Salvador Pérez-Esteva

We study, for $1 \leq p \leq \infty$, the Hardy space $\bm{h}_e^p(\B)$, the elastic analogue of the classical Hardy spaces of harmonic functions in the unit ball of $\mathbb{R}^3$. The space consists of vector-field solutions of the Lamé system satisfying the standard integrability condition on concentric spheres centered at the origin. Using the elastic Poisson kernel, we establish a Fatou-type theorem and show that $\bm{h}_e^p(\B)$ is isomorphic to the $\mathbb{R}^3$-valued Lebesgue space $L^p$ on the unit sphere for $1 < p \leq \infty$, while $\bm{h}_e^1(\B)$ corresponds to the space of $\mathbb{R}^3$-valued Borel measures on the unit sphere. For $1 < p < \infty$, we prove that $\bm{h}_e^p(\B)$ decomposes as the direct sum of three subspaces. The main contribution of this paper is to describe each of these subspaces along with the corresponding spaces of boundary values. In particular, two of these spaces consist of solutions of the Lamé equation for all eligible choices of the Lamé constants: one of them is the space of Riesz fields (solutions of the generalized Cauchy--Riemann equations) in $\bm{h}_e^p(\B)$; the second is the space of fields given by the cross product of $x$ with such Riesz fields. The results rely on the classical decomposition of $L^2$ vector fields on the sphere into the direct sum of three spaces of vector spherical harmonics, which we extend to $L^p$.

We study the spectrum of the Dirichlet to Neumann operator of the two-sphere associated to a Schrödinger operator in the unit ball. The spectrum forms clusters of size $O(1/k)$ around the sequence of natural numbers $k=1,2,\ldots$, and we compute the first three terms in the asymptotic distribution of the eigenvalues within the clusters, as $k\to\infty$ (band invariants). There are two independent aspects of the proof. The first is a study of the Berezin symbol of the Dirichlet to Neumann operator, which arises after one applies the averaging method. The second is the use of a symbolic calculus of Berezin-Toeplitz operators on the manifold of closed geodesics of the sphere.

The purpose of this paper is to characterize all the entire solutions of the homogeneous Helmholtz equation (solutions in $\mathbb{R}^d$) arising from the Fourier extension operator of distributions in Sobolev spaces of the sphere $H^\alpha(\mathbb{S}^{d-1}),$ with $\alpha\in \mathbb{R}$. We present two characterizations. The first one is written in terms of certain $L^2$-weighted norms involving real powers of the spherical Laplacian. The second one is in the spirit of the classical description of the Herglotz wave functions given by P. Hartman and C. Wilcox. For $\alpha>0$ this characterization involves a multivariable square function evaluated in a vector of entire solutions of the Helmholtz equation, while for $\alpha<0$ it is written in terms of an spherical integral operator acting as a fractional integration operator. Finally, we also characterize all the solutions that are the Fourier extension operator of distributions in the sphere.

We consider the asymptotic behavior of the spectrum of the Landau Hamiltonian plus a rapidly decaying potential, as the magnetic field strength, $B$, tends to infinity. After a suitable rescaling, this becomes a semiclassical problem where the role of Planck's constant is played by $1/B$. The spectrum of the operator forms eigenvalue clusters. We obtain a Szeg\H{o} limit theorem for the eigenvalues in the clusters as a suitable cluster index and $B$ tend to infinity with a fixed ratio $\calE$. The answer involves the averages of the potential over circles of radius $\sqrt{\calE/2}$ (circular Radon transform). We also discuss related inverse spectral results.

We prove a characterization of the Sobolev spaces $H^\alpha$ on the unit sphere $\mathbb{S}^{d-1}$, where the smoothness index $\alpha$ is any positive real number and $d\geq 2$. This characterization does not use differentiation and it is given in terms of $([\alpha/2]+1)$-multidimensional square functions $S_\alpha$. For $[\alpha/2]=0,$ a function $f\in L^2(\mathbb{S}^{d-1})$ belongs to $H^\alpha(\mathbb{S}^{d-1})$ if and only if $S_\alpha (f)\in L^2(\mathbb{S}^{d-1})$. If $n=[\alpha/2]>0$, the membership of $f$ is equivalent to the existence of $g_1,\cdots,g_n$ in $L^2(\mathbb{S}^{d-1})$ such that $S_\alpha(f,g_1,\ldots,g_n)\in L^2(\mathbb{S}^{d-1})$ and in this case, $g_j=T_j((-\Delta_S)^j f)$, where $T_j$ is a zonal Fourier multiplier in the sphere and $\Delta_S$ is the Laplace-Beltrami operator. The square functions $S_\alpha $ are based on averaging operators over euclidean balls (caps) in the sphere that may be viewed as zonal multipliers. The results in the paper are in the spirit of the characterization of fractional Sobolev spaces given in $\mathbb{R}^d$ proved in \cite{AMV}. The development of the theory is fully based on zonal Fourier multipliers and special functions.

We consider Berezin-Toeplitz operators whose multipliers are compactly supported densities carried by a submanifold of ${\mathbb C}^N$ . We compute asymptotically the moments of their spectral measures, and we prove Szeg\"o limit theorems in cases when the submanifold is isotropic or co-isotropic, from which Weyl estimates follow. We also obtain asymptotics of the Schatten norms of such operators. Rescaled versions of these operators can be thought of as quantum mechanical mixed states, and our results give the semi-classical limit of their entropy.

In this paper we consider the phase retrieval problem for Herglotz functions, that is, solutions of the Helmholtz equation $\Delta u+\lambda^2u=0$ on domains $\Omega\subset\mathbb{R}^d$, $d\geq2$. In dimension $d=2$, if $u,v$ are two such solutions then $|u|=|v|$ implies that either $u=cv$ or $u=c\bar v$ for some $c\in\mathbb{C}$ with $|c|=1$. In dimension $d\geq3$, the same conclusion holds under some restriction on $u$ and $v$: either they are real valued or zonal functions or have non vanishing mean.