Quadratic estimates and perturbations of Dirac type operators on doubling measure metric spaces Lashi Bandara maths.anu.edu.au/~bandara Mathematical Sciences Institute Australian National University February 10, 2011 AMSI International Conference on Harmonic Analysis and Applications Macquarie University Lashi Bandara (ANU) Quad. Est. on measure metric spaces February 10, 2011 1 / 26
Bisectorial operators Let H be a Hilbert space. We say that an operator T : D(T ) H H is bi-sectorial if (i) T is closed and densely defined, (ii) σ(t ) S ω for some ω < π 2 where S ω = {ζ C : arg(±ζ) ω}, and (iii) For all µ > ω, there exists C µ > 0 such that (1 + ξt ) 1 C µ for all ξ C \ {0} with arg ξ µ. σ(t ) σ(t ) Lashi Bandara (ANU) Quad. Est. on measure metric spaces February 10, 2011 2 / 26
The Ψ-functional calculus Let S o µ = {ζ C : ζ 0, arg(±ζ) < ω} for ω < µ < π 2 and Ψ(So µ) to be the collection of holomorphic functions ψ : S o µ C such that for some s > 0. ψ(ζ) C ζ s 1 + ζ 2s As in [McIntosh], we can define a bounded operator ψ(t ) by ψ(t ) = 1 ψ(ζ)(ζ T ) 1 dζ 2πı γ where γ = { ±e ±θ : ω < θ < µ } parametrised counterclockwise around S ω. Lashi Bandara (ANU) Quad. Est. on measure metric spaces February 10, 2011 3 / 26
H bounded functional calculi We say that T has a bounded S o µ functional calculus if for all ψ Ψ(S o µ). C > 0 such that ψ(t ) C ψ In this case, we can define f(t ) for every bounded f : S o µ {0} C holomorphic on S o µ by f(t ) = f(0)p 0 u + lim n ψ n(t )u where u H, P 0 : H N (T ) is the bounded projection corresponding to the decomposition H = N (T ) R(T ), and ψ n f uniformly on compact subsets of S o µ. See [McIntosh]. Lashi Bandara (ANU) Quad. Est. on measure metric spaces February 10, 2011 4 / 26
Connection to Harmonic analysis Theorem A bisectorial operator T has a bounded H functional calculus if and only if tt (1 + t 2 T 2 ) 1 u 2 dt t u 2 for each u R(T ). 0 A version of this Theorem can be found as Theorem F in [ADM]. Lashi Bandara (ANU) Quad. Est. on measure metric spaces February 10, 2011 5 / 26
Motivation In [AKM], Axelsson, Keith and McIntosh consider H = L 2 (R n, ) such that B L (R n, L( )) with B invertible and strictly accretive. They consider the perturbation of the Hodge-Dirac operator They show D( DB 2 ) = D(D B) and for all u D(D B ). D B = d + B 1 d B. DB 2 u D B u du + dbu This is an extension of the Kato Square Root problem for Differential forms on R n. Lashi Bandara (ANU) Quad. Est. on measure metric spaces February 10, 2011 6 / 26
A more general problem To prove this, [AKM] consider a more general setup. (H1) The operator Γ : D(Γ) H is closed, densely defined and nilpotent. (H2) The operators B 1, B 2 L(H ) satisfy Re B 1 u, u κ 1 u whenever u R(Γ ) Re B 2 u, u κ 2 u where κ 1, κ 2 > 0 are constants. whenever u 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 1 Γ B 2, Π B = Γ + Γ B and Π = Γ + Γ. Lashi Bandara (ANU) Quad. Est. on measure metric spaces February 10, 2011 7 / 26
Quadratic estimates and Kato Given quadratic estimates and hence a bounded H functional calculus, they conclude D( Π 2 B ) = D(Π B) = D(Γ) D(Γ B) with Π 2 B u Π B u Γu + Γ Bu. [AKM] give additional necessary conditions in order to prove quadratic estimates for Π B on H = L 2 (R n, C N ). Lashi Bandara (ANU) Quad. Est. on measure metric spaces February 10, 2011 8 / 26
Our setting We let (X, d) be a complete, connected metric space. Let µ be a Borel measure on X satisfying the doubling condition: C > 1 such that µ(b(x, 2r)) Cµ(B(x, r) for all x X, r > 0. We further assume that 0 < µ(b(x, r)) < for all x X and r > 0. Such a space exhibits polynomial growth in the sense µ(b(x, tr)) Ct p µ(b(x, r)) for all t > 1 and where p = log 2 C. Lashi Bandara (ANU) Quad. Est. on measure metric spaces February 10, 2011 9 / 26
