Nontangential limits and Fatou-type theorems on post-critically finite self-similar sets

Transcription

1 Nontangential limits and on post-critically finite self-similar sets 4th Cornell Conference on Analysis, Probability, and Mathematical Physics on Fractals Universidad de Colima

2 Setting Boundary limits K a PCF self-similar set with regular harmonic structure (D, r) Hausdorff dimension d with respect to effective resistence metric R(x, y) µ the Bernoulli measure on K with weights µ i = r d i

3 Setting Boundary limits K a PCF self-similar set with regular harmonic structure (D, r) Hausdorff dimension d with respect to effective resistence metric R(x, y) µ the Bernoulli measure on K with weights µ i = r d i µ(b ε (x)) ε d for small ε

4 Setting Boundary limits K a PCF self-similar set with regular harmonic structure (D, r) Hausdorff dimension d with respect to effective resistence metric R(x, y) µ the Bernoulli measure on K with weights µ i = r d i µ(b ε (x)) ε d for small ε Maximal function 1 Mf (x) = sup f dµ ε>0 µ(b ε (x)) B ε(x) is of weak type (1, 1) and bounded on L p

5 Dirichlet problem Boundary limits Let be the Laplacian associated to (D, r) and µ.

6 Dirichlet problem Boundary limits Let be the Laplacian associated to (D, r) and µ. Consider the Dirichlet problem 2 u t 2 + u = 0, (x, t) K R + u(x, 0) = f (x), f defined in K

7 Boundary limits P b (t, x, y) = e λ b n t φ b n(x)φ b n(y), n=1 where {φ b n} is a complete o.n. system of Dirichlet or Neumann (b = D or N) eigenfunctions of

8 Boundary limits P b (t, x, y) = e λ b n t φ b n(x)φ b n(y), n=1 where {φ b n} is a complete o.n. system of Dirichlet or Neumann (b = D or N) eigenfunctions of Subordination principle P b (t, x, y) = where H b (t, x, y) is the heat kernel t 2 e t2 /4s H b (s, x, y) ds π 0 s 3/2

9 Boundary limits P b (t, x, y) is nonegative, continuous and P b (t, x, ) dom K 2 P b (t, x, y) t 2 + P b (t, x, y) = 0 P b (t, x, z)p b (s, z, y)dµ(z) = P b (t + s, x, y)

10 Poisson semigroup Boundary limits Pt b f (x) = K P b (t, x, y)f (y)dµ(y)

11 Poisson semigroup Boundary limits Pt b f (x) = K P b (t, x, y)f (y)dµ(y) P b t : L p (K, µ) C(K) is bounded

12 Poisson semigroup Boundary limits P b t Pt b f (x) = : L p (K, µ) C(K) is bounded If u(t, x) = P b t f (x), K P b (t, x, y)f (y)dµ(y) 2 u t 2 + u = 0

13 Poisson semigroup Boundary limits P b t Pt b f (x) = : L p (K, µ) C(K) is bounded If u(t, x) = P b t f (x), K P b (t, x, y)f (y)dµ(y) 2 u t 2 + u = 0 For f C(K), P N t f f 0 as t 0

14 Poisson semigroup Boundary limits P b t Pt b f (x) = : L p (K, µ) C(K) is bounded If u(t, x) = P b t f (x), K P b (t, x, y)f (y)dµ(y) 2 u t 2 + u = 0 For f C(K), P N t f f 0 as t 0 For f C(K), f V0 0, P D t f f 0 as t 0

15 Boundary limits Boudary limits of Poisson integrals Theorem If f L p (K, dµ), 1 p, u(t, x) = P t f (x) There exists A > 0 s.t. u(t, x) AMf (x) u(t, ) f in L p (K, µ), if 1 p < ; lim t 0 u(t, x) = f (x) for a.e. x K.

16 Boundary limits Boudary limits of Poisson integrals Theorem If f L p (K, dµ), 1 p, u(t, x) = P t f (x) There exists A > 0 s.t. u(t, x) AMf (x) u(t, ) f in L p (K, µ), if 1 p < ; lim t 0 u(t, x) = f (x) for a.e. x K. We use the estimate P(t, x, y) C min {t 2d t } d+1,. R(x, y) 3d+1 2 Follows from the well-known estimates for the heat kernel and the subordination principle.

