# Analysis, Probability and Mathematical Physics on Fractals

edited by Patricia Alonso Ruiz, Joe P Chen, Luke G Rogers, Robert S Strichartz and Alexander Teplyaev

In the 50 years since Mandelbrot identified the fractality of coastlines, mathematicians and physicists have developed a rich and beautiful theory describing the interplay between analytic, geometric and probabilistic aspects of the mathematics of fractals. Continue reading

# Besov class via heat semigroup on Dirichlet spaces I: Sobolev type inequalities

by Patricia Alonso-Ruiz, Fabrice Baudoin, Li Chen, Luke Rogers, Nageswari Shanmugalingam, Alexander Teplyaev

We introduce heat semigroup-based Besov classes in the general framework of Dirichlet spaces. General properties of those classes are studied and quantitative regularization estimates for the heat semigroup in this scale of spaces are obtained. Continue reading

# The Strong Maximum Principle for Schrödinger operators on fractals

by Marius V. Ionescu, Kasso A. Okoudjou, Luke G. Rogers

We prove a strong maximum principle for Schrödinger operators defined on a class of fractal sets and their blowups without boundary. Our primary interest is in weaker regularity conditions than have previously appeared in the literature; in particular we permit both the fractal Laplacian and the potential to be Radon measures on the fractal. As a consequence of our results, we establish a Harnack inequality for solutions of these operators.

Arxiv preprint version

Published version in Demonstratio Mathematica

# Fractal AC circuits and propagating waves on fractals

by Eric Akkermans, Joe P. Chen, Gerald Dunne, Luke G. Rogers and Alexander Teplyaev

We extend Feynman’s analysis of the infinite ladder AC circuit to fractal AC circuits. We show that the characteristic impedances can have positive real part even though all the individual impedances inside the circuit are purely imaginary. Continue reading

# Besov class via heat semigroup on Dirichlet spaces II: BV functions and Gaussian heat kernel estimates

by Patricia Alonso-Ruiz, Fabrice Baudoin, Li Chen, Luke Rogers, Nageswari Shanmugalingam, Alexander Teplyaev

We introduce the class of bounded variation (BV) functions in a general framework of strictly local Dirichlet spaces with doubling measure. Continue reading

# Embedding convex geometries and a bound on convex dimension

by Michael Richter, Luke G. Rogers

The notion of an abstract convex geometry offers an abstraction of the standard notion of convexity in a linear space. Continue reading

# Sobolev Algebra Counterexamples

by Thierry Coulhon, Luke G. Rogers

In the Euclidean setting the Sobolev spaces $$W^{α,p}\cap L^\infty$$ are algebras for the pointwise product when α>0 and p∈(1,∞). This property has recently been extended to a variety of geometric settings. We produce a class of fractal examples where it fails for a wide range of the indices α,p.

Arxiv prepint version

Published version in JFG

# Power dissipation in fractal AC circuits

by Joe P Chen, Luke G. Rogers, Loren Anderson, Ulysses Andrews, Antoni Brzoska, Aubrey Coffey, Hannah Davis, Lee Fisher, Madeline Hansalik, Stephen Loew and Alexander Teplyaev.

We extend Feynman’s analysis of an infinite ladder circuit to fractal circuits, providing examples in which fractal circuits constructed with purely imaginary impedances can have characteristic impedances with positive real part. Continue reading

# Magnetic Laplacians of locally exact forms on the Sierpinski Gasket

by Jessica Hyde, Daniel Kelleher, Jesse Moeller, Luke G. Rogers, Luis Seda

We give an explicit construction of a magnetic Schrödinger operator corresponding to a field with flux through finitely many holes of the Sierpinski Gasket. Continue reading

# Some spectral properties of pseudo-differential operators on the Sierpinski Gasket

by Marius Ionescu, Kasso A. Okoudjou and Luke G. Rogers.

