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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 and L −α/2 is defined by Mathematics subject classification (2010): 42B25, 42B20,42B35.doi:10.7153/jmi-07-30 fatcat:bswhpbyv3jdchfzsuzsqiqztbm