17 Proof Boundary limits A n (x) = {y K : R(x, y) 2 n t 2 d+1 }

18 Proof Boundary limits A n(x) A n (x) = {y K : R(x, y) 2 n t 2 d+1 } P b (t, x, y) f (y) dµ(y) A n(x) t ( 2 n t 2 d+1 t R(x, y) 3d+1 2 ) 3d d+1 2 n Mf (x). f (y) dµ(y) B 2 n t 2/(d+1) (x) f (y) dµ(y)

19 Nontangential limits Nontangential limits of Poisson integrals Boundary values of harmonic functions Local Fatou theorem in nested fractals Cone Γ α (x) = {(t, y) R + K : R(x, y) d+1 < αt 2 }

20 Nontangential limits Nontangential limits of Poisson integrals Boundary values of harmonic functions Local Fatou theorem in nested fractals Cone Γ α (x) = {(t, y) R + K : R(x, y) d+1 < αt 2 } Theorem f L p (K, dµ), 1 p, u(t, x) = P b t f (x), α > 0 There exists A α > 0 such that For a. e. x K, sup u(t, y) A α Mf (x). (t,y) Γ α(x) lim (t,y) (0,x) (t,y) Γ α(x) u(t, y) = f (x).

21 Proof Nontangential limits of Poisson integrals Boundary values of harmonic functions Local Fatou theorem in nested fractals For the first part: for (t, y) Γ α (x) P b (t, y, z) C α min {t 2d t } d+1,, R(x, z) 3d+1 2

22 Proof Nontangential limits of Poisson integrals Boundary values of harmonic functions Local Fatou theorem in nested fractals For the first part: for (t, y) Γ α (x) P b (t, y, z) C α min {t 2d t } d+1,, R(x, z) 3d+1 2 For the second: for x K \ V 0, lim P D (t, x, y)dµ(y) = 1. t 0 K

23 Maximum principle Nontangential limits of Poisson integrals Boundary values of harmonic functions Local Fatou theorem in nested fractals u on R + K is harmonic if, for (t, x) R + (K \ V 0 ), 2 u(t, x) t 2 + µ u(t, x) = 0

24 Maximum principle Nontangential limits of Poisson integrals Boundary values of harmonic functions Local Fatou theorem in nested fractals u on R + K is harmonic if, for (t, x) R + (K \ V 0 ), 2 u(t, x) t 2 + µ u(t, x) = 0 Theorem u cannot take a maximum in R + (K \ V 0 ).

25 Maximum principle Nontangential limits of Poisson integrals Boundary values of harmonic functions Local Fatou theorem in nested fractals u on R + K is harmonic if, for (t, x) R + (K \ V 0 ), 2 u(t, x) t 2 + µ u(t, x) = 0 Theorem u cannot take a maximum in R + (K \ V 0 ). Uses Lemma Let u D µ. If u(x) = max{u(y) : y K} for some x K \ V 0, then µ u(x) 0.

26 Nontangential limits of Poisson integrals Boundary values of harmonic functions Local Fatou theorem in nested fractals Harmonic functions and boundary limits Theorem Let u be a Dirichlet harmonic function on R + K such that sup u(t, ) L (K,dµ) <. t>0 Then u is the Dirichlet Poisson integral of a function f L (K, dµ).

27 Proof Nontangential limits of Poisson integrals Boundary values of harmonic functions Local Fatou theorem in nested fractals Use maximal principle to show u ( t + 1 n, x) = P D (t, x, y)u(1/n, y)dµ(y). K

28 Proof Nontangential limits of Poisson integrals Boundary values of harmonic functions Local Fatou theorem in nested fractals Use maximal principle to show u ( t + 1 n, x) = P D (t, x, y)u(1/n, y)dµ(y). K Weak-* compactness: u(1/n, y) f weakly Verify u(t, x) = P D t f (x)

29 Proof Nontangential limits of Poisson integrals Boundary values of harmonic functions Local Fatou theorem in nested fractals Use maximal principle to show u ( t + 1 n, x) = P D (t, x, y)u(1/n, y)dµ(y). K Corollary Weak-* compactness: u(1/n, y) f weakly Verify u(t, x) = P D t f (x) A bounded Dirichlet harmonic function on R + K has nontangential limit at x as t 0 for almost every x K.

