Carleson Embeddings into Weighted Outer Measure Spaces

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.