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

Marcinkiewicz interpolation

Updated May 18, 2020

Plan

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

Interpolation for quasinorms

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Lemma Let p0, p1 P p0, 8s and given θ P r0, 1s define p by 1 p “ 1 ´ θ p0 ` θ p1 Then @f P L0 : rfsp ď rfs1´θ

p0

rfsθ

p1

Moreover, if p0 ă p1, then @f P L0 : }f}p ď ” p p ´ p0 ` p p1 ´ p ı1{p rfs1´θ

p0

rfsθ

p1

Proof of Lemma

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Assume p0 ă p1 ă 8. Then for t ą 0 and f P L0, tµ ` |f| ą t ˘1{pθ “ ´ tµ ` |f| ą t ˘1{p0¯1´θ´ tµ ` |f| ą t ˘1{p1¯θ Now RHS ď rfs1´θ

p0

rfsθ

• p1. Optimize over t ą 0. For p1 “ 8 only

need t ď }f}8. Second part: }f}p

p “

ż a p tp´1µp|f| ą tqdt ` ż 8

a

p tp´1µp|f| ą tqdt ď prfsp0

p0

ż a tp´p0´1dt ` prfsp1

p1

ż 8

a

tp´p1´1dt “ p p ´ p0 rfsp0

p0ap´p0 `

p p1 ´ prfsp1

p1ap1´p

Now optimize over a ą 0.

Sublinear operators

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Definition An operator T: DompTq Ñ L0 on a linear subspace DompTq Ď L0 is said to be sublinear if @f, g P DompTq: |Tpf ` gq| ď |Tf| ` |Tg| and @f P DompTq @c P R: |Tpcfq| “ |c||Tf| If the space L0 is over C, then this holds with c P C. Example: Hardy-Littlewood max function f ‹pxq :“ sup

rą0

1 µpBpx, rqq ż

Bpx,rq

|f|dµ

• n pRd, BpRdq, µq with µ Radon.

Diagonal Marcinkiewicz interpolation theorem

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Theorem Let p0, p1 P p0, 8s obey p0 ă p1 and let T: pLp0 ` Lp1qpX, F, µq Ñ L0pY, G, νq be sublinear with DC0 P p0, 8q @f P Lp0 : rTfsp0 ď C0}f}p0 and DC1 P p0, 8q @f P Lp1 : rTfsp1 ď C1}f}p1 Then for all p P pp0, p1q we have @f P Lp : }Tf}p ď 2 ” p p ´ p0 ` p p1 ´ p ı1{p C1´θ Cθ

1 }f}p

where θ P p0, 1q is the unique number such that 1

p “ 1´θ p0 ` θ p1 .

Note: Same structure as Riesz-Thorin. Numerical prefactor blows up as p Ó p0 or p Ò p1.

Proof of Theorem

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Take f bounded with µpsupppfqq ă 8. Set f0 :“ f 1t|f|ąatu ^ f1 :“ f 1t|f|ďatu Sublinearity gives ν ` |Tf| ą t ˘ ď ν ` |Tf0| ą t{2 ˘ ` ν ` |Tf1| ą t{2 ˘ Assumptions show ν ` |Tf0| ą t{2 ˘ ď Cp0 2p0 tp0 ż

t|f|ąatu

|f|p0dµ and ν ` |Tf1| ą t{2 ˘ ď Cp1

1

2p1 tp1 ż

t|f|ďatu

|f|p1dµ Since }Tf}p

p “

ż 8 p tp´1ν ` |Tf| ą t ˘ dt we need to compute . . .

Proof of Theorem continued ...

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. . . using p0 ă p ă p1 that ż 8 tp´1´ 1 tp0 ż

t|f|ąatu

|f|p0dµ ¯ dt “ ap0´p p ´ p0 ż |f|pdµ and ż 8 tp´1´ 1 tp1 ż

t|f|ďatu

|f|p1dµ ¯ dt “ ap1´p p1 ´ p ż |f|pdµ Putting these together }Tf}p

p ď

„ p2C0qp0ap0´p p p ´ p0 ` p2C1qp1ap1´p p p1 ´ p  }f}p

p

Now optimize over a ą 0 as before. For p1 “ 8 we use that rTf1s8 “ }Tf1}8 to get }Tf1}8 ď C1at Set a :“

1 2C1 to get νp|Tf1| ą t{2q “ 0. Same as taking p Ò p1.

