Weak Type Estimates of the Maximal Quasiradial Bochner-riesz Operator on Certain Hardy Spaces

LetM be a real-valued n×nmatrix whose eigenvalues have positive real parts. Then we consider the dilation group {At}t>0 in R generated by the infinitesimal generator M , where At = exp(M log t) for t > 0. We introduce At-homogeneous distance functions ̺ defined on R; that is, ̺ : R → [0,∞) is a continuous function satisfying ̺(Atξ) = t̺(ξ) for all ξ ∈ R. One can refer to [3] and [11] for its fundamental properties. In what follows we shall denote by Σ̺ ; {ξ ∈ R| ̺(ξ) = 1} the unit sphere of ̺ and denote by R0 = R n \ {0}. We use the polar coordinates; given x ∈ R, we write x = rθ where r = |x| and θ = (θ1, θ2, · · · , θn) ∈ S. Given two quantities A and B, we write A . B or B & A if there is a positive constant c ( possibly depending on the dimension n and the index p to be given ) such that A ≤ cB. We also write A ∼ B if A . B and B . A.