Densely defined non-closable curl on topologically one-dimensional Dirichlet metric measure spaces

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1 Densely defined non-closable curl on topologically one-dimensional Dirichlet metric measure spaces Kansai Probability Seminar, Kyoto March 11, 2016 Universität Bielefeld

2 joint with Alexander Teplyaev (University of Connecticut). Long term goals: Geometric analysis on fractal metric measure spaces and intrinsic geometry, in particular vector analysis and differential forms Motivations: Items of Riemannian flavor already studied (e.g. by Bakry, Cheeger, Emery, Gigli, Hino, Kajino, Kigami, Koskela, Ledoux, Sturm, Zhou and others) Items of derham or Hodge type flavor hardly looked at, but in principle accessible using first order derivations (Cipriani, Sauvageot, Weaver and others) Potential applications in physics (magnetic fields, fluid dynamics, optical waveguides) and data science

3 Smooth manifold case dimension dim M of M defined as the dimension of the Euclidean space containing its charat images as open sets dim M equals its topological dimension dim topo M (Separable and metrizable X has dim topo X = n if n is minimal value s.t. any finite open cover of X has refinement s.t. each x X is contained in at most n + 1 sets of the refinement.)

4 tangent space T x M at every x M is n-dim vector space, similarly for cotangent space T x M in particular, Λ k T x M = {0} for k > n (there are no nontrivial k-forms) dim T x M = dim topo M for all x M

5 Dirichlet metric measure spaces DMMS=locally compact separable metric space X, nonnegative Radon measure µ with full support, regular symmetric Dirichlet form (E, F) Here: (E, F) strongly local Topological dimension defined as before Generally no smooth charts, no classical (co-)tangent spaces

6 Can talk about dimension of (co-)tangent spaces using concepts of AF-martingale dimension dim mart (Motoo, Watanabe,...) resp. index of Dirichlet form (Hino): There is an equiv class of (mutually abs. cont.) minimal energy dominant measures m The index p of (E, F) is the smallest integer such that for any N N and any f 1,..., f N F, ( ) dγ(fi, f j ) N rank dm (x) p for m-a.e. x X. i,j=1 Hino 08, 10: p well def (indep of choice of m) and dim mart = p

7 Energy dominant : For any f F C c (X), energy measure Γ(f ) satisfies Γ(f ) << m; recall ϕ dγ(f ) = E(f ϕ, f ) 1 2 E(f 2, ϕ), ϕ F C c (X). X Minimal : If m has same property, then m << m.

8 Kusuoka 89: dim mart = 1 for d-dim standard Sierpinski gasket (although dim H = log(d+1) log 2 may be very large) Hino 08, 10, 13: P.c.f. self-similar fractals: dim mart = 1 Self-similar generalized Sierpinski carpets: 1 dim mart d s

9 dim mart may be interpreted as m-essential supremum of dimensions of tangent spaces in a measurable bundle sense (papers of Hino, also Eberle 99, H./Röckner/Teplyaev 13) Examples For X = R n with E(f ) = R n ( f ) 2 dx, f H 1 (R n ), have dim mart = n. Examples For X = M compact RMf with E(f ) = M ( f )2 dvol, f H 1 (M), have dim mart = n.

10 Generally dim topo and dim mart may differ In particular: Topo one-dim spaces might carry nontrivial 2-forms... Somehow counterintuitive (would expect dim mart dim topo )... What happens? Connected to behaviour of (analogs of) the exterior derivation d : L 2 (M, T M, dvol) L 2 (M, Λ 2 T M, dvol) taking 1-forms into 2-forms, a i dx i a i x j dx j dx i Phenomenon does not occur in classical theory For simplicity, illustrate issue for curl-operator

11 Curl of vector fields U R 3 open, connected, v = (v 1, v 2, v 3 ) : U R 3 vector field ( ) curl v = v = v3 y v 2 z, v 1 z v 3 x, v 2 x v 1 y If v velocity field of a fluid flow Small ball is made rotate by flow axis points in direction of vector field curl v (right hand rule) Angular speed is 1 2 of length of curl v

12 Connection to differential forms by duality argument: Given v = (v 1, v 2, v 3 ), consider ω := v i dx i. Then dω = v i x j dx j dx i ( v3 = x 2 v ) 2 x 3 dx 2 dx Two-dim curl: U R 2 open, connected and vector field u = (u 1, u 2 ) : U R 2 consider v := (u 1, u 2, 0) then function curl u : U R, curl u = u 2 x (x, y) u 1 (x, y), y is third component of curl v = (0, 0, curl u)

13 In terms of differential forms, d(u 1 (x, y)dx + u 2 (x, y)dy) = curl u(x, y) dx dy Can consider curl : L 2 (U, R 2 ) L 2 (U) as closed unbounded operator Remark: L 2 -perspective interesting because of qm models (e.g. for magnetic fields) Next idea: Replace U by generalized Sierpinski carpet with 2 = dim mart > dim topo = 1

