Related papers: Rigidity of harmonic functions on the supercritica…
We prove quantitative homogenization results for harmonic functions on supercritical continuum percolation clusters--that is, Poisson point clouds with edges connecting points which are closer than some fixed distance. We show that, on…
We study the space of bandlimited Lipschitz functions in one variable. In particular we provide a geometrical description of the natural interpolating and sampling sequences for this space. We also find a description of the trace of such…
We study harmonic functions on random environments with particular emphasis on the case of the infinite cluster of supercritical percolation on $\mathbb{Z}^d$. We prove that the vector space of harmonic functions growing at most linearly is…
In the present paper, a systematic study is made of quantitative semicontinuity (a.k.a. Lipschitzian) properties of certain multifunctions, which are defined as a solution map associated to a family of parameterized ``split" feasibility…
We establish quantitative homogenization, large-scale regularity and Liouville results for the random conductance model on a supercritical (Bernoulli bond) percolation cluster. The results are also new in the case that the conductivity is…
We derive asymptotic estimates at infinity for positive harmonic functions in a large class of non-smooth unbounded domains. These include domains whose sections, after rescaling, resemble a Lipschitz cylinder or a Lipschitz cone, e.g.,…
This paper is concerned with boundary regularity estimates in the homogenization of elliptic equations with rapidly oscillating and high-contrast coefficients. We establish uniform nontangential-maximal-function estimates for the Dirichlet,…
We establish asymptotic bounds on the L^p norms of spectrally localized functions in the case of two-dimensional Dirichlet forms with coefficients of Lipschitz regularity. These bounds are new for the range p>6. A key step in the proof is…
Recently, it is proven that positive harmonic functions defined in the unit disc or the upper half-plane in $\mathbb{C}$ are contractions in hyperbolic metrics \cite{Markovic}. Furthermore, the same result does not hold in higher dimensions…
We study various properties of the gradients of solutions to harmonic functions on Lipschitz surfaces. We improve an exponential bound of Naber and Valtorta on the size of the superlevel sets for the frequency function to a sharp quadratic…
It is well-known that measures whose density is the form $e^{-V}$ where $V$ is a uniformly convex potential on $\RR^n$ attain strong concentration properties. In search of a notion of log-concavity on the discrete hypercube, we consider…
We prove that a positive function on the unit disk admits a harmonic majorant if and only if a certain logarithmic Lipschitz upper envelope of it (relevant because of the Harnack inequality) admits a superharmonic majorant. We discuss the…
In the context of a finite measure metric space whose measure satisfies a growth condition, we prove "T1" type necessary and sufficient conditions for the boundedness of fractional integrals, singular integrals, and hypersingular integrals…
To minimize or upper-bound the value of a function "robustly", we might instead minimize or upper-bound the "epsilon-robust regularization", defined as the map from a point to the maximum value of the function within an epsilon-radius. This…
An application of the Zalcman renormalization theorem to harmonic functions shows that the limit functions are nonconstant affine. Extensions of this method are given for maps with values in a torus or in a complex Lie groups. As an…
We show that the elasticity Hilbert complex with mixed boundary conditions on bounded strong Lipschitz domains is closed and compact. The crucial results are compact embeddings which follow by abstract arguments using functional analysis…
We explore the question whether Lipschitz functions of random variables under various forms of negative correlation satisfy concentration bounds similar to McDiarmid's inequality for independent random variables. We prove such a…
We consider an i.i.d. supercritical bond percolation on $\mathbb{Z}^d$, every edge is open with a probability $p > p_c (d)$, where $p_c (d)$ denotes the critical parameter for this percolation. We know that there exists almost surely a…
We present a construction of harmonic functions on bounded domains for the spectral fractional Laplacian operator and we classify them in terms of their divergent profile at the boundary. This is used to establish and solve boundary value…
We introduce different classical characteristics used to regularize a subharmonic function and compare them. As an application we give a complete proof of a useful characterization of the modulus of continuity of such functions in terms of…