribution of the eigenvalues of A is estimated. This is done by applying the method of approximate spectral projectors of Tulovskiĭ and Shubin. ...

We prove an inequality of the type $$\parallel |x{|}^{r_f}{\parallel }_{L^p({\mathbf{R}}^n)}\le c(n,p,q,r)\parallel |x{|}^{\tau +\mu }\Delta f{\parallel }_{L^q({\mathbf{R}}^n)}.$$ This is then used to derive the unique continuation property for the differential inequality $\left|\Delta f\right(x\left)\right|\le \left|v\right(x\left)\right|\left|f\right(x\left)\right|$ under suitable local integrability assumptions on the function $v$.

Pseudodifferential operators $P(x,D)\sim {\sum }_{j=0}^{+\infty }{P}_{m-j}(x,D)$ are studied, from the viewpoint of local solvability and under the assumption that, micro-locally, the principal symbol factorizes as $P_m=Q{L}^2$ with $Q$ elliptic, homogeneous of degree $m-2$, and $L$ homogeneous of degree one, satisfying the following condition : there is a point $(x_0,{\xi }^0)$ in the characteristic variety $L=0$ and a complex number $z$ such that $d_{\xi }\mathrm{Re}\left(zL\right)\ne 0$ at $(x_0,{\xi }^0)$ and such that the restriction of $\mathrm{Im}\left(zL\right)$ to the bicharacteristic strip of $\mathrm{Re}\left(zL\right)$ vanishes of order $k\&lt;+\infty$ at $(x_0,{\xi }^0)$, changing sign there from...