Additional hypothesis To get the quadratic estimates, additional hypothesis are needed. The following three hypothesis easily translate to our context. (H4) Then set H = L 2 (X, C N ; dµ). (H5) B i L (X, L(C N )) for i = 1, 2. (H7) For B an open ball, Γu dµ = 0 and B B Γ v dµ = 0 for all u D(Γ) with spt u B and for all v D(Γ ) with spt v B. Lashi Bandara (ANU) Quad. Est. on measure metric spaces February 10, 2011 10 / 26
Upper gradients One of the key components in the proof of [AKM] is to use cutoff functions with good gradient bounds. In our situation the lack of a differential structure means that we do not have the luxury of a gradient and smooth functions. Motivated by Cheeger in [Cheeger], we define: Definition (Lipschitz pointwise constant) For ξ : X C N Lipschitz, define Lip ξ : X R by Lip ξ(x) = lim sup y x ξ(x) ξ(y). d(x, y) We let Lip ξ denote the Lipschitz constant ξ. Lashi Bandara (ANU) Quad. Est. on measure metric spaces February 10, 2011 11 / 26
The Leibniz hypothesis (H6) For every bounded Lipschitz function ξ : X C, multiplication by ξ preserves D(Γ) and M ξ = [Γ, ξi] is a multiplication operator. Furthermore, there exists a constant m > 0 such that M ξ (x) m Lip ξ(x) for almost all x X. In [AKM], ξ was used in place of Lip ξ. Lashi Bandara (ANU) Quad. Est. on measure metric spaces February 10, 2011 12 / 26
The Poincaré hypothesis Since we do not have a gradient, we replace the Coercivity hypothesis of [AKM] as a Poincaré hypothesis in terms of the operator Π. (H8) There exist a C > 0 and c > 0 such that for all balls B X, u(x) m B u 2 dµ(x) C rad(b) 2 Πu(x) 2 dµ(x) B for all u D(Π) R(Π). cb Lashi Bandara (ANU) Quad. Est. on measure metric spaces February 10, 2011 13 / 26
The main result Theorem If (Γ, B 1, B 2 ) satisfies the hypothesis (H1)-(H8), then for all u R(Π B ) L 2 (X, C N ). 0 tπ B (1 + t 2 Π 2 B) 1 u 2 dt t u 2 Our work closely follows that of [Morris], where this Theorem is proved as a local estimate on a complete Riemannian Manifold with at most exponential volume growth. Lashi Bandara (ANU) Quad. Est. on measure metric spaces February 10, 2011 14 / 26
Christ s dyadic cubes Theorem (Michael Christ s dyadic cubes [Christ]) There exists a collection of open subsets { Q k α X : k Z, α I k } with each z k α Q k α, where I k are index sets (possibly finite), and constants δ (0, 1), a 0 > 0, η > 0 and C 1, C 2 < satisfying: (i) For all k Z, µ(x \ α Q k α) = 0, (ii) If l k, either Q l β Qk α or Q l β Qk α =, (iii) For each (k, α) and each l < k there exists a unique β such that Q k α Q l β, (iv) diam Q k α C 1 δ k, (v) B(z k α, a 0 δ k ) Q k α, (vi) For all k, α and for all t > 0, µ { x Q k α : d(x, X \ Q k α) tδ k} C 2 t η µ(q k α). Lashi Bandara (ANU) Quad. Est. on measure metric spaces February 10, 2011 15 / 26
Important associated bounded operators As in [AKM], for t 0 we define the following bounded operators Rt B Pt B Q B t = (1 + ıtπ B ) 1 = (1 + t 2 Π 2 B) 1 = tπ B (1 + t 2 Π 2 B) 1 Θ B t = tγ B(1 + t 2 Π 2 B) 1. The operators R t, P t, Q t are defined by setting B 1, B 2 = I. Lashi Bandara (ANU) Quad. Est. on measure metric spaces February 10, 2011 16 / 26
Dyadic averaging and the principal part Let Q t = Q j when δ j+1 < t δ j, where Q j = { } Q j α. We also need the dyadic averaging operator. Let Q Q t be the unique Q x and define A t (x) = 1 u dµ. µ(q) Q Furthermore, we define the principal part of Θ B t the function w(x) = w by for w C N considered as γ t (x)w = (Θ B t w)(x). Lashi Bandara (ANU) Quad. Est. on measure metric spaces February 10, 2011 17 / 26
