Square roots of perturbed sub-elliptic operators on Lie groups

Transcription

1 Square roots of perturbed sub-elliptic operators on Lie groups Lashi Bandara (Joint work with Tom ter Elst, Auckland and Alan McIntosh, ANU) Centre for Mathematics and its Applications Australian National University August 13, 2012 POSTI/Mprime Seminar University of Calgary Lashi Bandara (ANU) Square roots of operators on Lie groups August 13, / 28

2 Lie groups Let G be a Lie group of dimension n and g is Lie algebra. Lashi Bandara (ANU) Square roots of operators on Lie groups August 13, / 28

3 Lie groups Let G be a Lie group of dimension n and g is Lie algebra. We let dµ denote the left invariant Haar measure. Lashi Bandara (ANU) Square roots of operators on Lie groups August 13, / 28

4 Algebraic basis and vectorfields A set {a 1,..., a k } g is an algebraic basis if we can recover a basis for g by multi-commutation. Lashi Bandara (ANU) Square roots of operators on Lie groups August 13, / 28

5 Algebraic basis and vectorfields A set {a 1,..., a k } g is an algebraic basis if we can recover a basis for g by multi-commutation. We assume that the {a i } are linearly independent. Lashi Bandara (ANU) Square roots of operators on Lie groups August 13, / 28

6 Algebraic basis and vectorfields A set {a 1,..., a k } g is an algebraic basis if we can recover a basis for g by multi-commutation. We assume that the {a i } are linearly independent. Let A i denote the left translation of a i. Lashi Bandara (ANU) Square roots of operators on Lie groups August 13, / 28

7 Algebraic basis and vectorfields A set {a 1,..., a k } g is an algebraic basis if we can recover a basis for g by multi-commutation. We assume that the {a i } are linearly independent. Let A i denote the left translation of a i. The vectorfields {A i } are linearly independent and global. Lashi Bandara (ANU) Square roots of operators on Lie groups August 13, / 28

8 Distance Theorem of Carathéodory-Chow tells us that for any two points x, y G, we can find a curve γ : [0, 1] G such that γ(t) = i γ i (t)a i (γ(t)) span {A i (γ(t))}. Lashi Bandara (ANU) Square roots of operators on Lie groups August 13, / 28

9 Distance Theorem of Carathéodory-Chow tells us that for any two points x, y G, we can find a curve γ : [0, 1] G such that γ(t) = i γ i (t)a i (γ(t)) span {A i (γ(t))}. The length of such a curve then is given by l(γ) = ˆ 1 0 ( i γ i (t) 2 ) 1 2 dt Lashi Bandara (ANU) Square roots of operators on Lie groups August 13, / 28

10 Distance Theorem of Carathéodory-Chow tells us that for any two points x, y G, we can find a curve γ : [0, 1] G such that γ(t) = i γ i (t)a i (γ(t)) span {A i (γ(t))}. The length of such a curve then is given by l(γ) = ˆ 1 0 ( i γ i (t) 2 ) 1 2 dt Define distance d(x, y) as the infimum over the length of all such curves. Lashi Bandara (ANU) Square roots of operators on Lie groups August 13, / 28

11 Distance Theorem of Carathéodory-Chow tells us that for any two points x, y G, we can find a curve γ : [0, 1] G such that γ(t) = i γ i (t)a i (γ(t)) span {A i (γ(t))}. The length of such a curve then is given by l(γ) = ˆ 1 0 ( i γ i (t) 2 ) 1 2 dt Define distance d(x, y) as the infimum over the length of all such curves. The measure dµ is Borel-regular with respect to d and we consider (G, d, dµ) as a measure metric space. Lashi Bandara (ANU) Square roots of operators on Lie groups August 13, / 28

12 Sub-Laplacian Define an associated sub-laplacian by: = i A 2 i. Lashi Bandara (ANU) Square roots of operators on Lie groups August 13, / 28

13 Sub-Laplacian Define an associated sub-laplacian by: = i A 2 i. This is a densely-defined, self-adjoint operator on L 2 (G). Lashi Bandara (ANU) Square roots of operators on Lie groups August 13, / 28

14 We say that a Lie group is nilpotent if g 1 = [g, g], g 2 = [g, g 1 ], g 3 = [g 1, g 2 ],... is eventually 0. That is, there is a k such that g k = 0. Lashi Bandara (ANU) Square roots of operators on Lie groups August 13, / 28

