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On Kato’s Square Root Problem
Moritz Egert
WIAS Berlin, February 11, 2015
T. Kato 1960s: Non-autonomous parabolic evo-lution equation
ddt
u(t)(x) + A(t)u(t)(x) = 0 (t > 0, x ∈ Ω),
u(0) = u0 ∈ L2(Ω).
I A(t) ∼ −∇x · µ(t , x)∇x via elliptic form a(t) : V(Ω)× V(Ω)→ C.I u(t)(x) = e−tAu0(x) if A(t) = A for all t > 0.
Kato Square Root Problem (1961)
“We do not know whether or not D(A1/2) = D(A∗1/2). This is perhaps not truein general. But the question is open even when A is regularly accretive. Inthis case it appears reasonable to suppose that both D(A1/2) and D(A∗1/2)coincide with D(a), where a is the regular sesquilinear form which defines A.”
I Counterexamples by Lions 1962, McIntosh 1982I Specialize to divergence-form operators.
T. Kato 1960s: Non-autonomous parabolic evo-lution equation
ddt
u(t)(x) + A(t)u(t)(x) = 0 (t > 0, x ∈ Ω),
u(0) = u0 ∈ L2(Ω).
I A(t) ∼ −∇x · µ(t , x)∇x via elliptic form a(t) : V(Ω)× V(Ω)→ C.I u(t)(x) = e−tAu0(x) if A(t) = A for all t > 0.
Kato Square Root Problem (1961)
“We do not know whether or not D(A1/2) = D(A∗1/2). This is perhaps not truein general. But the question is open even when A is regularly accretive. Inthis case it appears reasonable to suppose that both D(A1/2) and D(A∗1/2)coincide with D(a), where a is the regular sesquilinear form which defines A.”
I Counterexamples by Lions 1962, McIntosh 1982I Specialize to divergence-form operators.
T. Kato 1960s: Non-autonomous parabolic evo-lution equation
ddt
u(t)(x) + A(t)u(t)(x) = 0 (t > 0, x ∈ Ω),
u(0) = u0 ∈ L2(Ω).
I A(t) ∼ −∇x · µ(t , x)∇x via elliptic form a(t) : V(Ω)× V(Ω)→ C.I u(t)(x) = e−tAu0(x) if A(t) = A for all t > 0.
Kato Square Root Problem (1961)
“We do not know whether or not D(A1/2) = D(A∗1/2). This is perhaps not truein general. But the question is open even when A is regularly accretive. Inthis case it appears reasonable to suppose that both D(A1/2) and D(A∗1/2)coincide with D(a), where a is the regular sesquilinear form which defines A.”
I Counterexamples by Lions 1962, McIntosh 1982I Specialize to divergence-form operators.
T. Kato 1960s: Non-autonomous parabolic evo-lution equation
ddt
u(t)(x) + A(t)u(t)(x) = 0 (t > 0, x ∈ Ω),
u(0) = u0 ∈ L2(Ω).
I A(t) ∼ −∇x · µ(t , x)∇x via elliptic form a(t) : V(Ω)× V(Ω)→ C.I u(t)(x) = e−tAu0(x) if A(t) = A for all t > 0.
Kato Square Root Problem (1961)
“We do not know whether or not D(A1/2) = D(A∗1/2). This is perhaps not truein general. But the question is open even when A is regularly accretive. Inthis case it appears reasonable to suppose that both D(A1/2) and D(A∗1/2)coincide with D(a), where a is the regular sesquilinear form which defines A.”
I Counterexamples by Lions 1962, McIntosh 1982I Specialize to divergence-form operators.
Setup
LetI Ω ⊆ Rd domain, D ⊆ ∂ Ω closed, µ ∈ L∞(Ω)
I A ∼ −∇ · µ∇ accretive operator on L2(Ω)associated with
W1,2D (Ω)×W1,2
D (Ω)→ C, (u, v) 7→∫
Ωµ∇u · ∇v .
I A1/2 square root of A defined by e.g.
A1/2u =1π
∫ ∞0
t−1/2A(t + A)−1 dt .
D
Kato conjecture
It holds D(A1/2) = W1,2D (Ω) with equivalent norms.
