Two characterizations of central BMO space via the commutators of Hardy operators
Zunwei Fu, Shanzhen Lu, Shaoguang Shi
Abstract
Abstract This article addresses two characterizations of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>BMO</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mi>n</m:mi> </m:msup> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {\mathrm{BMO}(\mathbb{R}^{n})} -type space via the commutators of Hardy operators with homogeneous kernels on Lebesgue spaces: (i) characterization of the central <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>BMO</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mi>n</m:mi> </m:msup> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {\mathrm{BMO}(\mathbb{R}^{n})} space by the boundedness of the commutators; (ii) characterization of the central <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>BMO</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mi>n</m:mi> </m:msup> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {\mathrm{BMO}(\mathbb{R}^{n})} -closure of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msubsup> <m:mi>C</m:mi> <m:mi>c</m:mi> <m:mi mathvariant="normal">∞</m:mi> </m:msubsup> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mi>n</m:mi> </m:msup> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {C_{c}^{\infty}(\mathbb{R}^{n})} space via the compactness of the commutators. This is done by exploiting the center symmetry of Hardy operator deeply and by a more explicit decomposition of the operator and the kernel function.