Marcinkiewicz interpolation Updated May 18, 2020 Plan 2 Outline: - PowerPoint PPT Presentation

Marcinkiewicz interpolation Updated May 18, 2020

Plan 2 Outline: Interpolation of quasinorms Diagonal Marcinkiewicz interpolation theorem General version Applications: Schur test, Hardy-Littlewood-Sobolev

Interpolation for quasinorms 3 Lemma Let p 0 , p 1 P p 0, 8s and given θ P r 0, 1 s define p by 1 p “ 1 ´ θ ` θ p 0 p 1 Then @ f P L 0 : r f s p ď r f s 1 ´ θ r f s θ p 0 p 1 Moreover, if p 0 ă p 1 , then ” ı 1 { p p p @ f P L 0 : r f s 1 ´ θ r f s θ } f } p ď ` p 0 p 1 p ´ p 0 p 1 ´ p

Proof of Lemma 4 Assume p 0 ă p 1 ă 8 . Then for t ą 0 and f P L 0 , ´ ˘ 1 { p 0 ¯ 1 ´ θ ´ ˘ 1 { p 1 ¯ θ ` ˘ 1 { p θ “ ` ` t µ | f | ą t t µ | f | ą t t µ | f | ą t Now RHS ď r f s 1 ´ θ r f s θ p 1 . Optimize over t ą 0. For p 1 “ 8 only p 0 need t ď } f } 8 . Second part: ż a ż 8 } f } p p t p ´ 1 µ p| f | ą t q d t ` p t p ´ 1 µ p| f | ą t q d t p “ 0 a ż a ż 8 ď p r f s p 0 t p ´ p 0 ´ 1 d t ` p r f s p 1 t p ´ p 1 ´ 1 d t p 0 p 1 0 a p p r f s p 0 p 0 a p ´ p 0 ` p 1 ´ p r f s p 1 p 1 a p 1 ´ p “ p ´ p 0 Now optimize over a ą 0.

Sublinear operators 5 Definition An operator T : Dom p T q Ñ L 0 on a linear subspace Dom p T q Ď L 0 is said to be sublinear if @ f , g P Dom p T q : | T p f ` g q| ď | Tf | ` | Tg | and @ f P Dom p T q @ c P R : | T p cf q| “ | c || Tf | If the space L 0 is over C , then this holds with c P C . Example: Hardy-Littlewood max function ż 1 f ‹ p x q : “ sup | f | d µ µ p B p x , r qq r ą 0 B p x , r q on p R d , B p R d q , µ q with µ Radon.

Diagonal Marcinkiewicz interpolation theorem 6 Theorem Let p 0 , p 1 P p 0, 8s obey p 0 ă p 1 and let T : p L p 0 ` L p 1 qp X , F , µ q Ñ L 0 p Y , G , ν q be sublinear with D C 0 P p 0, 8q @ f P L p 0 : r Tf s p 0 ď C 0 } f } p 0 and D C 1 P p 0, 8q @ f P L p 1 : r Tf s p 1 ď C 1 } f } p 1 Then for all p P p p 0 , p 1 q we have ” ı 1 { p p p @ f P L p : C 1 ´ θ C θ } Tf } p ď 2 ` 1 } f } p 0 p ´ p 0 p 1 ´ p where θ P p 0, 1 q is the unique number such that 1 p “ 1 ´ θ p 0 ` θ p 1 . Note: Same structure as Riesz-Thorin. Numerical prefactor blows up as p Ó p 0 or p Ò p 1 .

Proof of Theorem 7 Take f bounded with µ p supp p f qq ă 8 . Set f 0 : “ f 1 t| f |ą at u ^ f 1 : “ f 1 t| f |ď at u Sublinearity gives ` ˘ ` ˘ ` ˘ | Tf | ą t ď ν | Tf 0 | ą t { 2 ` ν | Tf 1 | ą t { 2 ν Assumptions show ż ` ˘ 2 p 0 ď C p 0 | f | p 0 d µ ν | Tf 0 | ą t { 2 0 t p 0 t| f |ą at u and ż ` ˘ 2 p 1 ď C p 1 | f | p 1 d µ ν | Tf 1 | ą t { 2 1 t p 1 t| f |ď at u Since ż 8 ` ˘ } Tf } p p t p ´ 1 ν p “ | Tf | ą t d t 0 we need to compute . . .

Proof of Theorem continued ... 8 . . . using p 0 ă p ă p 1 that ż 8 ż ż t p ´ 1 ´ 1 ¯ d t “ a p 0 ´ p | f | p 0 d µ | f | p d µ t p 0 p ´ p 0 0 t| f |ą at u and ż 8 ż ż t p ´ 1 ´ 1 ¯ d t “ a p 1 ´ p | f | p 1 d µ | f | p d µ t p 1 p 1 ´ p 0 t| f |ď at u Putting these together „  p p } Tf } p } f } p p 2 C 0 q p 0 a p 0 ´ p ` p 2 C 1 q p 1 a p 1 ´ p p ď p p ´ p 0 p 1 ´ p Now optimize over a ą 0 as before. For p 1 “ 8 we use that r Tf 1 s 8 “ } Tf 1 } 8 to get } Tf 1 } 8 ď C 1 at 1 Set a : “ 2 C 1 to get ν p| Tf 1 | ą t { 2 q “ 0. Same as taking p Ò p 1 .

