Geometry and the Kato square root problem

Transcription

1 Geometry and the Kato square root problem Lashi Bandara Centre for Mathematics and its Applications Australian National University 29 July 2014 Geometric Analysis Seminar Beijing International Center for Mathematical Research Lashi Bandara Geometry and the Kato problem 1/34

2 History of the problem In the 1960 s, Kato considered the following abstract evolution equation t u(t) + A(t)u(t) = f(t), t [0, T ]. on a Hilbert space H. Lashi Bandara Geometry and the Kato problem 2/34

3 History of the problem In the 1960 s, Kato considered the following abstract evolution equation t u(t) + A(t)u(t) = f(t), t [0, T ]. on a Hilbert space H. 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 Geometry and the Kato problem 2/34

4 Typically A(t) is defined by an associated sesquilinear form where W H. J t : W W C Lashi Bandara Geometry and the Kato problem 3/34

5 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 Lashi Bandara Geometry and the Kato problem 3/34

6 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, Lashi Bandara Geometry and the Kato problem 3/34

7 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}, Lashi Bandara Geometry and the Kato problem 3/34

8 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 2 + Re J t [u, u]. Lashi Bandara Geometry and the Kato problem 3/34

9 T : D(T ) H is called ω-accretive if (i) T is densely-defined and closed, Lashi Bandara Geometry and the Kato problem 4/34

10 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 ), Lashi Bandara Geometry and the Kato problem 4/34

11 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+. Lashi Bandara Geometry and the Kato problem 4/34

12 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 Geometry and the Kato problem 4/34

13 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. Lashi Bandara Geometry and the Kato problem 5/34

14 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. Lashi Bandara Geometry and the Kato problem 5/34

15 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 ) Lashi Bandara Geometry and the Kato problem 5/34

16 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 Geometry and the Kato problem 5/34

17 Kato asked two questions. For ω < π/2, (K1) Does (K α ) hold for α = 1/2? Lashi Bandara Geometry and the Kato problem 6/34

18 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 t A(t)u u for u W? Lashi Bandara Geometry and the Kato problem 6/34

19 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 t A(t)u u for u W? In 1972, McIntosh provided a counter example in [Mc72] demonstrating that (K1) is false in such generality. Lashi Bandara Geometry and the Kato problem 6/34

20 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 t A(t)u u 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 Geometry and the Kato problem 6/34

21 The Kato square root problem then became the following. Lashi Bandara Geometry and the Kato problem 7/34

22 The Kato square root problem then became the following. Set J[u, v] = A u, v u, v W 1,2 (R n ), Lashi Bandara Geometry and the Kato problem 7/34

23 The Kato square root problem then became the following. Set J[u, v] = A u, v u, v W 1,2 (R n ), where A L is a pointwise matrix multiplication operator Lashi Bandara Geometry and the Kato problem 7/34

24 The Kato square root problem then became the following. Set J[u, v] = A u, v u, v W 1,2 (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. Lashi Bandara Geometry and the Kato problem 7/34

25 The Kato square root problem then became the following. Set J[u, v] = A u, v u, v W 1,2 (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 ) = W 1,2 (R n ) div A u u. (K1) Lashi Bandara Geometry and the Kato problem 7/34

26 The Kato square root problem then became the following. Set J[u, v] = A u, v u, v W 1,2 (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 ) = W 1,2 (R n ) 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 Geometry and the Kato problem 7/34

27 Kato square root problem for functions and forms Let M be a smooth, complete Riemannian manifold with metric g, Levi-Civita connection, and volume measure µ g. Lashi Bandara Geometry and the Kato problem 8/34

28 Kato square root problem for functions and forms Let M be a smooth, complete Riemannian manifold with metric g, Levi-Civita connection, and volume measure µ g. Write div g = in L 2. Lashi Bandara Geometry and the Kato problem 8/34

29 Kato square root problem for functions and forms Let M be a smooth, complete Riemannian manifold with metric g, Levi-Civita connection, and volume measure µ g. Write div g = in L 2. Let Ω(M) denote the algebra of differential forms over M. Lashi Bandara Geometry and the Kato problem 8/34

