A trace theorem for Dirichlet forms on fractals and its application Takashi Kumagai (RIMS, Kyoto University, Japan) http://www.kurims.kyoto-u.ac.jp/~kumagai/ November 17, 2005 at Bucharest Joint work with M. Hino (Kyoto)
Jonsson (Math Z. 05)
1 A quick view of the theory of Dirichlet forms General Theory (see Fukushima-Oshima-Takeda 94, Röckner-Ma 92 etc.) {X t } t : Sym. Hunt proc. on (K, µ) cont. path (diffusion), : non-neg. def. self-adj. op. on L 2 s.t. P t := exp(t ) Markovian local P t f(x) = E x [f(x t )], lim t 0 (P t I)/t =, (E, F) : regular Dirichlet form (i.e. sym. closed Markovian form) on L 2 Z E(u, v) = u vdµ, F = D( ) local K (E, F): regular, Def 9C Ω F C 0 (K) linear space which is dense i) in F w.r.t. E 1 -norm and ii) in C 0 (K) w.r.t. k k 1 -norm. (E, F): local Def, (u, v F, Supp u Supp v = ; ) E(u, v) = 0). Exam. BM on R n, Laplace op. on R n, E(f, f) = 1 2 R f 2 dx, F = H 1 (R n )
2 Dirichlet forms on fractals Sierpinski carpets {F i } N i=1 : Rn R n ; contraction maps (N = 3 n 1) d(f i (x), F i (y)) = d(x, y)/3 8x, y R n 91 non-void compact set K s.t. K = N i=1 F i(k). K: (n-dimensional) Sierpinski carpet d f := log N/ log 3: Hausdorff dimension of K (w.r.t. the Euclidean metric) µ: (normalized) Hausdorff measure on K, i.e. a Borel measure on K s.t. µ(f i1 i n (K)) = 3 n 8i 1,, i n S := {1,, N}. Theorem 2.1 (Barlow-Bass, Kusuoka-Zhou, etc.) 9(E, F): a local regular Dirichlet form on L 2 (K, µ) s.t. 9ρ > 0 E(f, g) = ρ X i S E(f F i, g F i ), 8f, g F (Self-similarity).
(Property I) Heat kernel estimates Theorem 2.2 (Barlow-Bass) Let d w = log(ρn)/ log 3 > 2. 9p t (x, y): jointly continuous sym. transition density of {X t } w.r.t. µ (P t f(x) = R K p t(x, y)f(y)µ(dy) 8x K, @ @t p t(x 0, x) = x p t (x 0, x) ) s.t. c 1 t d f/d w exp( c 2 ( d(x, y)d w t By integrating (2.1), we have E 0 [d(0, X t )] t 1/d w. As d w > 2, we say the process is sub-diffusive. Remark. ρ > 1 for n = 2, ρ < 1 for n 3. ) dw 1) 1 p t (x, y) c 3 t d f/d w exp( c 4 ( d(x, y)d w ) dw 1). 1 (2.1) t (Property II) Domains of the Dirichlet forms
For 1 p < 1, 1 q 1, β 0 and m N {0}, set Z Z a m (β, f) := L mβ (L md f f(x) f(y) p dµ(x)dµ(y)) 1/p, f L p (K, µ), x y <c 0 L m where 1 < L < 1, 0 < c 0 < 1. Λ β p,q(k): a set of f L p (K, µ) s.t. ā(β, f) := {a m (β, f)} 1 m=0 l q. Λ β p,q(k) is a Besov-Lipschitz space. It is a Banach space. p = 2 Λ β 2,q (Rn ) = B β 2,q (Rn ) if 0 < β < 1, = {0} if β > 1. p = 2, β = 1 Λ 1 2,1(R n ) = H 1 (R n ), Λ 1 2,2(R n ) = {0}. Theorem 2.3 (Jonsson 96 (for S. gasket), K, Paluba, Grigor yan-hu-lau, K-Sturm) Let (E, F) be the Dirichlet form on the gasket. Then, F = Λ d w/2 2,1 (K). (Q) Tr L F =?, L = [0, 1] n 1 {0}.
Recall that for α > 0 and k N where k < α k + 1, Bp,q α (Rn ) := {u L p (R n, m) : kuk B α p,q < 1}, where kuk B α p,q := X kd j uk L p + X Z k h D j q ukl ( p 0 j k j =k R n h n+q(α k) dh)1/q. ( ) Here, j = (j 1,..., j n ), j = j 1 + + j n, D j = @ j and h f(x) := f(x + h) f(x). @x j 1 1 @x j n n, When α N, h in (*) is changed to 2 h.