15 We say that a Lie group is nilpotent if g 1 = [g, g], g 2 = [g, g 1 ], g 3 = [g 1, g 2 ],... is eventually 0. That is, there is a k such that g k = 0. On such spaces, we consider the uniformly elliptic second order operator D H = b i,j A i b ij A j where b, b ij L (G). Lashi Bandara (ANU) Square roots of operators on Lie groups August 13, / 28

16 The main theorem for nilpotent Lie groups Theorem (B.-E.-Mc) Let G be a connected nilpotent and suppose there exist κ 1, κ 2 > 0 such that ˆ Re b(x) κ 1 and Re b ij A i ua j u κ 2 A i u 2 G for almost all x G and u H 1 (G). Then, (i) D( D H ) = m i=1 D(A i) = H 1 (G), and (ii) D H u m i=1 A iu for all u H 1 (G). i,j i Lashi Bandara (ANU) Square roots of operators on Lie groups August 13, / 28

17 Stability Theorem (B.-E.-Mc) Let 0 < η i < κ i and suppose that b, b ij L (G) such that b η 1 and ( b ij ) η 2. Then, D H u D H u ( b + ( b ij ) ) for u H 1 (G) and where D H = (b + b) k A i u, i=1 k A i (b ij + b ij )A j. i,j=1 Lashi Bandara (ANU) Square roots of operators on Lie groups August 13, / 28

18 Operator theory Procedure in [AKMc]. Lashi Bandara (ANU) Square roots of operators on Lie groups August 13, / 28

19 Operator theory Procedure in [AKMc]. Let H be a Hilbert space. Lashi Bandara (ANU) Square roots of operators on Lie groups August 13, / 28

20 Operator theory Procedure in [AKMc]. Let H be a Hilbert space. (H1) The operator Γ : D(Γ) H H is closed, densely-defined and nilpotent (Γ 2 = 0). Lashi Bandara (ANU) Square roots of operators on Lie groups August 13, / 28

21 Operator theory Procedure in [AKMc]. Let H be a Hilbert space. (H1) The operator Γ : D(Γ) H H is closed, densely-defined and nilpotent (Γ 2 = 0). (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) Lashi Bandara (ANU) Square roots of operators on Lie groups August 13, / 28

22 Operator theory Procedure in [AKMc]. Let H be a Hilbert space. (H1) The operator Γ : D(Γ) H H is closed, densely-defined and nilpotent (Γ 2 = 0). (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 ). Lashi Bandara (ANU) Square roots of operators on Lie groups August 13, / 28

23 Operator theory Procedure in [AKMc]. Let H be a Hilbert space. (H1) The operator Γ : D(Γ) H H is closed, densely-defined and nilpotent (Γ 2 = 0). (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 1Γ B 2, Π B = Γ + Γ B, and Π = Γ + Γ. Lashi Bandara (ANU) Square roots of operators on Lie groups August 13, / 28

24 Harmonic analysis and Kato square root type estimates Theorem (Kato square root type estimate) Suppose that (Γ, B 1, B 2 ) satisfy (H1)-(H3) and ˆ for all u R(Π B ) H. Then, 0 tπ B (1 + t 2 Π 2 B) 1 u 2 dt t u 2 (i) There is a spectral decomposition H = N (Π B ) E + B E B, where E ± B are spectral subspaces and the sum is in general non-orthogonal, and (ii) D(Γ) D(Γ B ) = D(Π B) = D( Π 2 B ) with Γu + Γ B u Π B u Π 2 B u for all u D(Π B). Lashi Bandara (ANU) Square roots of operators on Lie groups August 13, / 28

25 Homogeneous conditions Lashi Bandara (ANU) Square roots of operators on Lie groups August 13, / 28

26 Homogeneous conditions (H4) Let X be a complete, connected metric space and µ a Borel-regular measure on X that is doubling. Then set H = L 2 (X, C N ; dµ). Lashi Bandara (ANU) Square roots of operators on Lie groups August 13, / 28

27 Homogeneous conditions (H4) Let X be a complete, connected metric space and µ a Borel-regular measure on X that is doubling. Then set H = L 2 (X, C N ; dµ). (H5) The operators B i are matrix-valued pointwise multiplication operators such that the function x B i (x) are L (X, L(C N )). Lashi Bandara (ANU) Square roots of operators on Lie groups August 13, / 28

