TY - JOUR
T1 - Endpoint Estimates for Commutators of Fractional Integrals Associated to Operators with Heat Kernel Bounds
AU - Liu , Xianjun
AU - Li , Wenming
AU - Yan , Xuefang
JO - Communications in Mathematical Research 
VL - 1
SP - 73
EP - 84
PY - 2019
DA - 2019/12
SN - 33
DO - http://doi.org/10.13447/j.1674-5647.2017.01.08
UR - https://global-sci.org/intro/article_detail/cmr/13447.html
KW - fractional integral, commutator, $L{\rm log}L$ estimate, semigroup, sharp maximal function
AB - <p style="text-align: justify;">Let $L$ be the infinitesimal generator of an analytic semigroup on $L^2({\bf R}^n)$ with pointwise upper bounds on heat kernel, and denote by $L^{-\alpha/2}$ the fractional integrals of L. For a BMO function $b(x)$, we show a weak type $L{\rm log}L$ estimate of the commutators $[b,\ L^{-\alpha/2}](f)(x)=b(x)L^{-\alpha/2}(f)(x)-L^{-\alpha/2}(bf)(x)$. We give applications to large classes of differential operators such as the Schrödinger operators and second-order elliptic operators of divergence form.&nbsp;</p>