Schatten Classes and Commutator in the Two Weight Setting, I. Hilbert Transform
Michael T. Lacey, Ji Li, Brett D. Wick
Abstract
Abstract We characterize the Hilbert–Schmidt class membership of commutator with the Hilbert transform in the two weight setting. The characterization depends upon the symbol of the commutator being in a new weighted Besov space. This follows from a Schatten class S p result for dyadic paraproducts, where $1< p < \infty $ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mn>1</mml:mn> <mml:mo><</mml:mo> <mml:mi>p</mml:mi> <mml:mo><</mml:mo> <mml:mi>∞</mml:mi> </mml:math> . We discuss the difficulties in extending the dyadic result to the full range of Schatten classes for the Hilbert transform.
Topics & Concepts
CommutatorHilbert spaceMathematicsHilbert transformClass (philosophy)Mathematical analysisPure mathematicsArtificial intelligenceAlgebra over a fieldComputer scienceStatisticsSpectral densityLie conformal algebraAdvanced Harmonic Analysis ResearchMathematical Analysis and Transform MethodsHolomorphic and Operator Theory