Equivalent Characterizations for Boundedness of Maximal Singular Integrals on ax+b-Groups

摘要

Let (S,d,ρ) be the affine group ?n??+ endowed with the left-invariant Riemannian metric d and the right Haar measure ρ, which is of exponential growth at infinity. In this paper, for any linear operator T on (S,d,ρ) associated with a kernel K satisfying certain integral size condition and H?rmander's condition, the authors prove that the following four statements regarding the corresponding maximal singular integral T* are equivalent: T* is bounded from L∞c to BMO, T* is bounded on Lp for all p∈(1,∞), T* is bounded on Lp for some p∈(1,∞) and T* is bounded from L1 to L1,∞. As applications of these results, for spectral multipliers of a distinguished Laplacian on (S,d,ρ) satisfying certain Mihlin-H?rmander type condition, the authors obtain that their maximal singular integrals are bounded from L∞c to BMO, from L1 to L1,∞, and on Lp for all p∈(1,∞).