Bochner–Riesz means associated with conical surfaces

Let d 1 and d 2 be two nonnegative integers greater than 2. We study the Fourier multiplier T λ associated with a conical surface S = { ( ξ , τ ) ∈ R d 1 × R d 2 : | ξ | = | τ | } , defined by T λ f ̂ ( ξ , τ ) = ( 1 − | ξ | 2 | τ | 2 ) + λ Ψ ( | τ | ) f ̂ ( ξ , τ ) , ( ξ , τ ) ∈ R d 1 × R d 2 , where Ψ is a smooth function defined on R , that is supported in ( 1 / 2 , 2 ) .