Abstract
In this paper we show that b ∈ Lip β , μ if and only if the commutator [ b , T ] of the multiplication operator by b and the singular integral operator T is bounded from L p ( μ ) to L q ( μ 1 − q ) , where 1 < p < q < ∞ , 0 < β < 1 and 1 / q = 1 / p − β / n . Also we will obtain that b ∈ Lip β , μ if and only if the commutator [ b , I α ] of the multiplication operator by b and the fractional integral operator I α is bounded from L p ( μ ) to L r ( μ 1 − ( 1 − α / n ) r ) , where 1 < p < ∞ , 0 < β < 1 and 1 / r = 1 / p − ( β + α ) / n with 1 / p > ( β + α ) / n .
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