The Kato square root problem on vector bundles with generalised bounded geometry Lashi Bandara (Joint work with Alan McIntosh, ANU) Centre for Mathematics and its Applications Australian National University 7 August 2012 Recent trends in Geometric and Nonlinear Analysis Banff International Research Station, Canada Lashi Bandara (ANU) The Kato problem on GBG bundles 7 August 2012 1 / 30
History of the problem In the 1960 s, Kato considered the following abstract evolution equation du dt on a Hilbert space H. + A(t)u = f(t), t [0, T ]. This has a unique strict solution u = u(t) if D(A(t) α ) = const for some 0 < α 1 and A(t) and f(t) satisfy certain smoothness conditions. Lashi Bandara (ANU) The Kato problem on GBG bundles 7 August 2012 2 / 30
Typically A(t) is defined by an associated sesquilinear form J t : W W C where W H. Suppose 0 ω π/2. Then J t is ω-sectorial means that (i) W H is dense, (ii) J t [u, u] S ω+ = {ζ C : arg ζ ω} {0}, and (iii) W is complete under the norm u 2 W = u + Re J[u, u]. Lashi Bandara (ANU) The Kato problem on GBG bundles 7 August 2012 3 / 30
T : D(T ) H is called ω-accretive if (i) T is densely-defined and closed, (ii) T u, u S ω+ for u D(T ), and (iii) σ (T ) S ω+. A 0-accretive operator is non-negative and self-adjoint. Lashi Bandara (ANU) The Kato problem on GBG bundles 7 August 2012 4 / 30
Let A(t) : D(A(t)) H be defined as the operator with largest domain such that J t [u, v] = A(t)u, v u D(A(t)), v W. The theorem of Lax-Milgram guarantees that J t is ω-sectorial = A(t) is ω-accretive. In 1962, Kato showed in [Kato] that for 0 α < 1/2 and 0 ω π/2, D(A(t) α ) = D(A(t) α ) = D = const, and A(t) α u A(t) α u, u D. (K α ) Counter examples were known for α > 1/2 and for α = 1/2 when ω = π/2. Lashi Bandara (ANU) The Kato problem on GBG bundles 7 August 2012 5 / 30
Kato asked two questions. For ω < π/2, (K1) Does (K α ) hold for α = 1/2? (K2) For the case ω = 0, we know D( A(t)) = W and (K1) is true, but is d A(t)u u dt for u W? In 1972, McIntosh provided a counter example in [Mc72] demonstrating that (K1) is false in such generality. In 1982, McIntosh showed that (K2) also did not hold in general in [Mc82]. Lashi Bandara (ANU) The Kato problem on GBG bundles 7 August 2012 6 / 30
The Kato square root problem then became the following. Set J[u, v] = A u, v u, v H 1 (R n ), where A L is a pointwise matrix multiplication operator satisfying the following ellipticity condition: Re J[u, u] κ u, for some κ > 0. Under these conditions, is it true that D( div A ) = H 1 (R n ) div A u u (K1) This was answered in the positive in 2002 by Auscher, Hofmann, Lacey, McIntosh and Tchamitchian in [AHLMcT]. Lashi Bandara (ANU) The Kato problem on GBG bundles 7 August 2012 7 / 30
Setup Let M be a smooth, complete Riemannian manifold with metric g, Levi-Civita connection, and volume measure dµ. Write div = in L 2 and let S = (I, ). Consider the following uniformly elliptic second order differential operator L A : D(L A ) L 2 (M) L 2 (M) defined by L A u = as ASu = a div(a 11 u) a div(a 10 u) + aa 01 u + aa 00 u. That is, we assume a and A = (A ij ) are L multiplication operators and that there exist κ 1, κ 2 > 0 such that Re av, v κ 1 v 2, v L 2 Re ASu, Su κ 2 ( u 2 + u 2 ), u H 1 Lashi Bandara (ANU) The Kato problem on GBG bundles 7 August 2012 8 / 30
The problem The Kato square root problem on manifolds is to determine when the following holds: { D( LA ) = H 1 (M) L A u u + u = u H 1, u H 1 (M) Lashi Bandara (ANU) The Kato problem on GBG bundles 7 August 2012 9 / 30