14 Sierpinski carpets Consider non-self-similar generalized Sierpinski carpets studied by Mackay/Tyson/Wildrick 13. a = (a 1, a 2,...) sequence of reals a i > 0 s.t. 1 a i > 1 odd integer Rewrite S a,0 := [0, 1] 2 as union of congruent closed subsquares of side lengths a 1, touching only at boundaries, remove middle one to get a set S a,1 Rewrite S a,1 as union of congruent closed subsquares of side lengths a 1 a 2, touching only at boundaries, remove middle ones (w.r.t. the subsquares) to get a set S a,2 etc. S a := m 0 S a,m generalized Sierpinski carpet associated with sequence a

15 rting point for our investigations was the following well-known fact. ion 1.4. For each k, the carpet S 1/(2k+1), equipped with Euclidean metric and Haus its dimension Q k, does not support any Poincaré inequality. Standard Figure self-similar 1. S 1/3 carpet S a NonFigure self-similar 2. S carpet (1/3,1/5,1/7,...) S a with a = ( 1 3, 1 3, 1 3,...) with a = ( 1 3, 1 5, )

16 Proposition (Mackay/Tyson/Wildrick 13) If a l 2 then any nonempty open subset of S a has positive two dim Lebesgue measure. Examples a n := 1 2n+1. Let a l 2 be fixed, write S := S a and L 2 (S) for L 2 -space on S w.r.t. two dim Lebesgue.

17 Energy form Consider E S (f ) := ( f (x, y)) 2 d(x, y), f C 1 (R 2 ). S Polarization yields bilinear form. The form (E S, C 1 (R 2 )) is closable, and its closure (E S, D S ) is a strongly local regular Dirichlet form on L 2 (S). (Follows as in Koskela/Shanmugalingam/Tyson 04, Shanmugalingam 00; Newtonian Sobolev spaces; for f C 1 (R 2 ) the function f is minimal upper gradient of f.)

18 For a vector field u = (u 1, u 2 ) with u 1, u 2 C 1 (R 2 ), curl u is a continuous function and can be restricted to S, and (curl u) S L 2 (S) Therefore: May view curl as densely defined unbounded operator with domain C 1 (R 2, R 2 ) curl : L 2 (S, R 2 ) L 2 (S) Slightly reformulated: Endow curl with abstract domain dom curl and let curl be its adjoint with domain dom(curl )

19 Theorem (H./Teplyaev 15) Let a l 2 be such that a 1 a n 1 lim = 0 n a n and consider S = S a. If dom curl L 2 (S, R 2 ) contains all smooth vector fields, then dom(curl ) L 2 (S) is trivial. In particular, the operator (curl, dom curl) is not closable. ( a decays fast enough but not too fast.) Examples a n := 1 2n+1.

20 Proof (by contradiction) Suppose 0 u dom(curl ) L 2 (S) and curl u = w L 2 (S, R 2 ). Then ex. smooth function f such that u, f L 2 (S) > 0. Claim: Can construct sequence (v n ) n of smooth vector fields v n s.t. (a) lim n curl v n = f in L 2 (S) (b) lim n v n = 0 in L 2 (S, R 2 ). If so, then 0 = lim n w, v n L 2 (S,R 2 ) = lim n curl u, v n L 2 (S,R 2 ) = lim n u, curl v n L 2 (S) what cannot be true. Suffices to show claim. = u, f L 2 (S) > 0,

21 Cover S by compact subsets S n,k obtained by taking parallels to the axes through the midpoints of all holes of size δ n = a 1 a n. Intersections are Cantor sets and diam S n,k 2δ n 1. S n,k

22 Step 1: We show how to choose small nbhs U n,k of the boundaries of the sets S n,k and construct sequence of energy finite functions g n s.t. (i) g n arbitrarily close to vector field (0, 1) in L 2 (S, R 2 ) (ii) each g n is locally constant on each U n,k.

23 For fixed n, consider Cantor-set parts of boundaries of the S n,k parallel to x-axis, let F n be union of their vertical parallel sets Let ϕ n be a continuous function that is constant in x on S, constant in y on F n, and on each connected component of S \ F n differs from g(x, y) := y by an additive constant.

24 Each ϕ n is restriction to S of a Lipschitz function, hence of finite energy. Moreover lim n E(g ϕ n ) = lim n F n ( (g ϕ n )) 2 dλ 2 = lim n λ 2 (F n ) lim n δ n a 1 a n 1 = lim n a n = 0. Now consider vertical Cantor set parts of the boundaries of the sets S n,k. Connect two vertically adjacent holes by rectangles with horizontal side length δ n and vertical side length ε n := (1 a n )(a 1 a n 1 ). Inscribe trapezoids with lower edge length δ n and upper edge length δ n/2...

25 εn δn Let ψ n be the function on [0, 1] 2 created by putting little tents over each rectangle such that ψ n is zero on left, right and lower edge of each rectangle, has value ε n on the upper (short) edge of the trapezoid and is linear in between. For boundary pieces proceed similarly by mirroring to the outside. Number of such tents is 2 a 1 a n 1.

