We prove unique continuation for solutions of the inequality $\left|\Delta u\right(x\left)\right|\le V\left(x\right)\left|\nabla u\right(x\left)\right|$, $x\in \Omega$ a connected set contained in ${\mathbf{R}}^n$ and $V$ is in the Morrey spaces $F^{\alpha ,p}$, with $p\ge (n-2)/2(1-\alpha )$ and $\alpha \<1$. These spaces include $L^q$ for $q\ge (3n-2)/2$ (see [H], [BKRS]). If $p=(n-2)/2(1-\alpha )$, the extra assumption of $V$ being small enough is needed.

Hörmander’s famous Fourier multiplier theorem ensures the $L_p$-boundedness of $F(-{\Delta }_{ℝ}D)$ whenever $F\in \mathscr{H}\left(s\right)$ for some $s\>\frac{D}{2}$, where we denote by $\mathscr{H}\left(s\right)$ the set of functions satisfying the Hörmander condition for $s$ derivatives. Spectral multiplier theorems are extensions of this result to more general operators $A\ge 0$ and yield the $L_p$-boundedness of $F\left(A\right)$ provided $F\in \mathscr{H}\left(s\right)$ for some $s$ sufficiently large. The harmonic oscillator $A=-{\Delta }_{ℝ}+x^2$ shows that in general $s\>\frac{D}{2}$ is not sufficient even if $A$ has a heat kernel satisfying gaussian estimates. In...

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 G be a semisimple Lie group with Iwasawa decomposition G = KAN. Let X = G/K be the associated symmetric space and assume that X is of rank one. Let M be the centraliser of A in K and consider an orthonormal basis $Y_{\delta ,j}:\delta \in K̂₀,1\le j\le d_{\delta }$ of L²(K/M) consisting of K-finite functions of type δ on K/M. For a function f on X let f̃(λ,b), λ ∈ ℂ, be the Helgason Fourier transform. Let $h_t$ be the heat kernel associated to the Laplace-Beltrami operator and let $Q_{\delta }(i\lambda +\varrho )$ be the Kostant polynomials. We establish the following...

We consider the Pauli operator $\mathbf{H}\left(\mu \right):={\left({\sum }_{j=1}^m{\sigma }_j\left(-i\frac{\partial }{\partial x_j}-\mu A_j\right)\right)}^2+V$ selfadjoint in $L^2({ℝ}^m;{ℂ}^2)$, $m=2,3$. Here ${\sigma }_j$, $j=1,...,m$, are the Pauli matrices, $A:=(A_1,...,A_m)$ is the magnetic potential, $\mu \&gt;0$ is the coupling constant, and $V$ is the electric potential which decays at infinity. We suppose that the magnetic field generated by $A$ satisfies some regularity conditions; in particular, its norm is lower-bounded by a positive constant, and, in the case $m=3$, its direction is constant. We investigate the asymptotic behaviour as $\mu \to \infty$ of the number of the eigenvalues of $\mathbf{H}\left(\mu \right)$ smaller...

Ce travail est l’étude de divers cas particuliers d’un problème nouveau, semble-t-il, concernant les translatées de fonctions ou de distributions sur un groupe. Soit $\mathbf{E}$ un espace vectoriel topologique de fonctions ou de distributions sur un groupe abélien $G$ localement compact ; $\mathbf{E}$ est supposé invariant par les translations $a\to f_a\left(x\right)=f(x+a)(f\in \mathbf{E},a\in G)$. Si $f\in \mathbf{E}$ et si $A$ est un sous-ensemble non vide de $G$, $\mathbf{I}(f,A)=\mathbf{I}(f,A,\mathbf{E})$ désigne le sous-espace vectoriel fermé de $\mathbf{E}$ engendré par les translatées $f_a$ de $f$ avec $a\in A$. On dira qu’une $f\in \mathbf{E}$ a ses...

Writing $(T_R^{\lambda }f)\widehat{}\left(\xi \right)=(1-|\xi {|}^2/R^2{)}_{+}^{\lambda }\widehat{f}\left(\xi \right)$. E. Stein conjectured $$\parallel {\left(\sum _j|T_{R_j}^{\lambda }{f}_i{|}^2\right)}^{1/2}{\parallel }_p\le C\parallel {\left(\sum _j|f_j{|}^2\right)}^{1/2}{\parallel }_p$$ for $\lambda \&gt;0$, $\frac{4}{3}\le p\le 4$ and $C=C_{\lambda ,p}$. We prove this conjecture. We prove also $f\left(x\right)={lim}_{j\to \infty }{T}_{2^j}^{\lambda }f\left(x\right)$ a.e. We only assume $\frac{4}{3+2\lambda }\&lt;p\&lt;\frac{4}{1-2\lambda }$.