Spectral Multiplier Theorems

Transcription

1 Spectral Multiplier Theorems Lutz Weis KIT, University of Karlsruhe joint work with Christoph Kriegler Université Blaise-Pascal, Clermont-Ferand 0/23

2 Fourier Multipliers on L 2 (R d ): f ( )x = F 1 [f ( 2 )(Fx)( )] f ( ) 2B(L 2 (R d )) for all f 2 B b (R d ) on L p (R d ): spectral projections B( ) not bounded on L p (R d ) for all Borel sets B R + [0,r]( ) /2 B(L p (R d )ford 2 Hörmander s multiplier theorem: f ( ) bounded on L p (R d )if R 2R sup R R>0 for j =1, 2,...,k with k > n 1 p R j D j f (t) 2 dt R < /23

3 Fourier Multipliers on L 2 (R d ): f ( )x = F 1 [f ( 2 )(Fx)( )] f ( ) 2B(L 2 (R d )) for all f 2 B b (R d ) on L p (R d ): spectral projections B( ) not bounded on L p (R d ) for all Borel sets B R + [0,r]( ) /2 B(L p (R d )ford 2 Hörmander s multiplier theorem: f ( ) bounded on L p (R d )if R 2R sup R R>0 for j =1, 2,...,k with k > n 1 p R j D j f (t) 2 dt R < /23

4 Fourier Multipliers on L 2 (R d ): f ( )x = F 1 [f ( 2 )(Fx)( )] f ( ) 2B(L 2 (R d )) for all f 2 B b (R d ) on L p (R d ): spectral projections B( ) not bounded on L p (R d ) for all Borel sets B R + [0,r]( ) /2 B(L p (R d )ford 2 Hörmander s multiplier theorem: f ( ) bounded on L p (R d )if R 2R sup R R>0 for j =1, 2,...,k with k > n 1 p R j D j f (t) 2 dt R < /23

5 Laplace type Operators Sub-Laplacians on Lie groups of polynomial growth Sub-Laplacians on compact Riemannian mainfolds Elliptic operators on domains Schrödinger operators with certain singular potentials Di erential operators of Hermite and Laguerre type Bessel operators Laplace operators on graphs and fractals 2/23

6 Spectral Multiplier Theorems in the L p -Scale (X,, µ) metricmeasurespace Doubling property: V (x, s) apple C( s r )d V (x, r), s r 0 A 0 selfadjoint on L 2 (X,µ) e ta has Gaussian bounds k t (x, y) apple [V (x, t 1 /m )V (y, t 1 /m )] 1/2 exp[ C( (x,y)m t ) 1 m 1 ] Then for > d /2 kf (A)k L p (X )!L p (X ). kf k H 2. Hörmander, Alexopoulos, Christ, Cowling, Duong, McIntosh, Mauceri, Meda, Ouhabaz, Hebisch, Hulanicki More general results: Couhlon, Duong, Ouhabaz, Sikora, Yan, Yao, Blunck, Kunstmann, Uhl 3/23

7 Hörmander classes H p f 2 C b (R + ), 1 apple p apple1, > 1 /p, 2 C 1 ( 1 4, 4) cut-o f 2W p i kf k W p = kf (exp( ))k W,p (R +) < 1 2 N: kf k W p P n=0 R 1 0 t n D n f (t) p dt t 1/p f 2H p i kf k H p =supk ( )f ( )k W,p (R +) < 1 >0 2 N: P kf k H p sup n=0 R>0 R 2R R tn D n f (t) p dt t 1/p H p H q H p +r, p > q, r > 1 q 1 p p = 1 Mihlin condition, p = 2 Hörmander condition 4/23

8 Hörmander classes H p f 2 C b (R + ), 1 apple p apple1, > 1 /p, 2 C 1 ( 1 4, 4) cut-o f 2W p i kf k W p = kf (exp( ))k W,p (R +) < 1 2 N: kf k W p P n=0 R 1 0 t n D n f (t) p dt t 1/p f 2H p i kf k H p =supk ( )f ( )k W,p (R +) < 1 >0 2 N: P kf k H p sup n=0 R>0 R 2R R tn D n f (t) p dt t 1/p H p H q H p +r, p > q, r > 1 q 1 p p = 1 Mihlin condition, p = 2 Hörmander condition 4/23

