“a tribute to eric sawyer” carlos perez´ · cv b.sc. mcgill university, mathematics 1973 ph.d....

“A tribute to Eric Sawyer”

Carlos Perez

Universidad de Sevilla

Conference in honor of Eric Sawyer

Fields Institute, Toronto

July, 2011

Some theorems by E. Sawyer [email protected]

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B.Sc. McGill University, Mathematics 1973

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Ph.D. McGill University, Mathematics 1977

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Title of his thesis:

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“Inner Functions on Polydiscs”

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Advisor: Kohur N. Gowrisankaran

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Posdoctoral position at the University of Wisconsin, Madison

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McMaster University, 1980

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McKay Professor since 2000

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Two PhD students: Scott Rodney and Lu Qi Wang

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Two PhD students: Scott Rodney and Lu Qi Wang

Many posdoctoral researchers: Steve Hofman, Cristian Rios, Alex Iose-vich, Yong Sheng Han, Serban Costea.

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Co-authors

Andersen, Kenneth F.

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Co-authors

Andersen, Kenneth F.Arcozzi, Nicola

2


Co-authors


Baernstein, Albert

2


Co-authors


Baernstein, AlbertChanillo, Sagun

2


Co-authors


Baernstein, AlbertChanillo, SagunChen, Wengu

2


Co-authors



Costea, Serban

2


Co-authors



Costea, SerbanCsima, Jozsef

2


Co-authors



Costea, SerbanCsima, JozsefGuan, Pengfei

2


Co-authors




Han, Yong Sheng

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Co-authors




Han, Yong ShengIosevich, Alexander

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Co-authors





Kerman, Ronald

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Co-authors





Kerman, RonaldLacey, Michael T.

2


Co-authors






Martin-Reyes, Francisco Javier

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Co-authors






Martin-Reyes, Francisco JavierRios, Cristian

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Co-authors







Rochberg, Richard

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Co-authors







Rochberg, RichardSeeger, Andreas

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Co-authors








Uriarte-Tuero, Ignacio

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Co-authors








Uriarte-Tuero, IgnacioWheeden, Richard L.

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Co-authors









Wick, Brett D.

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Co-authors









Wick, Brett D.Zhao, Shi Ying

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More informationNumber of publications: 70

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Areas of research:

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Areas of research:

Convex and discrete geometryFourier analysisFunctional analysisFunctions of a complex variableMeasure and integrationNumber theoryOperator theoryPartial differential equationsPotential theoryReal functionsSeveral complex variablesAnalytic spaces

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Schrodinger operators and the theory of weights

4


Schrodinger operators and the theory of weightsFefferman-Phong problem

4



Let

−∆− V

be a Schrodinger operator with potential V .

4



Let

−∆− V


Consider the first eigenvalue defined by its quadratic, called the Rayleigh quo-tient

λ1 = infg∈C∞o

〈Hg, g〉〈g, g〉

4



Let

−∆− V



λ1 = infg∈C∞o

〈Hg, g〉〈g, g〉

By a formula due to of R. Kerman and E. Sawyer

4



Let

−∆− V



λ1 = infg∈C∞o

〈Hg, g〉〈g, g〉


−λ1 = infλ ≥ 0 : Cλ(V ) ≤ 1,

where Cλ(V ) is the smallest constant satisfying

4



Let

−∆− V



λ1 = infg∈C∞o

〈Hg, g〉〈g, g〉


−λ1 = infλ ≥ 0 : Cλ(V ) ≤ 1,

where Cλ(V ) is the smallest constant satisfying

∫Rng(y)2 V (y)dy ≤ Cλ(V )2

∫Rn

(∇g(y)2 + λ2g(y)2

)dy

4


The trace inequality I

5


The trace inequality IBy taking the Fourier transform, is easy to see that this is equivalent to,

5


The trace inequality IBy taking the Fourier transform, is easy to see that this is equivalent to,∫

Rn|J1,λf |2 V (y)dy ≤ Cλ(V )2

∫Rn|g(y)|2 dy λ > 0

5




∫Rn|g(y)|2 dy λ > 0

where Jα,λ, α, λ > 0 is the Bessel potential defined by

Jα,λf = Kα,λ ? f.