30 Kato square root problem for functions and forms Let M be a smooth, complete Riemannian manifold with metric g, Levi-Civita connection, and volume measure µ g. Write div g = in L 2. Let Ω(M) denote the algebra of differential forms over M. Let d be the exterior derivative as an operator on L 2 (Ω(M)) and d its adjoint, both of which are nilpotent operators. Lashi Bandara Geometry and the Kato problem 8/34

31 Kato square root problem for functions and forms Let M be a smooth, complete Riemannian manifold with metric g, Levi-Civita connection, and volume measure µ g. Write div g = in L 2. Let Ω(M) denote the algebra of differential forms over M. 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. Lashi Bandara Geometry and the Kato problem 8/34

32 Kato square root problem for functions and forms Let M be a smooth, complete Riemannian manifold with metric g, Levi-Civita connection, and volume measure µ g. Write div g = in L 2. Let Ω(M) denote the algebra of differential forms over M. 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. Lashi Bandara Geometry and the Kato problem 8/34

33 Let S = (I, ). Lashi Bandara Geometry and the Kato problem 9/34

34 Let S = (I, ). Assume a L (M) and A = (A ij ) L (M, L(L 2 (M) L 2 (T M)). Lashi Bandara Geometry and the Kato problem 9/34

35 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 Geometry and the Kato problem 9/34

36 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. The Kato square root problem for functions is then to determine: D( L A ) = W 1,2 (M) and L A u u + u = u W 1,2 for all u W 1,2 (M). Lashi Bandara Geometry and the Kato problem 9/34

37 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 Geometry and the Kato problem 10/34

38 For an invertible A L (L(Ω(M))), we consider perturbing D to obtain the operator D A = d +A 1 d A. The Kato square root problem for forms is then to determine the following whenever 0 β C: 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 Geometry and the Kato problem 10/34

39 Axelsson (Rosén)-Keith-McIntosh framework (H1) The operator Γ : D(Γ) H H is a closed, densely-defined and nilpotent operator, by which we mean R(Γ) N (Γ), Lashi Bandara Geometry and the Kato problem 11/34

40 Axelsson (Rosén)-Keith-McIntosh framework (H1) The operator Γ : D(Γ) H H is a closed, densely-defined and nilpotent operator, by which we mean R(Γ) N (Γ), (H2) B 1, B 2 L(H ) and there exist κ 1, κ 2 > 0 satisfying the accretivity conditions 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), and Lashi Bandara Geometry and the Kato problem 11/34

41 Axelsson (Rosén)-Keith-McIntosh framework (H1) The operator Γ : D(Γ) H H is a closed, densely-defined and nilpotent operator, by which we mean R(Γ) N (Γ), (H2) B 1, B 2 L(H ) and there exist κ 1, κ 2 > 0 satisfying the accretivity conditions 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), and (H3) B 1 B 2 R(Γ) N (Γ) and B 2 B 1 R(Γ ) N (Γ ). Lashi Bandara Geometry and the Kato problem 11/34

42 Axelsson (Rosén)-Keith-McIntosh framework (H1) The operator Γ : D(Γ) H H is a closed, densely-defined and nilpotent operator, by which we mean R(Γ) N (Γ), (H2) B 1, B 2 L(H ) and there exist κ 1, κ 2 > 0 satisfying the accretivity conditions 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), and (H3) B 1 B 2 R(Γ) N (Γ) and B 2 B 1 R(Γ ) N (Γ ). Let us now define Π B = Γ + B 1 Γ B 2 with domain D(Π B ) = D(Γ) D(B 1 Γ B 2 ). Lashi Bandara Geometry and the Kato problem 11/34

43 Quadratic estimates To say that Π B satisfies quadratic estimates means that ˆ 0 tπ B (I + t 2 Π 2 B) 1 u 2 dt t u 2, (Q) for all u R(Π B ). Lashi Bandara Geometry and the Kato problem 12/34

44 Quadratic estimates To say that Π B satisfies quadratic estimates means that ˆ 0 tπ B (I + t 2 Π 2 B) 1 u 2 dt t u 2, (Q) for all u R(Π B ). This implies that D( Π 2 B ) = D(Π B) = D(Γ) D(Γ B 2 ) Π 2 B u Π Bu Γu + Γ B 2 u Lashi Bandara Geometry and the Kato problem 12/34

