Spectral Multiplier Theorems - TU Dresden

Intro Tools Results Examples

Spectral Multiplier Theorems

Lutz WeisKIT, University of Karlsruhe

joint work with

Christoph KrieglerUniversite Blaise-Pascal, Clermont-Ferand

0 / 23


Fourier Multipliers

� on L2(Rd):

f (��)x = F�1[f (| · |2)(Fx)(·)]f (��) 2 B(L2(Rd)) for all f 2 Bb(Rd)

� on Lp(Rd):

spectral projections �B(��) not bounded on Lp(Rd)for all Borel sets B ⇢ R

+

�[0,r ](��) /2 B(Lp(Rd) for d � 2

Hormander’s multiplier theorem:

f (��) bounded on Lp(Rd) if

supR>0

R2RR

�R j |D j f (t)|�2 dtR < 1

for j = 1, 2, . . . , k with k > n| 1p � 1

2

|.1 / 23


Fourier Multipliers

� on L2(Rd):


� on Lp(Rd):


+

�[0,r ](��) /2 B(Lp(Rd) for d � 2



supR>0

R2RR

�R j |D j f (t)|�2 dtR < 1

for j = 1, 2, . . . , k with k > n| 1p � 1

2

|.1 / 23


Fourier Multipliers

� on L2(Rd):


� on Lp(Rd):


+

�[0,r ](��) /2 B(Lp(Rd) for d � 2



supR>0

R2RR

�R j |D j f (t)|�2 dtR < 1

for j = 1, 2, . . . , k with k > n| 1p � 1

2

|.1 / 23


Laplace type Operators

• Sub-Laplacians on Lie groups of polynomial growth

• Sub-Laplacians on compact Riemannian mainfolds

• Elliptic operators on domains

• Schrodinger operators with certain singular potentials

• Di↵erential operators of Hermite and Laguerre type

• Bessel operators

• Laplace operators on graphs and fractals

2 / 23


Spectral Multiplier Theorems in the Lp-Scale

• (X , ⇢, µ) metric measure space

• Doubling property: V (x , s) C ( sr )dV (x , r), s � r � 0

• A � 0 selfadjoint on L2(X , µ)

• e�tA has Gaussian bounds

kt(x , y) [V (x , t1/m)V (y , t1/m)]�1/2exp[�C (⇢(x ,y)m

t )1

m�1 ]

Then for ↵ > d/2 kf (A)kLp(X )!Lp(X )

. kf kH2

↵.

Hormander, Alexopoulos, Christ, Cowling, Duong, McIntosh, Mauceri, Meda,Ouhabaz, Hebisch, Hulanicki

More general results: Couhlon, Duong, Ouhabaz, Sikora, Yan, Yao,Blunck, Kunstmann, Uhl

3 / 23


Hormander classes Hp↵

f 2 Cb(R+

), 1 p 1, ↵ > 1/p, ⌘ 2 C1(14

, 4) cut-o↵

f 2 Wp↵ i↵ kf kWp

↵= kf (exp(·))kW↵,p

(R+

)

< 1

↵ 2 N: kf kWp↵⇡

↵Pn=0

� R10

|tnDnf (t)|p dtt

�1/p

f 2 Hp↵ i↵ kf kHp

↵= sup

�>0

k⌘(·)f (�·)kW↵,p(R

+

)

< 1

↵ 2 N: kf kHp↵⇡

↵Pn=0

supR>0

� R2RR |tnDnf (t)|p dt

t

�1/p

Hp↵ ⇢ Hq

↵ ⇢ Hp↵+r , p > q, r > 1

q � 1

p

p = 1 Mihlin condition, p = 2 Hormander condition4 / 23


Hormander classes Hp↵

f 2 Cb(R+

), 1 p 1, ↵ > 1/p, ⌘ 2 C1(14

, 4) cut-o↵

f 2 Wp↵ i↵ kf kWp

↵= kf (exp(·))kW↵,p

(R+

)

< 1

↵ 2 N: kf kWp↵⇡

↵Pn=0

� R10

|tnDnf (t)|p dtt

�1/p

f 2 Hp↵ i↵ kf kHp

↵= sup

�>0

k⌘(·)f (�·)kW↵,p(R

+

)