The main theorem Theorem (B.-Mc) Let M be a smooth, complete Riemannian manifold with Ric C and inj(m) κ > 0. Suppose there exist κ 1, κ 2 > 0 such that Re av, v κ 1 v 2 Re ASu, Su κ 2 u 2 H 1 for v L 2 (M) and u H 1 (M). Then, D( L A ) = D( ) = H 1 (M) and L A u u + u = u H 1 for all u H 1 (M). Lashi Bandara (ANU) The Kato problem on GBG bundles 7 August 2012 10 / 30
Stability Theorem (B.-Mc) Let M be a smooth, complete Riemannian manifold with Ric C and inj(m) κ > 0. Suppose that there exist κ 1, κ 2 > 0 such that Re av, v κ 1 v 2 Re ASu, Su κ 2 u 2 H 1 for v L 2 (M) and u H 1 (M). Then for every η i < κ i, whenever ã η 1, Ã η 2, the estimate L A u L A+ Ã u ( ã + Ã ) u H 1 holds for all u H 1 (M). The implicit constant depends in particular on A, a and η i. Lashi Bandara (ANU) The Kato problem on GBG bundles 7 August 2012 11 / 30
A more general problem We can consider the Kato square root problem on vector bundles by replacing M with V, a smooth, complex vector bundle of rank N over M with metric h and connection. These theorems are obtained as special cases of corresponding theorems on vector bundles. We use the adaptation of the first order systems approach in [AKMc], which captures the Kato problem (and some other results of harmonic analysis) in terms of perturbations of Dirac type operators. Lashi Bandara (ANU) The Kato problem on GBG bundles 7 August 2012 12 / 30
Axelsson (Rosén)-Keith-McIntosh framework (H1) Let Γ be a densely-defined, closed, nilpotent operator on a Hilbert space H, (H2) Suppose that B 1, B 2 L(H ) such that here exist κ 1, κ 2 > 0 satisfying Re B 1 u, u κ 1 u 2 and Re B 2 v, v κ 2 v 2 for u R(Γ ) and v R(Γ), (H3) The operators B 1, B 2 satisfy B 1 B 2 R(Γ) N (Γ) and B 2 B 1 R(Γ ) N (Γ ). Let Γ B = B 1Γ B 2, Π B = Γ + Γ B and Π = Γ + Γ. Lashi Bandara (ANU) The Kato problem on GBG bundles 7 August 2012 13 / 30
Growth restrictions We say M has exponential volume growth if there exists c 1, κ, λ 0 such that 0 < µ(b(x, tr)) ct κ e λtr µ(b(x, r)) < (E loc ) for all t 1, r > 0 and x M. For instance, if Ric ηg, for η R, then (E loc ) is satisfied. Lashi Bandara (ANU) The Kato problem on GBG bundles 7 August 2012 14 / 30
Generalised bounded geometry We want to set H = L 2 (V), but we need to assume more structure in V. Definition (Generalised Bounded Geometry) Suppose there exists ρ > 0, C 1 such that for each x M, there exists a trivialisation ψ : B(x, ρ) C N π 1 (B(x, ρ)) satisfying V C 1 I h CI in the basis { e i = ψ(x, ê i ) }, where { ê i} is the standard basis for C N. Then, we say that V has generalised bounded geometry or GBG. We call ρ the GBG radius. Lashi Bandara (ANU) The Kato problem on GBG bundles 7 August 2012 15 / 30
Further assumptions (H4) The bundle V has GBG, H = L 2 (V), and M grows at most exponentially. (H5) The operators B 1, B 2 are matrix valued pointwise multiplication operators. That is, B i L (M, L(V)) by which we mean that B i (x) L(π 1 V (x)) for every x M and there is a C B i > 0 so that B i (x) C for almost every x M. (H6) The operator Γ is a first order differential operator. That is, there exists a C Γ > 0 such that whenever η C c (M), we have that ηd(γ) D(Γ) and M η u(x) = [Γ, η(x)] u(x) is a multiplication operator satisfying M η u(x) C Γ η T M u(x) for all u D(Γ) and almost all x M. Lashi Bandara (ANU) The Kato problem on GBG bundles 7 August 2012 16 / 30
