Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
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 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 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 Typically A(t) is defined by an associated sesquilinear form Jt : W × W → C where W ⊂ H . Lashi Bandara Geometry and the Kato problem 3/34 Typically A(t) is defined by an associated sesquilinear form Jt : W × W → C where W ⊂ H . Suppose 0 ≤ ω ≤ π/2. Then Jt is ω-sectorial means that Lashi Bandara Geometry and the Kato problem 3/34 Typically A(t) is defined by an associated sesquilinear form Jt : W × W → C where W ⊂ H . Suppose 0 ≤ ω ≤ π/2. Then Jt is ω-sectorial means that (i) W ⊂ H is dense, Lashi Bandara Geometry and the Kato problem 3/34 Typically A(t) is defined by an associated sesquilinear form Jt : W × W → C where W ⊂ H . Suppose 0 ≤ ω ≤ π/2. Then Jt is ω-sectorial means that (i) W ⊂ H is dense, (ii) Jt [u, u] ∈ Sω+ = {ζ ∈ C : |arg ζ| ≤ ω} ∪ {0}, Lashi Bandara Geometry and the Kato problem 3/34 Typically A(t) is defined by an associated sesquilinear form Jt : W × W → C where W ⊂ H . Suppose 0 ≤ ω ≤ π/2. Then Jt is ω-sectorial means that (i) W ⊂ H is dense, (ii) Jt [u, u] ∈ Sω+ = {ζ ∈ C : |arg ζ| ≤ ω} ∪ {0}, and (iii) W is complete under the norm kuk2W = kuk2 + Re Jt [u, u]. Lashi Bandara Geometry and the Kato problem 3/34 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 T : D(T ) → H is called ω-accretive if (i) T is densely-defined and closed, (ii) hT u, ui ∈ Sω+ for u ∈ D(T ), Lashi Bandara Geometry and the Kato problem 4/34 T : D(T ) → H is called ω-accretive if (i) T is densely-defined and closed, (ii) hT u, ui ∈ Sω+ for u ∈ D(T ), and (iii) σ(T ) ⊂ Sω+ . Lashi Bandara Geometry and the Kato problem 4/34 T : D(T ) → H is called ω-accretive if (i) T is densely-defined and closed, (ii) hT u, ui ∈ Sω+ for u ∈ D(T ), and (iii) σ(T ) ⊂ Sω+ . A 0-accretive operator is non-negative and self-adjoint. Lashi Bandara Geometry and the Kato problem 4/34 Let A(t) : D(A(t)) → H be defined as the operator with largest domain such that Jt [u, v] = hA(t)u, vi Lashi Bandara u ∈ D(A(t)), v ∈ W. Geometry and the Kato problem 5/34 Let A(t) : D(A(t)) → H be defined as the operator with largest domain such that Jt [u, v] = hA(t)u, vi u ∈ D(A(t)), v ∈ W. The theorem of Lax-Milgram guarantees that Jt is ω-sectorial =⇒ A(t) is ω-accretive. Lashi Bandara Geometry and the Kato problem 5/34 Let A(t) : D(A(t)) → H be defined as the operator with largest domain such that Jt [u, v] = hA(t)u, vi u ∈ D(A(t)), v ∈ W. The theorem of Lax-Milgram guarantees that Jt 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 kA(t)α uk ' kA(t)∗α uk, Lashi Bandara u ∈ D. Geometry and the Kato problem (Kα ) 5/34 Let A(t) : D(A(t)) → H be defined as the operator with largest domain such that Jt [u, v] = hA(t)u, vi u ∈ D(A(t)), v ∈ W. The theorem of Lax-Milgram guarantees that Jt 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 kA(t)α uk ' kA(t)∗α uk, u ∈ D. (Kα ) 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 Kato asked two questions. For ω < π/2, (K1) Does (Kα ) hold for α = 1/2? Lashi Bandara Geometry and the Kato problem 6/34 Kato asked two questions. For ω < π/2, (K1) Does (Kα ) hold for α = 1/2? p (K2) For the case ω = 0, we know D( A(t)) = W and (K1) is true, but is p k∂t A(t)uk . kuk for u ∈ W? Lashi Bandara Geometry and the Kato problem 6/34 Kato asked two questions. For ω < π/2, (K1) Does (Kα ) hold