Weighted norm inequalities and Rubio de Francia extrapolation Jos - PowerPoint PPT Presentation

Weighted norm inequalities and Rubio de Francia extrapolation Jos´ e Mar´ ıa Martell Instituto de Ciencias Matem´ aticas CSIC-UAM-UC3M-UCM Spain Spring school in harmonic analysis and PDE 2008 Helsinki University of Technology June 2–6, 2008

Outline of Part I Muckenhoupt Weights Introduction Weak-type and Properties Strong-type and Extrapolation Other maximal operators Weighted norm inequalities Calder´ on-Zygmund Theory Coifman’s Inequality J.M. Martell (CSIC) Weights and Rubio de Francia extrapolation Helsinki, June 2008 2 / 91

Outline of Part II Extrapolation on Lebesgue spaces Rubio de Francia Extrapolation Extensions of the Rubio de Francia Extrapolation Consequences Extrapolation on Function Spaces Introduction Extrapolation on Banach Function Spaces Extrapolation on Modular Spaces J.M. Martell (CSIC) Weights and Rubio de Francia extrapolation Helsinki, June 2008 3 / 91

Outline of Part III Extrapolation for A ∞ weights Introduction Extrapolation for A ∞ weights Extrapolation for A ∞ weights on function spaces Applications Coifman’s Inequality: Extensions of Boyd and Lorentz-Shimogaki Commutators with CZO Further results Variable L p spaces Sawyer’s conjecture J.M. Martell (CSIC) Weights and Rubio de Francia extrapolation Helsinki, June 2008 4 / 91

Muckenhoupt Weights Weighted norm inequalities Part I Muckenhoupt Weights and Weighted Norm Inequalities J.M. Martell (CSIC) Weights and Rubio de Francia extrapolation Helsinki, June 2008 5 / 91

References: Weighted norm inequalities J. Duoandikoetexea, Fourier analysis , Graduate Studies in Mathematics 29, American Mathematical Society, Providence, RI, 2001. J. Garc´ ıa-Cuerva & J.L. Rubio de Francia, Weighted norm inequalities and related topics , North-Holland Mathematics Studies 116, North-Holland Publishing Co., Amsterdam, 1985. L. Grafakos, Classical and Modern Fourier Analysis , Pearson Education, Inc., Upper Saddle River, 2004.

Muckenhoupt Weights Weighted norm inequalities Introduction J.M. Martell (CSIC) Weights and Rubio de Francia extrapolation Helsinki, June 2008 7 / 91

Muckenhoupt Weights Weighted norm inequalities Weights and Extrapolation How much information is contained in the following inequalities? � � R n | Tf ( x ) | 2 w ( x ) dx � R n | f ( x ) | 2 w ( x ) dx , ∀ w ∈ A 2 1 � � R n | Tf ( x ) | p 0 w ( x ) dx � R n | f ( x ) | p 0 w ( x ) dx , ∀ w ∈ A p 0 2 (1 < p 0 < ∞ is fixed) � � R n | Tf ( x ) | 2 w ( x ) dx � R n | Sf ( x ) | 2 w ( x ) dx , ∀ w ∈ A ∞ 3 � � R n | Tf ( x ) | p 0 w ( x ) dx � R n | f ( x ) | p 0 w ( x ) dx , ∀ w ∈ A ∞ 4 (0 < p 0 < ∞ is fixed) J.M. Martell (CSIC) Weights and Rubio de Francia extrapolation Helsinki, June 2008 8 / 91

Muckenhoupt Weights Weighted norm inequalities Section 1 Muckenhoupt Weights J.M. Martell (CSIC) Weights and Rubio de Francia extrapolation Helsinki, June 2008 9 / 91

Muckenhoupt Weights Weighted norm inequalities Muckenhoupt weights Weights w ≥ 0 a.e., w ∈ L 1 loc ( R n ) � 1 � � R n | f ( x ) | p w ( x ) dx L p ( w ) = L p ( w ( x ) dx ) � � f � L p ( w ) = p 1 λ w { x ∈ R n : | f ( x ) | > λ } L p, ∞ ( w ) � � f � L p, ∞ ( w ) = sup p λ> 0 Muckenhoupt’s problem Characterize weights w so that M : L p ( w ) − → L p ( w ) � � R n Mf ( x ) p w ( x ) dx � R n | f ( x ) | p w ( x ) dx Characterize weights w so that M : L p ( w ) − → L p, ∞ ( w ) w { x ∈ R n : Mf ( x ) > λ } � 1 � R n | f ( x ) | p w ( x ) dx λ p J.M. Martell (CSIC) Weights and Rubio de Francia extrapolation Helsinki, June 2008 10 / 91

