Commutators on Weighted Morrey Spaces on Spaces of Homogeneous Type
Ruming Gong, Ji Li, Elodie Pozzi, Manasa N. Vempati
Abstract
Abstract In this paper, we study the boundedness and compactness of the commutator of Calderón– Zygmund operators T on spaces of homogeneous type ( X , d , µ ) in the sense of Coifman and Weiss. More precisely, we show that the commutator [ b , T ] is bounded on the weighted Morrey space <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:mrow> <m:msubsup> <m:mrow> <m:mi>L</m:mi> </m:mrow> <m:mi>ω</m:mi> <m:mrow> <m:mi>p</m:mi> <m:mo>,</m:mo> <m:mi>k</m:mi> </m:mrow> </m:msubsup> <m:mrow> <m:mo>(</m:mo> <m:mi>X</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> L_\omega ^{p,k}\left( X \right) with κ ∈ (0, 1) and ω ∈ A p ( X ), 1 < p < ∞, if and only if b is in the BMO space. We also prove that the commutator [ b , T ] is compact on the same weighted Morrey space if and only if b belongs to the VMO space. We note that there is no extra assumptions on the quasimetric d and the doubling measure µ .