Note on the compactness of commutators

The optimal sufficient conditions for the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript p"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">L^p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -to- <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript q"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>q</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">L^q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> compactness of commutators of singular integral operators of both Calderón-Zygmund and of rough type are shown in the different exponent ranges “ <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q greater-than p"> <mml:semantics> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mi>p</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">q&gt;p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ,” “ <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q equals p"> <mml:semantics> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>=</mml:mo> <mml:mi>p</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">q=p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ” and “ <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q greater-than p"> <mml:semantics> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mi>p</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">q&gt;p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ” to quickly follow from each other. The approach is through classical compactness interpolation methods. We also present a new elementary “off-diagonal to diagonal” extrapolation principle for the compactness of commutators of linear operators, which is of independent interest.