for α = 1/2? p (K2) For the case ω = 0, we know D( A(t)) = W and (K1) is true, but is p k∂t A(t)uk . kuk 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 Kato asked two questions. For ω < π/2, (K1) Does (Kα ) hold for α = 1/2? p (K2) For the case ω = 0, we know D( A(t)) = W and (K1) is true, but is p k∂t A(t)uk . kuk 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 The Kato square root problem then became the following. Lashi Bandara Geometry and the Kato problem 7/34 The Kato square root problem then became the following. Set J[u, v] = hA∇u, ∇vi Lashi Bandara u, v ∈ W1,2 (Rn ), Geometry and the Kato problem 7/34 The Kato square root problem then became the following. Set J[u, v] = hA∇u, ∇vi u, v ∈ W1,2 (Rn ), where A ∈ L∞ is a pointwise matrix multiplication operator Lashi Bandara Geometry and the Kato problem 7/34 The Kato square root problem then became the following. Set J[u, v] = hA∇u, ∇vi u, v ∈ W1,2 (Rn ), where A ∈ L∞ is a pointwise matrix multiplication operator satisfying the following ellipticity condition: Re J[u, u] ≥ κk∇uk, Lashi Bandara for some κ > 0. Geometry and the Kato problem 7/34 The Kato square root problem then became the following. Set J[u, v] = hA∇u, ∇vi u, v ∈ W1,2 (Rn ), where A ∈ L∞ is a pointwise matrix multiplication operator satisfying the following ellipticity condition: Re J[u, u] ≥ κk∇uk, for some κ > 0. Under these conditions, is it true that p D( div A∇) = W1,2 (Rn ) p k div A∇uk ' k∇uk. Lashi Bandara Geometry and the Kato problem (K1) 7/34 The Kato square root problem then became the following. Set J[u, v] = hA∇u, ∇vi u, v ∈ W1,2 (Rn ), where A ∈ L∞ is a pointwise matrix multiplication operator satisfying the following ellipticity condition: Re J[u, u] ≥ κk∇uk, for some κ > 0. Under these conditions, is it true that p D( div A∇) = W1,2 (Rn ) p k div A∇uk ' k∇uk. (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 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 µg . Lashi Bandara Geometry and the Kato problem 8/34 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 µg . Write divg = −∇ ∗ in L2 . Lashi Bandara Geometry and the Kato problem 8/34 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 µg . Write divg = −∇ ∗ in L2 . Let Ω(M) denote the algebra of differential forms over M. Lashi Bandara Geometry and the Kato problem 8/34 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 µg . Write divg = −∇ ∗ in L2 . Let Ω(M) denote the algebra of differential forms over M. Let d be the exterior derivative as an operator on L2 (Ω(M)) and d∗ its adjoint, both of which are nilpotent operators. Lashi Bandara Geometry and the Kato problem 8/34 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 µg . Write divg = −∇ ∗ in L2 . Let Ω(M) denote the algebra of differential forms over M. Let d be the exterior derivative as an operator on L2 (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 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 µg . Write divg = −∇ ∗ in L2 . Let Ω(M) denote the algebra of differential forms over M. Let d be the exterior derivative as an operator on L2 (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 D2 = d d∗ + d∗ d. Lashi Bandara Geometry and the Kato problem 8/34 Let S = (I, ∇). Lashi Bandara Geometry and the Kato problem 9/34 Let S = (I, ∇). Assume a ∈ L∞ (M) and A = (Aij ) ∈ L∞ (M, L(L2 (M) ⊕ L2 (T∗ M)). Lashi Bandara Geometry and the Kato problem 9/34 Let S = (I, ∇). Assume a ∈ L∞ (M) and A = (Aij ) ∈ L∞ (M, L(L2 (M) ⊕ L2 (T∗ M)). Consider the following second order differential operator LA : D(LA ) ⊂ L2 (M) → L2 (M) defined by: LA u = aS ∗ ASu = −a div(A11 ∇u) − a div(A10 u) + aA01 ∇u + aA00 u. Lashi Bandara Geometry and the Kato problem 9/34 Let S = (I, ∇). Assume a ∈ L∞ (M) and A = (Aij ) ∈ L∞ (M, L(L2 (M) ⊕ L2 (T∗ M)). Consider the following second order differential operator LA : D(LA ) ⊂ L2 (M) → L2 (M) defined by: LA u = aS ∗ ASu = −a div(A11 ∇u) − a div(A10 u) + aA01 ∇u + aA00 u. The Kato square root problem for functions is then to determine: √ √ D( LA ) = W1,2 (M) and k LA uk ' k∇uk + kuk = kukW1,2 for all u ∈ W1,2 (M). Lashi Bandara Geometry and the Kato problem 9/34 For an invertible A ∈ L∞ (L(Ω(M))), we consider perturbing D to obtain the operator DA = d +A−1 d∗ A. Lashi Bandara Geometry and the Kato problem 10/34 For an invertible A ∈ L∞ (L(Ω(M))), we consider perturbing D to obtain the operator DA = d +A−1 d∗ A. The Kato square root problem for forms is then to determine the following whenever 0 6= β ∈ C: q D( D2A + |β|2 ) = D(DA ) = D(d) ∩ D(d∗ A) and q k D2A + |β|2 uk ' k DA uk + kuk. Lashi Bandara Geometry and the Kato problem 10/34 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 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) B1 , B2 ∈ L(H ) and there exist κ1 , κ2 > 0 satisfying the accretivity conditions Re hB1 u, ui ≥ κ1 kuk2 and Re hB2 v, vi ≥ κ2 kvk2 , for u ∈ R(Γ∗ ) and v ∈ R(Γ), and Lashi Bandara Geometry and the Kato problem 11/34 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) B1 , B2 ∈ L(H ) and there exist κ1 , κ2 > 0 satisfying the accretivity conditions Re hB1 u, ui ≥ κ1 kuk2 and Re hB2 v, vi ≥ κ2 kvk2 , for u ∈ R(Γ∗ ) and v ∈ R(Γ), and (H3) B1 B2 R(Γ) ⊂ N (Γ) and B2 B1 R(Γ∗ ) ⊂ N (Γ∗ ). Lashi Bandara Geometry and the Kato problem 11/34 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) B1 , B2 ∈ L(H ) and there exist κ1 , κ2 > 0 satisfying the accretivity conditions Re hB1 u, ui ≥ κ1 kuk2 and Re hB2 v, vi ≥ κ2 kvk2 , for u ∈ R(Γ∗ ) and v ∈ R(Γ), and (H3) B1 B2 R(Γ) ⊂ N (Γ) and B2 B1 R(Γ∗ ) ⊂ N (Γ∗ ). Let us now define ΠB = Γ + B1 Γ∗ B2 with domain D(ΠB ) = D(Γ) ∩ D(B1 Γ∗ B2 ). Lashi Bandara Geometry and the Kato problem 11/34 Quadratic estimates To say that ΠB satisfies quadratic estimates means that ˆ ∞ dt ' kuk2 , ktΠB (I + t2 Π2B )−1 uk2 t 0 (Q) for all u ∈ R(ΠB ). Lashi Bandara Geometry and the Kato problem 12/34 Quadratic estimates To say that ΠB satisfies quadratic estimates means that ˆ ∞ dt ' kuk2 , ktΠB (I + t2 Π2B )−1 uk2 t 0 (Q) for all u ∈ R(ΠB ). This implies that q D( Π2B ) = D(ΠB ) = D(Γ) ∩ D(Γ∗ B2 ) q k Π2B uk ' kΠB uk ' kΓuk + kΓ∗ B2 uk Lashi Bandara Geometry and the Kato problem 12/34 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 κ1 , κ2 > 0 such that Re hav, vi ≥ κ1 kvk2 Re hASu, Sui ≥ κ2 kuk2W1,2 for √ v ∈ L2 (M) and u ∈ W1,2 (M). Then, 1,2 D( √ LA ) = D(∇) = W (M) and k LA uk ' k∇uk + kuk = kukW1,2 for all u ∈ W1,2 (M). Lashi Bandara Geometry and the Kato problem 13/34 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 κ1 , κ2 > 0 such that Re hav, vi ≥ κ1 kvk2 and Re hASu, Sui ≥ κ2 kuk2W1,2 for v ∈ L2 (M) and u ∈ W1,2 (M). Then for every ηi < κi , whenever kãk∞ ≤ η1 , kÃk∞ ≤ η2 , the estimate q p k LA u − LA+Ã uk . (kãk∞ + kÃk∞ )kukW1,2 holds for all u ∈ W1,2 (M). The implicit constant depends in particular on A, a and ηi . Lashi Bandara Geometry and the Kato problem 14/34 Curvature endomorphism for