Compact Differences of Composition Operators on Bergman Spaces Induced by Doubling Weights
Bin Liu, Jouni Rättyä, Fanglei Wu
Abstract
Abstract Bounded and compact differences of two composition operators acting from the weighted Bergman space $$A^p_\omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>A</mml:mi> <mml:mi>ω</mml:mi> <mml:mi>p</mml:mi> </mml:msubsup> </mml:math> to the Lebesgue space $$L^q_\nu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>L</mml:mi> <mml:mi>ν</mml:mi> <mml:mi>q</mml:mi> </mml:msubsup> </mml:math> , where $$0<q<p<\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo><</mml:mo> <mml:mi>q</mml:mi> <mml:mo><</mml:mo> <mml:mi>p</mml:mi> <mml:mo><</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:math> and $$\omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ω</mml:mi> </mml:math> belongs to the class "Equation missing" of radial weights satisfying two-sided doubling conditions, are characterized. On the way to the proofs a new description of q -Carleson measures for $$A^p_\omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>A</mml:mi> <mml:mi>ω</mml:mi> <mml:mi>p</mml:mi> </mml:msubsup> </mml:math> , with $$p>q$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>></mml:mo> <mml:mi>q</mml:mi> </mml:mrow> </mml:math> and "Equation missing", involving pseudohyperbolic discs is established. This last-mentioned result generalizes the well-known characterization of q -Carleson measures for the classical weighted Bergman space $$A^p_\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>A</mml:mi> <mml:mi>α</mml:mi> <mml:mi>p</mml:mi> </mml:msubsup> </mml:math> with $$-1<\alpha <\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo><</mml:mo> <mml:mi>α</mml:mi> <mml:mo><</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:math> to the setting of doubling weights. The case "Equation missing" is also briefly discussed and an open problem concerning this case is posed.