Search

We prove a strong maximum principle for Schrödinger operators defined on a class of postcritically finite 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.

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. This provides a physical setting for analyzing wave propagation of signals on fractals, by analogy with the Telegrapher's Equation, and generalizes the real resistance metric on a fractal, which provides a measure of distance on a fractal, to complex impedances.

In the Euclidean setting the Sobolev spaces $W^{\alpha,p}\cap L^\infty$ are algebras for the pointwise product when $\alpha>0$ and $p\in(1,\infty)$. 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 $\alpha,p$.

The 4 ⁢ N {4N} -carpets are a class of infinitely ramified self-similar fractals with a large group of symmetries. For a 4 ⁢ N {4N} -carpet F, let { F n } n ≥ 0 {\{F_{n}\}_{n\geq 0}} be the natural decreasing sequence of compact pre-fractal approximations with ⋂ n F n = F {\bigcap_{n}F_{n}=F} . On each F n {F_{n}} , let ℰ ⁢ ( u , v ) = ∫ F N ∇ ⁡ u ⋅ ∇ ⁡ v ⁢ d ⁢ x {\mathcal{E}(u,v)=\int_{F_{N}}\nabla u\cdot\nabla v\,dx} be the classical Dirichlet form and u n {u_{n}} be the unique harmonic function on F n {F_{n}} satisfying a mixed boundary value problem corresponding to assigning a constant potential between two specific subsets of the boundary. Using a method introduced by [M. T. Barlow and R. F. Bass, On the resistance of the Sierpiński carpet, Proc. Roy. Soc. Lond. Ser. A 431 (1990), no. 1882, 345–360], we prove a resistance estimate of the following form: there is ρ = ρ ⁢ ( N ) > 1 {\rho=\rho(N)>1} such that ℰ ⁢ ( u n , u n ) ⁢ ρ n {\mathcal{E}(u_{n},u_{n})\rho^{n}} is bounded above and below by constants independent of n. Such estimates have implications for the existence and scaling properties of Brownian motion on F.

We give a mathematically rigorous construction of a magnetic Schr\"odinger operator corresponding to a field with flux through finitely many holes of the Sierpinski Gasket. The operator is shown to have discrete spectrum accumulating at $\infty$, and it is shown that the asymptotic distribution of eigenvalues is the same as that for the Laplacian. Most eigenfunctions may be computed using gauge transformations corresponding to the magnetic field and the remainder of the spectrum may be approximated to arbitrary precision by using a sequence of approximations by magnetic operators on finite graphs., Comment: 20 pages, 5 figures

We consider criteria for the differentiability of functions with continuous Laplacian on the Sierpinski Gasket and its higher-dimensional variants $SG_N$, $N>3$, proving results that generalize those of Teplyaev. When $SG_N$ is equipped with the standard Dirichlet form and measure $��$ we show there is a full $��$-measure set on which continuity of the Laplacian implies existence of the gradient $\nabla u$, and that this set is not all of $SG_N$. We also show there is a class of non-uniform measures on the usual Sierpinski Gasket with the property that continuity of the Laplacian implies the gradient exists and is continuous everywhere, in sharp contrast to the case with the standard measure., 9 pages

We study the analogue of a convex interpolant of two sets on Sierpinski gaskets and an associated notion of measure transport. The structure of a natural family of interpolating measures is described and an interpolation inequality is established. A key tool is a good description of geodesics on these gaskets, some results on which have previously appeared in the literature.

We introduce the class of bounded variation (BV) functions in a general framework of strictly local Dirichlet spaces with doubling measure. Under the 2-Poincar\'e inequality and a weak Bakry-\'Emery curvature type condition, this BV class is identified with the heat semigroup based Besov class $\mathbf{B}^{1,1/2}(X)$ that was introduced in our previous paper. Assuming furthermore a quasi Bakry-\'Emery curvature type condition, we identify the Sobolev class $W^{1,p}(X)$ with $\mathbf{B}^{p,1/2}(X)$ for $p>1$. Consequences of those identifications in terms of isoperimetric and Sobolev inequalities with sharp exponents are given., Comment: The notes arXiv:1806.03428 will be divided in a series of papers. This is the second paper dealing with strictly local Dirichlet forms. v2 corrects typos and changes some terminology. To appear in Cal. Var & PDE

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. As a highlight of the paper, we obtain a far reaching $L^p$-analogue, $p \ge 1$, of the Sobolev inequality that was proved for $p=2$ by N. Varopoulos under the assumption of ultracontractivity for the heat semigroup. The case $p=1$ is of special interest since it yields isoperimetric type inequalities., Comment: The notes arXiv:1806.03428 will be divided in a series of papers. This is the first paper. V3: Some typos are corrected. V4. To appear in Journal of Functional Analysis

We give the first natural examples of Calderon-Zygmund operators in the theory of analysis on post-critically finite self-similar fractals. This is achieved by showing that the purely imaginary Riesz and Bessel potentials on nested fractals with $3$ or more boundary points are of this type. It follows that these operators are bounded on $L^{p}$, $1 < p < \infty$ and satisfy weak $1$-$1$ bounds. The analysis may be extended to infinite blow-ups of these fractals, and to product spaces based on the fractal or its blow-up.