Dyadic cubes Since we assume exponential growth of M, the work of Christ in [Christ] and subsequently of Morris in [Morris] allows us to perform a dyadic decomposition of the manifold below a fixed scale. In particular, we are able to choose arbitrarily large J N so that for every j J, Q j is an almost everywhere decomposition of M by open sets. If l > j then for every cube Q Q l, there is a unique cube Q Q j such that Q Q. Each such cube has a centre x Q. Each cube Q Q j also has a diameter of at most C 1 δ j, where C 1 > 0 and δ (0, 1) are fixed, uniform quantities. Lashi Bandara (ANU) The Kato problem on GBG bundles 7 August 2012 17 / 30
GBG coordinates Since we assume the bundle satisfies GBG, we recall the uniform ρ > 0 from this criterion. Choose J N such that C 1 δ J ρ 5. We call t S = δ J the scale. Call the system of trivialisations C = { ψ : B(x Q, ρ) C N π 1 V (B(x Q, ρ)) s.t. Q Q J} the GBG coordinates. Lashi Bandara (ANU) The Kato problem on GBG bundles 7 August 2012 18 / 30
GBG coordinates (cont.) Call the { set of a.e. trivialisations } C J = ϕ Q = ψ Q : Q C N π 1 V (Q) s.t. Q QJ the dyadic GBG coordinates. For any cube Q, the unique cube Q Q J satisfying Q Q we call the GBG cube of Q. The GBG coordinate system of Q is then ψ : B(x Q, ρ) C N π 1 V (B(x Q, ρ)). Lashi Bandara (ANU) The Kato problem on GBG bundles 7 August 2012 19 / 30
Machinery for the harmonic analysis For t t S, we write Q t = Q j whenever δ j+1 < t δ j. For j > J and Q Q j and u = u i e i L 1 loc (V) in the GBG coordinates associated to Q. Define, the cube integral ˆ (ˆ ) u = u i e i inside B(x Q, ρ). Q The cube average is then defined as u Q (y) = ffl Q u, for y B(x Q, ρ) and 0 otherwise. Let A t u(x) = u Q (x) whenever x Q Q t. For each w C N, let γ t (x)w = (Θ B t ω)(x) where ω(x) = w in the GBG coordinates of each Q. Q Lashi Bandara (ANU) The Kato problem on GBG bundles 7 August 2012 20 / 30
Cancellation assumption (H7) There exists c > 0 such that for all t t S and Q Q t, ˆ ˆ Γu dµ cµ(q) 1 2 u and Γ v dµ cµ(q) 1 2 v Q for all u D(Γ), v D(Γ ) satisfying spt u, spt v Q. Q Lashi Bandara (ANU) The Kato problem on GBG bundles 7 August 2012 21 / 30
Dyadic Poincaré assumption (H8) There exists C P, C C, c, c > 0 and an operator Ξ : D(Ξ) L 2 (V) L 2 (N ), where N is a normed bundle over M with norm N and D(Π) R(Π) D(Ξ) satisfying for all u D(Π) R(Π), -1 (Dyadic Poincaré ) ˆ ˆ ( u u Q 2 dµ C P (1 + r κ e λcrt )(rt) 2 Ξu 2 N + u 2) dµ B for all balls B = B(x Q, rt) with r C 1 /δ where Q Q t with t t S, and -2 (Coercivity) cb Ξu 2 L 2 (N ) + u 2 L 2 (V) C C Πu 2 L 2 (V). Lashi Bandara (ANU) The Kato problem on GBG bundles 7 August 2012 22 / 30
Kato square root type estimate Proposition Suppose M is a smooth, complete Riemannian manifold and V is a smooth vector bundle over M. If (H1)-(H8) are satisfied, then (i) D(Γ) D(Γ B ) = D(Π B) = D( Π 2 B ), and (ii) Γu + Γ B u Π B u Π 2 B u, for all u D(Π B). Lashi Bandara (ANU) The Kato problem on GBG bundles 7 August 2012 23 / 30
Kato square root problem on vector bundles Theorem (B.-Mc.) Suppose M grows at most exponentially and satisfies a local Poincaré inequality on functions. Further, suppose that both V and T M have GBG, and (i) the metric h and are compatible, (ii) there exists C > 0 such that in each GBG chart we have that e j, dx i, k h ij, k g ij C a.e., (iii) there exist κ 1, κ 2 > 0 such that Re au, u κ 1 u 2 and Re ASv, Sv κ 2 v 2 H 1 for all u L2 (V) and v H 1 (V), and (iv) we have that D( ) H 2 (V), and there exist C > 0 such that 2 u C (I + )u whenever u D( ). Then, D( L A ) = D( ) = H 1 (V) with L A u u + u = u H 1 for all u H 1 (V). Lashi Bandara (ANU) The Kato problem on GBG bundles 7 August 2012 24 / 30