9 H 1 and Norm Estimates If A has a H 1 -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 ) A it : t 2 R} is bounded { u (A) : u > 0} is bounded, u (t) =(1 t u ) + 5/23

10 H 1 and Norm Estimates If A has a H 1 -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 ) A it : t 2 R} is bounded { u (A) : u > 0} is bounded, u (t) =(1 t u ) + 5/23

11 H 1 and Norm Estimates If A has a H 1 -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 ) A it : t 2 R} is bounded { u (A) : u > 0} is bounded, u (t) =(1 t u ) + 5/23

12 Spectral Multipliers and Inversion Formulas µ = A f (A) = 1 R 2 f ( )( A) 1 d Cauchy f (A) = R 1 0 L 1 [f ](t)e ta dt Laplace f (A) = 1 p 2 R 1 1 F[f ](t)eita dt f (A) = 1 R M[f ](t)ait dt Fourier Mellin f (A) = 1 R 1 ( ) 0 f ( ) (t)(t A) + 1 dt integration by parts Norm estimates on N(t) only give a W p -calculus. 6/23

13 Spectral Multipliers and Inversion Formulas µ = A f (A) = 1 R 2 f ( )( A) 1 d Cauchy f (A) = R 1 0 L 1 [f ](t)e ta dt Laplace f (A) = 1 p 2 R 1 1 F[f ](t)eita dt f (A) = 1 R M[f ](t)ait dt Fourier Mellin f (A) = 1 R 1 ( ) 0 f ( ) (t)(t A) + 1 dt integration by parts Norm estimates on N(t) only give a W p -calculus. 6/23

14 Spectral Multipliers and Inversion Formulas µ = A f (A) = 1 R 2 f ( )( A) 1 d Cauchy f (A) = R 1 0 L 1 [f ](t)e ta dt Laplace f (A) = 1 p 2 R 1 1 F[f ](t)eita dt f (A) = 1 R M[f ](t)ait dt Fourier Mellin f (A) = 1 R 1 ( ) 0 f ( ) (t)(t A) + 1 dt integration by parts Norm estimates on N(t) only give a W p -calculus. 6/23

15 Our Point of View A on a L p -scale A on a fixed Banach space X e.g. X = L p (R d, E) A selfadjoint on L 2 A is 0-sectorial and has a bounded H 1 ( )-calculus f (ta), t 2 R +,has kernel bounds f (ta), t 2 R + is R-bounded Goal: Characterize the H p -calculus in terms of R-bounds for families {f (ta) : t > 0}. 7/23

16 Our Point of View A on a L p -scale A on a fixed Banach space X e.g. X = L p (R d, E) A selfadjoint on L 2 A is 0-sectorial and has a bounded H 1 ( )-calculus f (ta), t 2 R +,has kernel bounds f (ta), t 2 R + is R-bounded Goal: Characterize the H p -calculus in terms of R-bounds for families {f (ta) : t > 0}. 7/23

17 H 1 -calculus A a sectorial operator on a Banach space X, f 2 H 1 ( ) f (A)x = 1 R 2 A has bounded H 1 ( )-calculus if f ( )R(, A)xd, x 2 D(A) \ R(A) kf (A)k applec sup 2 f ( ). Theorem: Let A be the generator of a bounded analytic semigroup on L p (U) for some p 2 (1, 1) s.th.e ta is positive and contractive for t > 0. Then A has a bounded H 1 ( )-calculus for some < 2. Stein, Cowling, Coifman-Weiss, Kalton-W. 8/23

18 H 1 -calculus A a sectorial operator on a Banach space X, f 2 H 1 ( ) f (A)x = 1 R 2 A has bounded H 1 ( )-calculus if f ( )R(, A)xd, x 2 D(A) \ R(A) kf (A)k applec sup 2 f ( ). Theorem: Let A be the generator of a bounded analytic semigroup on L p (U) for some p 2 (1, 1) s.th.e ta is positive and contractive for t > 0. Then A has a bounded H 1 ( )-calculus for some < 2. Stein, Cowling, Coifman-Weiss, Kalton-W. 8/23