We define and study pseudo-differential operators on a class of fractals that include the post-critically finite self-similar sets and Sierpinski carpets. Using the sub-Gaussian estimates of the heat operator we prove that our operators have kernels that decay and, in the constant coefficient case, are smooth off the diagonal. Our analysis can be extended to products of fractals. While our results are applicable to a larger class of metric measure spaces with Laplacian, we use them to study elliptic, hypoelliptic, and quasi-elliptic operators on p.c.f. fractals, answering a few open questions posed in a series of recent papers. We extend our class of operators to include the so called Hormander hypoelliptic operators and we initiate the study of wavefront sets and microlocal analysis on p.c.f. fractals.

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. Using (weak) self-similarity of our fractal structures, we provide algorithms for studying the equilibrium distribution of energy on these circuits. This extends the analysis of self-similar resistance networks introduced by Fukushima, Kigami, Kusuoka, and more recently studied by Strichartz et al., v2: 16 pages, 8 figures. See also the recent preprint arXiv:1701.08039

One of the main features of analysis on post-critically finite self-similar (pcfss) sets is that it is possible to understand the behavior of the Laplacian and its inverse, the Green operator, in terms of the self-similar structure of the set. Indeed, a major step in the approach to analysis on self-similar fractals via Dirichlet forms was Kigami’s proof [8, 10] that for a self-similar Dirichlet form the Green kernel can be written explicitly as a series in which each term is a rescaling of a single expression via the self-similar structure. In [6] this result was extended to show that the resolvent kernel of the Laplacian, meaning the kernel of (z − ∆)−1, can also be written as a self-similar series for suitable values of z ∈ C. Part of the motivation for that work was that it gives a new understanding of functions of the Laplacian (such as the heat operator et∆) by writing them as integrals of the resolvent. The purpose of the present work is to establish estimates that permit the above approach to be carried out. We study the functions occurring in the series decomposition from [6] (see Theorem 3.3 below for this decomposition) and give estimates on their decay. From this we determine estimates on the resolvent kernel and on kernels of operators defined as integrals of the resolvent kernel. In particular we recover the sharp upper estimates for the heat kernel (see Theorem 10.2) that were proved for pcfss sets by Hambly and Kumagai [4] by probabilistic methods (see also [1, 13, 3] for earlier results of this type on less general classes of sets). It is important to note that the preceding authors were able to prove not just upper estimates but also lower bounds for the heat kernel, and therefore were able to prove sharpness of their bounds. Our methods permit sharp bounds for the resolvent kernel on the positive real axis, but we do not know how to obtain these globally in the complex plane or how to obtain lower estimates for the heat kernel from them. Therefore in this direction our results are not as strong as those obtained in [4]. However in other directions we obtain more information than that known from heat kernel estimates, and we hope that our approach will complement the existing probabilistic methods. In particular we are able to obtain resolvent bounds on any ray in C other than the negative real axis (where the spectrum lies), while standard calculations from heat kernel bounds only give these estimates in a half-plane. A further consequence of our approach is that we extend (in Theorem 9.7) the decomposition from [6] to the case of blowups, which are non-compact sets with local structure equivalent to that of the underlying self-similar sets. The blowup of a pcfss set bears the same relation to the original set as the real line bears to the unit interval, see [17] for details. The structure of the paper is as follows. In Section 3 we recall some basic features of analysis on pcfss sets, as well as the main result of [6], which is the decomposition of the resolvent as a weighted sum of piecewise eigenfunctions. Section 4 then discusses

For the Laplacian Δ \Delta 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 and also with Neumann boundary conditions. That is, we construct a symmetric function G ( λ ) G^{(\lambda )} which solves ( λ I − Δ ) − 1 f ( x ) = ∫ G ( λ ) ( x , y ) f ( y ) d μ ( y ) (\lambda \mathbb {I} - \Delta )^{-1} f(x) = \int G^{(\lambda )}(x,y) f(y) \, d\mu (y) . The method is similar to Kigami’s construction of the Green kernel and G ( λ ) G^{(\lambda )} is expressed as a sum of scaled and “translated” copies of a certain function ψ ( λ ) \psi ^{(\lambda )} which may be considered as a fundamental solution of the resolvent equation. Examples of the explicit resolvent kernel formula are given for the unit interval, standard Sierpinski gasket, and the level-3 Sierpinski gasket S G 3 SG_3 .

