Singular Integral Operators of Convolution Type on Jacobi Hypergroup

Abstract

We shall define a Calderon-Zygmund class \(\mathrm{CZ}^p(\Delta _{\alpha ,\beta })\) on the Jacobi hypergroup \((\mathbf{R}_+,\Delta _{\alpha ,\beta }, *)\) such that, if a function g on \(\mathbf{R}_+\) belongs to \(\mathrm{CZ}^p(\Delta _{\alpha ,\beta })\), then the convolution operator \(g*\) is bounded from \(L^q(\Delta _{\alpha ,\beta })\) to itself for \(p\le q\le 2\). Actually, we shall obtain a relation between the \(L^p\) norms of g and the Abel transform \(\mathcal{A}g\) and a transference principle between the \(L^p\) operator norms of \(g*\) and the Euclidean operator \(\phi \circledast \), where \(\phi (x)=e^{(\frac{2}{p}-1)\rho x}\mathcal{A}g(x)\). Therefore, to define the Calderon-Zygmund class \(\mathrm{CZ}^p(\Delta _{\alpha ,\beta })\), we shall obtain some conditions on g under which \(\phi \) belongs to \(\mathrm{CZ}(\mathbf{R})\). Then, \(\phi \circledast \) is bounded on \(L^q({\mathbf{R}})\) and, by the transference principle, \(g*\) is bounded on \(L^q(\Delta _{\alpha ,\beta })\).