Teoría de pesos para operadores multilineales, extensiones vectoriales y extrapolación

Transcription

1 Teoría de pesos para operadores multilineales, extensiones vectoriales y extrapolación Encuentro Análisis Funcional Sheldy Ombrosi Universidad Nacional del Sur y CONICET 8 de marzo de 208

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3 If f is a locally integrable function we define the Hardy-Littlewood maximal operator as

4 If f is a locally integrable function we define the Hardy-Littlewood maximal operator as Mf (x) = sup f (y) dy. Q x Q Q

5 If f is a locally integrable function we define the Hardy-Littlewood maximal operator as Mf (x) = sup f (y) dy. Q x Q Q Theorem (B. Muckenhoupt, 972)

6 If f is a locally integrable function we define the Hardy-Littlewood maximal operator as Mf (x) = sup f (y) dy. Q x Q Q Theorem (B. Muckenhoupt, 972) Let < p < then

7 If f is a locally integrable function we define the Hardy-Littlewood maximal operator as Mf (x) = sup f (y) dy. Q x Q Q Theorem (B. Muckenhoupt, 972) Let < p < then (Mf ) p w(x)dx C p,w f p w(x)dx

8 If f is a locally integrable function we define the Hardy-Littlewood maximal operator as Mf (x) = sup f (y) dy. Q x Q Q Theorem (B. Muckenhoupt, 972) Let < p < then (Mf ) p w(x)dx C p,w f p w(x)dx if and only if ( ) ( p sup Q w(x)dx w(x) dx) /(p ) <. Q Q Q Q

9 If f is a locally integrable function we define the Hardy-Littlewood maximal operator as Mf (x) = sup f (y) dy. Q x Q Q Theorem (B. Muckenhoupt, 972) Let < p < then (Mf ) p w(x)dx C p,w f p w(x)dx if and only if ( ) ( p sup Q w(x)dx w(x) dx) /(p ) <. Q Q Q Q if the last inequality holds we say that w belongs to A p. If Mw w L we say w A

10 The Hilbert transform: Hf (x) = π lim ε 0 x y >ε x y f (y)dy. Hunt, Muckenhoupt and Wheeden (973) proved that ( < p < )

11 The Hilbert transform: Hf (x) = π lim ε 0 x y >ε x y f (y)dy. Hunt, Muckenhoupt and Wheeden (973) proved that ( < p < ) Hf L p (w) C f L p (w)

12 The Hilbert transform: Hf (x) = π lim ε 0 x y >ε x y f (y)dy. Hunt, Muckenhoupt and Wheeden (973) proved that ( < p < ) if and only if w A p Hf L p (w) C f L p (w)

13 Definition Calderón-Zygmund operators A Calderón-Zygmund operator T (CZO) is an operator bounded on L 2 (R n ) that admits the following representation Tf (x) = K(x, y)f (y)dy with f Cc (R n ) and x supp f and where K : R n R n \ {(x, x) : x R n } R has the following properties Size condition: K(x, y) C 2 x y x 0. n Smoothness condition (Hölder-Lipschitz): K(x, y) K(x, z) C K(x, y) K(z, y) C y z δ x y n+δ 2 x z δ x y n+δ 2 x y > y z x y > x z where C > 0 and C 2 > 0 are constants independent of x, y, z.

14 R. Coifman y C. Fefferman (974) proved that ( < p < ) if w A p then any CZO operator T safisfies

15 R. Coifman y C. Fefferman (974) proved that ( < p < ) if w A p then any CZO operator T safisfies Tf L p (w) C T,w f L p (w)

16 R. Coifman y C. Fefferman (974) proved that ( < p < ) if w A p then any CZO operator T safisfies Tf L p (w) C T,w f L p (w) If T is the Riesz transform then A p is also a necessary condition for the L p (w) boundedness.

17 R. Coifman y C. Fefferman (974) proved that ( < p < ) if w A p then any CZO operator T safisfies Tf L p (w) C T,w f L p (w) If T is the Riesz transform then A p is also a necessary condition for the L p (w) boundedness. Coifman-Fefferman estimate, if 0 < r < and v is a good weight, then Tf r v(x)dx C p,v Mf r v(x)dx

18 Rubio de Francia s extrapolation theorem Theorem (Rubio de Francia, 984) Fixed p 0 <, if T is a bounded operator on L p 0 (w) for every w A p0. Then for every < p < and for all w A p ; T is bounded on L p (w).