< 1

↵ 2 N: kf kHp↵⇡

↵Pn=0

supR>0

� R2RR |tnDnf (t)|p dt

t

�1/p

Hp↵ ⇢ Hq

↵ ⇢ Hp↵+r , p > q, r > 1

q � 1

p

p = 1 Mihlin condition, p = 2 Hormander condition4 / 23


H1↵ and Norm Estimates

If A has a H1↵ -functional calculus and ↵ > 1/2 then

• {(⇡2

� |✓|)↵e�zA : |arg(z)| = ✓} uniformly bounded for |✓| ! ⇡2

• {(1 + |s|)�↵(1 + A)�↵e isA : s 2 R} is bounded

• {|✓|↵(z � A)�1 : |arg(z)| = ✓} uniformly bounded for |✓| ! 0

• {(1 + |t|)�↵Ait : t 2 R} is bounded

• {�↵u (A) : u > 0} is bounded, �↵u (t) = (1� t

u )↵+

5/ 23




• {(⇡2






u )↵+

5/ 23




• {(⇡2






u )↵+

5/ 23


Spectral Multipliers and Inversion Formulas

µ = A

f (A) = 1

2⇡i

R@⌃ f (�)(�� A)�1d� Cauchy

f (A) =R10

L�1[f ](t)e�tAdt Laplace

f (A) = 1p2⇡

R1�1F [f ](t)e itAdt Fourier

f (A) = 1

2⇡

R1�1M[f ](t)Aitdt Mellin

f (A) = 1

�(⌫)

R10

f (⌫)(t)(t � A)⌫�1

+

dt integration by parts

Norm estimates on N(t) only give a Wp↵-calculus.

6 / 23



µ = A

f (A) = 1

2⇡i

R@⌃ f (�)(�� A)�1d� Cauchy

f (A) =R10


f (A) = 1p2⇡


f (A) = 1

2⇡


f (A) = 1

�(⌫)

R10

f (⌫)(t)(t � A)⌫�1

+



6 / 23



µ = A

f (A) = 1

2⇡i

R@⌃ f (�)(�� A)�1d� Cauchy

f (A) =R10


f (A) = 1p2⇡


f (A) = 1

2⇡


f (A) = 1

�(⌫)

R10

f (⌫)(t)(t � A)⌫�1

+



6 / 23


Our Point of View

A on a Lp-scale A on a fixedBanach space Xe.g. X = Lp(Rd ,E )

A selfadjoint on L2 A is 0-sectorial and hasa bounded H1(⌃�)-calculus

f (tA), t 2 R+

, haskernel bounds

f (tA), t 2 R+

is R-bounded

Goal: Characterize the Hp↵-calculus in terms of R-bounds for families

{f (tA) : t > 0}.7 / 23


Our Point of View

A on a Lp-scale A on a fixedBanach space Xe.g. X = Lp(Rd ,E )

A selfadjoint on L2 A is 0-sectorial and hasa bounded H1(⌃�)-calculus

f (tA), t 2 R+

, haskernel bounds

f (tA), t 2 R+

is R-bounded

Goal: Characterize the Hp↵-calculus in terms of R-bounds for families

{f (tA) : t > 0}.7 / 23


H1-calculus

A a sectorial operator on a Banach space X , f 2 H1(⌃�)

f (A)x = 1

2⇡i

R@⌃�

f (�)R(�,A)xd� , x 2 D(A) \ R(A)

A has bounded H1(⌃�)-calculus if

kf (A)k C sup�2⌃�

|f (�)|.

Theorem: Let A be the generator of a bounded analytic semigroup onLp(U) for some p 2 (1,1) s.th. e�tA is positive and contractive fort > 0.Then A has a bounded H1(⌃�)-calculus for some � < ⇡

2

.

Stein, Cowling, Coifman-Weiss, Kalton-W.

8 / 23


H1-calculus

A a sectorial operator on a Banach space X , f 2 H1(⌃�)

f (A)x = 1

2⇡i

R@⌃�

f (�)R(�,A)xd� , x 2 D(A) \ R(A)

A has bounded H1(⌃�)-calculus if

kf (A)k C sup�2⌃�

|f (�)|.