We prove versions of the strong Szëgo limit theorem for certain classes of pseudodifferential operators defined on the Sierpinski gasket. Our results uses in a fundamental way the existence of localized eigenfunctions for the Laplacian on this fractal. Continue reading

# Pseudo-differential Operators on Fractals

by Marius Ionescu, Luke G. Rogers and Robert S. Strichartz

We define and study pseudo-differential operators on a class of fractals that include the post-critically finite self-similar sets and Sierpinski carpets. Continue reading

# Derivations and Dirichlet forms on fractals

by Marius Ionescu, Luke G. Rogers and Alexander Teplyaev

We study derivations and Fredholm modules on metric spaces with a local regular conservative Dirichlet form. Continue reading

# Estimates for the resolvent kernel of the Laplacian on p.c.f. self similar fractals and blowups

by Luke G. Rogers

We provide a method for obtaining upper estimates of the resolvent kernel of the Laplacian on a post-critically finite self-similar fractal that relies on a self-similar series decomposition of the resolvent. Continue reading

# Complex Powers of the Laplacian on Affine Nested Fractals as Calderón-Zygmund operators

by Marius Ionescu and Luke G. Rogers

We give the first natural examples of Calderón-Zygmund operators in the theory of analysis on post-critically finite self-similar fractals. Continue reading

# Smooth bumps, a Borel theorem and partitions of smooth functions on p.c.f. fractals.

by Luke G. Rogers, Robert S. Strichartz and Alexander Teplyaev

We provide two methods for constructing smooth bump functions and for smoothly cutting off smooth
functions on fractals, one using a probabilistic approach and sub-Gaussian estimates for the heat
operator, and the other using the analytic theory for p.c.f. fractals and a fixed point argument. Continue reading

# The resolvent kernel for PCF self-similar fractals

by Marius Ionescu, Erin P. J. Pearse, Luke G. Rogers, Huo-Jun Ruan, Robert S. Strichartz

For the Laplacian Δ defined on a p.c.f. self-similar fractal, we give an explicit formula for the resolvent kernel of the Laplacian with Dirichlet boundary conditions, Continue reading

# Distribution theory on p.c.f. fractals

by Luke G. Rogers, Robert S. Strichartz

We construct a theory of distributions in the setting of analysis on post-critically finite self-similar fractals, and on fractafolds and products based on such fractals. Continue reading

# Szegö limit theorems on the Sierpinski gasket

by Kasso A. Okoudjou, Luke G. Rogers, Robert S. Strichartz

We use the existence of localized eigenfunctions of the Laplacian on the Sierpinski gasket to formulate and prove analogues of the strong Szego limit theorem in this fractal setting. Furthermore, we recast some of our results in terms of equally distributed sequences. Continue reading

# Laplacians on the basilica Julia set

by Luke G Rogers and Alexander Teplyaev.

We consider the basilica Julia set of the polynomial $$P(z)=z^{2}-1$$ and construct all possible
resistance (Dirichlet) forms, and the corresponding Laplacians, for which the topology in the
effective resistance metric coincides with the usual topology. Continue reading

by Jessica L. DeGrado, Luke G. Rogers, Robert S. Strichartz

We use spectral decimation to provide formulae for computing the harmonic gradients of Laplacian eigenfunctions on the Sierpinski Gasket. These formulae are given in terms of special functions that are defined as infinite products. Continue reading

# Unimodular Multipliers on Modulation Spaces.

by Árpád Bényi, Kasso A. Okoudjou, Karlheniz Gröchenig and Luke G. Rogers.

We investigate the boundedness of unimodular Fourier multipliers on modulation spaces. Surprisingly, the multipliers with general symbol  $$e^{i|\xi|^\alpha}$$ where $$\alpha\in[0,2]$$, are bounded on all modulation spaces, Continue reading

# Generalized Eigenfunctions and a Borel Theorem on the Sierpinski Gasket

by Kasso A Okoudjou, Luke G. Rogers and Robert S. Strichartz

We prove there exist exponentially decaying generalized eigenfunctions on a blow-up of the Sierpinski gasket with boundary. These are used to show a Borel-type theorem, specifically that for a prescribed jet at the boundary point there is a smooth function having that jet. Continue reading

# Degree-independent Sobolev extension on locally uniform domains

by Luke G. Rogers

We consider the problem of constructing extensions $$L^{p}_{k}(\Omega)\rightarrow L^{p}_{k}(\mathbb{R}^{n})$$, where $$L^{p}_{k}$$ is the Sobolev space of functions with $$k$$ derivatives in $$L^{p}$$ and $$\Omega\subset\mathbb{R}^{n}$$ is a domain. Continue reading