19 Rubio de Francia s extrapolation theorem Theorem (Rubio de Francia, 984) Fixed p 0 <, if T is a bounded operator on L p 0 (w) for every w A p0. Then for every < p < and for all w A p ; T is bounded on L p (w). New proofs, variants and very useful extensions have been widely studied by J. García-Cuerva, J. L. Torrea, J. Duoandikoetxea, D. Cruz-Uribe, J. M. Martell, and C. Pérez between others...

20 Rubio de Francia s extrapolation theorem Theorem (Rubio de Francia, 984) Fixed p 0 <, if T is a bounded operator on L p 0 (w) for every w A p0. Then for every < p < and for all w A p ; T is bounded on L p (w). New proofs, variants and very useful extensions have been widely studied by J. García-Cuerva, J. L. Torrea, J. Duoandikoetxea, D. Cruz-Uribe, J. M. Martell, and C. Pérez between others... - It can be consider a pair of functions (f, g), where, in particular, g could be Tf...

21 Rubio de Francia extrapolation trick f L p (w) = sup { g L p (σ) =} f g

22 Rubio de Francia extrapolation trick f L p (w) = sup { g L p (σ) =} f g Given g 0 with g L p (σ) =, define Rg(x) = k=0 M k g(x) (2 M L 2 (σ)) k.

23 Rubio de Francia extrapolation trick f L p (w) = sup { g L p (σ) =} f g Given g 0 with g L p (σ) =, define Rg(x) = k=0 M k g(x) (2 M L 2 (σ)) k. Then g Rg, Rg L 2 (σ) 2, and Rg A.

24 Rubio de Francia extrapolation trick f L p (w) = sup { g L p (σ) =} f g Given g 0 with g L p (σ) =, define Rg(x) = k=0 M k g(x) (2 M L 2 (σ)) k. Then g Rg, Rg L 2 (σ) 2, and Rg A. Tf g Tf Rg C f L (Rg) C f L p (w) Rg L p (σ) 2C f L p (w).

25 Bilinear Calderón-Zygmund operators let T : S(R n ) S(R n ) S (R n ). T is an bilinear Calderón-Zygmund operator if, for some q, q 2 < and 2 q < satisfying q = q + q 2, it extends to a bounded bilinear operator from L q L q2 to L q, and if there exists K defined off the diagonal x = y = y 2 in (R n ) 3 satisfying

26 Bilinear Calderón-Zygmund operators let T : S(R n ) S(R n ) S (R n ). T is an bilinear Calderón-Zygmund operator if, for some q, q 2 < and 2 q < satisfying q = q + q 2, it extends to a bounded bilinear operator from L q L q2 to L q, and if there exists K defined off the diagonal x = y = y 2 in (R n ) 3 satisfying T (f, f 2 )(x) = K(x, y, y 2 )f (y )f 2 (y 2 ) dy dy 2 (R n ) 2 for all x / 2 j= suppf j;

27 Bilinear Calderón-Zygmund operators let T : S(R n ) S(R n ) S (R n ). T is an bilinear Calderón-Zygmund operator if, for some q, q 2 < and 2 q < satisfying q = q + q 2, it extends to a bounded bilinear operator from L q L q2 to L q, and if there exists K defined off the diagonal x = y = y 2 in (R n ) 3 satisfying T (f, f 2 )(x) = K(x, y, y 2 )f (y )f 2 (y 2 ) dy dy 2 (R n ) 2 for all x / 2 j= suppf j; A K(y 0, y, y 2 ) ( 2 ) 2n ; y k y l k,l=0

28 Bilinear Calderón-Zygmund operators let T : S(R n ) S(R n ) S (R n ). T is an bilinear Calderón-Zygmund operator if, for some q, q 2 < and 2 q < satisfying q = q + q 2, it extends to a bounded bilinear operator from L q L q2 to L q, and if there exists K defined off the diagonal x = y = y 2 in (R n ) 3 satisfying T (f, f 2 )(x) = K(x, y, y 2 )f (y )f 2 (y 2 ) dy dy 2 (R n ) 2 for all x / 2 j= suppf j; A K(y 0, y, y 2 ) ( 2 ) 2n ; y k y l k,l=0 K(y 0, y, y 2 ) K(y 0, y, A y y y 2 ) ( ɛ 2 ) 2n+ɛ, k,l=0 y k y l for some ɛ > 0 and all 0 j m, whenever y y 2 max 0 k 2 y j y k.

