My research interests lie in the area of real harmonic analysis and Singular Integral Operators (SIOs). I specialize in techniques from time-scale-frequency analysis and have works on intrinsic questions in that field, applications to functional analysis and Banach space geometry, and applications to probabilistic dispersive PDEs.

Outer Lebesgue spaces

Outer Lebesgue spaces have found great use in formalizing, cleaning up, and streamlining quite complex arguments in time-scale-frequency proofs and beyond.

The theory was first introduced by Do and Thiele in $L^{p}$ theory for outer measures and two themes of Lennart Carleson united. I expanded this theory in my PhD thesis and applied it to the variational Carleson operator. Since then I have made use of this framework to prove many results in Banach space-valued harmonic analysis, uniform bounds in time-scale-frequency analysis, as well as to encompass harmonic analysis results like sparse domination of the variational Carleson operator.

Outer Lebesgue spaces provide a consistent functional analytic framework for dealing with embedding maps. Many abstract questions (duality, interpolation etc.) about outer Lebesgue spaces remain open.

Duality in Outer Lebesgue theory has been studied abstractly by Marco Fraccaroli 🔗 🔗. Interesting endpoint results have been obtained by Di Plinio and Fragos in “The weak-type Carleson theorem via wave packet estimates”.

I am currently interested in Outer Lebesgue spaces in the context of higher order (polynomial frequency) phenomena.

Banach space-valued harmonic analysis

Most results about singular integral operators were originally formulated for functions valued in $\mathbb{C}$. Many applications in PDEs, ergodic theory, probability, and geometry require analogous results functions valued in Banach spaces:

Calderón-Zygmund theory, a cornerstone of harmonic analysis, has been successfully adapted to the Banach-space-valued setting. Few results were known for time-scale-frequency analysis techniques in this setting. Outer Lebesgue spaces admit generalizations to this setting that with A. Amenta we investigated in a sequence of papers 🔗 🔗 🔗.

I am interested in multi-linear operators valued in Banach spaces as well as non-commutative time-scale-frequency analysis. This can provide interesting insights into the intrinsic geometry of Banach spaces both on their own (linear theory) and when related to each other (multi-linear Banach space geometry).

Uniform bounds singular Brascamp-Lieb

I have given a full answer to the question of the full range of uniform bounds for the bilinear Hilbert transform. This question has been open for a long time with partial results of high impact by Thiele, Lacey, Li, Grafakos, Oberlin 🔗, 🔗, 🔗, 🔗.

Many multilinear singular integral operators in harmonic analysis fall under the umbrella of singular Brascamp-Lieb operators. My result opens the possibility of showing stability singular Brascamp-Lieb operators constants under geometric perturbations.

Uniform bounds unify the study of Calderón-Zygmund and time-scale-frequency operators.

Stochastic dispersive PDEs

Nonlinear Schrödinger equations arise when studying light propagation in nonlinear mediums, Bose-Einstein condensates, etc. Because of its “dispersive” nature different frequencies (colors) of solutions travel at different speeds. This motivates using Fourier analysis to understand this equation.

Existence and properties of solutions for short times have been well understood. In particular, solutions may fail to exist if the intial data is has “energy supercritical” regularity.

In these circumstances one studies probabilistic well posedness: how probable is it that a solution exists even in this “energy supercritical” regime. With J. Földes and J.B. Casteras we recover and generalize current results for the Schrödinger equation and significantly improve results for more general differential operators.