Geometry and the Kato square root problem

Geometry and the Kato square root problem Lashi Bandara Centre for Mathematics and its Applications Australian National University 7 June 2013 Geometric Analysis Seminar University of Wollongong Lashi Bandara (ANU) Geometry and the Kato problem 7 June 2013 1 / 28

Outline Brief overview of the Kato square root problem on R n. A motivating application to hyperbolic PDE. Recent progress on the Kato square root problem on smooth manifolds by McIntosh and B. Recent progress on subelliptic Kato square root problems on Lie groups by ter Elst, McIntosh, and B. Kato square root problem on smooth manifolds with non-smooth metrics, connection to geometric flows and PDEs. Lashi Bandara (ANU) Geometry and the Kato problem 7 June 2013 2 / 28

The Kato square root problem Let A L (R n, L(C N )) and a L (R n ). Suppose that there exists κ 1, κ 2 > 0 such that for all u W 1,2 (R n ), Re a(x) κ 1 and Re A u, u κ 2 u 2. The Kato square root problem on R n is the statement that D( a div A ) = W 1,2 (R n ) a div A u u. (K1) This was answered in the positive in 2002 by Pascal Auscher, Steve Hofmann, Michael Lacey, Alan McIntosh and Phillipe Tchamitchian in [AHLMcT]. Lashi Bandara (ANU) Geometry and the Kato problem 7 June 2013 3 / 28

If further A = A, (K1) is a trivial consequence of the Lax-Milgram Theorem. Solution to (K1) implies that D( div A ) = D( div A ). We can ask a more abstract question for accretive operators L on a Hilbert space H. There, the question is whether D( L ) = D( L). In general, this is not true by a counterexample of McIntosh in 1972 in [Mc72]. A second related question is the following. Suppose that J t is a family of closed, densely-defined, Hermitian forms on H with domain W and L(t) the associated self-adjoint operators to J t with domain W. If t J t extends to holomorphic family (for small z), then is t L(t) : V H a bounded operator? Counterexample to this second question by McIntosh in 1982 in [Mc82]. Lashi Bandara (ANU) Geometry and the Kato problem 7 June 2013 4 / 28

Motivations from PDE For k = 1, 2, let L k = div A k where A k L (R n, L(C n )) non-negative self-adjoint and L k uniformly elliptic. As aforementioned, D( L k ) = W 1,2 (R n ) and L k u u for u W 1,2 (R n ). Let u k be solutions to the wave equation with respect to L k with the same initial data. That is, 2 t u k + L k u k = 0 t u k t=0 = g L 2 (R n ) u k (0) = f W 1,2 (R n ). Lashi Bandara (ANU) Geometry and the Kato problem 7 June 2013 5 / 28

Suppose there exists a C > 0 L 1 u L 2 u C A 1 A 2 u. (P) Then, whenever t > 0, the following estimate holds: u 1 (t) u 2 (t) + ˆ t 0 (u 1 (s) u 2 (s)) ds Ct A 1 A 2 ( f + g ). See [Aus]. Lashi Bandara (ANU) Geometry and the Kato problem 7 June 2013 6 / 28

The estimate (P) is related to the second question of Kato. By solving the Kato square root problem (K1) for complex coefficients A, we are able to automatically obtain (P) from (K1). Lashi Bandara (ANU) Geometry and the Kato problem 7 June 2013 7 / 28

Kato square root problem on manifolds Let M be a smooth, complete Riemannian manifold with metric g, Levi-Civita connection, and volume measure µ. Write div = in L 2 and let S = (I, ). Assume a L (M) and A = (A ij ) L (M, L(L 2 (M) L 2 (T M)). Consider the following 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. Lashi Bandara (ANU) Geometry and the Kato problem 7 June 2013 8 / 28

The main theorem on manifolds Theorem (B.-Mc, 2012) Let M be a smooth, complete Riemannian manifold with Ric C and inj(m) κ > 0. Suppose the following ellipticity condition holds: there exists κ 1, κ 2 > 0 such that Re av, v κ 1 v 2 Re ASu, Su κ 2 u 2 W 1,2 for v L 2 (M) and u W 1,2 (M). Then, D( L A ) = D( ) = W 1,2 (M) and L A u u + u = u W 1,2 for all u W 1,2 (M). Lashi Bandara (ANU) Geometry and the Kato problem 7 June 2013 9 / 28

Lipschitz estimates Since we allow the coefficients a and A to be complex, we obtain the following stability result as a consequence: Theorem (B.-Mc, 2012) 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 and Re ASu, Su κ 2 u 2 W for v L 2 (M) and 1,2 u W 1,2 (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 W 1,2 holds for all u W 1,2 (M). The implicit constant depends in particular on A, a and η i. Lashi Bandara (ANU) Geometry and the Kato problem 7 June 2013 10 / 28