Why do we care about the Kato conjecture?
Philosophy
I Elliptic non-regularity results D(A) * W2,2(Ω).I Kato Conjecture ∼ optimal Sobolev regularity for A1/2.
Ex. 1: Elliptic equations on Rd+
∂2
∂t2 u(t)(x) +∇ · µ(x)∇u(t , x) = 0 (t > 0, x ∈ Rd ),
u(0, x) = u0(x) ∈W1,2(Rd ).
I Solution u(t , x) = e−tA1/2u0(x).
I Kato conjecture ∼ Rellich inequality “‖∂tu|t=0‖2 ∼ ‖∇u0‖2”.
Why do we care about the Kato conjecture?
Philosophy
I Elliptic non-regularity results D(A) * W2,2(Ω).I Kato Conjecture ∼ optimal Sobolev regularity for A1/2.
Ex. 1: Elliptic equations on Rd+
∂2
∂t2 u(t)(x) +∇ · µ(x)∇u(t , x) = 0 (t > 0, x ∈ Rd ),
u(0, x) = u0(x) ∈W1,2(Rd ).
I Solution u(t , x) = e−tA1/2u0(x).
I Kato conjecture ∼ Rellich inequality “‖∂tu|t=0‖2 ∼ ‖∇u0‖2”.
Why do we care about the Kato conjecture?
Philosophy
I Elliptic non-regularity results D(A) * W2,2(Ω).I Kato Conjecture ∼ optimal Sobolev regularity for A1/2.
Ex. 1: Elliptic equations on Rd+
∂2
∂t2 u(t)(x) +∇ · µ(x)∇u(t , x) = 0 (t > 0, x ∈ Rd ),
u(0, x) = u0(x) ∈W1,2(Rd ).
I Solution u(t , x) = e−tA1/2u0(x).
I Kato conjecture ∼ Rellich inequality “‖∂tu|t=0‖2 ∼ ‖∇u0‖2”.
Ex. 2: Maximal parabolic regularity (e.g. Haller-Dintelmann-Rehberg)
I In Lp-setting study parabolic equation ddt
u(t) + Au(t) = f (0 < t < T ),
u(0) = 0.
I Goal: Transport Max. Reg. from Lp(Ω) to W−1,pD (Ω).
I Lp′-Kato conjecture ∼ (−∇ · µ>∇)1/2 : W1,p′
D (Ω)→ Lp′(Ω) isom.
I Adjoint (−∇ · µ∇)1/2 : Lp(Ω)→W−1,pD (Ω) isomorphism that
commutes with parabolic solution operator( ddt
+ A)−1
.
Many further examples, e.g. Cauchy-Integral along Lipschitz curve,hyperbolic wave equations, . . . .
Ex. 2: Maximal parabolic regularity (e.g. Haller-Dintelmann-Rehberg)
I In Lp-setting study parabolic equation ddt
u(t) + Au(t) = f (0 < t < T ),
u(0) = 0.
I Goal: Transport Max. Reg. from Lp(Ω) to W−1,pD (Ω).
I Lp′-Kato conjecture ∼ (−∇ · µ>∇)1/2 : W1,p′
D (Ω)→ Lp′(Ω) isom.
I Adjoint (−∇ · µ∇)1/2 : Lp(Ω)→W−1,pD (Ω) isomorphism that
commutes with parabolic solution operator( ddt
+ A)−1
.
Many further examples, e.g. Cauchy-Integral along Lipschitz curve,hyperbolic wave equations, . . . .
Ex. 2: Maximal parabolic regularity (e.g. Haller-Dintelmann-Rehberg)
I In Lp-setting study parabolic equation ddt
u(t) + Au(t) = f (0 < t < T ),
u(0) = 0.
I Goal: Transport Max. Reg. from Lp(Ω) to W−1,pD (Ω).
I Lp′-Kato conjecture ∼ (−∇ · µ>∇)1/2 : W1,p′
D (Ω)→ Lp′(Ω) isom.
I Adjoint (−∇ · µ∇)1/2 : Lp(Ω)→W−1,pD (Ω) isomorphism that
commutes with parabolic solution operator( ddt
+ A)−1
.
