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Page 1: On Kato's Square Root Problem - Université Paris-Sudegert/WIASVortragEgert.pdfOn Kato’s Square Root Problem Moritz Egert WIAS Berlin, February 11, 2015. T. Kato 1960s: Non-autonomous

On Kato’s Square Root Problem

Moritz Egert

WIAS Berlin, February 11, 2015

Page 2: On Kato's Square Root Problem - Université Paris-Sudegert/WIASVortragEgert.pdfOn Kato’s Square Root Problem Moritz Egert WIAS Berlin, February 11, 2015. T. Kato 1960s: Non-autonomous

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.

Page 3: On Kato's Square Root Problem - Université Paris-Sudegert/WIASVortragEgert.pdfOn Kato’s Square Root Problem Moritz Egert WIAS Berlin, February 11, 2015. T. Kato 1960s: Non-autonomous

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.

Page 4: On Kato's Square Root Problem - Université Paris-Sudegert/WIASVortragEgert.pdfOn Kato’s Square Root Problem Moritz Egert WIAS Berlin, February 11, 2015. T. Kato 1960s: Non-autonomous

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.

Page 5: On Kato's Square Root Problem - Université Paris-Sudegert/WIASVortragEgert.pdfOn Kato’s Square Root Problem Moritz Egert WIAS Berlin, February 11, 2015. T. Kato 1960s: Non-autonomous

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.

Page 6: On Kato's Square Root Problem - Université Paris-Sudegert/WIASVortragEgert.pdfOn Kato’s Square Root Problem Moritz Egert WIAS Berlin, February 11, 2015. T. Kato 1960s: Non-autonomous

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.

Page 7: On Kato's Square Root Problem - Université Paris-Sudegert/WIASVortragEgert.pdfOn Kato’s Square Root Problem Moritz Egert WIAS Berlin, February 11, 2015. T. Kato 1960s: Non-autonomous

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”.

Page 8: On Kato's Square Root Problem - Université Paris-Sudegert/WIASVortragEgert.pdfOn Kato’s Square Root Problem Moritz Egert WIAS Berlin, February 11, 2015. T. Kato 1960s: Non-autonomous

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”.

Page 9: On Kato's Square Root Problem - Université Paris-Sudegert/WIASVortragEgert.pdfOn Kato’s Square Root Problem Moritz Egert WIAS Berlin, February 11, 2015. T. Kato 1960s: Non-autonomous

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”.

Page 10: On Kato's Square Root Problem - Université Paris-Sudegert/WIASVortragEgert.pdfOn Kato’s Square Root Problem Moritz Egert WIAS Berlin, February 11, 2015. T. Kato 1960s: Non-autonomous

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, . . . .

Page 11: On Kato's Square Root Problem - Université Paris-Sudegert/WIASVortragEgert.pdfOn Kato’s Square Root Problem Moritz Egert WIAS Berlin, February 11, 2015. T. Kato 1960s: Non-autonomous

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, . . . .

Page 12: On Kato's Square Root Problem - Université Paris-Sudegert/WIASVortragEgert.pdfOn Kato’s Square Root Problem Moritz Egert WIAS Berlin, February 11, 2015. T. Kato 1960s: Non-autonomous

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, . . . .

Page 13: On Kato's Square Root Problem - Université Paris-Sudegert/WIASVortragEgert.pdfOn Kato’s Square Root Problem Moritz Egert WIAS Berlin, February 11, 2015. T. Kato 1960s: Non-autonomous

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, . . . .

Page 14: On Kato's Square Root Problem - Université Paris-Sudegert/WIASVortragEgert.pdfOn Kato’s Square Root Problem Moritz Egert WIAS Berlin, February 11, 2015. T. Kato 1960s: Non-autonomous

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).

Page 15: On Kato's Square Root Problem - Université Paris-Sudegert/WIASVortragEgert.pdfOn Kato’s Square Root Problem Moritz Egert WIAS Berlin, February 11, 2015. T. Kato 1960s: Non-autonomous

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).

