Carleson Embeddings into Weighted Outer Measure Spaces

Lewers, Mark, Mathematics - Graduate School of Arts and Sciences, University of Virginia
Do, Yen, AS-Mathematics, University of Virginia

Recent progress in harmonic analysis in obtaining Lp norm inequalities for modulation invariant operators has been in part due to the formalization of time-frequency analysis methods under an outer measure framework developed in \cite{dt2015}. The framework codifies the underlying nature of such an analysis and shifts the difficulty in proofs to obtaining Lp norm estimates on projection operators mapping from a classic Lp space into an outer measure Lp space; such maps are referred to as (generalized) Carleson embeddings. This dissertation seeks to extend known generalized Carleson embeddings in outer Lp theory from non-weighted settings to weighted settings. The highlight estimate of this work is a generalized Carleson embedding for the wave packet transform
of a function f in Lp(R,w) into a weighted outer Lp space situated in upper 3-space for exponents p>2 and Muckenhoupt weights w in A_{p/2}. The wave packet transform is a projection of modulation invariant operators into upper 3-space as mentioned in \cite{dt2015,dpo2018,uraltsev2016} and generalized Carleson embeddings of this transform are known in non-weighted settings. The proof utilizes weighted phase plane techniques adapted to a continuum along with new restriction L2 estimates for the wave packet transform.

PHD (Doctor of Philosophy)
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