Quantitative Weighted Estimates of the Lq-Type Rough Singular Integral Operator and Its Commutator

Let Ω be a homogeneous function of degree zero on Rn, n≥2, integrable and having mean value zero on the unit sphere Sn−1, and let TΩ be the homogeneous convolution singular integral operator with kernel Ω(x)|x|n. By introducing reasonable refined decomposition and approximation techniques, together with sparse domination and variable measure interpolation methods, we establish the quantitative A1−A∞ weighted estimates for TΩ under the rough condition Ω∈Lq(Sn−1) for some q∈(1,∞). The results of the paper improve the previous works for the case Ω∈L∞(Sn−1). We also give the quantitative A1−A∞ weighted estimates for the commutator [b,TΩ] with BMO symbol b.