Bilinear Calder\'{o}n-Zygmund operators on Vilenkin groups

Abstract

In this article, we study bilinear Calder\'on--Zygmund operators on a Vilenkin group $G$. As a preliminary step, we establish a Grafakos--Torres-type endpoint weak-type result in our setting. Furthermore, we prove that such operators extend to bounded bilinear mappings from $L^{p_1}(G)\times L^{p_2}(G)$ into $L^p(G)$ under the natural condition $\frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2}.$ We then obtain a corresponding boundedness result in Morrey spaces, showing that these operators extend to bounded bilinear mappings from $\mathcal{M}_{p_1,u_1}(G)\times \mathcal{M}_{p_2,u_2}(G)$ into $\mathcal{M}_{p,u}(G)$ under suitable assumptions. These results generalize the classical bilinear estimates to the setting of Vilenkin groups.