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

Let n ∈ C∞(Rd \{0}) be a non-radial homogeneous distance function of degree n ∈ N satisfying n(tξ ) = tn n(ξ ). For f ∈ S(Rd+1) and δ > 0, we consider convolution operator T δ n associated with the smooth cone type multipliers defined by T δ n f (ξ,τ) = 1− n(ξ ) |τ |n δ + fˆ(ξ, τ), (ξ, τ) ∈ Rd ×R. If the unit sphere Σ n {ξ ∈ Rd : n(ξ ) = 1} is a convex hypersurface of finite type, then we prove that the operator T δ(p) n maps from Hp(Rd+1), 0 < p < 1, into weak-Lp(Rd+1) for the critical index δ(p) = d(1/p − 1/2) − 1/2. In addition, we discuss some relation between this result and some PDE’s like the wave equation and the Schrödinger equation.  2005 Elsevier Inc. All rights reserved