Weak type pp, qq

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Definition (Weak type-pp, qq) Given p, q P r1, 8s, an opeartor T: DompTq Ñ L0 defined on a dense linear subspace DompTq Ď Lp, is said to be weak type-pp, qq if DC P p0, 8q @f P DompTq: rTfsq ď C}f}p Note that T strong type pp, qq ñ T weak type pp, qq. Above theorem: If T is sublinear and weak type pp0, p0q and pp1, p1q, then it is strong type pp, pq for all p P pp0, p1q

Lp-continuity of max function

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Besicovich covering ñ f ‹ obeys weak L1-estimate µ ` f ‹ ą tq ď cpdq t }f}1 so f ÞÑ f ‹ weak type p1, 1q. Obvious bound }f ‹}8 ď }f}8 so weak type p8, 8q. Above theorem: Dc P p0, 8q @p P p1, 8s @f P Lp : }f ‹

p } ď

´ cp p ´ 1 ¯1{p }f}p where c depends only on dimension.

Constraints on indices

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In general we want T: Lpi Ñ Lqi. Q: Are there restrictions on p0, p1, q0, q1? Lemma Set X “ Y :“ N, F “ G “ 2N and, given any β ą 0, consider the measures µ and ν defined by µptnuq :“ 2n ^ νptnuq :“ 2βn Let Tf :“ f be the identity map L0pµq Ñ L0pνq. Then T is weak type pp, βpq for each p ą 0 For β ă 1, T is not strong type pp, βpq for any p ą 0 Appears that we will need p0 ď q0 ^ p1 ď q1

Proof of Lemma

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Key point: µp|f| ą tq — 2maxtnPN: |fpnq|ątu νp|f| ą tq — 2β maxtnPN: |fpnq|ątu So for all p ą 0 @t ą 0: ν ` |f| ą t ˘ 1

βp ď cµ

` |f| ą t ˘1{p and thus rTfsβp ď crfsp ď c}f}p. T is weak type pp, βpq. For fpnq :“ n´α{p2´n{p with α ą 1 s.t. αβ ă 1 }f}p

p “

ÿ

ně1

n´α ă 1 yet }Tf}βp

βp “

ÿ

ně1

n´αβ “ 8 So T is not strong type pp, βpq for any p ą 0.

Marcinkiewicz interpolation theorem, full version

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Theorem Let p0, p1, q0, q1 P r1, 8s obey p0 ď q0 ^ p1 ď q1 ^ q0 ‰ q1 Let T: Lp0 ` Lp1 Ñ L0 be sublinear and set, for θ P r0, 1s, 1 pθ :“ 1 ´ θ p0 ` θ p1 ^ 1 qθ :“ 1 ´ θ q0 ` θ q1 If T is weak type pp0, q0q and pp1, q1q then T is strong type ppθ, qθq for all θ P p0, 1q. Explicitly, for all C0, C1 P p0, 8q and all θ P p0, 1q there is Cθ P p0, 8q such that @f P Lp0 ` Lp1 : rTfsq0 ď C0}f}p0 ^ rTfsq1 ď C1}f}p1 implies @f P Lp0 ` Lp1 @θ P p0, 1q: }Tf}qθ ď Cθ}f}pθ

Proof of Theorem

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Will only treat q0, q1 ă 8. For p0 “ p1 invoke interpolation for quasinorms to get }Tf}qθ ď rTfs1´θ

q0 rTfsθ q1 ď C1´θ

Cθ

1}f}p0

so assume p0 ‰ p1 and WLOG q0 ă q1. Let f : X Ñ Y be simple satisfying (using homogeneity of T) rfspθ ď 1 Then DN ě 1 such that f “

N

ÿ

m“´N

fm where fm :“ f1t2mă|f|ď2m`1u Subadditivity gives |Tf| ď řN

m“´N |Tfm| and so, for any n P Z

and any tamuN

m“´N positive with řN m“´N am “ 1,

ν ` |Tf| ą 2n˘ ď

N

ÿ

m“´N

ν ` |Tfm| ą am2n˘ This now feeds . . .