29 -Multilinear Calderón-Zygmund were widely studied by R. Coifman and Y. Meyer in the 70 th and 80 th

30 -Multilinear Calderón-Zygmund were widely studied by R. Coifman and Y. Meyer in the 70 th and 80 th -L. Grafakos and R. Torres in several works (since 2000)

31 -Multilinear Calderón-Zygmund were widely studied by R. Coifman and Y. Meyer in the 70 th and 80 th -L. Grafakos and R. Torres in several works (since 2000) If we denote w = (w, w 2 ); ν w = w q/q w q/q 2 2 and then if < q i and w i A qi i =, 2, then bilinear Calderón-Zygmund operator T maps

32 -Multilinear Calderón-Zygmund were widely studied by R. Coifman and Y. Meyer in the 70 th and 80 th -L. Grafakos and R. Torres in several works (since 2000) If we denote w = (w, w 2 ); ν w = w q/q w q/q 2 2 and then if < q i and w i A qi i =, 2, then bilinear Calderón-Zygmund operator T maps L q (w ) L q 2 (w 2 ) L q (ν w )

33 -Multilinear Calderón-Zygmund were widely studied by R. Coifman and Y. Meyer in the 70 th and 80 th -L. Grafakos and R. Torres in several works (since 2000) If we denote w = (w, w 2 ); ν w = w q/q w q/q 2 2 and then if < q i and w i A qi i =, 2, then bilinear Calderón-Zygmund operator T maps L q (w ) L q 2 (w 2 ) L q (ν w ) As a consequence of a control of the way... T (f, f 2 ) Mf Mf 2

34 Theorem (Grafakos and Martell, 2004) Let < r, r 2 < and r = r + r 2. Assume that T (f, f 2 ) L r (ν w ) C holds for all (w, w 2 ) (A r, A r2 ). Then T (f, f 2 ) L p (ν w ) C 2 f i L r i (wi ) i= 2 f i L p i (wi ) holds for all (w, w 2 ) (A p, A p2 ) with < p, p 2 < and p = p + p 2. i=

35 In 2009 jointly with Lerner, Pérez, Torres and Trujillo-Gonzalez [LOPTT] introduced M(f, f 2 )(x) = sup x Q 2 i= f i (y i ) dy i. Q Q

36 In 2009 jointly with Lerner, Pérez, Torres and Trujillo-Gonzalez [LOPTT] introduced M(f, f 2 )(x) = sup x Q 2 i= T (f, f 2 ) M(f, f 2 ) f i (y i ) dy i. Q Q

37 Multilinear Muckenhoupt weights Let ν w = w q/q w q/q 2 2. Let q, q 2 < and q such that q = q + q 2. We say that w = (w, w 2 ) satisfies the multilinear A q condition if

38 Multilinear Muckenhoupt weights Let ν w = w q/q w q/q 2 2. Let q, q 2 < and q such that q = q + q 2. We say that w = (w, w 2 ) satisfies the multilinear A q condition if ( ) /q sup ν w Q Q Q 2 i= ( Q Q ) /q w q i i i <

39 Multilinear Muckenhoupt weights Let ν w = w q/q w q/q 2 2. Let q, q 2 < and q such that q = q + q 2. We say that w = (w, w 2 ) satisfies the multilinear A q condition if when q i =, ( ) /q sup ν w Q Q Q 2 i= ( Q Q ) /q w q i i i < ( ) /q Q Q w q i i i is understood as (inf Q w i ).

