Weak type estimates on certain Hardy spaces for smooth cone type multipliers

Let ϱn∈C∞(Rd∖{0})ϱn∈C∞(Rd∖{0}) be a non-radial homogeneous distance function of degree n∈Nn∈N satisfying ϱn(tξ)=tnϱn(ξ)ϱn(tξ)=tnϱn(ξ). For f∈S(Rd+1)f∈S(Rd+1) and δ>0δ>0, we consider convolution operator Tϱnδ associated with the smooth cone type multipliers defined byTϱnδfˆ(ξ,τ)=(1−ϱn(ξ)|τ|n)+δfˆ(ξ,τ),(ξ,τ)∈Rd×R. If the unit sphere Σϱn≒{ξ∈Rd:ϱn(ξ)=1} is a convex hypersurface of finite type, then we prove that the operator Tϱnδ(p) maps from Hp(Rd+1)Hp(Rd+1), 0