## Mid-Term Report Summary - HAPDEGMT (Harmonic Analysis, Partial Differential Equations and Geometric Measure Theory)

This project lies between the interface of three areas: Harmonic Analysis, Partial Differential Equations and Geometric Measure theory. The fruitful interaction between these three fields allows us to tackle problems which are motivated by elliptic PDEs and which also require the use of techniques from Harmonic Analysis and Geometric Measure Theory. In some problems our operators are nice (e.g., the Laplacian) but the domains are not topologically friendly and their boundaries are rough. Other times, even for nice domains, we treat rough complex coefficients, systems or even allow the ellipticity to have some degeneracy.

We have studied the link that exists between the harmonic (or elliptic) measure associated with an open set and the rectifiability of the boundary. These results are motivated by the classical F. and M. Riesz theorem (and its quantitative version proved by Lavrentiev) who showed that for a simply connected domain in the complex plane with a rectifiable boundary, harmonic measure is absolutely continuous with respect to arclength measure. In this direction we have obtained the following results:

- For open sets with locally finite surface measure, the fact that surface measure is absolutely continuous with respect to harmonic measure implies rectifiability of the boundary, and the converse holds assuming some weak lower ADR condition and some infinitesimal interior thickness.

- For domains with ADR boundary, weak-$A_\infty$ of harmonic measure (or $p$-harmonic measure) implies uniform rectifiability of the boundary.

- In the domain given by the complement of a UR set, all bounded harmonic functions satisfy Carleson measure estimates and are $\epsilon$-approximable.

- For 1-sided CAD domains (ADR boundary and interior corkscrew and Harnack chain) we have shown the following:

- Uniform rectifiability implies exterior corkscrews.

- Harmonic measure (or certain elliptic measures) in $A_\infty$ implies exterior corkscrews.

- Rectifiability implies qualitative exterior corkscrews.

- $A_\infty$ is preserved by Carleson perturbations of the coefficients.

In the upper-half space we have studied boundary value problems for elliptic systems with constant complex coefficients (e.g., Lamé system of elasticity) obtaining well-posedness of the Dirichlet problem on very general Köthe function spaces such as Lebesgue spaces, variable exponent Lebesgue spaces, Lorentz spaces, Zygmund spaces, as well as their weighted versions. We have shown that the associated Poisson kernel generates a strongly continuous semigroup on $L^p$ and we have identified its infinitesimal generator (which is the Dirichlet-to-Normal map) and its domain (the subspace of the $L^p$-Sobolev space for which the regularity problem is solvable). We have also obtained the well-posedness of the BMO and VMO Dirichlet problems and as a consequence we have found nice classes of dense functions in VMO, improving Sarason’s classical result.

For Semmes-Kenig-Toro domains we have studied BVP with data in weighted Lebesgue spaces. For higher order Sobolev regularity problems we have linked the well-posedness and the nature of the outward unit normal exhibited through its multiplier properties and oscillatory behavior.

For second order divergence form elliptic operators with complex coefficients we have obtained weighted norm inequalities for the associated conical square functions which allow us to characterize the corresponding weighted Hardy spaces in various forms. We have also considered weighted norm inequalities for the operators associated with degenerate elliptic operators. As a consequence of the developed theory, we have identified a class of degenerate elliptic operators for which the $L^2$-Kato problem can be solved, that is, the square root of the operator is comparable to the gradient on $L^2$.

We have studied the link that exists between the harmonic (or elliptic) measure associated with an open set and the rectifiability of the boundary. These results are motivated by the classical F. and M. Riesz theorem (and its quantitative version proved by Lavrentiev) who showed that for a simply connected domain in the complex plane with a rectifiable boundary, harmonic measure is absolutely continuous with respect to arclength measure. In this direction we have obtained the following results:

- For open sets with locally finite surface measure, the fact that surface measure is absolutely continuous with respect to harmonic measure implies rectifiability of the boundary, and the converse holds assuming some weak lower ADR condition and some infinitesimal interior thickness.

- For domains with ADR boundary, weak-$A_\infty$ of harmonic measure (or $p$-harmonic measure) implies uniform rectifiability of the boundary.

- In the domain given by the complement of a UR set, all bounded harmonic functions satisfy Carleson measure estimates and are $\epsilon$-approximable.

- For 1-sided CAD domains (ADR boundary and interior corkscrew and Harnack chain) we have shown the following:

- Uniform rectifiability implies exterior corkscrews.

- Harmonic measure (or certain elliptic measures) in $A_\infty$ implies exterior corkscrews.

- Rectifiability implies qualitative exterior corkscrews.

- $A_\infty$ is preserved by Carleson perturbations of the coefficients.

In the upper-half space we have studied boundary value problems for elliptic systems with constant complex coefficients (e.g., Lamé system of elasticity) obtaining well-posedness of the Dirichlet problem on very general Köthe function spaces such as Lebesgue spaces, variable exponent Lebesgue spaces, Lorentz spaces, Zygmund spaces, as well as their weighted versions. We have shown that the associated Poisson kernel generates a strongly continuous semigroup on $L^p$ and we have identified its infinitesimal generator (which is the Dirichlet-to-Normal map) and its domain (the subspace of the $L^p$-Sobolev space for which the regularity problem is solvable). We have also obtained the well-posedness of the BMO and VMO Dirichlet problems and as a consequence we have found nice classes of dense functions in VMO, improving Sarason’s classical result.

For Semmes-Kenig-Toro domains we have studied BVP with data in weighted Lebesgue spaces. For higher order Sobolev regularity problems we have linked the well-posedness and the nature of the outward unit normal exhibited through its multiplier properties and oscillatory behavior.

For second order divergence form elliptic operators with complex coefficients we have obtained weighted norm inequalities for the associated conical square functions which allow us to characterize the corresponding weighted Hardy spaces in various forms. We have also considered weighted norm inequalities for the operators associated with degenerate elliptic operators. As a consequence of the developed theory, we have identified a class of degenerate elliptic operators for which the $L^2$-Kato problem can be solved, that is, the square root of the operator is comparable to the gradient on $L^2$.