19 H 1 -calculus and Spectral Multiplier Theorem A a 0-sectorial operator on a Banach space X with a H 1 ( )-calculus for some 2 (0, ). A has a H p -calculus if for f 2 H 1 ( ) kf (A)k appleckf R+ k H p Theorem: A has a H 1 -calculus ) kf (A)k apple C! kf k H 1 (!) for! & 0 ) A has a H 1 + -calculus >0 Cowling, Doust, McIntosh, Yagi 9/23

20 H 1 -calculus and Spectral Multiplier Theorem A a 0-sectorial operator on a Banach space X with a H 1 ( )-calculus for some 2 (0, ). A has a H p -calculus if for f 2 H 1 ( ) kf (A)k appleckf R+ k H p Theorem: A has a H 1 -calculus ) kf (A)k apple C! kf k H 1 (!) for! & 0 ) A has a H 1 + -calculus >0 Cowling, Doust, McIntosh, Yagi 9/23

21 R-boundedness X = L p (U), 1 apple p < 1, B(X ) (1) For all T 1,...,T n 2, x 1,...,x n 2 X np np k T j x j 2 1 /2k L p apple Ck x j 2 1 /2k L p j=1 j=1 Marcinkiewicz-Zygmund, ( j ) Bernoulli Random Variables P (2) Ek n P j T j x j kappleek n j x j k j=1 j=1 X Banach space, B(X ) R-bounded if 9 C < 1 such that (2) holds. R( ) :=infc. Bonami-Clerc 1985, Stempak: Marcinkiewicz Zygmund property If X = L p (U) thenr-boundedness follows from (generalized) Gaussian bounds for T 2 Fe erman-stein maximal functions Extrapolation via A p -weights, interpolation 10 / 23

22 R-boundedness X = L p (U), 1 apple p < 1, B(X ) (1) For all T 1,...,T n 2, x 1,...,x n 2 X np np k T j x j 2 1 /2k L p apple Ck x j 2 1 /2k L p j=1 j=1 Marcinkiewicz-Zygmund, ( j ) Bernoulli Random Variables P (2) Ek n P j T j x j kappleek n j x j k j=1 j=1 X Banach space, B(X ) R-bounded if 9 C < 1 such that (2) holds. R( ) :=infc. Bonami-Clerc 1985, Stempak: Marcinkiewicz Zygmund property If X = L p (U) thenr-boundedness follows from (generalized) Gaussian bounds for T 2 Fe erman-stein maximal functions Extrapolation via A p -weights, interpolation 10 / 23

23 R-boundedness X = L p (U), 1 apple p < 1, B(X ) (1) For all T 1,...,T n 2, x 1,...,x n 2 X np np k T j x j 2 1 /2k L p apple Ck x j 2 1 /2k L p j=1 j=1 Marcinkiewicz-Zygmund, ( j ) Bernoulli Random Variables P (2) Ek n P j T j x j kappleek n j x j k j=1 j=1 X Banach space, B(X ) R-bounded if 9 C < 1 such that (2) holds. R( ) :=infc. Bonami-Clerc 1985, Stempak: Marcinkiewicz Zygmund property If X = L p (U) thenr-boundedness follows from (generalized) Gaussian bounds for T 2 Fe erman-stein maximal functions Extrapolation via A p -weights, interpolation 10 / 23

24 R-boundedness and H 1 -calculus X has Pisier s property ( ), e.g. X L p (L q ), 1 apple p, q < 1 Theorem: If A has a bounded H 1 ( )-calculus on X then {f (A) : kf k H 1 ( ) apple 1} is R-bounded Kalton-W. Theorem: If X = L p (R d, E) anda = Then Id E with E a UMD-space. {f (A) : kf k H 2 apple 1}, > d /2 is R-bounded Girardi-W., scalar case: e.g. A p -extrapolation In this case the H 1 or H p -calculus is called R-bounded. 11 / 23