45 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 Geometry and the Kato problem 13/34

46 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) 1,2 and 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 Geometry and the Kato problem 14/34

47 Curvature endomorphism for forms Let { θ i} be an orthonormal frame at x for Ω 1 (M) = T M. Lashi Bandara Geometry and the Kato problem 15/34

48 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. Lashi Bandara Geometry and the Kato problem 15/34

49 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))) Lashi Bandara Geometry and the Kato problem 15/34

50 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. Lashi Bandara Geometry and the Kato problem 15/34

51 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 R. Lashi Bandara Geometry and the Kato problem 15/34

52 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 Geometry and the Kato problem 16/34

53 The Kato problem for functions are captured in the AKM framework on 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 = ( ) A Lashi Bandara Geometry and the Kato problem 17/34

54 The Kato problem for functions are captured in the AKM framework on 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 = ( ) A For the case of forms, the setup takes the form, H = L 2 (Ω(M)) L 2 (Ω(M)) and ( ) d 0 Γ =, Γ = β d ( ) ( ) δ β A 1 0, B 0 δ 1 = 0 A 1, B 2 = ( ) A 0. 0 A Lashi Bandara Geometry and the Kato problem 17/34

55 Geometry and harmonic analysis Harmonic analytic methods are used to prove quadratic estimates (Q). Lashi Bandara Geometry and the Kato problem 18/34

56 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. Lashi Bandara Geometry and the Kato problem 18/34

57 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. Lashi Bandara Geometry and the Kato problem 18/34

58 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. Lashi Bandara Geometry and the Kato problem 18/34

59 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. Lashi Bandara Geometry and the Kato problem 18/34

60 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 Geometry and the Kato problem 18/34

61 Elements of the proofs Similar in structure to the proof of [AKMc] which is inspired from the proof in [AHLMcT]. Lashi Bandara Geometry and the Kato problem 19/34

62 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 Lashi Bandara Geometry and the Kato problem 19/34

63 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) Lashi Bandara Geometry and the Kato problem 19/34

64 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 Lashi Bandara Geometry and the Kato problem 19/34

65 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 Geometry and the Kato problem 19/34

66 Rough metrics Definition (Rough metric) Let g be a (2, 0) symmetric tensor field with measurable coefficients and that for each x M, there is some chart (U, ψ) near x and a constant C 1 such that C 1 u ψ δ(y) u g(y) C u ψ δ(y), for almost-every y U and where δ is the Euclidean metric in ψ(u). Then we say that g is a rough metric, and such a chart (U, ψ) is said to satisfy the local comparability condition. Lashi Bandara Geometry and the Kato problem 20/34

67 Metric perturbations Definition We say that two rough metrics g and g are C-close if C 1 u g(x) u g(x) C u g(x) for almost-every x M where C 1. Two such metrics are said to be C-close everywhere if this inequality holds for every x M. Lashi Bandara Geometry and the Kato problem 21/34

68 Metric perturbations Definition We say that two rough metrics g and g are C-close if C 1 u g(x) u g(x) C u g(x) for almost-every x M where C 1. Two such metrics are said to be C-close everywhere if this inequality holds for every x M. We also say that g and g are close if there exists some C 1 for which they are C-close. Lashi Bandara Geometry and the Kato problem 21/34

69 Metric perturbations Definition We say that two rough metrics g and g are C-close if C 1 u g(x) u g(x) C u g(x) for almost-every x M where C 1. Two such metrics are said to be C-close everywhere if this inequality holds for every x M. We also say that g and g are close if there exists some C 1 for which they are C-close. For two continuous metrics, C-close and C-close everywhere coincide. Lashi Bandara Geometry and the Kato problem 21/34

70 Proposition Let g and g be two rough metrics that are C-close. Then, there exists B Γ(T M TM) such that it is symmetric, almost-everywhere positive and invertible, and g x (B(x)u, v) = g x (u, v) for almost-every x M. Furthermore, for almost-every x M, C 2 u g(x) B(x)u g(x) C 2 u g(x), and the same inequality with g and g interchanged. If g C k and g C l (with k, l 0), then the properties of B are valid for all x M and B C min{k,l} (T M TM). Lashi Bandara Geometry and the Kato problem 22/34