Theorem: Let A be the generator of a bounded analytic semigroup onLp(U) for some p 2 (1,1) s.th. e�tA is positive and contractive fort > 0.Then A has a bounded H1(⌃�)-calculus for some � < ⇡

2

.

Stein, Cowling, Coifman-Weiss, Kalton-W.

8 / 23


H1-calculus and Spectral Multiplier Theorem

A a 0-sectorial operator on a Banach space X with a H1(⌃�)-calculusfor some � 2 (0,⇡).

A has a Hp↵-calculus if for f 2 H1(⌃�)

kf (A)k Ckf |R+

kHp↵

Theorem: A has a H1↵ -calculus

) kf (A)k C!↵ kf kH1

(⌃!)for ! & 0

) A has a H1↵+✏-calculus ✏ > 0

Cowling, Doust, McIntosh, Yagi

9 / 23


H1-calculus and Spectral Multiplier Theorem

A a 0-sectorial operator on a Banach space X with a H1(⌃�)-calculusfor some � 2 (0,⇡).

A has a Hp↵-calculus if for f 2 H1(⌃�)

kf (A)k Ckf |R+

kHp↵

Theorem: A has a H1↵ -calculus

) kf (A)k C!↵ kf kH1

(⌃!)for ! & 0

) A has a H1↵+✏-calculus ✏ > 0

Cowling, Doust, McIntosh, Yagi

9 / 23


R-boundedness

X = Lp(U), 1 p < 1, ⌧ ⇢ B(X )

(1) For all T1

, . . . ,Tn 2 ⌧ , x1

, . . . , xn 2 X

k�nP

j=1

|Tjxj |2�1/2kLp Ck�

nPj=1

|xj |2�1/2kLp

Marcinkiewicz-Zygmund, (✏j) Bernoulli Random Variables

(2) EknP

j=1

✏jTjxjk EknP

j=1

✏jxjk

X Banach space, ⌧ ⇢ B(X ) R-bounded if 9 C < 1 such that (2) holds.R(⌧) := inf C .Bonami-Clerc 1985, Stempak: Marcinkiewicz Zygmund property

If X = Lp(U) then R-boundedness follows from

• (generalized) Gaussian bounds for T 2 ⌧

• Fe↵erman-Stein maximal functions

• Extrapolation via Ap-weights, interpolation10 / 23


R-boundedness

X = Lp(U), 1 p < 1, ⌧ ⇢ B(X )

(1) For all T1

, . . . ,Tn 2 ⌧ , x1

, . . . , xn 2 X

k�nP

j=1


nPj=1

|xj |2�1/2kLp


(2) EknP

j=1

✏jTjxjk EknP

j=1

✏jxjk







R-boundedness

X = Lp(U), 1 p < 1, ⌧ ⇢ B(X )

(1) For all T1

, . . . ,Tn 2 ⌧ , x1

, . . . , xn 2 X

k�nP

j=1


nPj=1

|xj |2�1/2kLp


(2) EknP

j=1

✏jTjxjk EknP

j=1

✏jxjk







R-boundedness and H1-calculus

X has Pisier’s property (↵), e.g. X ⇢ Lp(Lq), 1 p, q < 1

Theorem: If A has a bounded H1(⌃�)-calculus on X then

{f (A) : kf kH1(⌃�)

1}is R-bounded

Kalton-W.

Theorem: If X = Lp(Rd ,E ) and A = ��⌦ IdE with E a UMD-space.Then

{f (A) : kf kH2

↵ 1} ,↵ > d/2

is R-bounded

Girardi-W. , scalar case: e.g. Ap-extrapolation

In this case the H1 or Hp↵-calculus is called R-bounded.

11 / 23


R-boundedness and H1-calculus

X has Pisier’s property (↵), e.g. X ⇢ Lp(Lq), 1 p, q < 1

Theorem: If A has a bounded H1(⌃�)-calculus on X then

{f (A) : kf kH1(⌃�)

1}is R-bounded

Kalton-W.

Theorem: If X = Lp(Rd ,E ) and A = ��⌦ IdE with E a UMD-space.Then

{f (A) : kf kH2

↵ 1} ,↵ > d/2

is R-bounded

Girardi-W. , scalar case: e.g. Ap-extrapolation

In this case the H1 or Hp↵-calculus is called R-bounded.