Many further examples, e.g. Cauchy-Integral along Lipschitz curve,hyperbolic wave equations, . . . .
Ex. 2: Maximal parabolic regularity (e.g. Haller-Dintelmann-Rehberg)
I In Lp-setting study parabolic equation ddt
u(t) + Au(t) = f (0 < t < T ),
u(0) = 0.
I Goal: Transport Max. Reg. from Lp(Ω) to W−1,pD (Ω).
I Lp′-Kato conjecture ∼ (−∇ · µ>∇)1/2 : W1,p′
D (Ω)→ Lp′(Ω) isom.
I Adjoint (−∇ · µ∇)1/2 : Lp(Ω)→W−1,pD (Ω) isomorphism that
commutes with parabolic solution operator( ddt
+ A)−1
.
Many further examples, e.g. Cauchy-Integral along Lipschitz curve,hyperbolic wave equations, . . . .
Positive answers
Self-adjoint operators
XWhole space Ω = Rd
X
I d = 1: Coifman - McIntosh - Meyer ’82.
I d ≥ 2: Auscher-Hofmann-Lacey-McIntosh-Tchamitchian ’01,Axelsson-Keith-McIntosh ’06.
Bounded domainsI Ω Lipschitz, D ∈ ∅, ∂ Ω: Auscher-Tchamitchian ’03, ’01 (p 6= 2).
I Ω smooth, smooth D ! ∂ Ω \ D interface: Axelsson-Keith-McIntosh ’06.
I Ω Lipschitz around ∂ Ω \ D:Auscher-Badr-Haller-Dintelmann-Rehberg ’12 (p 6= 2).
Positive answers
Self-adjoint operatorsX
Whole space Ω = Rd
X
I d = 1: Coifman - McIntosh - Meyer ’82.
I d ≥ 2: Auscher-Hofmann-Lacey-McIntosh-Tchamitchian ’01,Axelsson-Keith-McIntosh ’06.
Bounded domainsI Ω Lipschitz, D ∈ ∅, ∂ Ω: Auscher-Tchamitchian ’03, ’01 (p 6= 2).
I Ω smooth, smooth D ! ∂ Ω \ D interface: Axelsson-Keith-McIntosh ’06.
I Ω Lipschitz around ∂ Ω \ D:Auscher-Badr-Haller-Dintelmann-Rehberg ’12 (p 6= 2).
Positive answers
Self-adjoint operatorsXWhole space Ω = Rd
X
I d = 1: Coifman - McIntosh - Meyer ’82.
I d ≥ 2: Auscher-Hofmann-Lacey-McIntosh-Tchamitchian ’01,Axelsson-Keith-McIntosh ’06.
Bounded domainsI Ω Lipschitz, D ∈ ∅, ∂ Ω: Auscher-Tchamitchian ’03, ’01 (p 6= 2).
I Ω smooth, smooth D ! ∂ Ω \ D interface: Axelsson-Keith-McIntosh ’06.
I Ω Lipschitz around ∂ Ω \ D:Auscher-Badr-Haller-Dintelmann-Rehberg ’12 (p 6= 2).
Positive answers
Self-adjoint operatorsXWhole space Ω = Rd X
I d = 1: Coifman - McIntosh - Meyer ’82.
I d ≥ 2: Auscher-Hofmann-Lacey-McIntosh-Tchamitchian ’01,Axelsson-Keith-McIntosh ’06.
Bounded domainsI Ω Lipschitz, D ∈ ∅, ∂ Ω: Auscher-Tchamitchian ’03, ’01 (p 6= 2).
I Ω smooth, smooth D ! ∂ Ω \ D interface: Axelsson-Keith-McIntosh ’06.
I Ω Lipschitz around ∂ Ω \ D:Auscher-Badr-Haller-Dintelmann-Rehberg ’12 (p 6= 2).
Positive answers
Self-adjoint operatorsXWhole space Ω = Rd X
I d = 1: Coifman - McIntosh - Meyer ’82.
I d ≥ 2: Auscher-Hofmann-Lacey-McIntosh-Tchamitchian ’01,Axelsson-Keith-McIntosh ’06.