25 R-boundedness and H 1 -calculus X has Pisier s property ( ), e.g. X L p (L q ), 1 apple p, q < 1 Theorem: If A has a bounded H 1 ( )-calculus on X then {f (A) : kf k H 1 ( ) apple 1} is R-bounded Kalton-W. Theorem: If X = L p (R d, E) anda = Then Id E with E a UMD-space. {f (A) : kf k H 2 apple 1}, > d /2 is R-bounded Girardi-W., scalar case: e.g. A p -extrapolation In this case the H 1 or H p -calculus is called R-bounded. 11 / 23

26 Su cient Conditions for a H p -calculus A a 0-sectorial operator on a Banach space X, > 1 /2. A has a bounded H 1 ( )-calculus for some 2 (0, ), {e za : arg(z) = } apple C ( 2 ), bounded for! 2 {(1 + s A) e isa : s 2 R} bounded {(1 + t ) A it : t 2 R} bounded Theorem: Each of these conditions implies, A has a H r -calculus where > + 1 r and r 2 (1, 2) with 1 r > 1 1 typex cotypex. 12 / 23

27 Su cient Conditions for a H p -calculus A a 0-sectorial operator on a Banach space X, > 1 /2. A has a bounded H 1 ( )-calculus for some 2 (0, ), {e za : arg(z) = } apple C ( 2 ), bounded for! 2 {(1 + s A) e isa : s 2 R} bounded {(1 + t ) A it : t 2 R} bounded Theorem: Each of these conditions implies, A has a H r -calculus where > + 1 r and r 2 (1, 2) with 1 r > 1 1 typex cotypex. 12 / 23

28 Paley-Littlewood Theory for Ḣ,p (R d )=D(( ) /2 ) Ḃ p,q(r d )= D(( ) n /2 ), D(( ) m /2 ),q =(1 )n + m ' 2 C 1 ( 1 2, 2), ' n(t) :='(2 n t ), P n2z ' n (t) 1, supp ' n B 2 n+1 \ B 2 n 1 kxkḣ,p (R d ) = k kxkḃ p,q (Rd ) = P 2 n ˇ' n x 2 1 /2k L p (R d ) n2z P (2 n k ˇ' n xk L p) q 1 /q n2z Advantages: ˇ' n x analytic function with supp \ˇ' n x B 2 n+1 \ B 2 n 1 D ˇ' n x =( i) F 1 [' n (u)u ˆx(u)] 2 n ˇ' n x Bernstein s inequality Note: ˇ' n x = ' n (( ) 1 /2 )x 13 / 23

29 Paley-Littlewood Theory for Ḣ,p (R d )=D(( ) /2 ) Ḃ p,q(r d )= D(( ) n /2 ), D(( ) m /2 ),q =(1 )n + m ' 2 C 1 ( 1 2, 2), ' n(t) :='(2 n t ), P n2z ' n (t) 1, supp ' n B 2 n+1 \ B 2 n 1 kxkḣ,p (R d ) = k kxkḃ p,q (Rd ) = P 2 n ˇ' n x 2 1 /2k L p (R d ) n2z P (2 n k ˇ' n xk L p) q 1 /q n2z Advantages: ˇ' n x analytic function with supp \ˇ' n x B 2 n+1 \ B 2 n 1 D ˇ' n x =( i) F 1 [' n (u)u ˆx(u)] 2 n ˇ' n x Bernstein s inequality Note: ˇ' n x = ' n (( ) 1 /2 )x 13 / 23

30 Paley-Littlewood Decomposition for A A 0-sectorial on a uniformly convex Banach space X. X =(D(A ), ka k) ' 2H 1, ' n (t) ='(2 n P t), ' n 1 on R + R n2z 2H 1 1, t (t) 2 dt 0 t < 1, sup t k (k) (t). min(t, t ) kapple +1 Theorem: Let A have a H 1 -calculus for some <.Then (a) kxk X P = k 2 n ' n (A)x 2 1 /2k L p for X L p (U) n2z = Ek P n2z n 2 n ' n (A)xk X for general X R (b) kxk X 1 = k t (ta)x 2 dt 0 t = kt (ta)xk (R+, dt t,x ) 1/2 k L p for X L p (U) for general X Remarks: X complex interpolation scale If A has a R-bounded W 1 -calculus and (a) holds then A has a H 1 -calculus Similar results for inhomogeneous scale D((1 + A) ). 14 / 23