3 A trace theorem for Dirichlet forms on fractals Theorem 3.1 (Hino-K 05) Let d be the Hdff dim. for L (in this case n 1). Then, Tr L F = Λ β 2,2 (L), β = d w 2 d f d. 2 What is Tr L? 8f F, 9 f: quasi-cont. modification of f (I.e. f = f, µ-a.e., 8 > 0, 9G Ω K, CapG < s.t. f K\G is finite and cont.) Let be the Hdff meas. on L (in this case (n 1)-dim. Lebesgue meas.). Since d w > d f d, charges no set of 0-capacity (i.e. CapA = 0 ) (A) = 0). So, f is determined -a.e. on L. Further, f L L 2 (L, ). Thus, Tr L F := { f L : f F}.
More generally, (X, d, µ) nice metric meas. space {F i } i S : X X, 9α > 1 s.t. d(f i (x), F i (y)) = α 1 d(x, y), 8x, y X 91 non-void compact set K s.t. K = i S F i (K), µ: Hdff meas., d f : Hdff dim. Assume 9(E, F): loc. reg. D-form on L 2 (K, µ) that satisfies Self-similarity Elliptic Harnack ineq. Poincaré ineq. L: self-similar subset {F i } i I, I Ω S, L = i I F i (L), :Hdff meas., d: Hdff dim. s.t. (1) d w > d f d & (D) ccap(d) 8D: compact in K. (2) Technical geometric assumptions on K and L Assumption A f F, E S m \Im(f, f) = 0 8m ) f const. Here, E A (f, f) := ρ m P w A E(f F w, f F w ) for A Ω S m. Theorem 3.2 (Hino-K 05) Under above conditions, Tr L F = Λ β 2,2 (L) C 0(L) k k Λ β 2,2 (L), β = d w 2 d f d. (3.1) 2
Remarks. The latter half of (1) holds if K Ω R n & F = Λ d w/2 2,1 (K) or the diffusion for (E, F) satisfies the heat kernel estimates (2.1). (RHS of (3.1))= Λ β 2,2 (L) if (a) L Ω RD, β < 1 or (b) β > d/2. By the general theory of D-forms, we have the reg. D-form (Ě, ˇF) on L 2 (L, ), the trace of (E, F) to L by. The corresponding process is in general a jump-type process. (Classical case: Tr R n H 1 (R n ) = B 2,2 1/2 (Rn 1 ) domain of the D-form for the Cauchy proc.) The above theorems characterize the domain ˇF = Tr L F.
Examples. K: n-dim. Sierpinski gaskets, L: (n 1)-dim. gasket on the bottom (Jonsson (Math Z. 05), n = 2 case) β = log(n + 3) 2 log 2 log(1 + 1/n). 2 log 2 Nested fractals E.g., K: Penta-kun, L either the Cantor set or the Koch-like curve in K. K Vicsek set, L: diagonal line Assumption A does not hold! Tr L F = Λ 1 2,1(L) : Brownian motion on L In general, Λ β 2,2 (L) Ω Tr L F Ω Λ β 2,1 (L), but we cannot say anymore.
Trace from R n to a d-set, β > 0 Triebel ( 97 book) No extension thm B β p,q (K) := tr KB β+(n d)/2 p,q (R n ) Ω L p (K, µ) kf B β p,q(k)k := inf trk g=f kg Bp,q β+(n d)/2 (R n )k, Jonsson-Wallin ( 84 book) Both restriction and extension thm B p,q β,jw (K) := {{f (j) } j k : } Def. quite involved, Vector space B p,q β,jw f (j) : formal j-th derivative of f, j: multi index, (K) = Bβ++(n d)/2 p,q (R n ) K Λ β p,q (K) Ω Bp,q β,jw (K) in the sense f 7 (f, 0, 0,..., 0) When β < 1, both def. coincides and B β p,q (K) = Bp,q β,jw (K) = Λβ p,q (K). k < β k + 1, k N
4 Application: Penetrating processes K i : fractal (i = 1, 2), G = K 1 K 2 Assume 9(E Ki, F Ki ) on L 2 (K i, µ i ): loc. reg. D-form on K i with nice properties. (Q) Construct diffusion on G which behave as the appropriate diff. on each K i. Superposition of D-forms µ = µ 1 + µ 2 Ẽ(u, v) := E K1 (u K1, v K1 ) + E K2 (u K2, v K2 ), D(Ẽ) := {u C 0(G) : u Ki F Ki, i = 1, 2, Ẽ(u, u) < 1}. (Q) Enough functions in D(Ẽ)? Esp., D(Ẽ) dense in C 0(G)? ( Using trace thm, YES! Once solved, (Ẽ, F) loc. reg. D-form on L 2 (G, µ) where F 1. = D(Ẽ)Ẽ Various properties of the diffusion can be obtained. (K, JFA 00; Hambly-K, PTRF 03) Penetrating P x (σ B < 1) > 0 q.e. x G, 8B: pos. cap.
