Weak type estimates for cone type multipliers associated with a convex polygon

Let P be a convex polygon in ℝ² which contains the origin in its interior. Let p be the associated Minkowski functional defined by ρ(ξ) = inf{ε > 0: ε^-1 ξ ∈ P), ξ ≠ 0. We consider the family of convolution operators T^δ associated with cone type multipliers (1- ρ(ξ) ²/τ²)^δ₊, (ξ, τ) ∈ ℝ² × ℝ, and show that T^δ is of weak type (p, p) on H^p (ℝ³), 1/2 < p < 1 for the critical value 5 = 2 (1 / p - 1).