Weak type p p , q q 9 Definition (Weak type- p p , q q ) Given p , q P r 1, 8s , an opeartor T : Dom p T q Ñ L 0 defined on a dense linear subspace Dom p T q Ď L p , is said to be weak type- p p , q q if D C P p 0, 8q @ f P Dom p T q : r Tf s q ď C } f } p Note that T strong type p p , q q ñ T weak type p p , q q . Above theorem: If T is sublinear and weak type p p 0 , p 0 q and p p 1 , p 1 q , then it is strong type p p , p q for all p P p p 0 , p 1 q

L p -continuity of max function 10 Besicovich covering ñ f ‹ obeys weak L 1 -estimate ` f ‹ ą t q ď c p d q t } f } 1 µ so f ÞÑ f ‹ weak type p 1, 1 q . Obvious bound } f ‹ } 8 ď } f } 8 so weak type p8 , 8q . Above theorem: ´ cp ¯ 1 { p D c P p 0, 8q @ p P p 1, 8s @ f P L p : } f ‹ p } ď } f } p p ´ 1 where c depends only on dimension.

Constraints on indices 11 In general we want T : L p i Ñ L q i . Q: Are there restrictions on p 0 , p 1 , q 0 , q 1 ? Lemma Set X “ Y : “ N , F “ G “ 2 N and, given any β ą 0 , consider the measures µ and ν defined by µ pt n uq : “ 2 n ν pt n uq : “ 2 β n ^ Let Tf : “ f be the identity map L 0 p µ q Ñ L 0 p ν q . Then T is weak type p p , β p q for each p ą 0 For β ă 1 , T is not strong type p p , β p q for any p ą 0 Appears that we will need p 0 ď q 0 ^ p 1 ď q 1

Proof of Lemma 12 Key point: µ p| f | ą t q — 2 max t n P N : | f p n q|ą t u ν p| f | ą t q — 2 β max t n P N : | f p n q|ą t u So for all p ą 0 ` ˘ 1 ` ˘ 1 { p β p ď c µ @ t ą 0: ν | f | ą t | f | ą t and thus r Tf s β p ď c r f s p ď c } f } p . T is weak type p p , β p q . For f p n q : “ n ´ α { p 2 ´ n { p with α ą 1 s.t. αβ ă 1 ÿ n ´ α ă 1 } f } p p “ n ě 1 yet ÿ n ´ αβ “ 8 } Tf } β p β p “ n ě 1 So T is not strong type p p , β p q for any p ą 0.

Marcinkiewicz interpolation theorem, full version 13 Theorem Let p 0 , p 1 , q 0 , q 1 P r 1, 8s obey p 0 ď q 0 ^ p 1 ď q 1 ^ q 0 ‰ q 1 Let T : L p 0 ` L p 1 Ñ L 0 be sublinear and set, for θ P r 0, 1 s , : “ 1 ´ θ : “ 1 ´ θ 1 ` θ 1 ` θ ^ p θ p 0 p 1 q θ q 0 q 1 If T is weak type p p 0 , q 0 q and p p 1 , q 1 q then T is strong type p p θ , q θ q for all θ P p 0, 1 q . Explicitly, for all C 0 , C 1 P p 0, 8q and all θ P p 0, 1 q there is C θ P p 0, 8q such that @ f P L p 0 ` L p 1 : r Tf s q 0 ď C 0 } f } p 0 ^ r Tf s q 1 ď C 1 } f } p 1 implies @ f P L p 0 ` L p 1 @ θ P p 0, 1 q : } Tf } q θ ď C θ } f } p θ

Proof of Theorem 14 Will only treat q 0 , q 1 ă 8 . For p 0 “ p 1 invoke interpolation for quasinorms to get } Tf } q θ ď r Tf s 1 ´ θ q 1 ď C 1 ´ θ q 0 r Tf s θ C θ 1 } f } p 0 0 so assume p 0 ‰ p 1 and WLOG q 0 ă q 1 . Let f : X Ñ Y be simple satisfying (using homogeneity of T ) r f s p θ ď 1 Then D N ě 1 such that ÿ N f “ f m where f m : “ f 1 t 2 m ă| f |ď 2 m ` 1 u m “´ N Subadditivity gives | Tf | ď ř N m “´ N | Tf m | and so, for any n P Z m “´ N positive with ř N and any t a m u N m “´ N a m “ 1, N ÿ ` | Tf | ą 2 n ˘ ` | Tf m | ą a m 2 n ˘ ν ď ν m “´ N This now feeds . . .