The Hodge-Dirac operator Let Ω(M) denote the algebra of differential forms over M under the exterior product. Let d be the exterior derivative as an operator on L 2 (Ω(M)) and d its adjoint, both of which are nilpotent operators. The Hodge-Dirac operator is then the self-adjoint operator D = d +d. The Hodge-Laplacian is then D 2 = d d + d d. For an invertible A L (L(Ω(M))), we consider perturbing D to obtain the operator D A = d +A 1 d A. Lashi Bandara (ANU) Geometry and the Kato problem 7 June 2013 11 / 28

Curvature endomorphism for forms Let { θ i} be an orthonormal frame at x for Ω 1 (M) = T M. Denote the components of the curvature tensor in this frame by Rm ijkl. The curvature endomorphism is then the operator for ω Ω x (M). R ω = Rm ijkl θ i (θ j (θ k (θ l ω))) This can be seen as an extension of Ricci curvature for forms, since g(r ω, η) = Ric(ω, η ) whenever ω, η Ω 1 x(m) and where : T M TM is the flat isomorphism through the metric g. The Weitzenböck formula then asserts that D 2 = tr 12 2 + R. Lashi Bandara (ANU) Geometry and the Kato problem 7 June 2013 12 / 28

Theorem (B., 2012) Let M be a smooth, complete Riemannian manifold and let β C \ {0}. Suppose there exist η, κ > 0 such that Ric η and inj(m) κ. Furthermore, suppose there is a ζ R satisfying g(r u, u) ζ u 2, for u Ω x (M) and A L (L(Ω(M))) and κ 1 > 0 satisfying Re Au, u κ 1 u 2. Then, D( D 2 A + β 2 ) = D(D A ) = D(d) D(d A) and D 2 A + β 2 u D A u + u. Lashi Bandara (ANU) Geometry and the Kato problem 7 June 2013 13 / 28

Lie groups Let G be a Lie group of dimension n with Lie algebra g and equipped with the left-invariant Haar measure µ. We say that a linearly independent a = {a 1,..., a k } g is an algebraic basis if we can recover a basis for g through multi-commutation. Let A i denote the right-translation of a i and A i = A i. Let span {A 1,..., A k } = A TG be the bundle obtained through the right-translation of a and A = { A 1,..., A k} the dual of A. Lashi Bandara (ANU) Geometry and the Kato problem 7 June 2013 14 / 28

Subelliptic distance Theorem of Carathéodory-Chow tells us that for any two points x, y G, we can find an absolutely continuous curve γ : [0, 1] G such that γ(t) = i γ i (t)a i (γ(t)) A. The length of such a curve then is given by l(γ) = ˆ 1 0 ( i γ i (t) 2 ) 1 2 dt Define distance d(x, y) as the infimum over the length of all such curves. The measure µ is Borel-regular with respect to d. Lashi Bandara (ANU) Geometry and the Kato problem 7 June 2013 15 / 28

Subelliptic operators For f C (G), define f = A i f A i. This defines an sub-connection on C (M). Each vector field A i is a skew-adjoint differential operator. We consider it as a unbounded operator on L 2 (G) with domain D(A i ). By also considering as a closed, densely-defined operator on L 2 (M), we obtain the first-order Sobolev space W 1,2 (G) = D( ) = k i=1 D(A i). We write the divergence as div =. Then, the subelliptic Laplacian associated to A is k = div = A 2 i. i=1 Lashi Bandara (ANU) Geometry and the Kato problem 7 June 2013 16 / 28

Nilpotent Lie groups The Lie group G is nilpotent if the inductively defined sequence g 1 = [g, g], g 2 = [g 1, g],... is eventually zero. Theorem (B.-E.-Mc., 2012) Let (G, d, µ) be a connected, nilpotent Lie group with a an algebraic basis, d the associated sub-elliptic distance, and µ the left Haar measure. Suppose that a, A L and that there exist κ 1, κ 2 > 0 satisfying Re av, v κ 1 v 2, and Re A u, u κ 2 u 2. for every v L 2 (G) and u W 1,2 (G). Then, D( a div A ) = W 1,2 (G) and a div A u u for u W 1,2 (G). Lashi Bandara (ANU) Geometry and the Kato problem 7 June 2013 17 / 28

General Lie groups Let S = (I, ) as in the manifold case. Theorem (B.-E.-Mc., 2012) Let (G, d, µ) be a connected Lie group, a an algebraic basis, d the associated sub-elliptic distance, and µ the left Haar measure. Let a, A L such that Re av, v κ 1 v 2, and Re ASu, Su κ 2 u W 1,2 for every v L 2 (G) and u W 1,2 (G). Then, D( as AS) = W 1,2 (G) with as ASu u W 1,2 = u + u. Lashi Bandara (ANU) Geometry and the Kato problem 7 June 2013 18 / 28