11 / 23


Su�cient Conditions for a Hp↵-calculus

A a 0-sectorial operator on a Banach space X , ↵ > 1/2.

A has a bounded H1(⌃�)-calculus for some � 2 (0,⇡),

• {e�zA : |arg(z)| = ✓} C(

⇡2

�✓)↵ , bounded for |✓| ! ⇡2

• {(1 + |s|A)�↵e isA : s 2 R} bounded

• {(1 + |t|)�↵Ait : t 2 R} bounded

Theorem: Each of these conditions implies, A has a Hr�-calculus where

� > ↵+ 1

r and r 2 (1, 2) with 1

r > 1

typeX � 1

cotypeX .

12 / 23


Su�cient Conditions for a Hp↵-calculus

A a 0-sectorial operator on a Banach space X , ↵ > 1/2.

A has a bounded H1(⌃�)-calculus for some � 2 (0,⇡),

• {e�zA : |arg(z)| = ✓} C(

⇡2

�✓)↵ , bounded for |✓| ! ⇡2

• {(1 + |s|A)�↵e isA : s 2 R} bounded

• {(1 + |t|)�↵Ait : t 2 R} bounded

Theorem: Each of these conditions implies, A has a Hr�-calculus where

� > ↵+ 1

r and r 2 (1, 2) with 1

r > 1

typeX � 1

cotypeX .

12 / 23


Paley-Littlewood Theory for �

H↵,p

(Rd) = D((��)↵/2)B

↵p,q(Rd) =

�D((��)n/2),D((��)m/2)

�✓,q

↵ = (1� ✓)n + ✓m

' 2 C1( 12

, 2), 'n(t) := '(2�n|t|), Pn2Z

'n(t) ⌘ 1, supp'n ⇢ B2

n+1 \ B2

n�1

kxk˙H↵,p

(Rd)

⇠= k� Pn2Z

|2n'n ⇤ x |2�1/2kLp(Rd

)

kxk˙B↵p,q(Rd

)

⇠= � Pn2Z

(2nk'n ⇤ xkLp)q�1/q

Advantages:

• 'n ⇤ x analytic function with supp \'n ⇤ x ⇢ B2

n+1

\ B2

n�1

• D↵'n ⇤ x = (�i)|↵|F�1['n(u)u↵x(u)] ⇠ 2n|↵|'n ⇤ x• Bernstein’s inequality

Note: 'n ⇤ x = 'n((��)1/2)x13 / 23


Paley-Littlewood Theory for �

H↵,p

(Rd) = D((��)↵/2)B

↵p,q(Rd) =

�D((��)n/2),D((��)m/2)

�✓,q

↵ = (1� ✓)n + ✓m

' 2 C1( 12

, 2), 'n(t) := '(2�n|t|), Pn2Z

'n(t) ⌘ 1, supp'n ⇢ B2

n+1 \ B2

n�1

kxk˙H↵,p

(Rd)

⇠= k� Pn2Z

|2n'n ⇤ x |2�1/2kLp(Rd

)

kxk˙B↵p,q(Rd

)

⇠= � Pn2Z

(2nk'n ⇤ xkLp)q�1/q

Advantages:

• 'n ⇤ x analytic function with supp \'n ⇤ x ⇢ B2

n+1

\ B2

n�1

• D↵'n ⇤ x = (�i)|↵|F�1['n(u)u↵x(u)] ⇠ 2n|↵|'n ⇤ x• Bernstein’s inequality

Note: 'n ⇤ x = 'n((��)1/2)x13 / 23


Paley-Littlewood Decomposition for A

A 0-sectorial on a uniformly convex Banach space X . X✓ = (D(A✓), kA✓ · k)⇠' 2 H1

� , 'n(t) = '(2�nt),Pn2Z

'n ⌘ 1 on R+

2 H1� ,

R10

|t�✓ (t)|2 dtt < 1, sup

k�+1

|tk�✓ (k)(t)| . min(t✏, t�✏)

Theorem: Let A have a H1↵ -calculus for some ↵ < �. Then

(a) kxkX✓⇠= k� P

n2Z|2n✓'n(A)x |2

�1/2kLp for X ⇢ Lp(U)