31 Paley-Littlewood Decomposition for A A 0-sectorial on a uniformly convex Banach space X. X =(D(A ), ka k) ' 2H 1, ' n (t) ='(2 n P t), ' n 1 on R + R n2z 2H 1 1, t (t) 2 dt 0 t < 1, sup t k (k) (t). min(t, t ) kapple +1 Theorem: Let A have a H 1 -calculus for some <.Then (a) kxk X P = k 2 n ' n (A)x 2 1 /2k L p for X L p (U) n2z = Ek P n2z n 2 n ' n (A)xk X for general X R (b) kxk X 1 = k t (ta)x 2 dt 0 t = kt (ta)xk (R+, dt t,x ) 1/2 k L p for X L p (U) for general X Remarks: X complex interpolation scale If A has a R-bounded W 1 -calculus and (a) holds then A has a H 1 -calculus Similar results for inhomogeneous scale D((1 + A) ). 14 / 23

32 Paley-Littlewood Decomposition for A A 0-sectorial on a uniformly convex Banach space X. X =(D(A ), ka k) ' 2H 1, ' n (t) ='(2 n P t), ' n 1 on R + R n2z 2H 1 1, t (t) 2 dt 0 t < 1, sup t k (k) (t). min(t, t ) kapple +1 Theorem: Let A have a H 1 -calculus for some <.Then (a) kxk X P = k 2 n ' n (A)x 2 1 /2k L p for X L p (U) n2z = Ek P n2z n 2 n ' n (A)xk X for general X R (b) kxk X 1 = k t (ta)x 2 dt 0 t = kt (ta)xk (R+, dt t,x ) 1/2 k L p for X L p (U) for general X Remarks: X complex interpolation scale If A has a R-bounded W 1 -calculus and (a) holds then A has a H 1 -calculus Similar results for inhomogeneous scale D((1 + A) ). 14 / 23

33 Paley-Littlewood Decomposition for A A 0-sectorial on a uniformly convex Banach space X. X =(D(A ), ka k) ' 2H 1, ' n (t) ='(2 n P t), ' n 1 on R + R n2z 2H 1 1, t (t) 2 dt 0 t < 1, sup t k (k) (t). min(t, t ) kapple +1 Theorem: Let A have a H 1 -calculus for some <.Then (a) kxk X P = k 2 n ' n (A)x 2 1 /2k L p for X L p (U) n2z = Ek P n2z n 2 n ' n (A)xk X for general X R (b) kxk X 1 = k t (ta)x 2 dt 0 t = kt (ta)xk (R+, dt t,x ) 1/2 k L p for X L p (U) for general X Remarks: X complex interpolation scale If A has a R-bounded W 1 -calculus and (a) holds then A has a H 1 -calculus Similar results for inhomogeneous scale D((1 + A) ). 14 / 23

34 Besov-Type Scale A, ' and as before Ḃ q =(X 0, X 1 ) #,q, =(1 #) 0 + # 1 Theorem: Let A have W 1 -calculus <.Then (a) kxkḃ q P n2z 2 n q k' n (A)xk q X 1/q (b) kxkḃ R 1 q 0 t q k (ta)xk q dt t 1/q (c) A has H 1 -calculus on all Ḃ q Remark: Similar result for the inhomogeneous case Liegroups: Furioli, Melzi, Veneruso, Liu, Ma Schrödinger Operators: Olafsson, Zheng 15 / 23

35 Besov-Type Scale A, ' and as before Ḃ q =(X 0, X 1 ) #,q, =(1 #) 0 + # 1 Theorem: Let A have W 1 -calculus <.Then (a) kxkḃ q P n2z 2 n q k' n (A)xk q X 1/q (b) kxkḃ R 1 q 0 t q k (ta)xk q dt t 1/q (c) A has H 1 -calculus on all Ḃ q Remark: Similar result for the inhomogeneous case Liegroups: Furioli, Melzi, Veneruso, Liu, Ma Schrödinger Operators: Olafsson, Zheng 15 / 23