forms Let θi be an orthonormal frame at x for Ω1 (M) = T∗ M. Lashi Bandara Geometry and the Kato problem 15/34 Curvature endomorphism for forms Let θi be an orthonormal frame at x for Ω1 (M) = T∗ M. Denote the components of the curvature tensor in this frame by Rmijkl . Lashi Bandara Geometry and the Kato problem 15/34 Curvature endomorphism for forms Let θi be an orthonormal frame at x for Ω1 (M) = T∗ M. Denote the components of the curvature tensor in this frame by Rmijkl . The curvature endomorphism is then the operator R ω = Rmijkl θi ∧ (θj x (θk ∧ (θl x ω))) for ω ∈ Ωx (M). Lashi Bandara Geometry and the Kato problem 15/34 Curvature endomorphism for forms Let θi be an orthonormal frame at x for Ω1 (M) = T∗ M. Denote the components of the curvature tensor in this frame by Rmijkl . The curvature endomorphism is then the operator R ω = Rmijkl θi ∧ (θj x (θk ∧ (θl x ω))) for ω ∈ Ωx (M). This can be seen as an extension of Ricci curvature for forms, since g(R ω, η) = Ric(ω [ , η [ ) whenever ω, η ∈ Ω1x (M) and where [ : T∗ M → TM is the flat isomorphism through the metric g. Lashi Bandara Geometry and the Kato problem 15/34 Curvature endomorphism for forms Let θi be an orthonormal frame at x for Ω1 (M) = T∗ M. Denote the components of the curvature tensor in this frame by Rmijkl . The curvature endomorphism is then the operator R ω = Rmijkl θi ∧ (θj x (θk ∧ (θl x ω))) for ω ∈ Ωx (M). This can be seen as an extension of Ricci curvature for forms, since g(R ω, η) = Ric(ω [ , η [ ) whenever ω, η ∈ Ω1x (M) and where [ : T∗ M → TM is the flat isomorphism through the metric g. The Weitzenböck formula then asserts that D2 = tr12 ∇ 2 + R . Lashi Bandara Geometry and the Kato problem 15/34 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 ∈ Ωx (M) and A ∈ L∞ (L(Ω(M))) and κ1 > 0 satisfying Re hAu, ui ≥ κ1 kuk2 . q Then, D( D2A + |β|2 ) = D(DA ) = D(d) ∩ D(d∗ A) and q k D2A + |β|2 uk ' k DA uk + kuk. Lashi Bandara Geometry and the Kato problem 16/34 The Kato problem for functions are captured in the AKM framework on letting H = L2 (M) ⊕ (L2 (M) ⊕ L2 (T∗ M)) and letting 0 0 0 S∗ a 0 0 0 ∗ Γ= , Γ = , B1 = , B2 = . S 0 0 0 0 0 0 A Lashi Bandara Geometry and the Kato problem 17/34 The Kato problem for functions are captured in the AKM framework on letting H = L2 (M) ⊕ (L2 (M) ⊕ L2 (T∗ M)) and letting 0 0 0 S∗ a 0 0 0 ∗ Γ= , Γ = , B1 = , B2 = . S 0 0 0 0 0 0 A For the case of forms, the setup takes the form, H = L2 (Ω(M)) ⊕ L2 (Ω(M)) and A−1 0 A 0 d 0 δ β ∗ , B2 = . Γ= ,Γ = , B1 = 0 A−1 0 A β −d 0 −δ Lashi Bandara Geometry and the Kato problem 17/34 Geometry and harmonic analysis Harmonic analytic methods are used to prove quadratic estimates (Q). Lashi Bandara Geometry and the Kato problem 18/34 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 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 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 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 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 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 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 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 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 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 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 Metric perturbations Definition We say that two rough metrics g and g̃ are C-close if C −1 |u|g̃(x) ≤ |u|g(x) ≤ C |u|g̃(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 Metric perturbations Definition We say that two rough metrics g and g̃ are C-close if C −1 |u|g̃(x) ≤ |u|g(x) ≤ C |u|g̃(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 g̃ are close if there exists some C ≥ 1 for which they are C-close. Lashi Bandara Geometry and the Kato problem 21/34 Metric perturbations Definition We say that two rough metrics g and g̃ are C-close if C −1 |u|g̃(x) ≤ |u|g(x) ≤ C |u|g̃(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 g̃ 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 Proposition Let g and g̃ 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 g̃x (B(x)u, v) = gx (u, v) for almost-every x ∈ M. Furthermore, for almost-every x ∈ M, C −2 |u|g̃(x) ≤ |B(x)u|g̃(x) ≤ C 2 |u|g̃(x) , and the same inequality with g̃ and g interchanged. If g̃ ∈ Ck and g ∈ Cl (with k, l ≥ 0), then the properties of B are valid for all x ∈ M and B ∈ Cmin{k,l} (T∗ M ⊗ TM). Lashi Bandara Geometry and the Kato problem 22/34 The measure µg (x) = θ(x) dµg̃ (x), where θ(x) = Lashi Bandara Geometry and the Kato problem p det B(x). 23/34 The measure µg (x) = θ(x) dµg̃ (x), where θ(x) = Consequently, p det B(x). (i) whenever p ∈ [1, ∞), Lp (T (r,s) M, g) = Lp (T (r,s) M, g̃) with C Lashi Bandara n − r+s+ 2p kukp,g̃ ≤ kukp,g ≤ C n r+s+ 2p Geometry and the Kato problem kukp,g̃ , 23/34 The measure µg (x) = θ(x) dµg̃ (x), where θ(x) = Consequently, p det B(x). (i) whenever p ∈ [1, ∞), Lp (T (r,s) M, g) = Lp (T (r,s) M, g̃) with C n − r+s+ 2p kukp,g̃ ≤ kukp,g ≤ C n r+s+ 2p kukp,g̃ , (ii) for p = ∞, L∞ (T (r,s) M, g) = L∞ (T (r,s) M, g̃) with C −(r+s) kuk∞,g̃ ≤ kuk∞,g ≤ C r+s kuk∞,g̃ , Lashi Bandara Geometry and the Kato problem 23/34 (iii) the Sobolev spaces W1,p (M, g) = W1,p (M, g̃) and W01,p (M, g) = W01,p (M, g̃) with C Lashi Bandara n − 1+ 2p kukW1,p ,g̃ ≤ kukW1,p ,g ≤ C Geometry and the Kato problem n 1+ 2p kukW1,p ,g̃ , 24/34 (iii) the Sobolev spaces W1,p (M, g) = W1,p (M, g̃) and W01,p (M, g) = W01,p (M, g̃) with C n − 1+ 2p kukW1,p ,g̃ ≤ kukW1,p ,g ≤ C n 1+ 2p kukW1,p ,g̃ , (iv) the Sobolev spaces Wd,p (M, g) = Wd,p (M, g̃) and W0d,p (M, g) = W0d,p (M, g̃) with C Lashi Bandara n − n+ 2p kukWd,p ,g̃ ≤ kukWd,p ,g ≤ C Geometry and the Kato problem n n+ 2p kukWd,p ,g̃ , 24/34 (iii) the Sobolev spaces W1,p (M, g) = W1,p (M, g̃) and W01,p (M, g) = W01,p (M, g̃) with C n − 1+ 2p kukW1,p ,g̃ ≤ kukW1,p ,g ≤ C n 1+ 2p kukW1,p ,g̃ , (iv) the Sobolev spaces Wd,p (M, g) = Wd,p (M, g̃) and W0d,p (M, g) = W0d,p (M, g̃) with C n − n+ 2p kukWd,p ,g̃ ≤ kukWd,p ,g ≤ C n n+ 2p kukWd,p ,g̃ , (v) the divergence operators satisfy divD,g = θ−1 divD,g̃ θB and divN,g = θ−1 divN,g̃ θB. Lashi Bandara Geometry and the Kato problem 24/34 Case of functions Theorem (B, 2014) Let g̃ be a smooth, complete metric and suppose that there exists κ > 0 and η > 0 such that (i) inj(M, g̃) ≥ κ and, (ii) |Ric(g̃)| ≤ 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 Case of forms Theorem (B, 2014) Let g be a rough metric close to g̃, a smooth, complete metric, and suppose that: (i) there exists κ > 0 such that inj(M, g̃) ≥ κ, (ii) there exists η > 0 such that |Ric(g̃)| ≤ η, and (iii) there exists ζ ∈ R such that g̃(R ω, ω) ≥ ζ |ω|g̃2 . Then, the Kato square root problem for forms has a solution on (M, g). Lashi Bandara Geometry and the Kato problem 26/34 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 Cones and induced metrics n be the n-cone of height h > 0 and radius