Weak type estimates of square functions associated with quasiradial Bochner–Riesz means on certain Hardy spaces

Let d ∈ C∞(Rn \ {0}) be a non-radial homogeneous distance function of degree d ∈ N satisfying d (tξ ) = td d (ξ ). For f ∈ S(Rn), we define square functions Gδ d f (x) associated with quasiradial Bochner–Riesz means Rδ d ,tf of index δ by Gδ d f (x) = ∞ 0 Rδ+1 d ,tf (x) −Rδ d ,tf (x) 2 dt t 1/2 where Rδ d ,tf (x) = F−1[(1 − d /td )δ +fˆ](x). If {ξ ∈ Rn: d (ξ ) = 1} is a smooth convex hypersurface of finite type, then we prove in an extremely easy way that Gδ d is well-defined on Hp(Rn) when δ = n(1/p − 1/2) − 1/2 and 0