71 The measure µ g (x) = θ(x) dµ g (x), where θ(x) = det B(x). Lashi Bandara Geometry and the Kato problem 23/34

72 The measure µ g (x) = θ(x) dµ g (x), where θ(x) = det B(x). Consequently, (i) whenever p [1, ), L p (T (r,s) M, g) = L p (T (r,s) M, g) with ( ) C r+s+ n 2p u p, g u p,g C r+s+ n 2p u p, g, Lashi Bandara Geometry and the Kato problem 23/34

73 The measure µ g (x) = θ(x) dµ g (x), where θ(x) = det B(x). Consequently, (i) whenever p [1, ), L p (T (r,s) M, g) = L p (T (r,s) M, g) with ( ) C r+s+ n 2p u p, g u p,g C r+s+ n 2p u p, g, (ii) for p =, L (T (r,s) M, g) = L (T (r,s) M, g) with C (r+s) u, g u,g C r+s u, g, Lashi Bandara Geometry and the Kato problem 23/34

74 (iii) the Sobolev spaces W 1,p (M, g) = W 1,p (M, g) and (M, g) = W1,p(M, g) with W 1,p 0 0 ( ) C 1+ n 2p u W 1,p, g u W 1,p,g C 1+ n 2p u W 1,p, g, Lashi Bandara Geometry and the Kato problem 24/34

75 (iii) the Sobolev spaces W 1,p (M, g) = W 1,p (M, g) and (M, g) = W1,p(M, g) with W 1,p 0 0 ( ) C 1+ n 2p u W 1,p, g u W 1,p,g C 1+ n 2p u W 1,p, g, (iv) the Sobolev spaces W d,p (M, g) = W d,p (M, g) and (M, g) = Wd,p(M, g) with W d,p 0 0 ( ) C n+ n 2p u W d,p, g u W d,p,g C n+ n 2p u W d,p, g, Lashi Bandara Geometry and the Kato problem 24/34

76 (iii) the Sobolev spaces W 1,p (M, g) = W 1,p (M, g) and (M, g) = W1,p(M, g) with W 1,p 0 0 ( ) C 1+ n 2p u W 1,p, g u W 1,p,g C 1+ n 2p u W 1,p, g, (iv) the Sobolev spaces W d,p (M, g) = W d,p (M, g) and (M, g) = Wd,p(M, g) with W d,p 0 0 ( ) C n+ n 2p u W d,p, g u W d,p,g C n+ n 2p u W d,p, g, (v) the divergence operators satisfy div D,g = θ 1 div D, g θb and div N,g = θ 1 div N, g θb. Lashi Bandara Geometry and the Kato problem 24/34

77 Case of functions Theorem (B, 2014) Let g be a smooth, complete metric and suppose that there exists κ > 0 and η > 0 such that (i) inj(m, g) κ and, (ii) Ric( g) η. Then, for any rough metric g that is close, the Kato square root problem for functions has a solution on (M, g). Lashi Bandara Geometry and the Kato problem 25/34

78 Case of forms Theorem (B, 2014) Let g be a rough metric close to g, a smooth, complete metric, and suppose that: (i) there exists κ > 0 such that inj(m, g) κ, (ii) there exists η > 0 such that Ric( g) η, and (iii) there exists ζ R such that g(r ω, ω) ζ ω 2 g. Then, the Kato square root problem for forms has a solution on (M, g). Lashi Bandara Geometry and the Kato problem 26/34

79 Compact manifolds with rough metrics Theorem (B, 2014) Let M be a smooth, compact manifold and g a rough metric. Then, the Kato square root problem (on functions and forms, respectively) has a solution. Lashi Bandara Geometry and the Kato problem 27/34

80 Cones and induced metrics Let Cr,h n be the n-cone of height h > 0 and radius r > 0. Lashi Bandara Geometry and the Kato problem 28/34

81 Cones and induced metrics Let Cr,h n be the n-cone of height h > 0 and radius r > 0. The cone can be realised as the image of the graph function ( ( F r,h (x) = x, h 1 x )). r Lashi Bandara Geometry and the Kato problem 28/34

