Powers of the Laplacian on PCF Fractals

Transcription

1 Based on joint work with Luke Rogers and Robert S. Strichartz AMS Fall Meeting, Riverside November 8, 2009

2 PCF Fractals Definitions Let {F 1, F 2,..., F N } be an iterated function system of contractive similarities in R n. We let K be the invariant set of the i.f.s. K = F i (K). i K is called postcritically finite (pcf) if K is connected, and there exists a finite set V 0 K called the boundary, such that for w w with w = w. F w K F w K F w V 0 Fw V 0

3 Energy and Laplacian Definitions We assume the existence of a self similar measure µ on K µ(a) = i µ i µ(f 1 i A). We also assume the existence of a self-similar energy E on K E(u) = i r i E(u F i ). The Laplacian is defined using the weak formulation: u = f if ˆ E(u, v) = fvdµ for all v dom 0 E.

4 Fractal Blow-ups and Fractafolds Definitions Let w {1,..., N} be an infinite word. Then we define a fractal blow-up K via K = m=1 Fw Fw 1 m K. A fractafold based on K is a set X such that every point in X has a neighborhood that is similar to a neighborhood of a point in K.

5 Set up Let X denote either a fractal blow-up without boundary or a compact fractafold without boundary. Then one can extend the definition of the energy E and the measure µ to X. We define the Laplacian on X via the weak formulation. has compact resolvent if X is a compact fractafold. has pure point spectrum if X is a fractal blow-up. The effective resistance metric R(x, y) on X is defined via R(x, y) 1 = min{e(u) : u(x) = 0 and u(x) = 1}.

6 Singular Integral Operators on Fractals Theorem Suppose that T : L 2 (µ) L 2 (µ) is an operator that is given by integration with respect to a kernel K(x, y) that is smooth off the diagonal and satisfies the estimates and K(x, y) R(x, y) d 2 K(x, y) R(x, y) 2d 1. Then T is a singular integral operator on (X, µ).that is, K(x, y) satisfies also ( ) R(y, y) 1/2 K(x, y) K(x, y) R(x, y) d, R(x, y) for all x, y, y X such that R(x, y) cr(y, y), for some c > 1.

7 Sketch of the proof For y, y, x X with x far away from y and y, then we can find a large cell C that contains both y and y and does not contain x. By a large cell we mean that C is an m-cell such that r m r k R(x, y). Then u x (z) := K(x, z) is smooth on C and u x (y) u x (y) = h C x (y) h C x (y) + ˆ (G C (y, z) G C (y, z) ) u x (z)dµ(z).

8 Heat kernels We assume that the heat operator e t is given by integration with respect to a positive heat kernel h t (x, y) ˆ e t u = h t (x, y)u(y)dµ(y). X We also assume that the heat kernel satisfies the following sub-gaussian estimates ( ( ) R(x, y) h t (x, y) t β d+1 γ ) exp c. t These estimates hold for a large class of PCF fractals with β = d/(d + 1), d is the Hausdorff dimension of the fractal, and γ = 1/(d w 1) with d ω a constant depending on the fractal.

9 Heat Kernels are Integrable Theorem If y X we have that ˆ ( e c R(x,y) d+1 ) γ t dµ(x) t d d+1 for all t > 0. X

10 Imaginary powers of the Laplacian Definitions Let α R and let D be the set of finite linear combinations of eigenfunctions of. 1 We define (ˆ ( ) iα u = C( ) e t ut iα dt ), for u D, where C > 0 is a constant. 2 The kernel of ( ) iα is K iα (x, y) = C ˆ 0 0 ( 1 )h t (x, y)t iα dt.

11 Properties Theorem For any α R, ˆ ( ) iα u = X K iα (x, y)u(y)dµ(y) extends to a bounded operator on L 2 (µ). Moreover, the kernel K iα (x, y) is smooth off the diagonal and satisfies the following estimates K iα (x, y) R(x, y) d and 2 K iα (x, y) R(x, y) 2d 1.

12 L p -boundedness Corollary ( ) iα extends to a bounded operator on L p (µ) for all 1 < p <.

13 Laplace transforms Definitions A function p : [0, ) R is said to be of Laplace transform type if p(λ) = λ ˆ 0 m(t)e tλ dt, where m is uniformly bounded. For such a p we can define an operator p( )u = ( ) ˆ 0 m(t)e t udt.

14 Laplace Transforms Operators are Singular Integral Operators Theorem Suppose that p is of Laplace transform type. Then the operator p( ) extends to a bounded operator on L 2 (µ) and it is a singular integral operator. It is given by integration with respect to the kernel K p (x, y) = ˆ 0 ( 1 )h t (x, y)m(t)dt.

15 Bibliography Jun Kigami, Analysis on fractals, Cambridge Tracts in Mathematics, vol. 143, Cambridge University Press, Cambridge, Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993, Robert S. Strichartz, Function spaces on fractals, J. Funct. Anal. 198 (2003), no. 1, Robert S. Strichartz, Differential equations on fractals, Princeton University Press, Princeton, NJ, 2006, A tutorial.