The weighted estimate for the commutator of the generalized fractional integral

Abstract

Abstract. Let L be the infinitesimal generator of an analytic semigroup on L 2 (R n ) with Gaussian kernel bound, and let L −α/2 be the fractional integral of L for 0 < α < n . Suppose that b is a locally integral function, then the commutator generated by b andMathematics subject classification (2010): 42B25, 42B20,42B35.

Weighted Norm Inequalities for Toeplitz Type Operator Related to Singular Integral Operator with Variable Kernel

“…When b belongs to the weighted Lipschitz spaces Lip β,ω , Hu and Gu [10] proved that [b, I α ] is bounded from L p (ω) to L q (ω 1−(1−α/n)q ) for 1/q = 1/p − (α + β)/n with 1 < p < n/(α + β). A similar result obtained when I α is replaced by the generalized fractional integral operator [7].…”

“…Chanillo [4] obtained a similar result when Calderón-Zygmund singular integral operators are replaced by the fractional integral operators. Recently, some Toeplitz type operators related to the singular integral operators are introduced, and the boundedness for the operators generated by singular integral operators and BMO functions or Lipschitz functions are obtained (see [5][6][7][8]).…”

Weighted norm inequalities for Toeplitz type operators associated to generalized Calderón–Zygmund operators

“…Paluszyński (1995) showed that (homogeneous Lipschitz space) if and only if is bounded from to where and When b belongs to the weighted Lipschitz spaces Hu and Gu (2008) proved that is bounded from to for with A similar result obtained when is replaced by the generalized fractional integral operator (Hu et al. 2013). …”