Proof of Theorem continued ... 15 . . . into ´ C i ¯ q i } f m } q i ` ˘ ν | Tf m | ą a m t ď p i a m t ´ C i ¯ q i } f m } q i ` ˘ q i { p i ď 8 µ supp p f m q a m t ´ C i ¯ q i 2 p m ` 1 q q i µ p| f | ą 2 m q q i { p i ď a m t Hereby we get ż N ÿ ÿ ` ˘ } Tf } q θ 2 n ă t { a m ď 2 n ` 1 q θ t q θ ´ 1 ν q θ ď | Tf m | ą t d t m “´ N n P Z ÿ N ÿ ` | Tf m | ą a m 2 n ˘ ď 2 q θ p 2 n a n , m q q θ ν m “´ N n P Z ˆ ˙ ´ 2 m ´ n ¯ q i µ N ď 2 q θ ÿ ÿ ` | f | ą 2 m ˘ q i { p i a q θ 2 nq θ p 2 C i q q i m min a m i “ 0,1 n P Z m “´ N Now comes the time . . .

Proof of Theorem continued ... 16 . . . to use the conditions q i ě p i : ` ˘ qi 2 ´ mp θ r f s p θ pi ´ 1 µ p| f | ą 2 m q q i { p i ď µ p| f | ą 2 m q p θ which using the normalization r f s p θ ď 1 gives N ÿ ` | f | ą 2 m ˘ } Tf } q θ q θ ď 2 q θ 2 mp θ µ R m m “´ N with ˆ´ 2 C i ˙ ¯ q i 2 n p q θ ´ q i q 2 m p 1 ´ p θ { p i q q i ÿ R m : “ a q θ min m a m i “ 0,1 n P Z Our goal is to show sup m P Z R m ă 8 for suitable t a m u N m “´ N .

Proof of Theorem, finished 17 Recall q 0 ă q θ ă q 1 . Pick u ą 0 and use i “ 0 for n with 2 n ă u and i “ 1 for 2 n ą u . Then ˆ´ 2 C 0 ˙ 2 m p 1 ´ p θ { p 0 q ¯ q 0 u q θ ´ q 0 ` ´ 2 C 1 2 m p 1 ´ p θ { p 1 q ¯ q 1 u q θ ´ q 1 c a q θ R m ď ˆ m a m a m ř n ě´ 1 2 ´ n p q θ ´ q i q . Now optimize over u ą 0 where ˆ c : “ max i “ 0,1 using ` Au α ` Bu ´ γ ˘ γ α α ` γ A inf “ Γ p α , γ q A α ` γ u ą 0 where Γ p α , γ q numerical constant, to get ´ 2 C 0 2 m p 1 ´ p θ { p 0 q ¯ q 0 q 1 ´ q 0 ´ 2 C 1 2 m p 1 ´ p θ { p 1 q ¯ q 1 q 1 ´ q θ q θ ´ q 0 c Γ p . . . q a q θ q 1 ´ q 0 R m ď ˆ m a m a m Exponents equal p 1 ´ θ q q θ and q θ θ , so c C 1 ´ θ C θ } f } p θ ď 1 ñ } Tf } q θ ď r 1 0 where r c depends only on p 0 , p 1 , q 0 , q 1 , θ .

Restricted weak type 18 Definition (Restricted weak type) We say that T is restricted weak type p p , q q , if there exists C P p 0, 8q such that (for q ă 8 ) ` ˘ ` ˘ q { p ď Ct ´ q } f } q @ t ą 0: | Tf | ą t supp p f q ν 8 µ If q “ 8 then same as the weak/strong type p p , q q . Weak type ñ restricted weak type: ` ˘ p ď p C { t q q } f } q ď p C { t q q } f } q 8 µ p supp p f qq q { p | Tf | ą t ν Restricted weak type sufficient for Marcinkiewicz.

Applications: Schur test 19 ş Integral operator T K f p x q : “ K p x , y q f p y q µ p d y q with kernel K . C ñ T maps L p Ñ L p Recall: } K p x , ¨q} 1 ď C ^ } K p¨ , y q} 1 ď r Proposition (Schur test extended) Let p X , F , µ q and p Y , G , ν q be σ -finite measure spaces and let K : X Ñ Y Ñ R be F b G -measurable. Suppose, for some r , s ě 1 , D C P p 0, 8q : } K p x , ¨q} L r p ν q ď C for µ -a.e. x P X and D r } K p¨ , y q} L s p µ q ď r C P p 0, 8q : for ν -a.e. y P Y . C Then T K is strong type p p , q q for every p and q with r ´ 1 ^ s ď q ď 8 ^ 1 r p ` 1 r “ 1 ` s 1 1 ď p ď r q