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 ${ʃ}_{{ℍ}^n}\left|\varphi \left(x\right)\right|su{p}_{0<t\le 1}{\left|e^{(\surd -1)t\Delta }{f}_n\left(x\right)\right|}^{2}dx\ge C_n\parallel f_n{\parallel }_{W^s}^2,li{m}_{n\to \infty }{C}_n=+\infty$.

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 $v$.

In this paper we show an asymptotic formula for the number of eigenvalues of a pseudodifferential operator. As a corollary we obtain a generalization of the result by Shubin and Tulovskiĭ about the Weyl asymptotic formula. We also consider a version of the Weyl formula for the quasi-classical asymptotics.

Let $P(\lambda ,D)={\sum }_{\left|\alpha \right|\le m}{a}_{\alpha }\left(\lambda \right)D^{\alpha }$ be a differential operator with constant coefficients $a_{\alpha }$ depending analytically on a parameter $\lambda$. Assume that the family $\{$ P($\lambda$,D)$\}$ is of constant strength. We investigate the equation $P(\lambda ,D){𝔣}_{\lambda }\equiv g_{\lambda }$ where ${𝔤}_{\lambda }$ is a given analytic function of $\lambda$ with values in some space of distributions and the solution ${𝔣}_{\lambda }$ is required to depend analytically on $\lambda$, too. As a special case we obtain a regular fundamental solution of P($\lambda$,D) which depends analytically on $\lambda$. This result answers a question of L. Hörmander. ...

For $a\in {\mathbf{R}}^n\setminus \left\{0\right\}$ and $\Omega$ an open bounded subset of ${\mathbf{R}}^n$ definie $L^p(\Omega ,a)$ as the closed subset of $L^p\left(\Omega \right)$ consisting of all functions that are constant almost everywhere on almost all lines parallel to $a$. For a given set of directions $a^{\nu }\in {\mathbf{R}}^n\setminus \left\{0\right\}$, $\nu =1,...,m$, we study for which $\Omega$ it is true that the vector space $$(*)L^p(\Omega ,a^1)+\cdots +L^p(\Omega ,a^m)\text{is}\text{a}\text{closed}\text{subspace}\text{of}{L}^p\left(\Omega \right).$$ This problem arizes naturally in the study of image reconstruction from projections (tomography). An essentially equivalent problem is to decide whether a certain matrix-valued differential operator...

Let A be a closed linear operator in a Banach space E. In the study of the nth-order abstract Cauchy problem $u^{\left(n\right)}\left(t\right)=Au\left(t\right)$, t ∈ ℝ, one is led to considering the linear Volterra equation (AVE) $u\left(t\right)=p\left(t\right)+A{ʃ}_0^{t}a(t-s)u\left(s\right)ds$, t ∈ ℝ, where $a\left(·\right)\in L_{loc}^1\left(ℝ\right)$ and p(·) is a vector-valued polynomial of the form $p\left(t\right)={\sum }_{j=0}^{n}1/(j!)x_j{t}^j$ for some elements $x_j\in E$. We describe the spectral properties of the operator A through the existence of slowly growing solutions of the (AVE). The main tool is the notion of Carleman spectrum of a vector-valued function. Moreover, an extension...

A recent result of Bahouri shows that continuation from an open set fails in general for solutions of $ℒu=Vu$ where $V\in C^{\infty }$ and $ℒ={\sum }_{j=1}^{N-1}{X}_j^2$ is a (nonelliptic) operator in ${\mathbf{R}}^N$ satisfying Hörmander’s condition for hypoellipticity. In this paper we study the model case when $ℒ$ is the subelliptic Laplacian on the Heisenberg group and $V$ is a zero order term which is allowed to be unbounded. We provide a sufficient condition, involving a first order differential inequality, for nontrivial solutions of $ℒu=Vu$ to have a...

We consider the Volterra integral operator $T:L^p\left({ℝ}^{+}\right)\to L^p\left({ℝ}^{+}\right)$ defined by $\left(Tf\right)\left(x\right)=v\left(x\right){ʃ}_0^{x}u\left(t\right)f\left(t\right)dt$. Under suitable conditions on u and v, upper and lower estimates for the approximation numbers $a_n\left(T\right)$ of T are established when 1 < p < ∞. When p = 2 these yield $li{m}_{n\to \infty }n{a}_n\left(T\right)={\pi }^{-1}{ʃ}_0^{\infty }\left|u\left(t\right)v\left(t\right)\right|dt$. We also provide upper and lower estimates for the ${\ell }^{\alpha }$ and weak ${\ell }^{\alpha }$ norms of (an(T)) when 1 < α < ∞.

The main part of the paper is devoted to the problem of the existence of absolutely representing systems of exponentials with imaginary exponents in the spaces $C^{\infty }\left(G\right)$ and $C^{\infty }\left(K\right)$ of infinitely differentiable functions where G is an arbitrary domain in ${ℝ}^p$, p≥1, while K is a compact set in ${ℝ}^p$ with non-void interior K̇ such that $\overline{K}̇=K$. Moreover, absolutely representing systems of exponents in the space H(G) of functions analytic in an arbitrary domain $G\subseteq {ℂ}^p$ are also investigated.

The group SU(1,d) acts naturally on the Hilbert space $L²\left(Bd{\mu }_{\alpha }\right)(\alpha >-1)$, where B is the unit ball of ${ℂ}^d$ and $d{\mu }_{\alpha }$ the weighted measure ${(1-|z\left|²\right)}^{\alpha }dm\left(z\right)$. It is proved that the irreducible decomposition of the space has finitely many discrete parts and a continuous part. Each discrete part corresponds to a zero of the generalized Harish-Chandra c-function in the lower half plane. The discrete parts are studied via invariant Cauchy-Riemann operators. The representations on the discrete parts are equivalent to actions on some...