Outline Boundedness of Operators Littlewood-Paley Theory Multipliers Littlewood-Paley Theory and Multipliers George Kinnear September 11, 2009 George Kinnear Littlewood-Paley Theory and Multipliers
Outline Boundedness of Operators Littlewood-Paley Theory Multipliers Boundedness of Operators 1 Littlewood-Paley Theory 2 Multipliers 3 George Kinnear Littlewood-Paley Theory and Multipliers
Outline Boundedness of Operators Littlewood-Paley Theory Multipliers Definitions How does an operator T behave with respect to function spaces, e.g. L p ? Definition We say that T is bounded from L p to L p if � Tf � p � C � f � p . We may also say T satisfies a strong ( p , p ) inequality. A weaker condition is the weak ( p , p ) inequality, � � f � p � p |{ x : | Tf ( x ) | > α }| � C . α Indeed, if T is strong ( p , p ) , then it is weak ( p , p ) . George Kinnear Littlewood-Paley Theory and Multipliers
Outline Boundedness of Operators Littlewood-Paley Theory Multipliers Interpolation We can often deduce that T is bounded for intermediate values of p just by considering end points. e.g. Marcinkiewicz interpolation: Theorem weak ( p 0 , p 0 ) and ( p 1 , p 1 ) = ⇒ strong ( p , p ) , p 0 < p < p 1 George Kinnear Littlewood-Paley Theory and Multipliers
Outline Boundedness of Operators Littlewood-Paley Theory Multipliers Maximal functions � 1 Mf ( x ) = sup | f ( y ) | dy | B ( x , r ) | r > 0 B ( x , r ) M is weak ( 1, 1 ) . M is weak ( ∞ , ∞ ) . Theorem M is strong ( p , p ) , 1 < p < ∞ George Kinnear Littlewood-Paley Theory and Multipliers
Outline Boundedness of Operators Littlewood-Paley Theory Multipliers Singular integrals � Tf ( x ) = K ∗ f ( x ) = R n K ( x − y ) f ( y ) dy where K is locally integrable away from the origin, and (i) | ˆ K ( ξ ) | � B , � | x | > 2 | y | | K ( x − y ) − K ( x ) | dx � B , y ∈ R n . (ii) (or | ∇ K ( x ) | � C | x | − n − 1 ) T is strong ( 2, 2 ) , from (i). T is weak ( 1, 1 ) – use (ii) and the Calderón-Zygmund decomposition. Thus T is strong ( p , p ) for 1 < p < 2. By duality, also for 2 < p < ∞ . Theorem T is strong ( p , p ) , 1 < p < ∞ George Kinnear Littlewood-Paley Theory and Multipliers
Outline Boundedness of Operators Littlewood-Paley Theory Multipliers Littlewood-Paley Theorem Take ψ ∈ S ( R n ) with ψ ( 0 ) = 0 and define S j by � ( S j f )( ξ ) = ψ j ( ξ ) ˆ ψ j ( ξ ) = ψ ( 2 − j ξ ) . f ( ξ ) where Theorem For 1 < p < ∞ , � j | S j f | 2 � 1 / 2 � � � � � � � (a) � C � f � p . � � p (b) If for ξ � 0 we have � j | ψ ( 2 − j ξ ) | 2 = C, then also � j | S j f | 2 � 1 / 2 � � � � � � � � f � p � C . � � p George Kinnear Littlewood-Paley Theory and Multipliers
Outline Boundedness of Operators Littlewood-Paley Theory Multipliers Littlewood-Paley Theorem – Proof Consider the operator f �→ { S j f } . It is bounded from L 2 to L 2 ( ℓ 2 ) : � j | S j f | 2 � 1 / 2 � � � � 2 � � � f ( ξ ) | 2 d ξ � C � f � 2 � � | ψ j ( ξ ) | 2 | ˆ = 2 . � � R n 2 j Other L p follow from the Hörmander condition, which is satisfied since � � � � � ∇ ˇ ℓ 2 � C | x | − n − 1 . ψ j ( x ) � So we have part (a). George Kinnear Littlewood-Paley Theory and Multipliers
