In this article we give a complete proof in one dimension of an a priori inequality involving pseudo-differential operators: if $a$ and $b$ are symbols in $S_{1,0}^2$ such that $\left|a\right|\le b$, then for all $ϵ\>0$ we have the estimate $\parallel a^{w}u{\parallel }_s^2\le C_{ϵ}(\parallel b^{w}u{\parallel }_s^2+\parallel u{\parallel }_{s+ϵ}^2)$ for all $u$ in the Schwartz space, where $\parallel {\parallel }_t$ is the usual $H_t$ norm. We use microlocalization of levels I, II and III in the spirit of Fefferman’s SAK principle.

We give an explicit expression of a two-parameter family of Flensted-Jensen’s functions ${\Psi }_{\mu ,\alpha }$ on a concrete realization of the universal covering group of $U(1,1)$. We prove that these functions are, up to a phase factor, radial eigenfunctions of the Landau Hamiltonian on the hyperbolic disc with a magnetic field strength proportional to $\mu$, and corresponding to the eigenvalue $4\alpha (\alpha -1)$.

The paper deals with quarkonial decompositions and entropy numbers in weighted function spaces on hyperbolic manifolds. We use these results to develop a spectral theory of related Schrödinger operators in these hyperbolic worlds.

There has been much progress in recent years in understanding the existence problem for wave maps with small critical Sobolev norm (in particular for two-dimensional wave maps with small energy); a key aspect in that theory has been a renormalization procedure (either a geometric Coulomb gauge, or a microlocal gauge) which converts the nonlinear term into one closer to that of a semilinear wave equation. However, both of these renormalization procedures encounter difficulty if the energy of the...

Global maximal estimates are considered for solutions to an initial value problem for the Schrödinger equation.

Let A = -Δ + V be a Schrödinger operator on ${ℝ}^d$, d ≥ 3, where V is a nonnegative potential satisfying the reverse Hölder inequality with an exponent q > d/2. We say that f is an element of $H_A^p$ if the maximal function $su{p}_{t>0}\left|T_{t}f\left(x\right)\right|$ belongs to $L^p\left({ℝ}^d\right)$, where ${T_t}_{t>0}$ is the semigroup generated by -A. It is proved that for d/(d+1) < p ≤ 1 the space $H_A^p$ admits a special atomic decomposition.

Let ${K_t}_{t>0}$ be the semigroup of linear operators generated by a Schrödinger operator -L = Δ - V with V ≥ 0. We say that f belongs to $H{¹}_L$ if $\left|\right|su{p}_{t>0}|K_t{f\left(x\right)\left|\right||}_{L¹\left(dx\right)}<\infty$. We state conditions on V and $K_t$ which allow us to give an atomic characterization of the space $H{¹}_L$.

We prove Harnack's inequality for non-negative solutions of some degenerate elliptic operators in divergence form with the lower order term coefficients satisfying a Kato type contition.