Let $(M,d)$ be a metric space, equipped with a Borel measure $\mu$ satisfying suitable compatibility conditions. An amalgam $A_p^q\left(M\right)$ is a space which looks locally like $L^p\left(M\right)$ but globally like $L^q\left(M\right)$. We consider the case where the measure $\mu \left(B\right(x,\rho )$ of the ball $B(x,\rho )$ with centre $x$ and radius $\rho$ behaves like a polynomial in $\rho$, and consider the mapping properties between amalgams of kernel operators where the kernel $kerK(x,y)$ behaves like $d(x,y{)}^{-a}$ when $d(x,y)\le 1$ and like $d(x,y{)}^{-b}$ when $d(x,y)\ge 1$. As an application, we describe Hardy–Littlewood–Sobolev type regularity theorems...

In this paper we obtain the $L^p$-boundedness of Riesz transforms for the Dunkl transform for all $1\<p\<\infty$.

We show that a comeager Π₁¹ hereditary family of compact sets must have a dense $G_{\delta }$ subfamily which is also hereditary. Using this, we prove an “abstract” result which implies the existence of independent ℳ ₀-sets, the meagerness of ₀-sets with the property of Baire, and generalizations of some classical results of Mycielski. Finally, we also give some natural examples of true $F_{\sigma \delta }$ sets.

Let $H_n$ be the Heisenberg group of dimension $2n+1$. Let ${\mathcal{L}}_1,\mathrm{\dots },{\mathcal{L}}_n$ be the partial sub-Laplacians on $H_n$ and $T$ the central element of the Lie algebra of $H_n$. We prove that the kernel of the operator $m\left({\mathcal{L}}_1,\mathrm{\dots },{\mathcal{L}}_n,-iT\right)$ is in the Schwartz space $S\left(H_n\right)$ if $m\in S\left({\mathbb{R}}^{n+1}\right)$. We prove also that the kernel of the operator $h\left({\mathcal{L}}_1,\mathrm{\dots },{\mathcal{L}}_n\right)$ is in $S\left(H_n\right)$ if $h\in S\left({\mathbb{R}}^n\right)$ and that the kernel of the operator $g\left(\mathcal{L},-iT\right)$ is in $S\left(H_n\right)$ if $g\in S\left({\mathbb{R}}^2\right)$. Here $\mathcal{L}={\mathcal{L}}_1+\mathrm{\dots }+{\mathcal{L}}_n$ is the Kohn-Laplacian on $H_n$.