This paper establishes fundamental results for elliptic Riesz means \(E_{R}^{\delta }.\) We establish its boundedness on Hardy spaces \(H^{p}({\mathbb {R}}^{n})\) using both atomic decomposition and Riesz transform characterizations of \(H^{p}({\mathbb {R}}^{n}).\) Furthermore, we obtain the boundedness of \(E_{R}^{\delta }\) on Triebel–Lizorkin spaces. Under an additional mild condition on the kernel of \(E_{R}^{\delta },\) we study the associated maximal operator. At the critical index \(\delta _p = n/p - (n+1)/2,\) we obtain \(H^{p}({\mathbb {R}}^{n}) \rightarrow L^{p,\infty }({\mathbb {R}}^{n})\) boundedness for this maximal operator, thereby extending the Stein–Taibleson–Weiss theorem. Finally, we also analyze the almost everywhere convergence of a class of elliptic Riesz means \(\Omega _{R}^{\delta _{p}}\) on Hardy–Sobolev spaces \(I_{\lambda }(H^{p})({\mathbb {R}}^n),\) establishing the precise smoothness-convergence rate relationship.