40 Multilinear Muckenhoupt weights Let ν w = w q/q w q/q 2 2. Let q, q 2 < and q such that q = q + q 2. We say that w = (w, w 2 ) satisfies the multilinear A q condition if when q i =, ( ) /q sup ν w Q Q Q 2 i= ( Q Q ) /q w q i i i < ( ) /q Q Q w q i i i is understood as (inf Q w i ). Theorem (LOPTT, 2009) Let < q, q 2 < and q such that q = q + q 2 then w satisfies A q condition if and only if M maps L q (w ) L q 2 (w 2 ) into L q (ν w )

41 Multilinear Muckenhoupt weights Let ν w = w q/q w q/q 2 2. Let q, q 2 < and q such that q = q + q 2. We say that w = (w, w 2 ) satisfies the multilinear A q condition if when q i =, ( ) /q sup ν w Q Q Q 2 i= ( Q Q ) /q w q i i i < ( ) /q Q Q w q i i i is understood as (inf Q w i ). Theorem (LOPTT, 2009) Let < q, q 2 < and q such that q = q + q 2 then w satisfies A q condition if and only if M maps L q (w ) L q 2 (w 2 ) into L q (ν w ) Then, if w satisfies A q a bilinear Calderón-Zygmund operator T also maps L q (w ) L q 2 (w 2 ) into L q (ν w )

42 Some remarks on multiple A p weights -From the characterization, one can see that A q A q2 A q.

43 Some remarks on multiple A p weights -From the characterization, one can see that A q A q2 A q. It is easy to check that ( x n, ) A (,), however, x n is not even locally integrable, so of course x n / A q for any q.

44 Some remarks on multiple A p weights -From the characterization, one can see that A q A q2 A q. It is easy to check that ( x n, ) A (,), however, x n is not even locally integrable, so of course x n / A q for any q. -Moreover other general properties as monotonicity and (reasonable) factorization are not true for the clases A q.

45 Some remarks on multiple A p weights -From the characterization, one can see that A q A q2 A q. It is easy to check that ( x n, ) A (,), however, x n is not even locally integrable, so of course x n / A q for any q. -Moreover other general properties as monotonicity and (reasonable) factorization are not true for the clases A q. -All these facts kept open the extrapolation theorem related to multiple A q weights...

46 Extrapolation for multiple A p weights Theorem (K. Li, J. M. Martell, O., 208) Let F be a collection of 3-tuples of non-negative functions. Let p = (p, p 2 ), with p, p 2 <, such that given any w A p the inequality 2 f L p (w) C([ w] A p ) f i L p i (wi ) holds for every (f, f, f 2 ) F, where p := p + p 2 and w := 2 i= p i= w p i i.

47 Extrapolation for multiple A p weights Theorem (K. Li, J. M. Martell, O., 208) Let F be a collection of 3-tuples of non-negative functions. Let p = (p, p 2 ), with p, p 2 <, such that given any w A p the inequality 2 f L p (w) C([ w] A p ) f i L p i (wi ) holds for every (f, f, f 2 ) F, where p := p + p 2 and w := 2 Then for all exponents q = (q, q 2 ), with q i >, i =, 2, and for all weights v A q the inequality f L q (v) C([ v] A q ) i= 2 f i L q i (vi ) i= holds for every (f, f, f 2 ) F, q := q + q 2 and v := 2 q i= v q i i. p i= w p i i.

48 Moreover, for the same family of exponents and weights, and for all exponents s = (s, s 2 ) with s i >, i =, 2, ( (f j ) s) s C([ v] A q ) L q (v) j 2 ( i= for all {(f j, f j, f j 2 )} j F, where s := s + s 2. j (f j i )s i ) s i L qi (vi )

49 How can we build A (,) weights? New starting point...

50 New starting point... How can we build A (,) weights? It is known (w, w 2 ) A (,) if and only if w 2 A w 2 2 A and w 2 w 2 2 A.

51 New starting point... How can we build A (,) weights? It is known (w, w 2 ) A (,) if and only if w 2 A w 2 2 A and w 2 w 2 2 A. However, it is very useful the following: (w, w 2 ) A (,) if and only

52 New starting point... How can we build A (,) weights? It is known (w, w 2 ) A (,) if and only if w 2 A w 2 2 A and w 2 w 2 2 A. However, it is very useful the following: (w, w 2 ) A (,) if and only w 2 A and w 2 2 A (w 2 ).

53 A (very) rough idea of the proof

54 A (very) rough idea of the proof -Bearing in mind that, we can follow Duoandikoetxea s off-diagonal extrapolation theorem...

55 A (very) rough idea of the proof -Bearing in mind that, we can follow Duoandikoetxea s off-diagonal extrapolation theorem... We study the extrapolation from (p, p 2 ) to (q, q 2 ) by a two-step consideration: first (p, p 2 ) to (p, q 2 ) and then to (q, q 2 ).

