The Properties Of Commutators Generated By Lipschitz Functions And Bochner-Riesz Operators

This paper researches the almost everywhere convergence of the commutator T_λ;b^rgenerated by Lipschitz functions and Bochner-Riesz operator and the boundedness ofT_λ;b^r from Lp(Rⁿ) to Lq(Rⁿ) for some index p.First, we study the almost everywhere convergence of the commutator T_λ;b^r, whenthe sum indexλis under the critical order (n - 1)/2 and f∈Lp, 2n/(n - 1 - 2λ). Inorder to get this result, we research the (2, 2) boundedness and the two-weighted (2, 2)boundedness of the maximal operator of the commutator generated by the multipliers ofcompact smooth functions and Lipschitz functions.Next, the paper studies the boundedness from L^p(Rⁿ) to L^q(Rⁿ) of the commutatorT_λ;b^r, when the indexλ> 0, where 2n/(n+2β)≤p≤2 and 1/q = 1/p-β/n. In order todo this, by the support measure of the compact multiplier, we get the boundedness from(?)(Rⁿ) to L²(Rⁿ), then by the interpolation theorem, we finally get some new results.The paper indicates the relationship of the sum index of the Bochner-Riesz means,Lipschitz index and the index p of the integral spaces of order p profoundly and we getsome valuable results.