Bounded domainsI Ω Lipschitz, D ∈ ∅, ∂ Ω: Auscher-Tchamitchian ’03, ’01 (p 6= 2).
I Ω smooth, smooth D ! ∂ Ω \ D interface: Axelsson-Keith-McIntosh ’06.
I Ω Lipschitz around ∂ Ω \ D:Auscher-Badr-Haller-Dintelmann-Rehberg ’12 (p 6= 2).
Kato for mixed boundary conditions
Theorem (E.-Haller-Dintelmann-Tolksdorf ’14)Suppose
I Ω ⊆ Rd bounded d-Ahlfors regular domain,
I D ⊆ ∂ Ω closed and (d − 1)-Ahlfors regular,
I Ω Lipschitz around ∂ Ω \ D.
Then
D(A1/2) = W1,2D (Ω) with ‖A1/2u‖2 ∼ ‖∇u‖2.
I First formulated by J.-L. Lions 1962.I For rough (= L∞) coefficients new even on Lipschitz domains.
Some ideas of the proof
1 First-order approach via perturbed Dirac operators á la AKM ’06,H∞-functional calculus.
2 Getting rid of the coefficients via
Reduction-Theorem (E.-Haller-Dintelmann-Tolksdorf ’14)
In essence, the following holds: If D((−∆V)s) → H2s,2(Ω) for
some s > 12 , then D(A1/2) = W1,2
D (Ω).
Extrapolate Kato for −∆V =⇒ Kato property for general Ageometry, potential theoryharmonic analysis
Some ideas of the proof
1 First-order approach via perturbed Dirac operators á la AKM ’06,H∞-functional calculus.
2 Getting rid of the coefficients via
Reduction-Theorem (E.-Haller-Dintelmann-Tolksdorf ’14)
In essence, the following holds: If D((−∆V)s) → H2s,2(Ω) for
some s > 12 , then D(A1/2) = W1,2
D (Ω).
Extrapolate Kato for −∆V =⇒ Kato property for general Ageometry, potential theoryharmonic analysis
Some ideas of the proof
1 First-order approach via perturbed Dirac operators á la AKM ’06,H∞-functional calculus.
2 Getting rid of the coefficients via
Reduction-Theorem (E.-Haller-Dintelmann-Tolksdorf ’14)
In essence, the following holds: If D((−∆V)s) → H2s,2(Ω) for
some s > 12 , then D(A1/2) = W1,2
D (Ω).
Extrapolate Kato for −∆V =⇒ Kato property for general A
geometry, potential theoryharmonic analysis
Some ideas of the proof
1 First-order approach via perturbed Dirac operators á la AKM ’06,H∞-functional calculus.
2 Getting rid of the coefficients via
Reduction-Theorem (E.-Haller-Dintelmann-Tolksdorf ’14)
In essence, the following holds: If D((−∆V)s) → H2s,2(Ω) for
some s > 12 , then D(A1/2) = W1,2
D (Ω).
Extrapolate Kato for −∆V =⇒ Kato property for general Ageometry, potential theory ! harmonic analysis
3 D((−∆V)1/2) = W1,2D (Ω) by self-adjointness. Extrapolate by
Sneiberg’s stability theorem and the following result.
Theorem (E.-Haller-Dintelmann-Tolksdorf ’14)
Let θ ∈ (0,1) and s0, s1 ∈ (12 ,
32). Put sθ := (1− θ)s0 + θs1. Then,
W1,2D (Ω) = H1,2
D (Ω)
[Hs0,2
D (Ω),Hs1,2D (Ω)
]θ
= Hsθ,2D (Ω).
[L2(Ω),H1,2
D (Ω)]θ
=
Hθ,2D (Ω), if θ > 1
2 ,Hθ,2(Ω), if θ < 1
2 .
4 In fact, D((−∆V)s) = H2s,2D (Ω) for |12 − s| < ε.
3 D((−∆V)1/2) = W1,2D (Ω) by self-adjointness. Extrapolate by
Sneiberg’s stability theorem and the following result.