28 Homogeneous conditions (H4) Let X be a complete, connected metric space and µ a Borel-regular measure on X that is doubling. Then set H = L 2 (X, C N ; dµ). (H5) The operators B i are matrix-valued pointwise multiplication operators such that the function x B i (x) are L (X, L(C N )). (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. Lashi Bandara (ANU) Square roots of operators on Lie groups August 13, / 28

29 Homogeneous conditions (H4) Let X be a complete, connected metric space and µ a Borel-regular measure on X that is doubling. Then set H = L 2 (X, C N ; dµ). (H5) The operators B i are matrix-valued pointwise multiplication operators such that the function x B i (x) are L (X, L(C N )). (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. (H7) For each open ball B, we have ˆ Γu dµ = 0 B and ˆ 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) Square roots of operators on Lie groups August 13, / 28

30 (H8) -1 (Poincaré hypothesis) There exists C > 0, c > 0 and an operator Ξ : D(Ξ) L 2 (X, C N ) L 2 (X, C M ) such that D(Π) R(Π) D(Ξ) and ˆ ˆ u u B 2 dµ C r 2 Ξu 2 dµ B for all balls B = B(x, r) and u D(Π) R(Π). B Lashi Bandara (ANU) Square roots of operators on Lie groups August 13, / 28

31 (H8) -1 (Poincaré hypothesis) There exists C > 0, c > 0 and an operator Ξ : D(Ξ) L 2 (X, C N ) L 2 (X, C M ) such that D(Π) R(Π) D(Ξ) and ˆ ˆ u u B 2 dµ C r 2 Ξu 2 dµ B for all balls B = B(x, r) and u D(Π) R(Π). -2 (Coercivity hypothesis) There exists C > 0 such that for all u D(Π) R(Π), B Ξu C Πu. Lashi Bandara (ANU) Square roots of operators on Lie groups August 13, / 28

32 (H8) -1 (Poincaré hypothesis) There exists C > 0, c > 0 and an operator Ξ : D(Ξ) L 2 (X, C N ) L 2 (X, C M ) such that D(Π) R(Π) D(Ξ) and ˆ ˆ u u B 2 dµ C r 2 Ξu 2 dµ B for all balls B = B(x, r) and u D(Π) R(Π). -2 (Coercivity hypothesis) There exists C > 0 such that for all u D(Π) R(Π), B Ξu C Πu. This is slightly different from (H8) in [Bandara]. Lashi Bandara (ANU) Square roots of operators on Lie groups August 13, / 28

33 Theorem (B.) Let X, (Γ, B 1, B 2 ) satisfy (H1)-(H8). Then, Π B satisfies the quadratic estimate ˆ tπ B (1 + t 2 Π 2 B) 1 u 2 dt t u 2 for all u R(Π B ) L 2 (X, C N ). 0 Lashi Bandara (ANU) Square roots of operators on Lie groups August 13, / 28

34 Geometric setup Lashi Bandara (ANU) Square roots of operators on Lie groups August 13, / 28

35 Geometric setup Define the bundle W = span {A i } TG and complexify it. Lashi Bandara (ANU) Square roots of operators on Lie groups August 13, / 28

36 Geometric setup Define the bundle W = span {A i } TG and complexify it. Equip W with the inner product h(a i, A j ) = δ ij. Lashi Bandara (ANU) Square roots of operators on Lie groups August 13, / 28

37 Geometric setup Define the bundle W = span {A i } TG and complexify it. Equip W with the inner product h(a i, A j ) = δ ij. Equip G with the sub-connection f = A k fa k where A k = A k W. Lashi Bandara (ANU) Square roots of operators on Lie groups August 13, / 28

38 Geometric setup Define the bundle W = span {A i } TG and complexify it. Equip W with the inner product h(a i, A j ) = δ ij. Equip G with the sub-connection f = A k fa k where A k = A k W. Equip W with the sub-connection (u i A i ) = ( u i ) A i Lashi Bandara (ANU) Square roots of operators on Lie groups August 13, / 28

39 Geometric setup Define the bundle W = span {A i } TG and complexify it. Equip W with the inner product h(a i, A j ) = δ ij. Equip G with the sub-connection f = A k fa k where A k = A k W. Equip W with the sub-connection (u i A i ) = ( u i ) A i We have that W = C k and L 2 (G) L 2 (W) = L 2 (C k+1 ). Lashi Bandara (ANU) Square roots of operators on Lie groups August 13, / 28

