Periodic Reporting for period 2 - GAPT (Geometric Analysis and Potential Theory)
Reporting period: 2022-12-01 to 2024-05-31
The GAPT project aims to solve several long standing questions in geometric analysis by combining techniques from harmonic analysis, geometric measure theory, and free boundary problems. These questions deal with harmonic measure and caloric measure, square functions and rectifiability, and some related free boundary problems. A common feature is that their study involves multiscale methods from Littlewood-Paley theory and quantitative rectifiability.
Harmonic measure is a basic tool for the solution of the Dirichlet problem for the Laplace equation. The study of this notion is an old question which goes back to the 1910’s, at least. Recently there have been some striking advances on this topic, in part motivated by the deeper understanding of the connection between Riesz transforms and rectifiability. However, there are still related open compelling questions that this project aims to explore. The main one consists of finding a sharp bound for the Hausdorff dimension of harmonic measure. Other challenging questions arise in the parabolic setting, where the connection between the caloric measure associated with the heat equation and parabolic rectifiability is not well understood. Also, the study of the Lipschitz removability for the heat equation is more difficult than in the case of the Laplace equation.
Another exciting topic studied by this project deals with the relationship between rectifiability, square functions, and some free boundary problems. An important question concerns the characterization of the L2 boundedness of Riesz transforms in terms of the Jones-Wolff potential, essential to understand the behavior of Lipschitz harmonic capacity under bilipschitz maps. Square functions, techniques of quantitative rectifiability, and monotonicity formulas from free boundary problems also appear in other questions of interest to out project, such as the study of the blowups of the singular set in the one-phase and two-phase problems for harmonic measure and in problems of unique continuation at the boundary.
Harmonic measure is a basic tool for the solution of the Dirichlet problem for the Laplace equation. The study of this notion is an old question which goes back to the 1910’s, at least. Recently there have been some striking advances on this topic, in part motivated by the deeper understanding of the connection between Riesz transforms and rectifiability. However, there are still related open compelling questions that this project aims to explore. The main one consists of finding a sharp bound for the Hausdorff dimension of harmonic measure. Other challenging questions arise in the parabolic setting, where the connection between the caloric measure associated with the heat equation and parabolic rectifiability is not well understood. Also, the study of the Lipschitz removability for the heat equation is more difficult than in the case of the Laplace equation.
Another exciting topic studied by this project deals with the relationship between rectifiability, square functions, and some free boundary problems. An important question concerns the characterization of the L2 boundedness of Riesz transforms in terms of the Jones-Wolff potential, essential to understand the behavior of Lipschitz harmonic capacity under bilipschitz maps. Square functions, techniques of quantitative rectifiability, and monotonicity formulas from free boundary problems also appear in other questions of interest to out project, such as the study of the blowups of the singular set in the one-phase and two-phase problems for harmonic measure and in problems of unique continuation at the boundary.
Regarding our research about harmonic measure, one of the results we have obtained in the first half of the period of the ERC project GAPT shows that for a very general class of domains with fractal boundaries contained in a hyperplane, the dimension of harmonic measure is strictly smaller than the dimension of the boundary (i.e. there is a dimension drop). This result gives a partial positive answer to an old open problem.
A particularly outstanding connected question that we have solved (in collaboration with other mathematicians) is the L^p regularity problem for the Laplacian in rough domains, such as chord-arc domains. This problem was posed around 1990 by Carlos Kenig and it was up to now one of the main open problems in the area of elliptic PDE’s. Roughly speaking, it consists in proving that the gradient of a harmonic function in the interior of a domain is controlled, in the L^p space, by the tangential gradient of the function at the boundary of the domain.
In connection with the relationship between rectifiability, square functions, and some free boundary problems, we have already obtained very relevant results in the GAPT project. First, we have fully characterized the L2 boundedness of Riesz transforms in terms of a square function of geometric nature, the so-called Jones-Wolff potential. This result is essential to understand the behavior of Lipschitz harmonic capacity under bilipschitz maps and to describe removable singularities for Lipschitz harmonic functions in a geometric way. This completes a line of investigation which goes back to the formulation of the Painlevé problem for bounded analytic functions more than a hundred years ago.
Our second main result about rectifiability and square functions is the extension of Carleson's epsilon-conjecture to higher dimensions, The conjecture (now a theorem) characterizes rectifiability and existence of tangents in domains in terms of a square function analogous to the Carleson epsilon-square function in the plane. Further, our results show some surprising connections with the inequalities of Faber-Krahn and Friedland-Hayman involving quantitative estimates for the first Dirichlet eigenvalue of the Laplacian.
A particularly outstanding connected question that we have solved (in collaboration with other mathematicians) is the L^p regularity problem for the Laplacian in rough domains, such as chord-arc domains. This problem was posed around 1990 by Carlos Kenig and it was up to now one of the main open problems in the area of elliptic PDE’s. Roughly speaking, it consists in proving that the gradient of a harmonic function in the interior of a domain is controlled, in the L^p space, by the tangential gradient of the function at the boundary of the domain.
In connection with the relationship between rectifiability, square functions, and some free boundary problems, we have already obtained very relevant results in the GAPT project. First, we have fully characterized the L2 boundedness of Riesz transforms in terms of a square function of geometric nature, the so-called Jones-Wolff potential. This result is essential to understand the behavior of Lipschitz harmonic capacity under bilipschitz maps and to describe removable singularities for Lipschitz harmonic functions in a geometric way. This completes a line of investigation which goes back to the formulation of the Painlevé problem for bounded analytic functions more than a hundred years ago.
Our second main result about rectifiability and square functions is the extension of Carleson's epsilon-conjecture to higher dimensions, The conjecture (now a theorem) characterizes rectifiability and existence of tangents in domains in terms of a square function analogous to the Carleson epsilon-square function in the plane. Further, our results show some surprising connections with the inequalities of Faber-Krahn and Friedland-Hayman involving quantitative estimates for the first Dirichlet eigenvalue of the Laplacian.
The GAPT project has already obtained outstanding results, such as the solution of the L^p regularity problem of the Laplacian, the characterization of the L2 boundedness of Riesz transforms in terms of the Jones-Wolff potential, and the extension of Carleson’s epsilon conjecture to higher dimensions.
In the second half of the GAPT project we will study other mathematical problems beyond the state of the art. One of the main goals consists of finding a new sharper bound for the Hausdorff dimension of harmonic measure in higher dimensions. Other major questions deal with the connection between caloric measure, rectifiability, and the L^2 boundedness of the operator associated with the gradient of fundamental solution of the heat equation.
In the areas of unique continuation and free boundary problems there are also other compelling open questions which we plan to study. For example, it is still open if given a general Lipschitz domain, a harmonic function vanishing continuously in an open subset of the boundary and whose gradient vanishes in a subset of positive surface measure there, must vanish identically in the domain.
In the second half of the ERC project GAPT we expect to obtain major advances on some of the questions describes above.
In the second half of the GAPT project we will study other mathematical problems beyond the state of the art. One of the main goals consists of finding a new sharper bound for the Hausdorff dimension of harmonic measure in higher dimensions. Other major questions deal with the connection between caloric measure, rectifiability, and the L^2 boundedness of the operator associated with the gradient of fundamental solution of the heat equation.
In the areas of unique continuation and free boundary problems there are also other compelling open questions which we plan to study. For example, it is still open if given a general Lipschitz domain, a harmonic function vanishing continuously in an open subset of the boundary and whose gradient vanishes in a subset of positive surface measure there, must vanish identically in the domain.
In the second half of the ERC project GAPT we expect to obtain major advances on some of the questions describes above.