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Volume 33, Issue 1
Endpoint Estimates for Commutators of Fractional Integrals Associated to Operators with Heat Kernel Bounds

Xianjun Liu, Wenming Li & Xuefang Yan

DOI: 10.13447/j.1674-5647.2017.01.08

Commun. Math. Res., 33 (2017), pp. 73-84.

Published online: 2019-12

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  • Abstract

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. 

  • Keywords

fractional integral, commutator, $L{\rm log}L$ estimate, semigroup, sharp maximal function

  • AMS Subject Headings

42B20, 42B25, 47B38

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

liuxianjun@126.com (Xianjun Liu)

yanxuefang2008@163.com (Xuefang Yan)

  • BibTex
  • RIS
  • TXT
@Article{CMR-33-73, author = {Liu , XianjunLi , Wenming and Yan , Xuefang}, title = {Endpoint Estimates for Commutators of Fractional Integrals Associated to Operators with Heat Kernel Bounds}, journal = {Communications in Mathematical Research }, year = {2019}, volume = {33}, number = {1}, pages = {73--84}, abstract = {

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. 

}, issn = {2707-8523}, doi = {https://doi.org/10.13447/j.1674-5647.2017.01.08}, url = {http://global-sci.org/intro/article_detail/cmr/13447.html} }
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 -

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. 

Liu , XianjunLi , Wenming and Yan , Xuefang. (2019). Endpoint Estimates for Commutators of Fractional Integrals Associated to Operators with Heat Kernel Bounds. Communications in Mathematical Research . 33 (1). 73-84. doi:10.13447/j.1674-5647.2017.01.08
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