Page 16: On Kato's Square Root Problem - Université Paris-Sudegert/WIASVortragEgert.pdfOn Kato’s Square Root Problem Moritz Egert WIAS Berlin, February 11, 2015. T. Kato 1960s: Non-autonomous

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).

Page 17: On Kato's Square Root Problem - Université Paris-Sudegert/WIASVortragEgert.pdfOn Kato’s Square Root Problem Moritz Egert WIAS Berlin, February 11, 2015. T. Kato 1960s: Non-autonomous

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).

Page 18: On Kato's Square Root Problem - Université Paris-Sudegert/WIASVortragEgert.pdfOn Kato’s Square Root Problem Moritz Egert WIAS Berlin, February 11, 2015. T. Kato 1960s: Non-autonomous

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).

Page 19: On Kato's Square Root Problem - Université Paris-Sudegert/WIASVortragEgert.pdfOn Kato’s Square Root Problem Moritz Egert WIAS Berlin, February 11, 2015. T. Kato 1960s: Non-autonomous

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.

Page 20: On Kato's Square Root Problem - Université Paris-Sudegert/WIASVortragEgert.pdfOn Kato’s Square Root Problem Moritz Egert WIAS Berlin, February 11, 2015. T. Kato 1960s: Non-autonomous

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

Page 21: On Kato's Square Root Problem - Université Paris-Sudegert/WIASVortragEgert.pdfOn Kato’s Square Root Problem Moritz Egert WIAS Berlin, February 11, 2015. T. Kato 1960s: Non-autonomous

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

Page 22: On Kato's Square Root Problem - Université Paris-Sudegert/WIASVortragEgert.pdfOn Kato’s Square Root Problem Moritz Egert WIAS Berlin, February 11, 2015. T. Kato 1960s: Non-autonomous

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

Page 23: On Kato's Square Root Problem - Université Paris-Sudegert/WIASVortragEgert.pdfOn Kato’s Square Root Problem Moritz Egert WIAS Berlin, February 11, 2015. T. Kato 1960s: Non-autonomous

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

Page 24: On Kato's Square Root Problem - Université Paris-Sudegert/WIASVortragEgert.pdfOn Kato’s Square Root Problem Moritz Egert WIAS Berlin, February 11, 2015. T. Kato 1960s: Non-autonomous

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)

= 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.

Page 25: On Kato's Square Root Problem - Université Paris-Sudegert/WIASVortragEgert.pdfOn Kato’s Square Root Problem Moritz Egert WIAS Berlin, February 11, 2015. T. Kato 1960s: Non-autonomous

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)

= 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.

Page 26: On Kato's Square Root Problem - Université Paris-Sudegert/WIASVortragEgert.pdfOn Kato’s Square Root Problem Moritz Egert WIAS Berlin, February 11, 2015. T. Kato 1960s: Non-autonomous

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

Page 27: On Kato's Square Root Problem - Université Paris-Sudegert/WIASVortragEgert.pdfOn Kato’s Square Root Problem Moritz Egert WIAS Berlin, February 11, 2015. T. Kato 1960s: Non-autonomous

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

Page 28: On Kato's Square Root Problem - Université Paris-Sudegert/WIASVortragEgert.pdfOn Kato’s Square Root Problem Moritz Egert WIAS Berlin, February 11, 2015. T. Kato 1960s: Non-autonomous

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

Page 29: On Kato's Square Root Problem - Université Paris-Sudegert/WIASVortragEgert.pdfOn Kato’s Square Root Problem Moritz Egert WIAS Berlin, February 11, 2015. T. Kato 1960s: Non-autonomous

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.

Page 30: On Kato's Square Root Problem - Université Paris-Sudegert/WIASVortragEgert.pdfOn Kato’s Square Root Problem Moritz Egert WIAS Berlin, February 11, 2015. T. Kato 1960s: Non-autonomous

Thank you for your attention!

Kato Square Root Problem

Page 31: On Kato's Square Root Problem - Université Paris-Sudegert/WIASVortragEgert.pdfOn Kato’s Square Root Problem Moritz Egert WIAS Berlin, February 11, 2015. T. Kato 1960s: Non-autonomous

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.

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