We derive a priori second-order estimates for solutions of a class of fully nonlinear elliptic equations on Riemannian manifolds under structure conditions which are close to optimal. We treat both equations on closed manifolds and the Dirichlet problem on manifolds with boundary without any geometric restrictions to the boundary. These estimates yield regularity and existence results, some of which are new even for equations in Euclidean space.

Let $V$ be a projective hypersurface of fixed degree and dimension which has only isolated singular points. We show that, if the sum of the Milnor numbers at the singular points of $V$ is large, then $V$ cannot have a point of large multiplicity, unless $V$ is a cone. As an application, we give an affirmative answer to a conjecture of Dimca and Papadima.

For oscillatory functions on local fields coming from motivic exponential functions, we show that integrability over ${\mathbb{Q}}_p^n$ implies integrability over ${\mathbb{F}}_p\left(\right(t){)}^n$ for large $p$, and vice versa. More generally, the integrability only depends on the isomorphism class of the residue field of the local field, once the characteristic of the residue field is large enough. This principle yields general local integrability results for Harish-Chandra characters in positive characteristic as we show in other work. Transfer principles for related conditions such as boundedness and local integrability are also obtained. The proofs rely on a thorough study of loci of integrability, to which we give a geometric meaning by relating them to zero loci of functions of a specific kind.

We present the converse to a higher-dimensional, scale-invariant version of the classical F. and M. Riesz theorem, proved by the first two authors. More precisely, for $n\ge 2$, for an Ahlfors–David regular domain $\Omega \subset {\mathbb{R}}^{n+1}$ which satisfies the Harnack chain condition plus an interior (but not exterior) corkscrew condition, we show that absolute continuity of the harmonic measure with respect to the surface measure on $\partial \Omega$, with scale-invariant higher integrability of the Poisson kernel, is sufficient to imply quantitative rectifiability of $\partial \Omega$.