40 Operator setup Define: Γ : D(Γ) L 2 (G) L 2 (W ) L 2 (G) L 2 (W ) by ( ) 0 0 Γ =. 0 Lashi Bandara (ANU) Square roots of operators on Lie groups August 13, / 28

41 Operator setup Define: Γ : D(Γ) L 2 (G) L 2 (W ) L 2 (G) L 2 (W ) by ( ) 0 0 Γ =. 0 Then, Γ = ( ) 0 div 0 0 and Π = ( ) 0 div, 0 where we define div =. Lashi Bandara (ANU) Square roots of operators on Lie groups August 13, / 28

42 Operator setup Define: Γ : D(Γ) L 2 (G) L 2 (W ) L 2 (G) L 2 (W ) by ( ) 0 0 Γ =. 0 Then, Γ = ( ) 0 div 0 0 and Π = ( ) 0 div, 0 where we define div =. Let B = (b ij ). Then, define B 1 = ( ) b and B 2 = ( ) B Lashi Bandara (ANU) Square roots of operators on Lie groups August 13, / 28

43 Proof of the homogeneous problem Set X = G and H = L 2 (G) L 2 (W). Lashi Bandara (ANU) Square roots of operators on Lie groups August 13, / 28

44 Proof of the homogeneous problem Set X = G and H = L 2 (G) L 2 (W). (H1) The sub-connection is densely-defined and closed and so is Γ. Nilpotency is by construction. Lashi Bandara (ANU) Square roots of operators on Lie groups August 13, / 28

45 Proof of the homogeneous problem Set X = G and H = L 2 (G) L 2 (W). (H1) The sub-connection is densely-defined and closed and so is Γ. Nilpotency is by construction. (H2) By accretivity assumptions. Lashi Bandara (ANU) Square roots of operators on Lie groups August 13, / 28

46 Proof of the homogeneous problem Set X = G and H = L 2 (G) L 2 (W). (H1) The sub-connection is densely-defined and closed and so is Γ. Nilpotency is by construction. (H2) By accretivity assumptions. (H3) By construction. Lashi Bandara (ANU) Square roots of operators on Lie groups August 13, / 28

47 Proof (cont.) Lashi Bandara (ANU) Square roots of operators on Lie groups August 13, / 28

48 Proof (cont.) (H4) The measure dµ is Borel-regular and the nilpotency of G implies that it is doubling. Lashi Bandara (ANU) Square roots of operators on Lie groups August 13, / 28

49 Proof (cont.) (H4) The measure dµ is Borel-regular and the nilpotency of G implies that it is doubling. (H5) By assumption. Lashi Bandara (ANU) Square roots of operators on Lie groups August 13, / 28

50 Proof (cont.) (H4) The measure dµ is Borel-regular and the nilpotency of G implies that it is doubling. (H5) By assumption. (H6) It is an easy fact that for all bounded Lipschitz ξ : G C, for for almost all x G. M ξ (x) = [Γ, ξ(x)i] = ξ(x) k Lip ξ(x) Lashi Bandara (ANU) Square roots of operators on Lie groups August 13, / 28

51 Proof (cont.) (H4) The measure dµ is Borel-regular and the nilpotency of G implies that it is doubling. (H5) By assumption. (H6) It is an easy fact that for all bounded Lipschitz ξ : G C, for for almost all x G. M ξ (x) = [Γ, ξ(x)i] = ξ(x) k Lip ξ(x) (H7) By the left invariance of the measure dµ. Lashi Bandara (ANU) Square roots of operators on Lie groups August 13, / 28

52 Proof (cont.) Lashi Bandara (ANU) Square roots of operators on Lie groups August 13, / 28

53 Proof (cont.) (H8) -1 The nilpotency of G implies the following Poincaré inequality ˆ ˆ f f B 2 dµ r 2 f 2 dµ B B for all balls B, and f C (B). See [SC, (P.1), p118]. Lashi Bandara (ANU) Square roots of operators on Lie groups August 13, / 28

54 Proof (cont.) (H8) -1 The nilpotency of G implies the following Poincaré inequality ˆ ˆ f f B 2 dµ r 2 f 2 dµ B B for all balls B, and f C (B). See [SC, (P.1), p118]. Define Ξu = ( u 1, u 2 ). Lashi Bandara (ANU) Square roots of operators on Lie groups August 13, / 28