Proof of Theorem continued ...

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. . . into ν ` |Tfm| ą amt ˘ ď ´ Ci amt ¯qi}fm}qi

pi

ď ´ Ci amt ¯qi}fm}qi

8 µ

` supppfmq ˘qi{pi ď ´ Ci amt ¯qi2pm`1qqiµp|f| ą 2mqqi{pi Hereby we get }Tf}qθ

qθ ď N

ÿ

m“´N

ÿ

nPZ

ż

2năt{amď2n`1 qθtqθ´1ν

` |Tfm| ą t ˘ dt ď 2qθ

N

ÿ

m“´N

ÿ

nPZ

p2nan,mqqθν ` |Tfm| ą am2n˘ ď 2qθ ÿ

nPZ

2nqθ

N

ÿ

m“´N

aqθ

m min i“0,1

ˆ p2Ciqqi ´2m´n am ¯qiµ ` |f| ą 2m˘qi{pi ˙ Now comes the time . . .

Proof of Theorem continued ...

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. . . to use the conditions qi ě pi: µp|f| ą 2mqqi{pi ď µp|f| ą 2mq ` 2´mpθrfspθ

pθ

˘ qi

pi ´1

which using the normalization rfspθ ď 1 gives }Tf}qθ

qθ ď 2qθ N

ÿ

m“´N

2mpθµ ` |f| ą 2m˘ Rm with Rm :“ aqθ

m

ÿ

nPZ

min

i“0,1

ˆ´2Ci am ¯qi2npqθ´qiq 2mp1´pθ{piqqi ˙ Our goal is to show supmPZ Rm ă 8 for suitable tamuN

m“´N.

Proof of Theorem, finished

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Recall q0 ă qθ ă q1. Pick u ą 0 and use i “ 0 for n with 2n ă u and i “ 1 for 2n ą u. Then Rm ď ˆ c aqθ

m

ˆ´2C0 am 2mp1´pθ{p0q¯q0uqθ´q0 ` ´2C1 am 2mp1´pθ{p1q¯q1uqθ´q1 ˙ where ˆ c :“ maxi“0,1 ř

ně´1 2´npqθ´qiq. Now optimize over u ą 0

using inf

uą0

` Auα ` Bu´γ˘ “ Γpα, γqA

γ α`γ A α α`γ

where Γpα, γq numerical constant, to get Rm ď ˆ c Γp. . . qaqθ

m

´2C0 am 2mp1´pθ{p0q¯q0

q1´qθ q1´q0 ´2C1

am 2mp1´pθ{p1q¯q1

qθ´q0 q1´q0

Exponents equal p1 ´ θqqθ and qθθ, so }f}pθ ď 1 ñ }Tf}qθ ď r c C1´θ Cθ

1

where r c depends only on p0, p1, q0, q1, θ.

Restricted weak type

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Definition (Restricted weak type) We say that T is restricted weak type pp, qq, if there exists C P p0, 8q such that (for q ă 8) @t ą 0: ν ` |Tf| ą t ˘ ď Ct´q}f}q

8µ

` supppfq ˘q{p If q “ 8 then same as the weak/strong type pp, qq. Weak type ñ restricted weak type: ν ` |Tf| ą t ˘ ď pC{tqq}f}q

p ď pC{tqq }f}q 8 µpsupppfqqq{p

Restricted weak type sufficient for Marcinkiewicz.