Set H = L 2 (V) (L 2 (V) L 2 (T M V)). Define Γ = [ ] 0 0, B S 0 1 = [ ] a 0, and B 0 0 2 = [ ] 0 0. 0 A Then, [ ] Γ 0 S = 0 0 and Π 2 B = [ ] LA 0 0 Lashi Bandara (ANU) The Kato problem on GBG bundles 7 August 2012 25 / 30
Kato square root problem for tensors Theorem (B.-Mc.) Let M be a smooth, complete Riemannian manifold with Ric C and inj(m) κ > 0. Suppose that there exist C > 0 such that 2 u C (I + )u whenever u D( ) H 2 (T (p,q) M). Then, D( L A ) = D( ) = H 1 (T (p,q) M) and L A u u + u = u H 1 for all u H 1 (T (p,q) M). Lashi Bandara (ANU) The Kato problem on GBG bundles 7 August 2012 26 / 30
Ricci, injectivity bounds and GBG Proposition Suppose there is a κ, η > 0 such that inj(m) κ and Ric η. Then for A > 1 and α (0, 1), there exists r H (n, A, α, κ, η) > 0 such that for each x M, there is a coordinate system corresponding to B(x, r H ) satisfying: (i) A 1 δ ij g ij Aδ ij as bilinear forms and, (ii) l r H + l sup l g ij (y) y B(x,r H ) r 1+α H sup y z B(x,r H ) l g ij (z) l g ij (y) d(y, z) α A 1. See the observation following Theorem 1.2 in [Hebey]. Lashi Bandara (ANU) The Kato problem on GBG bundles 7 August 2012 27 / 30
The proposition guarantees GBG coordinates for V = T (p,q) M. The proposition gives e j, dx i, k h ij, k g ij C in each GBG coordinate chart { e j} for T (p,q) M. The Ricci bounds guarantee exponential volume growth and the local Poincaré inequality. By invoking Theorem 5 (on vector bundles), we obtain Theorem 6 (finite rank tensors). Theorem 1 follows from Theorem 6 since 2 u (I + )u is a consequence of the Bochner-Lichnerowicz-Weitzenböck identity, Ricci curvature bounds and uniform lower bounds on injectivity radius. See Proposition 3.3 in [Hebey]. Lashi Bandara (ANU) The Kato problem on GBG bundles 7 August 2012 28 / 30
References I [AHLMcT] Pascal Auscher, Steve Hofmann, Michael Lacey, Alan McIntosh, and Ph. Tchamitchian, The solution of the Kato square root problem for second order elliptic operators on R n, Ann. of Math. (2) 156 (2002), no. 2, 633 654. [AKMc-2] Andreas Axelsson, Stephen Keith, and Alan McIntosh, The Kato square root problem for mixed boundary value problems, J. London Math. Soc. (2) 74 (2006), no. 1, 113 130. [AKMc], Quadratic estimates and functional calculi of perturbed Dirac operators, Invent. Math. 163 (2006), no. 3, 455 497. [Christ] Michael Christ, A T (b) theorem with remarks on analytic capacity and the Cauchy integral, Colloq. Math. 60/61 (1990), no. 2, 601 628. Lashi Bandara (ANU) The Kato problem on GBG bundles 7 August 2012 29 / 30
References II [Hebey] Emmanuel Hebey, Nonlinear analysis on manifolds: Sobolev spaces and inequalities, Courant Lecture Notes in Mathematics, vol. 5, New York University Courant Institute of Mathematical Sciences, New York, 1999. [Kato] Tosio Kato, Fractional powers of dissipative operators, J. Math. Soc. Japan 13 (1961), 246 274. MR 0138005 (25 #1453) [Mc72] Alan McIntosh, On the comparability of A 1/2 and A 1/2, Proc. Amer. Math. Soc. 32 (1972), 430 434. MR 0290169 (44 #7354) [Mc82], On representing closed accretive sesquilinear forms as (A 1/2 u, A 1/2 v), Nonlinear partial differential equations and their applications. Collège de France Seminar, Vol. III (Paris, 1980/1981), Res. Notes in Math., vol. 70, Pitman, Boston, Mass., 1982, pp. 252 267. MR 670278 (84k:47030) [Morris] Andrew J. Morris, The Kato Square Root Problem on Submanifolds, ArXiv e-prints:1103.5089v1 (2011). Lashi Bandara (ANU) The Kato problem on GBG bundles 7 August 2012 30 / 30