56 A (very) rough idea of the proof -Bearing in mind that, we can follow Duoandikoetxea s off-diagonal extrapolation theorem... We study the extrapolation from (p, p 2 ) to (q, q 2 ) by a two-step consideration: first (p, p 2 ) to (p, q 2 ) and then to (q, q 2 ). We can actually rewrite g L p (w) C f L p (w ) f 2 L p 2 (w2 ) as g p L p p (w 2 2 ) p C f L p 2 (w2 ), where g = gw and f = f L p (w )f 2.

57 A (very) rough idea of the proof -Bearing in mind that, we can follow Duoandikoetxea s off-diagonal extrapolation theorem... We study the extrapolation from (p, p 2 ) to (q, q 2 ) by a two-step consideration: first (p, p 2 ) to (p, q 2 ) and then to (q, q 2 ). We can actually rewrite g L p (w) C f L p (w ) f 2 L p 2 (w2 ) as g p L p p (w 2 2 ) p C f L p 2 (w2 ), where g = gw and f = f L p (w )f 2. Since p and w are fixed, we can seek for some characterization of w 2 when assuming w A (p,p 2 )...

58 bilinear Marcinkiewicz-Zygmund inequalities Theorem (D. Carando, M. Mazzitelli, S.O., 206) Let T be a bilinear Calderón-Zygmund operator. Let < r 2 and let < q, q 2 < if r = 2 or < q, q 2 < r if < r < 2. Then for w = (w, w 2 ) A q there holds ( ) ( T (f i, g j ) r r ) C f i r r L q (w) i,j i L q (w ) ( ) g j r r L q 2 (w2 ), j where q = q + q 2 and w = w q q q w q 2 2.

59 bilinear Marcinkiewicz-Zygmund inequalities Theorem (D. Carando, M. Mazzitelli, S.O., 206) Let T be a bilinear Calderón-Zygmund operator. Let < r 2 and let < q, q 2 < if r = 2 or < q, q 2 < r if < r < 2. Then for w = (w, w 2 ) A q there holds ( ) ( T (f i, g j ) r r ) C f i r r L q (w) i,j i L q (w ) ( ) g j r r L q 2 (w2 ), j where q = q + q 2 and w = w q q q w q 2 2. Now, by using extrapolation we can remove the restriction q, q 2 < r

60 bilinear Marcinkiewicz-Zygmund inequalities Theorem (D. Carando, M. Mazzitelli, S.O., 206) Let T be a bilinear Calderón-Zygmund operator. Let < r 2 and let < q, q 2 < if r = 2 or < q, q 2 < r if < r < 2. Then for w = (w, w 2 ) A q there holds ( ) ( T (f i, g j ) r r ) C f i r r L q (w) i,j i L q (w ) ( ) g j r r L q 2 (w2 ), j where q = q + q 2 and w = w q q q w q 2 2. Now, by using extrapolation we can remove the restriction q, q 2 < r Corollary Let T be bilinear Calderón-Zygmund operator. Given < r 2 and < q, q 2 <, then previous inequality holds for all w = (w, w 2 ) A q.

61 - the results holds in the multilinear case (proofs a little more complicated...)

62 - the results holds in the multilinear case (proofs a little more complicated...) -Actually, the results also hold in the context of the classes A p, r (good weights for more singular operators as the bilinear Hilbert transform and commutators of the the bilinear Hilbert transform) f (x t)g(x + t) BH(f, g)(x) = p.v. dt t

63 - the results holds in the multilinear case (proofs a little more complicated...) -Actually, the results also hold in the context of the classes A p, r (good weights for more singular operators as the bilinear Hilbert transform and commutators of the the bilinear Hilbert transform) f (x t)g(x + t) BH(f, g)(x) = p.v. dt t -From A. Culiuc, F. Di Plinio and Y. Ou (206) we can go to the quasi-banach range and to recover several recent results of Benea-Muscalu.

64 Muchas gracias!

65 Muchas gracias! al Barcelona por tan buen futbol :) :)...

66 Muchas gracias! al Barcelona por tan buen futbol :) :)...

67 Muchas gracias! al Barcelona por tan buen futbol :) :)... y al Madrid también!!!