Applications: Schur test

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Integral operator TKfpxq :“ ş Kpx, yqfpyqµpdyq with kernel K. Recall: }Kpx, ¨q}1 ď C ^ }Kp¨, yq}1 ď r C ñ T maps Lp Ñ Lp Proposition (Schur test extended) Let pX, F, µq and pY, G, νq 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, DC P p0, 8q: }Kpx, ¨q}Lrpνq ď C for µ-a.e. x P X and Dr C P p0, 8q: }Kp¨, yq}Lspµq ď r C for ν-a.e. y P Y. Then TK is strong type pp, qq for every p and q with 1 ď p ď r r ´ 1 ^ s ď q ď 8 ^ 1 p ` 1 r “ 1 ` s r 1 q

Proof of extended Schur test

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H¨

• lder:

}TKf}8 ď C}f}

r r´1

so TK is strong type p

r r´1, 8q. Minkowski:

}TKf}s ď r C}f}1 and so TK is strong type p1, sq. If for θ P r0, 1s 1 p “ 1 ´ θ ` θ ´ 1 ´ 1 r ¯ ^ 1 q “ 1 ´ θ s Riesz-Thorin implies that TK is strong type pp, qq.

Schur test, weak-type version

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Proposition (Schur test, weak-type version) Let pX, F, µq and pY, G, νq be σ-finite measure spaces and let K: X Ñ Y Ñ R be F b G-measurable. Suppose, for some r, s ą 1, DC P p0, 8q: rKpx, ¨qsr ď C for µ-a.e. x P X and Dr C P p0, 8q: rKp¨, yqss ď r C for ν-a.e. y P Y. Then TK is strong type pp, qq for every p and q satisfying 1 ă p ă r r ´ 1 ^ s ă q ă 8 ^ 1 p ` 1 r “ 1 ` s r 1 q

Proof of weak-type Schur test

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Tonelli and ş

A hdµ ď s s´1rhssµpAq1´1{s gives

ż

A

´ ż ˇ ˇKpx, yq ˇ ˇˇ ˇfpyq ˇ ˇµpdyq ¯ νpdxq ď ż ´ż

A

ˇ ˇKpx, yq ˇ ˇνpdxq ¯ˇ ˇfpyq ˇ ˇµpdyq ď r C s s ´ 1µpAq1´1{s}f}1 This shows rTfss ď r C

s s´1}f}1 and so TK is weak type p1, sq. Next,

for any A P G, ż ˇ ˇKpx, yq ˇ ˇˇ ˇfpyq ˇ ˇµpdyq ď }f}8 ż

supppfq

ˇ ˇKpx, yq ˇ ˇµpdyq ď C}f}8µ ` supppfq ˘1´1{r so TK is restricted weak type p

r r´1, 8q. Marcinkiwicz gives that

TK is strong type pp, qq, for p, q as above.

Hardy-Littlewood-Sobolev inequality

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Corollary (Hardy-Littlewood-Sobolev inequality) For α P p0, dq and f P L1 define Tαfpxq :“ ż

Rd

1 |x ´ y|α fpyqdy Then Tα is strong type pp, qq for p P p1,

d d´αq and q P p d α, 8q such

that p´1 ` α{d “ 1 ` q´1. In particular, for each such p, q there is C P p0, 8q such that ż fpxq 1 |x ´ y|α gpyqdxdy ď C}f}p}g}

q q´1

holds for all measurable f, g: Rd Ñ r0, 8q.

More on Hardy-Littlewood-Sobolev

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Proof: Scaling properties of Lebesgue measure λp|x|´α ą tq “ ct´d{α so Schur (weak-type!) test applies with r “ s “ d{α. The double integral ż fpxq 1 |x ´ y|α fpyqdxdy is self-energy of charge distribution f with potential x ÞÑ |x|´α. Finiteness for all f P Lp occurs when p “

q q´1 which means

p “ 2d 2d ´ α Physical case: α “ d ´ 2 (d ě 2) gives p “

2d d`2. For these E. Lieb

proved value of sharp constant and characterized minimizers. Q: What happens when α “ d? A: Next time . . .