Operator theory We adapt the framework due to Axelsson (Rosén), Keith, McIntosh in [AKMc]. Let H be a Hilbert space and Γ : H H a closed, densely-defined, nilpotent operator. 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). Furthermore, suppose the operators B 1, B 2 satisfy B 1 B 2 R(Γ) N (Γ) and B 2 B 1 R(Γ ) N (Γ ). The primary operator we consider is Π B = Γ + B 1 Γ B 2. Lashi Bandara (ANU) Geometry and the Kato problem 7 June 2013 19 / 28

If the quadratic estimates ˆ 0 tπ B (1 + t 2 Π 2 B) 1 u 2 u (Q) hold for every u R(Π B ), then, H decomposes into the spectral subspaces of Π B as H = N (Π B ) E + E and D( Π 2 B ) = D(Π B) = D(Γ) D(Γ B 2 ) Π 2 B u Π Bu Γu + Γ B 2 u. The Kato problems are then obtained by letting H = L 2 (M) (L 2 (M) L 2 (T M)) and letting Γ = ( ) 0 0, Γ = S 0 ( 0 S 0 0 ), B 1 = ( ) a 0, B 0 0 2 = ( ) 0 0. 0 A Lashi Bandara (ANU) Geometry and the Kato problem 7 June 2013 20 / 28

Geometry and harmonic analysis Harmonic analytic methods are used to prove quadratic estimates (Q). The idea is to reduce the quadratic estimate (Q) to a Carleson measure estimate. This is achieved via a local T (b) argument. Geometry enters the picture precisely in the harmonic analysis. We need to perform harmonic analysis on vector fields, not just functions. One can show that this is not artificial - the Kato problem on functions immediately provides a solution to the dual problem on vector fields. Lashi Bandara (ANU) Geometry and the Kato problem 7 June 2013 21 / 28

Elements of the proofs Similar in structure to the proof of [AKMc] which is inspired from the proof in [AHLMcT]. A dyadic decomposition of the space A notion of averaging (in an integral sense) Poincaré inequality - on both functions and vector fields Control of 2 in terms of. Lashi Bandara (ANU) Geometry and the Kato problem 7 June 2013 22 / 28

The case of non-smooth metrics on manifolds We let M be a smooth, complete manifold as before but now let g be a C 0 metric. Let µ g denote the volume measure with respect to g. Let h C 0 (T (2,0) M). Then, define h op,g = sup x M sup h x (u, v). u g = v g =1 If g is another C 0 metric satisfying g g op,g δ < 1, then L 2 (M, g) = L 2 (M, g) and W 1,2 (M, g) = W 1,2 (M, g) with comparable norms. Lashi Bandara (ANU) Geometry and the Kato problem 7 June 2013 23 / 28

Π B under a change of metric The operator Γ g does not change under the change of metric. However, Γ g = C 1 Γ gc where C is the bounded, invertible, multiplication operator on L 2 (M) L 2 (M) L 2 (T M). Thus, Π B,g = Γ g + B 1 Γ gb 2 = Γ g + B 1 C 1 Γ gcb 2. This allows us to reduce the study of Π B,g for a C 0 metric g to the study of Π B, g = Γ g + B 1 Γ g B 2 where B 1 = B 1 C 1 and B 2 = CB 2, but now with a smooth metric g. Lashi Bandara (ANU) Geometry and the Kato problem 7 June 2013 24 / 28

Connection to geometric flows Given a C 0 metric g on a smooth compact manifold, we are able to always find C metric g. The metric g has inj(m, g) > κ and Ric( g) g η so we obtain a corresponding Kato square root estimate in this setting. The non-compact situation poses issues. Smooth the metric via mean curvature flow for, say, a C 2 imbedding? Smooth the metric via Ricci flow in the general case? Regularity of the initial metric? Lashi Bandara (ANU) Geometry and the Kato problem 7 June 2013 25 / 28

Application to PDE In the case we are able to find a suitable C metric near the C 0 one, then we have Lipschitz estimates. Possible application to hyperbolic PDE? Stability of geometries with Ricci bounds and injectivity radius bounds? Lashi Bandara (ANU) Geometry and the Kato problem 7 June 2013 26 / 28

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. [Aus] Pascal Auscher, Lectures on the Kato square root problem, Surveys in analysis and operator theory (Canberra, 2001), Proc. Centre Math. Appl. Austral. Nat. Univ., vol. 40, Austral. Nat. Univ., Canberra, 2002, pp. 1 18. Lashi Bandara (ANU) Geometry and the Kato problem 7 June 2013 27 / 28

References II [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. [Mc72] Alan McIntosh, On the comparability of A 1/2 and A 1/2, Proc. Amer. Math. Soc. 32 (1972), 430 434. [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. [Morris] Andrew J. Morris, The Kato square root problem on submanifolds, J. Lond. Math. Soc. (2) 86 (2012), no. 3, 879 910. Lashi Bandara (ANU) Geometry and the Kato problem 7 June 2013 28 / 28