⇠= Ek Pn2Z

✏n2n✓'n(A)xkX for general X

(b) kxkX✓⇠= k� R1

0

|t�✓ (tA)x |2 dtt


⇠= kt�✓ (tA)xk�(R+

, dtt ,X )

for general X

Remarks: • X✓ complex interpolation scale

• If A has a R-bounded W1↵ -calculus and (a) holds then A has

a H1↵ -calculus

• Similar results for inhomogeneous scale D((1 + A)✓).14 / 23




� , 'n(t) = '(2�nt),Pn2Z

'n ⌘ 1 on R+

2 H1� ,

R10

|t�✓ (t)|2 dtt < 1, sup

k�+1

|tk�✓ (k)(t)| . min(t✏, t�✏)



n2Z|2n✓'n(A)x |2


⇠= Ek Pn2Z



0



⇠= kt�✓ (tA)xk�(R+

, dtt ,X )

for general X



a H1↵ -calculus





� , 'n(t) = '(2�nt),Pn2Z

'n ⌘ 1 on R+

2 H1� ,

R10

|t�✓ (t)|2 dtt < 1, sup

k�+1

|tk�✓ (k)(t)| . min(t✏, t�✏)



n2Z|2n✓'n(A)x |2


⇠= Ek Pn2Z



0



⇠= kt�✓ (tA)xk�(R+

, dtt ,X )

for general X



a H1↵ -calculus





� , 'n(t) = '(2�nt),Pn2Z

'n ⌘ 1 on R+

2 H1� ,

R10

|t�✓ (t)|2 dtt < 1, sup

k�+1

|tk�✓ (k)(t)| . min(t✏, t�✏)



n2Z|2n✓'n(A)x |2


⇠= Ek Pn2Z



0



⇠= kt�✓ (tA)xk�(R+

, dtt ,X )

for general X



a H1↵ -calculus



Besov-Type Scale

A, ' and as beforeB✓q = (X✓

0

,X✓1

)#,q , ✓ = (1� #)✓0

+ #✓1

Theorem: Let A have W1↵ -calculus ↵ < �. Then

(a) kxk˙B✓q⇡ � P

n2Z2n✓qk'n(A)xkqX

�1/q

(b) kxk˙B✓q⇡ � R1

0

t�✓qk (tA)xkq dtt

�1/q

(c) A has H1↵ -calculus on all B✓

q

Remark: Similar result for the inhomogeneous case

Liegroups: Furioli, Melzi, Veneruso, Liu, MaSchrodinger Operators: Olafsson, Zheng

15 / 23


Besov-Type Scale

A, ' and as beforeB✓q = (X✓

0

,X✓1

)#,q , ✓ = (1� #)✓0

+ #✓1

Theorem: Let A have W1↵ -calculus ↵ < �. Then

(a) kxk˙B✓q⇡ � P

n2Z2n✓qk'n(A)xkqX

�1/q

(b) kxk˙B✓q⇡ � R1

0

t�✓qk (tA)xkq dtt

�1/q

(c) A has H1↵ -calculus on all B✓

q

Remark: Similar result for the inhomogeneous case

Liegroups: Furioli, Melzi, Veneruso, Liu, MaSchrodinger Operators: Olafsson, Zheng

15 / 23


Characterization of the H1↵-calculus

Let A have a W1

↵-calculus, ↵ > 1, on a Banach space with property (↵).For � > ↵� 1 define the Bochner Riesz means

��u (A) with ��u (t) = (1� tu )

�+

Theorem: In addition, let A have a bounded H1-calculus. Assume

⌧� = {��u (A) : u > 0} is R-bounded.

Then A has a H1

↵-calculus for ↵ > � +1. Conversely a H1

↵-calculus for Aimplies the R-boundedness of ⌧� for � > ↵� 1.

Special cases: Bonami, Clerc, StempakW1

↵-Calculus: Gale, Pytlik

16 / 23


Characterization of the H1↵-calculus

Let A have a W1

↵-calculus, ↵ > 1, on a Banach space with property (↵).For � > ↵� 1 define the Bochner Riesz means

��u (A) with ��u (t) = (1� tu )

�+

Theorem: In addition, let A have a bounded H1-calculus. Assume

⌧� = {��u (A) : u > 0} is R-bounded.