Muckenhoupt Weights Weighted norm inequalities Muckenhoupt’s problem: Weak-type Proposition Let 1 ≤ p < ∞ . M : L p ( w ) − → L p, ∞ ( w ) if and only if � p − 1 � � � � � w 1 − p ′ dx ( A p ) − w dx − ≤ C, p > 1 Q Q � ( A 1 ) − w dx ≤ C w ( y ) , a.e. y ∈ Q, p = 1 Q Scheme of the proof p > 1 � f = w 1 − p ′ χ Q , w 1 − p ′ dx � �  λ = − f = −   Q Q = ⇒ � f = | S | p = 1 � f = χ S , w χ S ≈ inf Q w, λ = −   | Q | Q ⇐ = H¨ older, Vitali. J.M. Martell (CSIC) Weights and Rubio de Francia extrapolation Helsinki, June 2008 11 / 91

Muckenhoupt Weights Weighted norm inequalities Muckenhoupt weights: Properties Definition � p − 1 � � � � � w 1 − p ′ dx w ∈ A p − w dx − ≤ C Q Q � w ∈ A 1 − w dx ≤ C w ( y ) , a.e. y ∈ Q Q � A ∞ = A p p ≥ 1 Properties A 1 ⊂ A p ⊂ A q , 1 < p < q w 1 − p ′ ∈ A p ′ w ∈ A p ⇐ ⇒ w 1 w 1 − p w 1 , w 2 ∈ A 1 = ⇒ ∈ A p Reverse Factorization 2 J.M. Martell (CSIC) Weights and Rubio de Francia extrapolation Helsinki, June 2008 12 / 91

Muckenhoupt Weights Weighted norm inequalities Muckenhoupt weights: Examples � p − 1 � � � � � w 1 − p ′ dx ( A p ) − w dx − ≤ C Q Q � w dx ≤ C w ( y ) , a.e. y ∈ Q ≡ Mw ( x ) ≤ C w ( x ) a.e. x ∈ R n ( A 1 ) − Q Examples w ( x ) = 1 ∈ A 1 � − n < α ≤ 0 p = 1 w ( x ) = | x | α ∈ A p ⇐ ⇒ − n < α < n ( p − 1) p > 1 w ( x ) = Mf ( x ) δ ∈ A 1 , for all 0 < δ < 1, f ∈ L 1 loc ( R n ), Mf < ∞ ⇒ w ( x ) ≈ Mf ( x ) δ Coifman: w ∈ A 1 = J.M. Martell (CSIC) Weights and Rubio de Francia extrapolation Helsinki, June 2008 13 / 91

Muckenhoupt Weights Weighted norm inequalities A 1 weights: The Rubio de Francia Algorithm Constructing A 1 weights Let 0 ≤ u ∈ L p ( R n ), 1 < p < ∞ . Find U ≥ 0 such that 1 0 ≤ u ( x ) ≤ U ( x ) a.e. x ∈ R n 2 � U � p � � u � p 3 U ∈ A 1 , that is, MU ( x ) � U ( x ) a.e. x ∈ R n The Rubio de Francia Algorithm Mχ Q 0 ( x ) ≈ (1 + | x | ) − n / U = Mu WRONG!!! ∈ A 1 1 U = M ( u r ) r , 1 < r < p 0 ≤ u ( x ) ≤ R u ( x )  ∞ M k u   � U = R u = �R u � p ≤ 2 � u � p 2 k � M � k L p  k =0  M ( R u )( x ) ≤ 2 � M � p R u ( x ) J.M. Martell (CSIC) Weights and Rubio de Francia extrapolation Helsinki, June 2008 14 / 91