26 Energy form E S does not see parts of ψ n over holes. On trapezoids between holes ψ n is constant in x-direction and has slope 1 in y-direction. Outside the trapezoids, slope ±4ε n /δ n in x-direction. Typical tent contributes energy 3 4 ε nδ n + 4 ε3 n δ n (and for situations close to the boundary this is upper bound). We obtain E S (ψ n ) a 1 a n + 8 a 1 a n 1 a n, and by hypothesis on a have lim n E S (ψ n ) = 0.

27 The functions g n := ϕ n ψ n now satisfy lim n E S (g g n ) 1/2 lim n E S (g ϕ n ) 1/2 + lim n E S (ψ n ) 1/2 = 0, what is (i). Each g n is locally constant on the neighborhood U n,k of S n,k consisting of two rectangles and two trapezoids (with modifications at the boundary of S), what shows (ii). S n,k

28 Step 2: Let f n,k be one of the values of the function f on S n,k, and let x n,k be one of the values of the x coordinate on S n,k. There exists a sequence of smooth functions h n such that h n sup a 1 a n 1 f sup and on each set S n,k \ U n,k we have Then we define h n (x, y) = f n,k (x x n,k ). v n = h n g n. Obviously (b) is satisfied, lim n v n = 0 in L 2 (S, R 2 ).

29 Moreover, curl v n = f n,k ( g n ) 2 + h n curl g n = f n,k ( g n ) 2 where ( g n ) 2 = gn(x,y) y denotes the second component of the vector field g n. construction this second component ( g n ) 2 converges in L 2 (S) to a function that is identically equal to 1 on S, and so curl v n converges to f in L 2 (S). This is (i).

30 Key element of our construction: g n vanishes on each U n,k and so we do not have to analyze the derivatives of h n on U n,k, although one can see that these derivatives can not be small. Using slightly different technique but similar ideology, can tackle general case.

31 Strongly local forms on compact spaces X compact metric space, µ finite Radon measure, full support, (E, F) strongly local regular Dirichlet form. We consider differential forms with respect to a coordinate sequence and an energy dominant measure. Assumption m is a finite energy dominant measure for (E, F) and (ϕ k ) k F C(X) a sequence of functions that is dense in F, separates the points of X and is s.t. dγ(ϕ k ) dm L (X, m) for all k (such (ϕ k ) k and m can always be found, folklore)

32 Let A be the algebra generated by (ϕ k ) k and 1 (dense in C(X)). Consider Hilbert space (H,, H ) of L 2 -differential 1-forms associated with (E, F) (Cipriani/Sauvageot 03, Ionescu/Rogers/Teplyaev 12) Recall construction: A A with Hilbert seminorm determined by f g 2 H = g 2 dγ(f ), factor out zero seminorm elements, complete to get H (Mokobodzki, LeJan, Ikeda, Manabe, Fukushima, Nakao, Eberle, Weaver, Guido, Isola,...) In terms of probability, H = M X

33 First order derivation 0 f := f 1 takes functions into 1-forms, 0 : A H and satisfies 0 f 2 H = E(f ), f A. Extends to closed unbounded 0 : L 2 (X, µ) H with domain F. By energy dominance can find measurable field of Hilbert spaces (fibers) (H x ) x X such that H = L 2 (X, (H x ) x X, m) and ω, η H = ω x, η x Hx m(dx) X for ω, η H. Note ( 0 f ) x, ( 0 g) x Hx = Γ(f, g)(x).

34 Examples M RMf with standard energy, then H = L 2 (M, TM, dvol) = L 2 (M, T M, dvol), H x = Tx M = T x M and 0 is resp. d 0. Examples On standard Sierpinski gasket, 0 is functional analytic version of Kusuoka-Kigami-Strichartz-Teplyaev gradient.

35 Consider space L 2 (X, (ˆΛ 2 H x ) x X, m) of L 2 -differential 2-forms, here ˆΛ 2 H x is completion of Λ 2 H x in scalar product determined by ω 1 x ηx, 1 ωx 2 ηx 2 ˆΛ 2 H x := ω 1 x, ωx 2 H x η 1 x, ηx 2 H x ωx, 1 ηx 2 H x η 1 x, ωx 2 Let A Lip be algebra generated by functions F g with g A and F : R R Lipschitz with F(0) = 0. Can introduce generalized exterior derivation such that 1 (f 0 g) = 0 f 0 g. 1 : F A Lip L 2 (X, (ˆΛ 2 H x ) x X, m) H x.

36 Examples For RMf M have ˆΛ 2 H x = Λ 2 H x = Λ 2 T x M and L 2 (X, (ˆΛ 2 H x ) x X, m) = L 2 (M, Λ 2 T M, dvol). Space F A Lip is contained in the span of 1-forms fd 0 g with g H 1 (M) and f H 1 (M) W 1, (M). Remark: Actually use two constructions that fit together: an algebraic definition of 1 (a bit similar to concepts of NCG, see Gracia-Bondia/Várilly/Figueroa) the definition 1 (g 0 f ) := 0 g 0 f via wedge product on fibers H x

37 Theorem (H./Teplyaev 15) Suppose that X is topologically one-dimensional. Then either the martingale dimension of (E, F) is one or ( 1, F A Lip ) is not closable. Self-criticism: Result is for specific (although reasonable) domain of definition, need investigation what happens for other / modified domains.

38 THANK YOU