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

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

liuxianjun@126.com (Xianjun Liu)

yanxuefang2008@163.com (Xuefang Yan)

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@Article{CMR-33-73, author = {Xianjun and Liu and liuxianjun@126.com and 5582 and College of Mathematics and Information Science, Hebei Normal University, Shijiazhuang, 050024 and Xianjun Liu and Wenming and Li and and 5583 and College of Mathematics and Information Science, Hebei Normal University, Shijiazhuang, 050024 and Wenming Li and Xuefang and Yan and yanxuefang2008@163.com and 5584 and College of Mathematics and Information Science, Hebei Normal University, Shijiazhuang, 050024 and Xuefang Yan}, 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.

Xianjun Liu, Wenming Li & Xuefang Yan. (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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