30 Nontangential limits of Poisson integrals Boundary values of harmonic functions Local Fatou theorem in nested fractals Harmonic functions and boundary limits Corollary Suppose u is a Dirichlet harmonic function on R + K. If 1 < p, then u is the Poisson integral of some f L p if and only if sup u(t, ) L p <. t>0 Moreover, u is the Poisson integral of some finite Borel measure on K if and only if sup u(t, ) L 1 <. t>0

31 Estimates for Nontangential limits of Poisson integrals Boundary values of harmonic functions Local Fatou theorem in nested fractals Theorem Let K be an affine nested fractal and α > 0. Then there exists c α > 0 such that P N (t, x, y)dµ(y) c α, B where B = B (αt 2 ) 1/(d+1)(x) is the ball of radius (αt2 ) 1 d+1 with center in x.

32 Estimates for Nontangential limits of Poisson integrals Boundary values of harmonic functions Local Fatou theorem in nested fractals Theorem Let K be an affine nested fractal and α > 0. Then there exists c α > 0 such that P N (t, x, y)dµ(y) c α, B where B = B (αt 2 ) 1/(d+1)(x) is the ball of radius (αt2 ) 1 d+1 with center in x. Proof: Subordination and estimates for heat kernel

33 Positive harmonic functions Nontangential limits of Poisson integrals Boundary values of harmonic functions Local Fatou theorem in nested fractals Corollary Let E K be a measurable set, α > 0, and Ω = Γ 1 α(x). Then x E there exists a positive harmonic function v on R + K such that 1 v 1 on ( Ω) (R + K); and 2 v has nontangential limit 0 at almost every point of E.

34 Positive harmonic functions Nontangential limits of Poisson integrals Boundary values of harmonic functions Local Fatou theorem in nested fractals Corollary Let E K be a measurable set, α > 0, and Ω = Γ 1 α(x). Then x E there exists a positive harmonic function v on R + K such that 1 v 1 on ( Ω) (R + K); and 2 v has nontangential limit 0 at almost every point of E. Proof: w(t, x) = K P N (t, x, y)χ K\E (y)dµ(y) + t,

35 Positive harmonic functions Nontangential limits of Poisson integrals Boundary values of harmonic functions Local Fatou theorem in nested fractals Corollary Let E K be a measurable set, α > 0, and Ω = Γ 1 α(x). Then x E there exists a positive harmonic function v on R + K such that 1 v 1 on ( Ω) (R + K); and 2 v has nontangential limit 0 at almost every point of E. Proof: w(t, x) = K P N (t, x, y)χ K\E (y)dµ(y) + t, Integral bounded from below by c α for x E

36 Local Fatou theorem Nontangential limits of Poisson integrals Boundary values of harmonic functions Local Fatou theorem in nested fractals u is nontangentially bounded at x K if u is bounded on some Γ h α(x).

37 Local Fatou theorem Nontangential limits of Poisson integrals Boundary values of harmonic functions Local Fatou theorem in nested fractals u is nontangentially bounded at x K if u is bounded on some Γ h α(x). Theorem Let u be harmonic on R + K and nontangentially bounded at each point in the set E K. Then u has a nontangential limit at almost every point of E.

38 Local Fatou theorem Nontangential limits of Poisson integrals Boundary values of harmonic functions Local Fatou theorem in nested fractals u is nontangentially bounded at x K if u is bounded on some Γ h α(x). Theorem Let u be harmonic on R + K and nontangentially bounded at each point in the set E K. Then u has a nontangential limit at almost every point of E. Proof: Uses the barrier function of previous corollary.

39 Future work Harmonic functions on R + K Mean value property Harnack inequality

40 Future work Harmonic functions on R + K Mean value property Harnack inequality Nondiagonal estimates for Pt N (t, x, y) for nested fractals

41 Future work Harmonic functions on R + K Mean value property Harnack inequality Nondiagonal estimates for Pt N (t, x, y) for nested fractals Local Fatou theorem for general PCF sets

42 Future work Harmonic functions on R + K Mean value property Harnack inequality Nondiagonal estimates for Pt N (t, x, y) for nested fractals Local Fatou theorem for general PCF sets Harmonic analysis Littlewood-Paley theory; square functions; area integrals

43 Future work Harmonic functions on R + K Mean value property Harnack inequality Nondiagonal estimates for Pt N (t, x, y) for nested fractals Local Fatou theorem for general PCF sets Harmonic analysis Littlewood-Paley theory; square functions; area integrals H p spaces; H 1 and BMO functions