Theorem (E.-Haller-Dintelmann-Tolksdorf ’14)
Let θ ∈ (0,1) and s0, s1 ∈ (12 ,
32). Put sθ := (1− θ)s0 + θs1. Then,
W1,2D (Ω) = H1,2
D (Ω)
[Hs0,2
D (Ω),Hs1,2D (Ω)
]θ
= Hsθ,2D (Ω).
[L2(Ω),H1,2
D (Ω)]θ
=
Hθ,2D (Ω), if θ > 1
2 ,Hθ,2(Ω), if θ < 1
2 .
4 In fact, D((−∆V)s) = H2s,2D (Ω) for |12 − s| < ε.
Elliptic BVPs on cylindrical domains
Elliptic mixed BVP
−divt ,xµ(x)∇t ,xU = 0 (R+ × Ω)
U = 0 (R+ × D)
∂νµU = 0 (R+ × N)
∂νµU = f ∈ L2 (0 × Ω)
l F ∼[∂νµU∇xU
]First order equation
∂tF +
[0 (−∇V)∗
−∇V 0
]︸ ︷︷ ︸
D
BF = 0 (t > 0)
F (0)⊥ = f L2loc(R+; L2(Ω)) setting
Elliptic BVPs on cylindrical domains
Elliptic mixed BVP
−divt ,xµ(x)∇t ,xU = 0 (R+ × Ω)
U = 0 (R+ × D)
∂νµU = 0 (R+ × N)
∂νµU = f ∈ L2 (0 × Ω)
l F ∼[∂νµU∇xU
]First order equation
∂tF +
[0 (−∇V)∗
−∇V 0
]︸ ︷︷ ︸
D
BF = 0 (t > 0)
F (0)⊥ = f
L2loc(R+; L2(Ω)) setting
Elliptic BVPs on cylindrical domains
Elliptic mixed BVP
−divt ,xµ(x)∇t ,xU = 0 (R+ × Ω)
U = 0 (R+ × D)
∂νµU = 0 (R+ × N)
∂νµU = f ∈ L2 (0 × Ω)
l F ∼[∂νµU∇xU
]First order equation
∂tF +
[0 (−∇V)∗
−∇V 0
]︸ ︷︷ ︸
D
BF = 0 (t > 0)
F (0)⊥ = f L2loc(R+; L2(Ω)) setting
Semigroup solutions via DB-formalism
DB has bounded H∞-calculus on H = R(DB) (Kato Technology).
Theorem (Auscher-E. ’14)1 For every F (0) ∈ H+ := 1C+(DB)H a solution to the first-order
system is
F (t) = e−tDBF (0) (t ≥ 0).
Via F ∼[∂νµU∇xU
]these functions are in one-to-one correspon-
dence with weak solutions U such that
N∗(|∇t ,xU|) ∈ L2(R+ × Ω).
2 If µ is either block-diagonal or Hermitean, then for each f ∈ L2(Ω)there exists a unique such solution u.
Thank you for your attention!
Kato Square Root Problem
Literature
[1] T. KATO. Fractional powers of dissipative operators. J. Math. Soc. Japan 13(1961), 246–274.
[2] P. AUSCHER, S. HOFMANN, M. LACEY, A. MC INTOSH, and P. TCHAMITCHIAN. Thesolution of the Kato square root problem for second order elliptic operators on Rn.Ann. of Math. (2) 156 (2002), no. 2, 633–654.
[3] A. AXELSSON, S. KEITH, and A. MC INTOSH. Quadratic estimates and functionalcalculi of perturbed Dirac operators. Invent. Math. 163 (2006), no. 3, 455–497.
[4] M. EGERT, R. HALLER-DINTELMANN, and P. TOLKSDORF. The Kato Square RootProblem follows from an extrapolation property of the Laplacian. Submitted.
[5] M. EGERT, R. HALLER-DINTELMANN, and P. TOLKSDORF. The Kato Square RootProblem for mixed boundary conditions. J. Funct. Anal. 267 (2014), no. 5,1419–1461.
[6] P. AUSCHER, A. AXELSSON, and A. MC INTOSH. Solvability of elliptic systems withsquare integrable boundary data. Ark. Mat. 48 (2010), no. 2, 253–287.