Harmonic Analysis, Several Complex Variables, and Partial Differential Equations

• Grant number: 25/03586-3
• Support Opportunities: Research Grants - Visiting Researcher Grant - International
• Start date: July 27, 2025
• End date: August 11, 2025
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Analysis
• Principal Investigator: Gustavo Hoepfner
• Grantee: Gustavo Hoepfner
• Visiting researcher: Andrew Seth Raich
• Visiting researcher institution: University of Arkansas, United States
• Host Institution: Centro de Ciências Exatas e de Tecnologia (CCET). Universidade Federal de São Carlos (UFSCAR). São Carlos , SP, Brazil

Abstract

The first part of this project can be considered a natural continuation of the recent work of Boggess and Raich [BR24], where the authors develop a robust theory $L^p$, $1\le p\le\infty$ of Hardy spaces for CR functions on embedded quadrics $M$ of$\mathbb C^n\times \mathbb C^m$.The main goal will be to extend the results to the range $p\le 1$. Note that this is not a simple adaptation of the results since the theory changes drastically for small values of $p$. A fundamental tool for the development and success of the project will be a semi-global or even global Baouendi-Treves approximation formula for these spaces.The second part is a new direction of our recent research that we have been working on the last 10 years and consists in to develop a new technique to recover estimates on inverses, heat operators, resolvents, and projections associated with a class of second order differential operators $L$ in $\R^n\times\R^m$ with $(x,t) \in \R^n\times\R^m$ where the coefficients $L$ are growing in $x$ at a controlled rate (e.g., polynomially) and translation invariant in $t$. Examples of such operators include the Grushin operator and its generalizations and the Kohn Laplacian on polynomial models. (AU)