Then A has a H1

↵-calculus for ↵ > � +1. Conversely a H1

↵-calculus for Aimplies the R-boundedness of ⌧� for � > ↵� 1.

Special cases: Bonami, Clerc, StempakW1

↵-Calculus: Gale, Pytlik

16 / 23


Characterization of H2↵: R-bounds

A strongly continuous function N : t 2 R ! B(X ) is R2

-bounded if thefollowing set is R-bounded

A(N) = {R R�R f (t)N(t)dt : kf kL2(R) 1, R > 0}

Theorem: Let A be 0-sectorial with a H1(⌃�) calculus for some� 2 (0,⇡) on a Banach space with property (↵) and ↵ > 1/2.Then A has bounded H2

↵-calculus i↵ one (all) of the following functionsare R

2

-bounded

• t 2 R+

! (⇡2

� |✓|)↵A1/2T (e i✓t), uniformly for |✓| ! ⇡2

• t 2 R+

! |✓|�↵A1/2R(e i✓t,A), uniformly for ✓ ! 0

• t 2 R ! (1 + |t|)�↵Ait

• t 2 R ! |t|↵A�↵+1

2

�e itA � 1

�mm > ↵� 1

2

(However, ↵! ↵+ ✏ for ✏ > 0 in some implications)Then {f (A) : kf kH2

↵ 1} is R-bounded.

17 / 23


Characterization of H2↵: Square functions

Let X ⇢ Lp(U), 1 < p < 1 , ↵ > 1/2.

Theorem: Let A be 0-sectorial with a bounded H1-calculus. Then Ahas a matricially bounded H2

↵-calculus i↵

k� R |N(tA)x |2dt�1/2kLp C (N)kxkLp

where N(t) is one of the functions

• N✓(t) = A1/2T (e i✓t), C (N✓) . (⇡2

� |✓|)�↵ for |✓| ! ⇡2

• N✓(t) = A1/2R(e i✓t,A), C (N✓) . |✓|�↵ for |✓| ! 0.

• N(t) = (1 + |t|)�↵Ait , t 2 R

• N(t) = |t|↵A�↵+1

2

�e itA � 1

�mm > ↵� 1

2

in some implications we need ↵! ↵+ ✏, ✏ > 018 / 23


Characterization of H2↵: Square functions

Let X ⇢ Lp(U), 1 < p < 1 , ↵ > 1/2.

Theorem: Let A be 0-sectorial with a bounded H1-calculus. Then Ahas a matricially bounded H2

↵-calculus i↵

k� R |N(tA)x |2dt�1/2kLp C (N)kxkLp

where N(t) is one of the functions

• N✓(t) = A1/2T (e i✓t), C (N✓) . (⇡2

� |✓|)�↵ for |✓| ! ⇡2

• N✓(t) = A1/2R(e i✓t,A), C (N✓) . |✓|�↵ for |✓| ! 0.

• N(t) = (1 + |t|)�↵Ait , t 2 R

• N(t) = |t|↵A�↵+1

2

�e itA � 1

�mm > ↵� 1

2

in some implications we need ↵! ↵+ ✏, ✏ > 018 / 23


Square Functions in Banach space

Let (hn) be a ONB of L2(J) and (�n) a sequence of i.i.d. standardGaussian random variables

For N : J ! X put yn =RJ N(t)x hn(t) dt 2 X

If X = Lp(U) then

k� RJ |N(t)x |2dt�1/2kLp(U)

= kNxkLp(U,L2(J))

= k(yn)kLp(U,l2) = k�Pn|yn|2

�1/2kLp(U)

Def kN(t)xk�(J,X )

:= EkPn�ynkX

Alternatively,

kN(t)xk�(J,X )

= Ek RJ N(t)x d�(t)kXk · k�(J,X )

has (almost) the same ’operational’ properties on X as in theclassical square functions in Lp.