Outline Boundedness of Operators Littlewood-Paley Theory Multipliers Littlewood-Paley Theorem – Proof Now if � j | ψ ( 2 − j ξ ) | 2 = C we actually have � j | S j f | 2 � 1 / 2 � � � � � √ � � = C � f � 2 . � � 2 So by polarization, � � √ � C R n fg = S j fS j g . R n j Hence � � � � � j | S j f | 2 � 1 / 2 � � j | S j g | 2 � 1 / 2 � � � � � � � C ′ fg � � j | S j f | 2 � 1 / 2 � � j | S j g | 2 � 1 / 2 � � � � � � � � � � � � � � C ′ � � � � p p ′ � j | S j f | 2 � 1 / 2 � � � � � � � � C ′ � g � p ′ . � � p George Kinnear Littlewood-Paley Theory and Multipliers
Outline Boundedness of Operators Littlewood-Paley Theory Multipliers Multipliers Given m ∈ L ∞ ( R n ) we can define an operator T m by � T m f ( x ) = m ( x ) ˆ f ( x ) . We say m is a multiplier for L p if T m is bounded on L p . The class of multiplers for L p is M p . Example M 2 = L ∞ . Theorem If 1 p + 1 p ′ = 1 , 1 � p � ∞ , then M p = M p ′ . George Kinnear Littlewood-Paley Theory and Multipliers
Outline Boundedness of Operators Littlewood-Paley Theory Multipliers Sobolev spaces For positive integers k , k = { f ∈ L p : D α f ∈ L p , | α | � k } L p k = � | α | � k � D α f � p . with norm � f � L p There is an alternative definition of L p a for general a > 0. When p = 2, this is a = { g ∈ L 2 : ( 1 + | ξ | 2 ) a / 2 ˆ L 2 g ( ξ ) ∈ L 2 } , � � � ( 1 + | · | 2 ) a / 2 ˆ � with norm � g � L 2 a = g 2 . Theorem If m ∈ L 2 a with a > n 2 then m ∈ M p for 1 � p � ∞ . George Kinnear Littlewood-Paley Theory and Multipliers
Outline Boundedness of Operators Littlewood-Paley Theory Multipliers Hörmander multiplier theorem Let ψ ∈ C ∞ be radial, supported on 1 2 � | x | � 2, and s.t. ∞ � | ψ ( 2 − j x ) | 2 = 1, x � 0. j =− ∞ Theorem If m is such that, for some k > n 2 , � � � m ( 2 j · ) ψ � sup k < ∞ L 2 j then m ∈ M p for all 1 < p < ∞ . George Kinnear Littlewood-Paley Theory and Multipliers
Outline Boundedness of Operators Littlewood-Paley Theory Multipliers Hörmander multiplier theorem – Proof Let ˜ ψ ∈ C ∞ , be supported on 1 4 � | ξ | � 4 and equal to 1 on supp ψ . The operators ˜ S j with multipliers ˜ ψ ( 2 − j ξ ) satisfy � S j f | 2 � 1 / 2 � � � � � � � j | ˜ � C p � f � p . � � p Now if S j has multiplier ψ ( 2 − j ξ ) , � j | S j Tf | 2 � 1 / 2 � � S j f | 2 � 1 / 2 � � � � � � � � � � � � � j | S j T ˜ � Tf � p � C = C . � � � � p p George Kinnear Littlewood-Paley Theory and Multipliers
Outline Boundedness of Operators Littlewood-Paley Theory Multipliers Hörmander multiplier theorem – Proof � � � � � 2 � m j R n | S j Tf | 2 u � R n | f ( y ) | 2 Mu ( y ) dy k C L 2 For p > 2, Riesz representation gives some u ∈ L ( p / 2 ) ′ s.t. � j | S j Tf j | 2 � 1 / 2 � � j | S j Tf j | 2 � � � � 2 � � � � � � � � � j | S j Tf j | 2 u . = p / 2 = � � � R n p Putting these together, � j | S j Tf j | 2 � 1 / 2 � � � � 2 � � � � � j | f j | 2 Mu � C � � R n p � j | f j | 2 � � � � p / 2 � Mu � ( p / 2 ) ′ � C � � � C ′ � j | f j | 2 � � � � � � p / 2 George Kinnear Littlewood-Paley Theory and Multipliers
Outline Boundedness of Operators Littlewood-Paley Theory Multipliers Hörmander multiplier theorem – Proof Combining this with the Littlewood-Paley estimates, � S j f | 2 � 1 / 2 � � � � � � � j | S j T ˜ � Tf � p � C � � p � S j f | 2 � 1 / 2 � � � � � � � j | ˜ � C � � p � C � f � p . So m ∈ M p for p > 2. The result for 1 < p < 2 follows by duality, and for p = 2 by interpolation. George Kinnear Littlewood-Paley Theory and Multipliers
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