We study the Hausdorff-Young transform for a commutative hypergroup K and its dual space K̂ by extending the domain of the Fourier transform so as to encompass all functions in $L^p(K,m)$ and $L^p(K̂,\pi )$ respectively, where 1 ≤ p ≤ 2. Our main theorem is that those extended transforms are inverse to each other. In contrast to the group case, this is not obvious, since the dual space K̂ is in general not a hypergroup itself.

We characterize the range of some spaces of functions by the Fourier transform associated with the spherical mean operator R and we give a new description of the Schwartz spaces. Next, we prove a Paley-Wiener and a Paley-Wiener-Schawrtz theorems.

We extend to the case 1 < p the results obtained by Geymonat and Krasucki for p = 2 on the characterization of the traces of W2,p(Ω) for a bounded Lipschitz domain.

Let G be a locally compact group. We use the canonical operator space structure on the spaces $L^p\left(G\right)$ for p ∈ [1,∞] introduced by G. Pisier to define operator space analogues $O{A}_p\left(G\right)$ of the classical Figà-Talamanca-Herz algebras $A_p\left(G\right)$. If p ∈ (1,∞) is arbitrary, then $A_p\left(G\right)\subset O{A}_p\left(G\right)$ and the inclusion is a contraction; if p = 2, then OA₂(G) ≅ A(G) as Banach spaces, but not necessarily as operator spaces. We show that $O{A}_p\left(G\right)$ is a completely contractive Banach algebra for each p ∈ (1,∞), and that $O{A}_q\left(G\right)\subset O{A}_p\left(G\right)$ completely contractively for amenable...

We study the maximal regularity on different function spaces of the second order integro-differential equations with infinite delay $\left(P\right)u^{\text{'}\text{'}}\left(t\right)+\alpha u^{\text{'}}\left(t\right)+d/dt\left({\int }_{-\infty }^{t}b(t-s)u\left(s\right)ds\right)=Au\left(t\right)-{\int }_{-\infty }^{t}a(t-s)Au\left(s\right)ds+f\left(t\right)$ (0 ≤ t ≤ 2π) with periodic boundary conditions u(0) = u(2π), u’(0) = u’(2π), where A is a closed operator in a Banach space X, α ∈ ℂ, and a,b ∈ L¹(ℝ₊). We use Fourier multipliers to characterize maximal regularity for (P). Using known results on Fourier multipliers, we find suitable conditions on the kernels a and b under which necessary and sufficient conditions...

For a smooth curve $\Gamma$ and a set $\Lambda$ in the plane ${ℝ}^2$, let $AC(\Gamma ;\Lambda )$ be the space of finite Borel measures in the plane supported on $\Gamma$, absolutely continuous with respect to the arc length and whose Fourier transform vanishes on $\Lambda$. Following [12], we say that $(\Gamma ,\Lambda )$ is a Heisenberg uniqueness pair if $AC(\Gamma ;\Lambda )=\left\{0\right\}$. In the context of a hyperbola $\Gamma$, the study of Heisenberg uniqueness pairs is the same as looking for uniqueness sets $\Lambda$ of a collection of solutions to the Klein-Gordon equation. In this work, we mainly address the...

We investigate whether the L¹-algebra of polynomial hypergroups has non-zero bounded point derivations. We show that the existence of such point derivations heavily depends on growth properties of the Haar weights. Many examples are studied in detail. We can thus demonstrate that the L¹-algebras of hypergroups have properties (connected with amenability) that are very different from those of groups.

We extend to locally compact abelian groups, Fejer's theorem on pointwise convergence of the Fourier transform. We prove that lim φU * f(y) = f (y) almost everywhere for any function f in the space (LP, l∞)(G) (hence in LP(G)), 2 ≤ p ≤ ∞, where {φU} is Simon's generalization to locally compact abelian groups of the summability Fejer Kernel. Using this result, we extend to locally compact abelian groups a theorem of F. Holland on the Fourier transform of unbounded measures of type q.