Spaces of \(D_{L^p}\) type and a convolution product associated with the spherical mean operator (Q2566590)

Summary: We define and study the spaces \({\mathcal M}_p(\mathbb R\times\mathbb R^n)\), \(1\leq p\leq\infty\), that are of \(D_{L^p}\) type. Using the harmonic analysis associated with the spherical mean operator, we give a new characterization of the dual space \({\mathcal M}_p'(\mathbb R\times\mathbb R^n)\) and describe its bounded subsets. Next, we define a convolution product in \({\mathcal M}_p'(\mathbb R\times\mathbb R^n)\times {\mathcal M}_r(\mathbb R\times\mathbb R^n)\), \(1\leq r\leq p\leq\infty\), and prove some new results.