36 Characterization of the H 1 -calculus Let A have a W 1 -calculus, >1, on a Banach space with property ( ). For > 1definetheBochnerRieszmeans u (A) with u (t) =(1 t u ) + Theorem: In addition, let A have a bounded H 1 -calculus. Assume = { u (A) : u > 0} is R-bounded. Then A has a H -calculus 1 for > + 1. Conversely a H -calculus 1 for A implies the R-boundedness of for > 1. Special cases: Bonami, Clerc, Stempak W 1 -Calculus: Galé, Pytlik 16 / 23

37 Characterization of the H 1 -calculus Let A have a W 1 -calculus, >1, on a Banach space with property ( ). For > 1definetheBochnerRieszmeans u (A) with u (t) =(1 t u ) + Theorem: In addition, let A have a bounded H 1 -calculus. Assume = { u (A) : u > 0} is R-bounded. Then A has a H -calculus 1 for > + 1. Conversely a H -calculus 1 for A implies the R-boundedness of for > 1. Special cases: Bonami, Clerc, Stempak W 1 -Calculus: Galé, Pytlik 16 / 23

38 Characterization of H 2 : R-bounds A strongly continuous function N : t 2 R!B(X )isr 2 -bounded if the following set is R-bounded A(N) ={ R RR f (t)n(t)dt : kf k L 2 (R) apple 1, R > 0} Theorem: Let A be 0-sectorial with a H 1 ( ) calculus for some 2 (0, ) on a Banach space with property ( ) and > 1 /2. Then A has bounded H 2 -calculus i one (all) of the following functions are R 2 -bounded t 2 R +! ( 2 ) A 1 /2 T (e i t), uniformly for! 2 t 2 R +! A 1 /2 R(e i t, A), uniformly for! 0 t 2 R! (1 + t ) A it t 2 R! t A e ita 1 m m > 1 2 (However,! + for >0 in some implications) Then {f (A) : kf k H 2 apple 1} is R-bounded. 17 / 23

39 Characterization of H 2 :Squarefunctions Let X L p (U), 1 < p < 1, > 1 /2. Theorem: Let A be 0-sectorial with a bounded H 1 -calculus. Then A has a matricially bounded H 2 -calculus i k R N(tA)x 2 dt 1 /2k L p apple C(N)kxk L p where N(t) is one of the functions N (t) =A 1 /2 T (e i t), C(N ). ( 2 ) for! 2 N (t) =A 1 /2 R(e i t, A), C(N ). for!0. N(t) =(1+ t ) A it, t 2 R N(t) = t A e ita 1 m m > 1 2 in some implications we need! +, >0 18 / 23

40 Characterization of H 2 :Squarefunctions Let X L p (U), 1 < p < 1, > 1 /2. Theorem: Let A be 0-sectorial with a bounded H 1 -calculus. Then A has a matricially bounded H 2 -calculus i k R N(tA)x 2 dt 1 /2k L p apple C(N)kxk L p where N(t) is one of the functions N (t) =A 1 /2 T (e i t), C(N ). ( 2 ) for! 2 N (t) =A 1 /2 R(e i t, A), C(N ). for!0. N(t) =(1+ t ) A it, t 2 R N(t) = t A e ita 1 m m > 1 2 in some implications we need! +, >0 18 / 23

41 Square Functions in Banach space Let (h n ) be a ONB of L 2 (J) and( n ) a sequence of i.i.d. standard Gaussian random variables For N : J! X put If X = L p (U) then y n = R J N(t)x h n(t) dt 2 X k R J N(t)x 2 dt 1 /2k L p (U) = knxk L p (U,L 2 (J)) = k(y n )k L p (U,l 2 ) = k P n y n 2 1 /2k L p (U) Def kn(t)xk (J,X ) := Ek P n y n k X Alternatively, kn(t)xk (J,X ) = Ek R J N(t)x d (t)k X k k (J,X ) has (almost) the same operational properties on X as in the classical square functions in L p. 19 / 23