r > 0. Let Cr,h Lashi Bandara Geometry and the Kato problem 28/34 Cones and induced metrics n be the n-cone of height h > 0 and radius r > 0. The cone Let Cr,h can be realised as the image of the graph function |x| . Fr,h (x) = x, h 1 − r Lashi Bandara Geometry and the Kato problem 28/34 Cones and induced metrics n be the n-cone of height h > 0 and radius r > 0. The cone Let Cr,h can be realised as the image of the graph function |x| . Fr,h (x) = x, h 1 − r Let U be an open set in Rn such that Br (0) ⊂ U . Then, define Gr,h : U → Rn+1 as the map Fr,h whenever x ∈ Br (0) and (x, 0) otherwise. Lashi Bandara Geometry and the Kato problem 28/34 Cones and induced metrics n be the n-cone of height h > 0 and radius r > 0. The cone Let Cr,h can be realised as the image of the graph function |x| . Fr,h (x) = x, h 1 − r Let U be an open set in Rn such that Br (0) ⊂ U . Then, define Gr,h : U → Rn+1 as the map Fr,h whenever x ∈ Br (0) and (x, 0) otherwise. Then we obtain that the map Gr,h satisfies p |x − y| ≤ |Gr,h (x) − Gr,h (y)| ≤ 1 + (hr−1 )2 |x − y| . Lashi Bandara Geometry and the Kato problem 28/34 Proposition Let γ : I → U be a smooth curve such that γ(0) 6∈ {0} ∪ ∂Br (0). Then, r 2 0 0 γ (0) ≤ (Gr,h ◦ γ) (0) ≤ 1 + h γ 0 (0) . r2 Moreover, for u ∈ Tx U , x 6∈ {0} ∪ ∂Br (0) (and in particular for almost-every x), r h2 |u|δ ≤ |u|g ≤ 1 + 2 |u|δ , r where δ is the usual inner product on U induced by Rn . Lashi Bandara Geometry and the Kato problem 29/34 Proposition Let γ : I → U be a smooth curve such that γ(0) 6∈ {0} ∪ ∂Br (0). Then, r 2 0 0 γ (0) ≤ (Gr,h ◦ γ) (0) ≤ 1 + h γ 0 (0) . r2 Moreover, for u ∈ Tx U , x 6∈ {0} ∪ ∂Br (0) (and in particular for almost-every x), r h2 |u|δ ≤ |u|g ≤ 1 + 2 |u|δ , r where δ is the usual inner product on U induced by Rn . A particular consequence is that the metrics g = G∗r,h δRn+1 and δRn p are 1 + (hr−1 )2 -close on U . Lashi Bandara Geometry and the Kato problem 29/34 Lemma Given ε > 0, there exists two points x, x0 and distinct minimising smooth geodesics γ1,ε and γ2,ε between x and x0 of length ε. Furthermore, there are two constants C1,r,h,ε , C2,r,h,ε > 0 depending on h, r and ε such that the geodesics γ1,ε and γ2,ε are contained in Gr,h (Aε ) where Aε is the Euclidean annulus {x ∈ Br (0) : C1,r,h,ε < |x| < C2,r,h,ε } . Lashi Bandara Geometry and the Kato problem 30/34 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(R2 , g) = 0. Furthermore, the Kato square root problem for functions can be solved for (R2 , g) under the. Lashi Bandara Geometry and the Kato problem 31/34 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 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 x0 ∈ M, there exists a rough metric h such that: (i) it induces a length structure and the metric dg preserves the topology of M, (ii) it is smooth everywhere except x0 , (iii) the geodesics through x0 are Lipschitz, (iv) it is C-close to g, (v) inj(M \ {x0 } , h) = 0. Lashi Bandara Geometry and the Kato problem 32/34 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 Rn , 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 Geometry and the Kato problem 33/34 References II [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 A1/2 and A∗1/2 , Proc. Amer. Math. Soc. 32 (1972), 430–434. MR 0290169 (44 #7354) [Mc82] , On representing closed accretive sesquilinear forms as (A1/2 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) Lashi Bandara Geometry and the Kato problem 34/34