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

摘要

Let (S,d,ρ) be the affine group ℝ n ⋉ℝ+ 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 Hörmander’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¥L_{c}^{infty} to BMO, T ∗ is bounded on L p for all p∈(1,∞), T ∗ is bounded on L p for some p∈(1,∞) and T ∗ is bounded from L 1 to L 1,∞. As applications of these results, for spectral multipliers of a distinguished Laplacian on (S,d,ρ) satisfying certain Mihlin-Hörmander type condition, the authors obtain that their maximal singular integrals are bounded from L_c¥L_{c}^{infty} to BMO, from L 1 to L 1,∞, and on L p for all p∈(1,∞).