The estimate break up It is enough to prove that there exists a C > 0 such that 0 Θ B t P t u 2 dt t C u. As in [AKM], we break up the integral in the following way 0 Θ B t P t u 2 dt t + + 0 0 0 (Θ B t γ t A t )P t u 2 dt t dt t γ t A t (P t I)u 2 A t (x) 2 γ t (x) 2 dµ(x) dt t. X Lashi Bandara (ANU) Quad. Est. on measure metric spaces February 10, 2011 18 / 26
Separation via Lipschitz functions Lemma (Lipschitz separation lemma) Let (X, d) be a metric space and suppose E, F X satisfying d(e, F ) > 0. Then, there exists a Lipschitz function η : X [0, 1], a Ẽ E with d(ẽ, F ) > 0 such that η E = 1, η X\ Ẽ = 0 and Lip η 4/d(E, F ). This is a crucial tool to obtain cutoff functions with good upper-gradient bounds. Lashi Bandara (ANU) Quad. Est. on measure metric spaces February 10, 2011 19 / 26
Off diagonal estimates Proposition (Off diagonal estimates) Let U t be either R B t for t R or P B t, Q B t, Θ B t for t > 0. Then, for each M N, there exists a constant C M > 0 (that depends only on M and H1-H6) such that U t u L 2 (E) C M dist(e, F ) M u H where E, F X are Borel sets and u H with spt u F. t Lashi Bandara (ANU) Quad. Est. on measure metric spaces February 10, 2011 20 / 26
Principle and second part approximation We prove 0 (Θ B t γ t A t )P t u 2 dt u t by application of the Off diagonal estimates and a Poincaré hypothesis (H8). The proof of 0 γ t A t (P t I)u 2 dt t u relies on the off diagonal estimates and the cancellation property (H7). Lashi Bandara (ANU) Quad. Est. on measure metric spaces February 10, 2011 21 / 26
Carleson estimate We define the Carleson box over a cube Q Q j as R Q = Q (0, δ j ]. δ j R Q Q We say that a Borel measure ν on X R + is Carleson if there exists a C > 0 such that dν Cµ(Q) and define R Q ν C = sup Q 1 ν(q) R Q dν Lashi Bandara (ANU) Quad. Est. on measure metric spaces February 10, 2011 22 / 26
Carleson estimate (cont.) Proposition For all u H, we have for every Carleson measure ν. X R + A t u(x) 2 dν(x, t) ν C u 2 The proof of this relies on obtaining a Carelson Theorem in the doubling setting. Lashi Bandara (ANU) Quad. Est. on measure metric spaces February 10, 2011 23 / 26
Carleson estimate (cont.) Proposition (Carleson Measure) For all Q Q, we have γ t (x) 2 dµ(x) dt R Q t µ(q). The tools we have constructed for the doubling setting allows us to repeat the argument found in 5.3 in [AKM] with minimal alteration to prove this Proposition. Lashi Bandara (ANU) Quad. Est. on measure metric spaces February 10, 2011 24 / 26
References I D. Albrecht, X. Duong, and A. McIntosh. Operator theory and harmonic analysis. In Instructional Workshop on Analysis and Geometry, Part III (Canberra, 1995), volume 34 of Proc. Centre Math. Appl. Austral. Nat. Univ., pages 77 136. Austral. Nat. Univ., Canberra, 1996. A. Axelsson, S. Keith, A. McIntosh. Quadratic estimates and functional calculi of perturbed Dirac operators. Invent. math., 163:455 497, 2006. J. Cheeger. Differentiability of Lipschitz functions on metric measure spaces. GAFA, Geom. funct. anal., 9:428 517, 1999. Lashi Bandara (ANU) Quad. Est. on measure metric spaces February 10, 2011 25 / 26
References II M. Christ. A T(b) theorem with remarks on analytic capacity and the Cauchy integral. Colloq. Math, 60:601 628, 1990. A. McIntosh. Operator theory - spectra and functional calculi. http://maths.anu.edu.au/~alan/lectures/optheory.pdf, 2009. A. Morris. Local Hardy Spaces and Quadratic Estimates for Dirac Type Operators on Riemannian Manifolds. PhD thesis, Australian National University, 2010. Lashi Bandara (ANU) Quad. Est. on measure metric spaces February 10, 2011 26 / 26