$B$-Riesz Transforms Generated by Generalized Translate Operator on $HM^p_{q,{\Delta_{\nu}}}$ Hardy-Morrey Spaces

We study the decomposition of Hardy-Morrey spaces via atoms and molecules, which have similar properties $H^{p}_{\Delta_{\nu}}(\mathbb{R}^{n}_{+})$ Hardy spaces. Then we introduce $HM^p_{q,{\Delta_{\nu}}}$ boundedness $ B $-Riesz transforms generated by a generalized translate operator that is associated to Laplace Bessel for $0<p\leq 1<q\leq \infty$ with $p\neq q$ through atomic molecular characterization.