82 Cones and induced metrics Let Cr,h n be the n-cone of height h > 0 and radius r > 0. The cone can be realised as the image of the graph function ( ( F r,h (x) = x, h 1 x )). r Let U be an open set in R n such that B r (0) U. Then, define G r,h : U R n+1 as the map F r,h whenever x B r (0) and (x, 0) otherwise. Lashi Bandara Geometry and the Kato problem 28/34

83 Cones and induced metrics Let Cr,h n be the n-cone of height h > 0 and radius r > 0. The cone can be realised as the image of the graph function ( ( F r,h (x) = x, h 1 x )). r Let U be an open set in R n such that B r (0) U. Then, define G r,h : U R n+1 as the map F r,h whenever x B r (0) and (x, 0) otherwise. Then we obtain that the map G r,h satisfies x y G r,h (x) G r,h (y) 1 + (hr 1 ) 2 x y. Lashi Bandara Geometry and the Kato problem 28/34

84 Proposition Let γ : I U be a smooth curve such that γ(0) {0} B r (0). Then, γ (0) (G r,h γ) (0) 1 + h2 r 2 γ (0). Moreover, for u T x U, x {0} B r (0) (and in particular for almost-every x), u δ u g 1 + h2 r 2 u δ, where δ is the usual inner product on U induced by R n. Lashi Bandara Geometry and the Kato problem 29/34

85 Proposition Let γ : I U be a smooth curve such that γ(0) {0} B r (0). Then, γ (0) (G r,h γ) (0) 1 + h2 r 2 γ (0). Moreover, for u T x U, x {0} B r (0) (and in particular for almost-every x), u δ u g 1 + h2 r 2 u δ, where δ is the usual inner product on U induced by R n. A particular consequence is that the metrics g = G r,h δ R n+1 are 1 + (hr 1 ) 2 -close on U. and δ R n Lashi Bandara Geometry and the Kato problem 29/34

86 Lemma Given ε > 0, there exists two points x, x and distinct minimising smooth geodesics γ 1,ε and γ 2,ε between x and x of length ε. Furthermore, there are two constants C 1,r,h,ε, C 2,r,h,ε > 0 depending on h, r and ε such that the geodesics γ 1,ε and γ 2,ε are contained in G r,h (A ε ) where A ε is the Euclidean annulus {x B r (0) : C 1,r,h,ε < x < C 2,r,h,ε }. Lashi Bandara Geometry and the Kato problem 30/34

87 Theorem (B., 2014) For any C > 1, there exists a smooth metric g which is C-close to the Euclidean metric δ for which inj(r 2, g) = 0. Furthermore, the Kato square root problem for functions can be solved for (R 2, g) under the. Lashi Bandara Geometry and the Kato problem 31/34

88 In higher dimensions, we obtain a similar result since the 2-dimensional cone can be realised as a totally geodesic submanifold. Lashi Bandara Geometry and the Kato problem 32/34

89 In higher dimensions, we obtain a similar result since the 2-dimensional cone can be realised as a totally geodesic submanifold. Theorem (B., 2014) Let M be a smooth manifold of dimension at least 2 and g a continuous metric. Given C > 1, and a point x 0 M, there exists a rough metric h such that: (i) it induces a length structure and the metric d g preserves the topology of M, (ii) it is smooth everywhere except x 0, (iii) the geodesics through x 0 are Lipschitz, (iv) it is C-close to g, (v) inj(m \ {x 0 }, h) = 0. Lashi Bandara Geometry and the Kato problem 32/34

90 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, [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, [AKMc], Quadratic estimates and functional calculi of perturbed Dirac operators, Invent. Math. 163 (2006), no. 3, [Christ] Michael Christ, A T (b) theorem with remarks on analytic capacity and the Cauchy integral, Colloq. Math. 60/61 (1990), no. 2, Lashi Bandara Geometry and the Kato problem 33/34

91 References II [Kato] Tosio Kato, Fractional powers of dissipative operators, J. Math. Soc. Japan 13 (1961), MR (25 #1453) [Mc72] Alan McIntosh, On the comparability of A 1/2 and A 1/2, Proc. Amer. Math. Soc. 32 (1972), MR (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 MR (84k:47030) Lashi Bandara Geometry and the Kato problem 34/34