In Aizenbud et al. (2010, Annals of Mathematics 172, 1407–1434), a multiplicity one theorem is proved for general linear groups, orthogonal groups, and unitary groups ( $GL, O,$ and U) over p-adic local fields. That is to say that when we have a pair of such groups $G_n{\subseteq } G_{n+1}$ , any restriction of an irreducible smooth representation of $G_{n+1}$ to $G_n$ is multiplicity-free. This property is already known for $GL$ over a local field of positive characteristic, and in this paper, we also give a proof for $O,U$ , and $SO$ over local fields of positive odd characteristic. These theorems are shown in Gan, Gross, and Prasad (2012, Sur les Conjectures de Gross et Prasad. I, Société Mathématique de France) to imply the uniqueness of Bessel models, and in Chen and Sun (2015, International Mathematics Research Notice 2015, 5849–5873) to imply the uniqueness of Rankin–Selberg models. We also prove simultaneously the uniqueness of Fourier–Jacobi models, following the outlines of the proof in Sun (2012, American Journal of Mathematics 134, 1655–1678).

By the Gelfand–Kazhdan criterion, the multiplicity one property for a pair $H\leq G$ follows from the statement that any distribution on G invariant to conjugations by H is also invariant to some anti-involution of G preserving H. This statement for $GL, O$ , and U over p-adic local fields is proved in Aizenbud et al. (2010, Annals of Mathematics 172, 1407–1434). An adaptation of the proof for $GL$ that works over of local fields of positive odd characteristic is given in Mezer (2020, Mathematische Zeitschrift 297, 1383–1396). In this paper, we give similar adaptations of the proofs of the theorems on orthogonal and unitary groups, as well as similar theorems for special orthogonal groups and for symplectic groups. Our methods are a synergy of the methods used over characteristic 0 (Aizenbud et al. [2010, Annals of Mathematics 172, 1407–1434]; Sun [2012, American Journal of Mathematics 134, 1655–1678]; and Waldspurger [2012, Astérisque 346, 313–318]) and of those used in Mezer (2020, Mathematische Zeitschrift 297, 1383–1396).

We introduce the notion of a distributional $k$-Hessian ($k=2,\ldots ,n$) associated with fractional Sobolev functions on $\unicode[STIX]{x1D6FA}$, a smooth bounded open subset in $\mathbb{R}^{n}$. We show that the distributional $k$-Hessian is weakly continuous on the fractional Sobolev space $W^{2-2/k,k}(\unicode[STIX]{x1D6FA})$ and that the weak continuity result is optimal, that is, the distributional $k$-Hessian is well defined in $W^{s,p}(\unicode[STIX]{x1D6FA})$ if and only if $W^{s,p}(\unicode[STIX]{x1D6FA})\subseteq W^{2-2/k,k}(\unicode[STIX]{x1D6FA})$.

The present paper concerns the system ut + [ϕ(u)]x = 0, vt + [ψ(u)v]x = 0 having distributions as initial conditions. Under certain conditions, and supposing ϕ, ψ: ℝ → ℝ functions, we explicitly solve this Cauchy problem within a convenient space of distributions u,v. For this purpose, a consistent extension of the classical solution concept defined in the setting of a distributional product (not constructed by approximation processes) is used. Shock waves, δ-shock waves, and also waves defined by distributions that are not measures are presented explicitly as examples. This study is carried out without assuming classical results about conservation laws. For reader's convenience, a brief survey of the distributional product is also included.

Let $F$ be a non-Archimedean local field or a finite field. Let $n$ be a natural number and $k$ be 1 or 2. Consider $G\,:=\,\text{G}{{\text{L}}_{n+k}}\left( F \right)$ and let $M\,:=\,\text{G}{{\text{L}}_{n}}\left( F \right)\,\times \,\text{G}{{\text{L}}_{k}}\left( F \right)\,<\,G$ be a maximal Levi subgroup. Let $U\,<\,G$ be the corresponding unipotent subgroup and let $P\,=\,MU$ be the corresponding parabolic subgroup. Let $J\,:=\,J_{M}^{G}\,:\,\mathcal{M}\left( G \right)\,\to \,\mathcal{M}\left( M \right)$ be the Jacquet functor, i.e., the functor of coinvariants with respect to $U$. In this paper we prove that $J$ is a multiplicity free functor, i.e.,$\dim\,\text{Ho}{{\text{m}}_{M}}\left( J\left( \pi \right),\,\rho \right)\,\le \,1$, for any irreducible representations $\pi $ of $G$ and $\rho $ of $M$. We adapt the classical method of Gelfand and Kazhdan, which proves the “multiplicity free” property of certain representations to prove the “multiplicity free” property of certain functors. At the end we discuss whether other Jacquet functors are multiplicity free.

A fundamental problem in Fourier analysis is to characterize the behaviour of a function (or distribution) whose Fourier transform vanishes in some particular set. Of course, this is, in general, a very difficult question and little seems to be known, except in some special cases. For example, a theorem of Paley-Wiener (Theorem XII in [6]) characterizes exactly the behaviour of the modulus of a function in L2(R) whose Fourier transform vanishes on a half-line.

In this note, we have shown that the set of conditions given by Gel'fand and Vilenkin for the positive definiteness of a functional L(ϕ) [1, Theorem 7, p. 285] though sufficient is not necessary, thereby providing an answer to an open problem posed by them [1, Theorem 7, p. 285].