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.

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. The heat semigroup (probabilistic) method is applicable to a more general class of metric measure spaces with Laplacian, including certain infinitely ramified fractals, however the cut off technique involves some loss in smoothness. From the analytic approach we establish a Borel theorem for p.c.f. fractals, showing that to any prescribed jet at a junction point there is a smooth function with that jet. As a consequence we prove that on p.c.f. fractals smooth functions may be cut off with no loss of smoothness, and thus can be smoothly decomposed subordinate to an open cover. The latter result provides a replacement for classical partition of unity arguments in the p.c.f. fractal setting., Comment: 26 pages. May differ slightly from published (refereed) version

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., Comment: LaTex, 8 pages, 4 figures

We investigate the boundedness of unimodular Fourier multipliers on modulation spaces. Surprisingly, the multipliers with general symbol e i | ξ | α , where α ∈ [ 0 , 2 ] , are bounded on all modulation spaces, but, in general, fail to be bounded on the usual L p -spaces. As a consequence, the phase-space concentration of the solutions to the free Schrodinger and wave equations are preserved. As a byproduct, we also obtain boundedness results on modulation spaces for singular multipliers | ξ | − δ sin ( | ξ | α ) for 0 ⩽ δ ⩽ α .

The notion of an abstract convex geometry offers an abstraction of the standard notion of convexity in a linear space. Kashiwabara, Nakamura and Okamoto introduce the notion of a generalized convex shelling into $\mathbb{R}$ and prove that a convex geometry may always be represented with such a shelling. We provide a new, shorter proof of their result using a recent representation theorem of Richter and Rubinstein, and deduce a different upper bound on the dimension of the shelling., Corrected attribution for Lemma 1 and Theorem 2 - Added an example related to generalized convex shellings of lower-bounded lattices and noted its relevance to convex dimension. - Added a section on embedding convex geometries as convex polygons, including a proof that any convex geometry may be embedded as convex polygons in R^2. - Extended the bibliography. Now 9 pages

We prove versions of the strong Sz\"ego limit theorem for certain classes of pseudodifferential operators defined on the Sierpi\'nski gasket. Our results used in a fundamental way the existence of localized eigenfunctions for the Laplacian on this fractal., Comment: Analysis on fractals, localized eigenfunctions, Sierpi\'nski gasket, Sz\"ego limit theorem

We study derivations and Fredholm modules on metric spaces with a local regular conservative Dirichlet form. In particular, on finitely ramified fractals, we show that there is a non-trivial Fredholm module if and only if the fractal is not a tree (i.e. not simply connected). This result relates Fredholm modules and topology, and refines and improves known results on p.c.f. fractals. We also discuss weakly summable Fredholm modules and the Dixmier trace in the cases of some finitely and infinitely ramified fractals (including non-self-similar fractals) if the so-called spectral dimension is less than 2. In the finitely ramified self-similar case we relate the p-summability question with estimates of the Lyapunov exponents for harmonic functions and the behavior of the pressure function., Comment: to appear in the Journal of Functional Analysis 2012

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. The results include basic properties of test functions and distributions, a structure theorem showing that distributions are locally-finite sums of powers of the Laplacian applied to continuous functions, and an analysis of the distributions with point support. Possible future applications to the study of hypoelliptic partial differential operators are suggested., 38 pages

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. Then we concentrate on two particular cases. One is a self-similar harmonic structure, for which the energy renormalization factor is 2, the spectral dimension is $\log9/\log6$, and we can compute all the eigenvalues and eigenfunctions by a spectral decimation method. The other is graph-directed self-similar under the map $z\mapsto P(z)$; it has energy renormalization factor $\sqrt2$ and spectral dimension 4/3, but the exact computation of the spectrum is difficult. The latter Dirichlet form and Laplacian are in a sense conformally invariant on the basilica Julia set., 24 pages, one figure in separate eps file. Replaced the theorem that was removed in the second version, with a corrected proof. Corrected an error in the description of the energy forms (we are grateful to Jun Kigami for pointing out this error). Other minor changes

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., Comment: 14 pages

We consider the problem of constructing extensions L k p ( Ω ) → L k p ( R n ) , where L k p is the Sobolev space of functions with k derivatives in L p and Ω ⊂ R n is a domain. In the case of Lipschitz Ω, Calderon gave a family of extension operators depending on k, while Stein later produced a single (k-independent) operator. For the more general class of locally-uniform domains, which includes examples with highly non-rectifiable boundaries, a k-dependent family of operators was constructed by Jones. In this work we produce a k-independent operator for all spaces L k p ( Ω ) on a locally uniform domain Ω.