Distributions with Decay and Restriction Problems

We develop the notion of distributions with decay and use it to define global wavefront sets of classes of function spaces, including \(L^p\) -Sobolev spaces on \(\mathbb {R}^d\) as well as global \(L^q\) -Denjoy–Carleman functions. We also introduce the corresponding notion of microglobal regularity. We prove a characterization of distributions (in a given function space) with decay in terms of microglobal regularity in every direction of their Fourier transforms. We conclude the chapter with an application to a restriction problem that we call the restriction problem with moments. We show that the surface area measure of the sphere satisfies the \(L^p\) - \(L^2\) restriction problem with moments if \(1 \leq p < \frac {2(d+1)}{d+3}\) and that the Frostman measure constructed by Salem satisfies the \(L^p\) - \(L^2\) restriction problem with moments if \(1 \leq p < \frac {2(2-2\alpha +\beta )}{4(1-\alpha )+\beta }\) for certain values of \(\alpha \) and \(\beta \) .