The Kato square root problem on vector bundles with generalised bounded geometry

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 ]. Lashi Bandara (ANU) The Kato problem on GBG bundles 7 August 2012 2 / 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 where W H. J t : W W C Lashi Bandara (ANU) The Kato problem on GBG bundles 7 August 2012 3 / 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 Lashi Bandara (ANU) The Kato problem on GBG bundles 7 August 2012 3 / 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, Lashi Bandara (ANU) The Kato problem on GBG bundles 7 August 2012 3 / 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}, Lashi Bandara (ANU) The Kato problem on GBG bundles 7 August 2012 3 / 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, Lashi Bandara (ANU) The Kato problem on GBG bundles 7 August 2012 4 / 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 ), Lashi Bandara (ANU) The Kato problem on GBG bundles 7 August 2012 4 / 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+. Lashi Bandara (ANU) The Kato problem on GBG bundles 7 August 2012 4 / 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. Lashi Bandara (ANU) The Kato problem on GBG bundles 7 August 2012 5 / 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 ) Lashi Bandara (ANU) The Kato problem on GBG bundles 7 August 2012 5 / 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? Lashi Bandara (ANU) The Kato problem on GBG bundles 7 August 2012 6 / 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? Lashi Bandara (ANU) The Kato problem on GBG bundles 7 August 2012 6 / 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. Lashi Bandara (ANU) The Kato problem on GBG bundles 7 August 2012 6 / 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. Lashi Bandara (ANU) The Kato problem on GBG bundles 7 August 2012 7 / 30

The Kato square root problem then became the following. Set J[u, v] = A u, v u, v H 1 (R n ), Lashi Bandara (ANU) The Kato problem on GBG bundles 7 August 2012 7 / 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 Lashi Bandara (ANU) The Kato problem on GBG bundles 7 August 2012 7 / 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) Lashi Bandara (ANU) The Kato problem on GBG bundles 7 August 2012 7 / 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. Lashi Bandara (ANU) The Kato problem on GBG bundles 7 August 2012 8 / 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, ). Lashi Bandara (ANU) The Kato problem on GBG bundles 7 August 2012 8 / 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. Lashi Bandara (ANU) The Kato problem on GBG bundles 7 August 2012 8 / 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: Lashi Bandara (ANU) The Kato problem on GBG bundles 7 August 2012 9 / 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, Lashi Bandara (ANU) The Kato problem on GBG bundles 7 August 2012 13 / 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), Lashi Bandara (ANU) The Kato problem on GBG bundles 7 August 2012 13 / 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 ). Lashi Bandara (ANU) The Kato problem on GBG bundles 7 August 2012 13 / 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. Lashi Bandara (ANU) The Kato problem on GBG bundles 7 August 2012 14 / 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. Lashi Bandara (ANU) The Kato problem on GBG bundles 7 August 2012 15 / 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. Lashi Bandara (ANU) The Kato problem on GBG bundles 7 August 2012 16 / 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. Lashi Bandara (ANU) The Kato problem on GBG bundles 7 August 2012 17 / 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. Lashi Bandara (ANU) The Kato problem on GBG bundles 7 August 2012 17 / 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. Lashi Bandara (ANU) The Kato problem on GBG bundles 7 August 2012 18 / 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. Lashi Bandara (ANU) The Kato problem on GBG bundles 7 August 2012 18 / 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. Lashi Bandara (ANU) The Kato problem on GBG bundles 7 August 2012 19 / 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. Lashi Bandara (ANU) The Kato problem on GBG bundles 7 August 2012 19 / 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. Lashi Bandara (ANU) The Kato problem on GBG bundles 7 August 2012 20 / 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. Q Lashi Bandara (ANU) The Kato problem on GBG bundles 7 August 2012 20 / 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. Q Lashi Bandara (ANU) The Kato problem on GBG bundles 7 August 2012 20 / 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), Lashi Bandara (ANU) The Kato problem on GBG bundles 7 August 2012 24 / 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( ). Lashi Bandara (ANU) The Kato problem on GBG bundles 7 August 2012 24 / 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)). Lashi Bandara (ANU) The Kato problem on GBG bundles 7 August 2012 25 / 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 Lashi Bandara (ANU) The Kato problem on GBG bundles 7 August 2012 25 / 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. Lashi Bandara (ANU) The Kato problem on GBG bundles 7 August 2012 27 / 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, Lashi Bandara (ANU) The Kato problem on GBG bundles 7 August 2012 27 / 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. Lashi Bandara (ANU) The Kato problem on GBG bundles 7 August 2012 27 / 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. Lashi Bandara (ANU) The Kato problem on GBG bundles 7 August 2012 28 / 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. Lashi Bandara (ANU) The Kato problem on GBG bundles 7 August 2012 28 / 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. Lashi Bandara (ANU) The Kato problem on GBG bundles 7 August 2012 28 / 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