55 Proof (cont.) (H8) -1 The nilpotency of G implies the following Poincaré inequality ˆ ˆ f f B 2 dµ r 2 f 2 dµ B B for all balls B, and f C (B). See [SC, (P.1), p118]. Define Ξu = ( u 1, u 2 ). -2 The crucial fact needed here is the regularity result [ERS, Lemma 4.2] which gives A i A j f f for f H 2 (G) = D( ). Lashi Bandara (ANU) Square roots of operators on Lie groups August 13, / 28

56 Inhomogeneous problem For general Lie groups, we need to consider operators with lower order terms. Lashi Bandara (ANU) Square roots of operators on Lie groups August 13, / 28

57 Inhomogeneous problem For general Lie groups, we need to consider operators with lower order terms. Let b, b ij, c k, d k, e L (G). Define the following uniformly elliptic second order operator m m m D I = b A i b ij A j u b A i c i u b d i A i u beu. ij=1 i=1 i=1 Lashi Bandara (ANU) Square roots of operators on Lie groups August 13, / 28

58 Theorem (B.-E.-Mc) Let G be a connected Lie group and suppose there exists κ 1, κ 2 > 0 such that Re b(x) κ 1, ˆ ( ) m Re eu + d i A i u u + G i=1 m c i u + i=1 for almost all x G and u H 1 (G). Then, m b ij A j u A i u dµ j=1 κ 2 ( u 2 + (i) D( D I ) = m i=1 D(A i) = H 1 (G), and (ii) D I u u + m i=1 A iu for all u H 1 (G). ) m A i u 2 i=1 Lashi Bandara (ANU) Square roots of operators on Lie groups August 13, / 28

59 Spaces of exponential growth (X, d, µ) an exponentially locally doubling measure metric space. That is: there exist κ, λ 0 and constant C 1 such that for all x X, r > 0 and t 1. 0 < µ(b(x, tr)) Ct κ e λtr µ(b(x, r)) Lashi Bandara (ANU) Square roots of operators on Lie groups August 13, / 28

60 Changes to (H7) and (H8) The following (H7) from [Morris]: (H7) There exist c > 0 such that for all open balls B X with r 1, ˆ ˆ Γu dµ cµ(b) 1 2 u and Γ v dµ cµ(b) 1 2 v B for all u D(Γ), v D(Γ ) with spt u, spt v B. B Lashi Bandara (ANU) Square roots of operators on Lie groups August 13, / 28

61 We introduce the following local (H8): (H8) -1 (Local Poincaré hypothesis) There exists C > 0, c > 0 and an operator Ξ : D(Ξ) L 2 (X, C N ) L 2 (X, C M ) such that D(Π) R(Π) D(Ξ) and ˆ ˆ u u B 2 dµ C r 2 ( Ξu 2 + u 2 ) dµ B for all balls B = B(x, r) and for u D(Π) R(Π). -2 (Coercivity hypothesis) There exists C > 0 such that for all u D(Π) R(Π), B Ξu + u C Πu. Lashi Bandara (ANU) Square roots of operators on Lie groups August 13, / 28

62 Theorem (Morris) Let X, (Γ, B 1, B 2 ) satisfy (H1)-(H8). Then, Π B satisfies the quadratic estimate ˆ tπ B (1 + t 2 Π 2 B) 1 u 2 dt t u 2 for all u R(Π B ) L 2 (X, C N ). 0 Lashi Bandara (ANU) Square roots of operators on Lie groups August 13, / 28

63 Setup Set X = G and H = L 2 (G) L 2 (G) L 2 (W) = L 2 (C k+2 ). Lashi Bandara (ANU) Square roots of operators on Lie groups August 13, / 28

64 Setup Set X = G and H = L 2 (G) L 2 (G) L 2 (W) = L 2 (C k+2 ). Let S = (I, ), S = [I div]. Lashi Bandara (ANU) Square roots of operators on Lie groups August 13, / 28

65 Setup Set X = G and H = L 2 (G) L 2 (G) L 2 (W) = L 2 (C k+2 ). Let S = (I, ), S = [I div]. Let Γ = ( ) 0 0, Γ = S 0 ( 0 S 0 0 ), and Π = ( 0 S S 0 ). Lashi Bandara (ANU) Square roots of operators on Lie groups August 13, / 28