Theorem 4.1 (Hambly-K 03) Short time heat kernel estimates Under the following strong assumption 2 d s (K i ) 2 d f (K i )d c (K i ) < i = 1, 2, (4.1) we have the following for all x, y G. p t (x, y) c 1 t (d s(x) d s (y))/2 Φ(d (1) (x, y), d (2) (x, y), c 2 t), 8t < 9t 0 (x) t 0 (y), where p t (x, y) c 3 t θ(x,y) (x, y, t)φ(d (1) (x, y), d (2) (x, y), c 4 t), 8t < 1, Φ(u 1, u 2, t) = exp( 2X i=1 (i) w ( ud (i) it )1/(d w 1) ). θ, : error terms. When x = y, (x, y, t) = 1 and θ(x, y) = (d s (x) d s (y))/2. Only nested fractals, needed strong assumption (4.1). ) The trace thm. generalizes this.
Fractal tiling (Hambly-K 03) SG(2) SG(4)
d (2) w < d (4) w d (2) w > d (l) w : l large Most probable path avoids to move on K i where d (i) w is small.
5 Ideas of the proof of the trace thm Basic idea: Use self-similarity instead of the differential structure! (Discrete Approximation) Q n : L 1 (L, ) R In, Q n f(w) = R F w (L) f(s)d (s). E (n) (g) := P v,w I n,v n ªw (g(v) g(w))2, 8g R In. Lemma 5.1 1) kā(β, f)k 2 l 2 P 1 n=1 α(2β d)n E (n) (Q n f) = P 1 n=1 ρn E (n) (Q n f) 2) E (n) (Q n f) cρ n E I n(f, f), 8f F, 8n. Let H I n := {h F : E(h, f) = 0, 8f F I n} where F I n = {f F : f = 0 on K S n \I n, Q nf 0}. Lemma 5.2 (Key Lemma) Under Assumption A, 9c 0 < 1 s.t. E I n+1(h, h) c 0 E I n(h, h), 8h H I n.
(Restriction thm) For each f F, let g n F be s.t. g n = f on K S n \I n, Q ng n = Q n f, g n H I n. Then g n f in F. Let f n := g n g n 1 (g 1 0). Then f = P n f n, E(f i, f j ) = 0, i 6= j (ortho. decomp.). So E(f, f) = P 1 n=0 E(f n, f n ). Using these, nx (E (n) (Q n f)) 1/2 = (E (n) (Q n g n )) 1/2 = (E (n) ( Q n f j )) 1/2 Minkowski nx (E (n) (Q n f j )) j=0 So we obtain 1/2 Lem 5.1 2) c j=0 nx (ρ n E I n(f j )) j=0 1X ρ n E (n) (Q n f) c n=0 ( ) c 0 1X c n 0( n=0 X 1 j=0 nx j=0 1/2 Lem 5.2 nx c 0 (ρ n c n j 0 E I j(f j )) 1/2. j=0 (c j 0 E I j(f j)) 1/2 ) 2 c j 0 c j 0 E(f j) = c 0 X j E(f j ) = c 0 E(f), where P 1 i=0 2ai ( P i j=0 a j) p c P 1 j=0 2aj a p j for a < 0, p > 0, a j 0 is used in (*)
Remark. The fact that F v : H(F v(k)) Fv (K) F is a compact operator is used (only) in Lem 5.2. (Here F v h = h F v.) Elliptic Harnack ineq. is used to guarantee this. (Extension thm) Similar to the classical ext. thm (Whitney decomp., assoc. partition of unity etc.), using self-similarity.