42 Steps in the Proof Use the assumed bounds on e za, R(z, A) ora it to establish a R-bounded W p -calculus Hytönen-Veraar: X abanachspace.let 1 r > 1 typex N(t) :(a, b)!b(x) a strongly continuous function with 1 cotypex and Then is R-bounded. R b a kn(t)kr B(X ) dt < 1. { R b a h(t)n(t)dt : khk Lr0 apple 1} If A has a H 1 ( )-calculus for some 2 (0, )andanr-bounded W -calculus, p then A has a H p -calculus for some >. 20 / 23

43 More Steps Use the Paley-Littlewood decomposition and a localization principle to get the right H p -calculus. If A has a H p -calculus for some (large) > 1 p and {f (2 n A) : f 2 C 1 c ( 1 2, 2), kf k W p is R-bounded. Then A has a H p -calculus. apple 1, n 2 Z} R 2 -boundedness is weaker than square function estimates. Le Merdy: Let N : J! X be strongly continuous and Then is R-bounded. kn( )xk (J,X ) apple Ckxk. { R J f (t)n(t)dt : kf k L2 apple 1} 21 / 23

44 Generalized Gaussian Bounds (X,, µ) metric measure space with doubling property A 0 selfadjoint on L 2 (X,µ) V (x, s) apple C( s r )d V (x, r), s r 0 k1 B(x,t 1/m ) e ta 1 B(y,t 1/m ) k p 0!p 0 0 apple CV (x, t 1 /m ) ( 1/p 0 1/p 0 0 ) exp[ b( (x,y) ) m t 1 m 1 /m ] for some p 0 2 [1, 2], m 2 Then A has a H -calculus on L p (X,µ)forp 2 (p 0, p0 0 ), >d 1 1 p Blunk, Duong, Ouhabaz, Sikora, Kunstmann, Uhl 22 / 23

45 Generalized Gaussian Bounds (X,, µ) metric measure space with doubling property A 0 selfadjoint on L 2 (X,µ) V (x, s) apple C( s r )d V (x, r), s r 0 k1 B(x,t 1/m ) e ta 1 B(y,t 1/m ) k p 0!p 0 0 apple CV (x, t 1 /m ) ( 1/p 0 1/p 0 0 ) exp[ b( (x,y) ) m t 1 m 1 /m ] for some p 0 2 [1, 2], m 2 Then A has a H -calculus on L p (X,µ)forp 2 (p 0, p0 0 ), >d 1 1 p Blunk, Duong, Ouhabaz, Sikora, Kunstmann, Uhl 22 / 23

46 Example: Maxwell Operators R 3 bounded with Lipschitz boundary P : L 2 (, C 3 )! L 2 ( ) Helmholtz Projection A defined on L 2 (, C 3 )bytheform a(u, v) = R ( ) 1 rotu rotvdx + R (divu)(divv)dx D(a) ={u 2 L 2 ( ) : divu 2 L 2 ( ), rotu 2 L 2 ( ), =0} M defined on L 2 ( ) by Mu = Au for u 2 D(M) =P(D(A)) Kunstamnn, Uhl: p 2 ( 3 /2, 3), >3 1 p A p, M p have a H -calculus on L p ( ), L p ( ), resp. 23 / 23

47 Example: Maxwell Operators R 3 bounded with Lipschitz boundary P : L 2 (, C 3 )! L 2 ( ) Helmholtz Projection A defined on L 2 (, C 3 )bytheform a(u, v) = R ( ) 1 rotu rotvdx + R (divu)(divv)dx D(a) ={u 2 L 2 ( ) : divu 2 L 2 ( ), rotu 2 L 2 ( ), =0} M defined on L 2 ( ) by Mu = Au for u 2 D(M) =P(D(A)) Kunstamnn, Uhl: p 2 ( 3 /2, 3), >3 1 p A p, M p have a H -calculus on L p ( ), L p ( ), resp. 23 / 23