66 Setup Set X = G and H = L 2 (G) L 2 (G) L 2 (W) = L 2 (C k+2 ). Let S = (I, ), S = [I div]. Let Γ = ( ) 0 0, Γ = S 0 ( 0 S 0 0 ), and Π = ( 0 S S 0 ). Let B 00 = e, B10 = (c 1,..., c m ), B01 = (d 1,..., d m ) tr, B11 = (b ij ), and B = ( B ij ). Then, we can write B 1 = ( ) b and B 2 = ( ) B Lashi Bandara (ANU) Square roots of operators on Lie groups August 13, / 28

67 Proof The proofs of (H1)-(H6) are similar to the homogeneous situation. (H7) The proof is the same as the homogeneous situation except the lower order term introduces the term µ(b) 1 2 u on the right. Lashi Bandara (ANU) Square roots of operators on Lie groups August 13, / 28

68 Proof The proofs of (H1)-(H6) are similar to the homogeneous situation. (H7) The proof is the same as the homogeneous situation except the lower order term introduces the term µ(b) 1 2 u on the right. (H8) -1 The existence of a local Poincaré inequality is guaranteed by [ER2, Proposition 2.4]: ˆ ˆ f f B 2 dµ r 2 ( f 2 + f 2 ) dµ B for all balls B = B(x, r) and where f C (B). B Lashi Bandara (ANU) Square roots of operators on Lie groups August 13, / 28

69 Proof The proofs of (H1)-(H6) are similar to the homogeneous situation. (H7) The proof is the same as the homogeneous situation except the lower order term introduces the term µ(b) 1 2 u on the right. (H8) -1 The existence of a local Poincaré inequality is guaranteed by [ER2, Proposition 2.4]: ˆ ˆ f f B 2 dµ r 2 ( f 2 + f 2 ) dµ B for all balls B = B(x, r) and where f C (B). Define Ξu = ( u 1, u 2, u 3 ). B Lashi Bandara (ANU) Square roots of operators on Lie groups August 13, / 28

70 Proof The proofs of (H1)-(H6) are similar to the homogeneous situation. (H7) The proof is the same as the homogeneous situation except the lower order term introduces the term µ(b) 1 2 u on the right. (H8) -1 The existence of a local Poincaré inequality is guaranteed by [ER2, Proposition 2.4]: ˆ ˆ f f B 2 dµ r 2 ( f 2 + f 2 ) dµ B for all balls B = B(x, r) and where f C (B). Define Ξu = ( u 1, u 2, u 3 ). -2 The crucial fact needed here is the regularity result in [ER, Theorem 7.2], for u H 2 (G) = D( ). B A i A j u 2 u 2 + u 2 Lashi Bandara (ANU) Square roots of operators on Lie groups August 13, / 28

71 References I [AKMc] Andreas Axelsson, Stephen Keith, and Alan McIntosh, Quadratic estimates and functional calculi of perturbed Dirac operators, Invent. Math. 163 (2006), no. 3, [Bandara] L. Bandara, Quadratic estimates for perturbed Dirac type operators on doubling measure metric spaces, ArXiv e-prints (2011). [Morris] Andrew Morris, Local hardy spaces and quadratic estimates for Dirac type operators on Riemannian manifolds, Ph.D. thesis, Australian National University, [SC] L. Saloff-Coste and D. W. Stroock, Opérateurs uniformément sous-elliptiques sur les groupes de Lie, J. Funct. Anal. 98 (1991), no. 1, MR (92k:58264) [ER] A. F. M. ter Elst and Derek W. Robinson, Subelliptic operators on Lie groups: regularity, J. Austral. Math. Soc. Ser. A 57 (1994), no. 2, MR (95e:22019) Lashi Bandara (ANU) Square roots of operators on Lie groups August 13, / 28

72 References II [ER2] A. F. M. Ter Elst and Derek W. Robinson, Second-order subelliptic operators on Lie groups. I. Complex uniformly continuous principal coefficients, Acta Appl. Math. 59 (1999), no. 3, MR (2001a:35043) [ERS] A. F. M. ter Elst, Derek W. Robinson, and Adam Sikora, Heat kernels and Riesz transforms on nilpotent Lie groups, Colloq. Math. 74 (1997), no. 2, MR (99a:35029) Lashi Bandara (ANU) Square roots of operators on Lie groups August 13, / 28