Nontangential limits and on post-critically finite self-similar sets 4th Cornell Conference on Analysis, Probability, and Mathematical Physics on Fractals Universidad de Colima
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
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
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
Dirichlet problem Boundary limits Let be the Laplacian associated to (D, r) and µ.
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
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
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
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)
Poisson semigroup Boundary limits Pt b f (x) = K P b (t, x, y)f (y)dµ(y)
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
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
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
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
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.
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.
Proof Boundary limits A n (x) = {y K : R(x, y) 2 n t 2 d+1 }
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+1 2 2 d+1 2 n Mf (x). f (y) dµ(y) B 2 n t 2/(d+1) (x) f (y) dµ(y)
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 }
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).
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
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
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
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 ).
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.
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).
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
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)
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.
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
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.
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
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.
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,
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
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).
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.
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.
Future work Harmonic functions on R + K Mean value property Harnack inequality
Future work Harmonic functions on R + K Mean value property Harnack inequality Nondiagonal estimates for Pt N (t, x, y) for nested fractals
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
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
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