We show a result on propagation of the anisotropic Gabor wave front set for linear operators with a tempered distribution Schwartz kernel. The anisotropic Gabor wave front set is parametrized by a positive parameter relating the space and frequency variables. The anisotropic Gabor wave front set of the Schwartz kernel is assumed to satisfy a graph type criterion. The result is applied to a class of evolution equations that generalizes the Schrödinger equation for the free particle. The Laplacian is replaced by any partial differential operator with constant coefficients, real symbol and order at least two.

The duality proof of sampling localization given by Loy, Newbury, Anderssen and Davies in 2001 contains an oversight, as the classes of functions chosen do not assume the compact support. Here, it is shown how a minor change to the argument there yields a precise conclusion.

In this paper we study Hankel transforms and Hankel convolution operators on spaces of entire functions of finite order and their duals.

We define embedding of an $n$-dimensional normed space into ${{L}_{-p}},\,0\,<\,p\,<\,n$ by extending analytically with respect to $p$ the corresponding property of the classical ${{L}_{p}}$-spaces. The well-known connection between embeddings into ${{L}_{p}}$ and positive definite functions is extended to the case of negative $p$ by showing that a normed space embeds in ${{L}_{-p}}$ if and only if $\parallel x{{\parallel }^{-p}}$ is a positive definite distribution. We show that the technique of embedding in ${{L}_{-p}}$ can be applied to stable processes in some situations where standard methods do not work. As an example, we prove inequalities of correlation type for the expectations of norms of stable vectors. In particular, for every $P\in [n-3,n),\mathbb{E}({{\max }_{i=1,...,n}}{{\left| {{X}_{i}} \right|}^{-p}})\ge \mathbb{E}({{\max }_{i=1,...,n}}{{\left| {{Y}_{i}} \right|}^{-p}})$, where ${{X}_{1}},...,{{X}_{n}}\,\text{and}\,{{Y}_{1}},...,{{Y}_{n}}$ are jointly $q$-stable symmetric random variables, $0\,<\,q\,\le \,2$, so that, for some $k\,\in \,\mathbb{N},\,1\,\le \,k\,<\,n$, the vectors $\left( {{X}_{1}},\,.\,.\,.\,,\,{{X}_{k}} \right)$ and $\left( {{X}_{k+1}},\,.\,.\,.\,,{{X}_{n}} \right)$ have the same distributions as $({{Y}_{1}},...,{{Y}_{k}})\,\,\text{and}\,\,({{Y}_{k+1}},...,{{Y}_{n}})$, respectively, but ${{Y}_{i}}\,\text{and}\,{{Y}_{j}}$ are independent for every choice of $1\,\le \,i\,\le \,k,\,k\,+\,1\,\le \,j\,\le \,n$.

Let be the Zemanian space of Hankel transformable generalized functions and let be the space of Hankel convolution operators for . This is the dual of a subspace of for which is also the space of Hankel convolutors. In this paper the elements of are characterized as those in and in that commute with Hankel translations. Moreover, necessary and sufficient conditions on the generalized Hankel transform are established in order that every such that in .

A distributional solution to the Hilbert problem in dimension > 1 is given.