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

Abstract

Let ϱ n ∈ C ∞ ( R d ∖ { 0 } ) be a non-radial homogeneous distance function of degree n ∈ N satisfying ϱ n ( t ξ ) = t n ϱ n ( ξ ) . For f ∈ S ( R d + 1 ) and δ > 0 , we consider convolution operator T ϱ n δ associated with the smooth cone type multipliers defined by T ϱ n δ f ˆ ( ξ , τ ) = ( 1 − ϱ n ( ξ ) | τ | n ) + δ f ˆ ( ξ , τ ) , ( ξ , τ ) ∈ R d × R . If the unit sphere Σ ϱ n ≒ { ξ ∈ R d : ϱ n ( ξ ) = 1 } is a convex hypersurface of finite type, then we prove that the operator T ϱ n δ ( p ) maps from H p ( R d + 1 ) , 0 < p < 1 , into weak- L p ( R d + 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.