19 / 23


Steps in the Proof

• Use the assumed bounds on e�zA, R(z ,A) or Ait to establish aR-bounded Wp

↵-calculus

Hytonen-Veraar: X a Banach space. Let 1

r > 1

typeX � 1

cotypeX andN(t) : (a, b) ! B(X ) a strongly continuous function with

R ba kN(t)krB(X )

dt < 1.

Then

{R ba h(t)N(t)dt : khkLr 0 1}

is R-bounded.

• If A has a H1(⌃�)-calculus for some � 2 (0,⇡) and an R-boundedWp

↵-calculus, then A has a Hp�-calculus for some � > ↵.

20 / 23


More Steps

• Use the Paley-Littlewood decomposition and a localization principleto get the ’right’ Hp

↵-calculus.

If A has a Hp�-calculus for some (large) � > 1

p and

{f (2nA) : f 2 C1c (1

2

, 2), kf kWp↵ 1, n 2 Z}

is R-bounded. Then A has a Hp↵-calculus.

• R2

-boundedness is weaker than square function estimates.

Le Merdy: Let N : J ! X be strongly continuous and

kN(·)xk�(J,X )

Ckxk.Then

{RJ f (t)N(t)dt : kf kL2 1}is R-bounded.

21 / 23


Generalized Gaussian Bounds

• (X , ⇢, µ) metric measure space with doubling property

V (x , s) C ( sr )dV (x , r), s � r � 0


• k1B(x ,t1/m)e�tA1B(y ,t1/m)kp0!p0

0

CV (x , t1/m)�(

1/p0

�1/p00

) exp[�b(⇢(x ,y)t1/m

)m

m�1 ]

for some p0

2 [1, 2], m � 2

Then A has a H↵-calculus on Lp(X , µ) for p 2 (p0

, p00

),↵ > d | 1p � 1

2

|+ 1

2

Blunk, Duong, Ouhabaz, Sikora, Kunstmann, Uhl

22 / 23


Generalized Gaussian Bounds

• (X , ⇢, µ) metric measure space with doubling property

V (x , s) C ( sr )dV (x , r), s � r � 0


• k1B(x ,t1/m)e�tA1B(y ,t1/m)kp0!p0

0

CV (x , t1/m)�(

1/p0

�1/p00

) exp[�b(⇢(x ,y)t1/m

)m

m�1 ]

for some p0

2 [1, 2], m � 2

Then A has a H↵-calculus on Lp(X , µ) for p 2 (p0

, p00

),↵ > d | 1p � 1

2

|+ 1

2

Blunk, Duong, Ouhabaz, Sikora, Kunstmann, Uhl

22 / 23


Example: Maxwell Operators

⌦ ⇢ R3 bounded with Lipschitz boundaryP : L2(⌦,C3) ! L2�(⌦) Helmholtz Projection

• A defined on L2(⌦,C3) by the form

a(u, v) =R⌦

✏(·)�1rotu · rotvdx +R⌦

(divu)(divv)dx

D(a) = {u 2 L2(⌦) : divu 2 L2(⌦), rotu 2 L2(⌦), ⌫u|@⌦ = 0}

• M defined on L2�(⌦) by

Mu = Au for u 2 D(M) = P(D(A))

Kunstamnn, Uhl: p 2 (3/2, 3), ↵ > 3| 1p � 1

2

|+ 1

2

.

Ap, Mp have a H↵-calculus on Lp(⌦), Lp�(⌦), resp.

23 / 23


Example: Maxwell Operators

⌦ ⇢ R3 bounded with Lipschitz boundaryP : L2(⌦,C3) ! L2�(⌦) Helmholtz Projection

• A defined on L2(⌦,C3) by the form

a(u, v) =R⌦

✏(·)�1rotu · rotvdx +R⌦

(divu)(divv)dx

D(a) = {u 2 L2(⌦) : divu 2 L2(⌦), rotu 2 L2(⌦), ⌫u|@⌦ = 0}

• M defined on L2�(⌦) by

Mu = Au for u 2 D(M) = P(D(A))

Kunstamnn, Uhl: p 2 (3/2, 3), ↵ > 3| 1p � 1

2

|+ 1

2

.

Ap, Mp have a H↵-calculus on Lp(⌦), Lp�(⌦), resp.

23 / 23

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Spectral Multiplier Theorems - TU Dresden