The regularity of convolution and restriction of distributions
[For the entire collection see Zbl 0742.00067.]Let be the set of hyperplanes in , the unit sphere of , the exterior of the unit ball, the set of hyperplanes not passing through the unit ball, the Radon transform, its dual. as operator from to is a closable, densely defined operator, denotes the operator given by if the integral exists for a.e. Then the closure of is the adjoint of . The author shows that the Radon transform and its dual can be linked by two operators...
Let be a compact manifold with boundary, and a scattering metric on , which may be either of short range or “gravitational” long range type. Thus, gives the geometric structure of a complete manifold with an asymptotically conic end. Let be an operator of the form , where is the Laplacian with respect to and is a self-adjoint first order scattering differential operator with coefficients vanishing at and satisfying a “gravitational” condition. We define a symbol calculus for...
We first discuss a class of inequalities of Onofri type depending on a parameter, in the two-dimensional Euclidean space. The inequality holds for radial functions if the parameter is larger than . Without symmetry assumption, it holds if and only if the parameter is in the interval . The inequality gives us some insight on the symmetry breaking phenomenon for the extremal functions of the Caffarelli-Kohn-Nirenberg inequality, in two space dimensions. In fact, for suitable sets of parameters (asymptotically...
We prove Strichartz estimates with fractional loss of derivatives for the Schrödinger equation on any riemannian compact manifold. As a consequence we infer global existence results for the Cauchy problem of nonlinear Schrödinger equations on surfaces in the case of defocusing polynomial nonlinearities, and on three-manifolds in the case of quadratic nonlinearities. We also discuss the optimality of these Strichartz estimates on spheres.
Let be a compact Riemannian manifold of dimension .We suppose that is a metric in the Sobolev space with and there exist a point and such that is smooth in the ball . We define the second Yamabe invariant with singularities as the infimum of the second eigenvalue of the singular Yamabe operator over a generalized class of conformal metrics to and of volume . We show that this operator is attained by a generalized metric, we deduce nodal solutions to a Yamabe type equation with...