Related papers: Coupling and Harnack inequalities for Sierpinski c…
We prove some sharp inequalities for complex harmonic functions on the unit disk. The results extend a M. Riesz conjugate function theorem and some well-known estimates for holomorphic functions. We apply some of results to the…
Fractal structures such as the Sierpi\'nski gasket have been predicted to enhance the critical temperature of s-wave superconductivity compared to regular crystals while maintaining macroscopic phase coherence of Cooper pairs. Here we…
This work is based on the double correlated linear aggregations of holes in checkerboard geometry. It is proved that the pairing function symmetry is - d(x*x-y*y), as been observed experimentally. It is also shown that there is a…
We prove a boundary Harnack inequality for jump-type Markov processes on metric measure state spaces, under comparability estimates of the jump kernel and Urysohn-type property of the domain of the generator of the process. The result holds…
We investigate several functional and geometric inequalities on the hyperbolic space $\mathbb{H}^N$, with a primary emphasis on logarithmic Sobolev inequalities, Poincar\'e inequalities, and Beckner-type inequalities, all studied within the…
We characterize Gaussian estimates for transition probability of a discrete time Markov chain in terms of geometric properties of the underlying state space. In particular, we show that the following are equivalent: (1) Two sided Gaussian…
We prove several differential Harnack inequalities for positive solutions to nonlinear backward heat equations with different potentials coupled with the Ricci flow. We also derive an interpolated Harnack inequality for the nonlinear heat…
We present a key algorithmic improvement to the generalized combinatorial Feynman--Vdovichenko method for calculating the critical temperature of the Ising model on Sierpinski carpets $SC_k(a,b)$, originally introduced in arxiv:1505.02699.…
In this paper, we give a few results on the local behavior of harmonic functions on the Sierpinski triangle - more precisely, of their restriction to a side of the triangle. First we present a general formula that gives the H\"older…
A Sierpi\'nski packing in the $2$-sphere is a countable collection of disjoint, non-separating continua with diameters shrinking to zero. We show that any Sierpi\'nski packing by continua whose diameters are square-summable can be…
We prove a uniform boundary Harnack inequality for nonnegative harmonic functions of the fractional Laplacian on arbitrary open set $D$. This yields a unique representation of such functions as integrals against measures on $D^c\cup…
We use the method of sliding paraboloids to establish a Harnack inequality for linear, degenerate and singular elliptic equation with unbounded lower order terms. The equations we consider include uniformly elliptic equations and linearized…
We establish Sobolev-Poincar\'e inequalities for piecewise $W^{1,p}$ functions over families of fairly general polytopic (thence also shape-regular simplicial and Cartesian) meshes in any dimension; amongst others, they cover the case of…
We prove that the critical probability for the Sierpinski carpet lattice in two dimensions is uniquely determined. The transition is sharp. This extends the Kumagai's result to the original Sierpinski carpet lattice.
We prove a Hardy inequality on convex sets, for fractional Sobolev-Slobodecki\u{\i} spaces of order $(s,p)$. The proof is based on the fact that in a convex set the distance from the boundary is a superharmonic function, in a suitable…
We study relations and characterizations of various elliptic Harnack inequalities for symmetric non-local Dirichlet forms on metric measure spaces. We allow the scaling function be state-dependent and the state space possibly disconnected.…
We construct symmetric self-similar diffusions with sub-Gaussian heat kernel estimates on two types of polygon carpets, which are natural generalizations of planner Sierpinski carpets (SC). The first ones are called perfect polygon carpets…
Generalised Sierpinski carpets are planar sets that generalise the well-known Sierpinski carpet and are defined by means of sequences of patterns. We study the structure of the sets at the kth iteration in the construction of the…
We provide a rich family of self-similar sets, called locally symmetric polygon-based self-similar sets, as examples of metric spaces having conductive homogeneity, which was introduced as a sufficient condition for the construction of…
We prove a critical Hardy inequality on the half-space by using the harmonic transplantation. Also we give an improvement of the subcritical Hardy inequality on the half-space, which converges to the critical Hardy inequality. Sobolev type…