Muckenhoupt Weights Weighted norm inequalities Muckenhoupt’s problem: Strong-type and Reverse H¨ older � w ∈ A q ≡ M : L q ( w ) − → L q, ∞ ( w ) � M : L r ( w ) − → L r ( w ) M : L ∞ ( w ) − → L ∞ ( w ) q < r < ∞ Can we move (a little) to the left? YES Theorem (Reverse H¨ older Inequality) Given w ∈ A p , there exists ǫ > 0 such that 1 � � 1+ ǫ ≤ C − � w ( x ) 1+ ǫ dx � ( RH 1+ ǫ ) − w ( x ) dx Q Q Consequently, w 1+ δ ∈ A p for some δ > 0 w ∈ A q for some 1 < q < p Theorem (Muckenhoupt’s Theorem) M : L p ( w ) − → L p ( w ) Let 1 < p < ∞ . ⇐ ⇒ w ∈ A p J.M. Martell (CSIC) Weights and Rubio de Francia extrapolation Helsinki, June 2008 15 / 91

Muckenhoupt Weights Weighted norm inequalities Muckenhoupt Weights: Properties P. Jones’ Factorization w = w 1 w 1 − p w ∈ A p , 1 < p < ∞ ⇐ ⇒ with w 1 , w 2 ∈ A 1 2 � A ∞ = A p can characterized by p ≥ 1 w ∈ RH 1+ ǫ for some ǫ > 0 � | S | � δ ∃ δ > 0 such that w ( S ) w ( Q ) ≤ C , S ⊂ Q | Q | ∃ 0 < α, β < 1 such that S ⊂ Q , | S | ⇒ w ( S ) | Q | < α = w ( Q ) < β � � log w − 1 dx � � � � − w dx exp − ≤ C Q Q J.M. Martell (CSIC) Weights and Rubio de Francia extrapolation Helsinki, June 2008 16 / 91

Muckenhoupt Weights Weighted norm inequalities Extrapolation at first glance Theorem (Rubio de Francia; Garc´ ıa-Cuerva) Let 0 < p 0 < ∞ . Assume that T satisfies � � R n | Tf ( x ) | p 0 w ( x ) dx ≤ C w R n | f ( x ) | p 0 w ( x ) dx, ∀ w ∈ A 1 . Then T is bounded on L p ( R n ) for all p > p 0 . Proof. Let r = p/p 0 > 1. By duality, ∃ h ≥ 0, � h � r ′ = 1 such that � � R n | Tf | p 0 h dx ≤ R n | Tf | p 0 R h dx � Tf � p 0 � | Tf | p 0 � � p = r = � � R n | f | p 0 R h dx ≤ � f � p 0 p �R h � r ′ � � f � p 0 � p ∞ M k h � R h = 2 k � M � k L r ′ k =0 J.M. Martell (CSIC) Weights and Rubio de Francia extrapolation Helsinki, June 2008 17 / 91

Muckenhoupt Weights Weighted norm inequalities Other Maximal operators � Q = { cubes in R n } M Q f ( x ) = sup − | f ( y ) | dy, Q ∈Q ,Q ∋ x Q � D = { dyadic cubes in R n } M D f ( x ) = sup − | f ( y ) | dy, Q ∈D ,Q ∋ x Q � R = { Rectangles in R n } M R f ( x ) = sup − | f ( y ) | dy R ∈R ,R ∋ x R � Z = { Rectangles ( s, t, s t ) in R 3 } M Z f ( x ) = sup − | f ( y ) | dy R ∈Z ,R ∋ x R J.M. Martell (CSIC) Weights and Rubio de Francia extrapolation Helsinki, June 2008 18 / 91

Muckenhoupt Weights Weighted norm inequalities Muckenhoupt Bases Definitions Basis: B collection of open sets B ⊂ R n � � Maximal operator: M B f ( x ) = sup − | f ( y ) | dy , x ∈ B B ∈B ,B ∋ x B B ∈B Weight: 0 < w ( B ) < ∞ for every B ∈ B � Muckenhoupt weights: A ∞ , B = A p, B p ≥ 1 � p − 1 � � � w 1 − p ′ dx � � w ∈ A p, B − w dx − ≤ C B B a.e. x ∈ R n w ∈ A 1 , B M B w ( x ) ≤ C w ( x ) , Muckenhoupt Basis: M B : L p ( w ) → L p ( w ) , ∀ w ∈ A p, B , 1 < p < ∞ J.M. Martell (CSIC) Weights and Rubio de Francia extrapolation Helsinki, June 2008 19 / 91