Related papers: Three circles theorems for harmonic functions
We prove two rigidity theorems for open (complete and noncompact) $n$-manifolds $M$ with nonnegative Ricci curvature and the infimum of volume growth order $<2$. The first theorem asserts that the Riemannian universal cover of $M$ has…
We study the connection between the dimension of certain spaces of harmonic functions on a group and its geometric and algebraic properties. Our main result shows that (for sufficiently "nice" random walk measures) a connected, compactly…
This work is devoted to the study of parabolic frequency for solutions of the heat equation on Riemannian manifolds. We show that the parabolic frequency functional is almost increasing on compact manifolds with nonnegative sectional…
This paper investigates the concept of harmonic functions of bounded mean oscillation, starting from John-Nirenberg's pioneering studies, under a renewed formalism, suitable for bringing out some fundamental properties inherent in it. In…
We prove that every entire solution of the minimal graph equation that is bounded from below and has at most linear growth must be constant on a complete Riemannian manifold $M$ with only one end if $M$ has asymptotically non-negative…
We give a proof of the Donnelly-Fefferman growth bound of Laplace-Beltrami eigenfunctions which is probably the easiest and the most elementary one. Our proof also gives new quantitative geometric estimates in terms of curvature bounds…
Three spheres type theorem is proved for the p-harmonic functions defined on the complement of k-balls in the Euclidean n-dimensional space.
Suppose $(M^{n},g)$ is a Riemannian manifold with nonnegative Ricci curvature, and let $h_{d}(M)$ be the dimension of the space of harmonic functions with polynomial growth of growth order at most $d$. Colding and Minicozzi proved that…
We consider certain estimates involving averaging operators over curves and hypersurfaces that can be cast into a combinatorial framework. We show that hypersurfaces with nonzero rotational curvature satisfy the usual restricted weak-type…
In considering a class of quasilinear elliptic equations on a Riemannian manifold with nonnegative Ricci curvature, we give a new proof of Tolksdorf's result on the construction of separable $p$-harmonic functions in a cone.
Let $M$ be a complete simply connected manifold which is in addition Gromov hyperbolic, coercive and roughly starlike. For a given harmonic function on $M$, a local Fatou Theorem and a pointwise criteria of non-tangential convergence coming…
It is developed the theory of the Dirichlet problem for harmonic functions. On this basis, for the nondegenerate Beltrami equations in the quasidisks and, in particular, in the smooth domains, it is proved the existence of regular solutions…
In this paper, we study volume growth, Liouville theorem and the local gradient estimate for $f$-harmonic functions, and volume comparison property of unit balls in complete noncompact gradient Ricci shrinkers. We also study integral…
We study the Dirichlet problem for discrete harmonic functions in unbounded product domains on multidimensional lattices. First we prove some versions of the Phragm\'en-Lindel\"of theorem and use Fourier series to obtain a discrete analog…
We review and give elementary proofs of Liouville type properties of harmonic and subharmonic functions in the plane endowed with a complete Riemannian metric, and prove a gap theorem for the possible growth of harmonic functions when this…
We prove the convergence case of Khintchine's theorem, with general approximation functions that are not necessarily monotonic, for analytic nonplanar manifolds over local fields of positive characteristic. Our approach is based on the…
We prove a Liouville property for any $f$-harmonic function with polynomial growth on a complete noncompact smooth metric measure space $(M,g,e^{-f}dv)$ when the Bakry-\'Emery Ricci curvature is nonnegative and its diameter of geodesic…
Spaces of harmonic functions in upper half-space with controlled growth near the boundary are described in terms of multiresolution approximations. The results are applied to prove the law of the iterated logarithm for the oscillation of…
Synchronization of coupled harmonic oscillators is investigated. Coupling considered here is pairwise, unidirectional, and described by a nonlinear function (whose graph resides in the first and third quadrants) of some projection of the…
We solve the ten martini problem (Cantor spectrum with no condition on irrational frequencies, previously only established for the almost Mathieu) for a large class of one-frequency quasiperiodic operators, including nonperturbative…