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∫Rn|g(y)|2 dy λ > 0



This is a what is called a trace inequality.

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∫Rn|g(y)|2 dy λ > 0




In other words Cλ(V ) = ‖J1,λ‖L2→L2(V )

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∫Rn|g(y)|2 dy λ > 0





The case λ = 0 is specially interesting and we have

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∫Rn|g(y)|2 dy λ > 0






C0(V ) = ‖I1‖L2→L2(V )

where now Iα , 0 < α < n , is the Riesz potential operator defined by

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∫Rn|g(y)|2 dy λ > 0






C0(V ) = ‖I1‖L2→L2(V )

where now Iα , 0 < α < n , is the Riesz potential operator defined by

Iαf(x) =∫Rn

f(y)

|x− y|n−αdy.

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The trace inequality II

6



Then, is interesting to find conditions on the positive measure dµ such that

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∫RnTΦf(x)p dµ(x) ≤ c

∫Rnf(x)p dx

6





∫Rnf(x)p dx

Kerman and Sawyer related these operators with their dual ones, namely

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∫Rnf(x)p dx


TµΦ(f)(x) =

∫Rn

Φ(x− y)f(y) dµ(y)

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∫Rnf(x)p dx


TµΦ(f)(x) =

∫Rn


and its associated maximal type fractional operators

6





∫Rnf(x)p dx


TµΦ(f)(x) =

∫Rn


and its associated maximal type fractional operators

MµΦ

(f)(x) = supx∈Q

Φ(`(Q))

|Q|

∫Qf(y) dµ(y).

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Kerman-Sawyer’s theorem for the trace inequality

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Φ is a decreasing function and µ a positive measure on Rn.

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Φ is a decreasing function and µ a positive measure on Rn.

theoremT.F.A.E.:i) The trace inequality holds:(∫

RnTΦf(x)p dµ(x)

)1/p≤ c

(∫Rnf(x)p dx

)1/pf ≥ 0

ii) (∫RnTµΦ(χQ)(x)p

′dx

)1/p′

≤ c µ(Q)1/p′ Q ∈ D

iii) (∫QMµΦ

(χQ)(x)p′dx

)1/p′

≤ c µ(Q)1/p′ Q ∈ D

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Some comments about of the proof

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Some comments about of the proofUse of duality:

8



(∫RnTµΦf(x)p

′dx

)1/p′

≤ Cµ(∫

Rnf(x)p

′dµ(x)

)1/p′

f ≥ 0

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(∫RnTµΦf(x)p

′dx

)1/p′

≤ Cµ(∫

Rnf(x)p

′dµ(x)

)1/p′

f ≥ 0

Control by the maximal function:

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(∫RnTµΦf(x)p

′dx

)1/p′

≤ Cµ(∫

Rnf(x)p

′dµ(x)

)1/p′

f ≥ 0


∥∥∥TµΦ(f)∥∥∥Lp′ ≈

∥∥∥MµΦ

(f)∥∥∥Lp′

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(∫RnTµΦf(x)p

′dx

)1/p′

≤ Cµ(∫

Rnf(x)p

′dµ(x)

)1/p′

f ≥ 0


∥∥∥TµΦ(f)∥∥∥Lp′ ≈

∥∥∥MµΦ

(f)∥∥∥Lp′

Proof by good-λ as in the work of Muckenhoupt-Wheeden

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(∫RnTµΦf(x)p

′dx

)1/p′

≤ Cµ(∫

Rnf(x)p

′dµ(x)

)1/p′

f ≥ 0


∥∥∥TµΦ(f)∥∥∥Lp′ ≈

∥∥∥MµΦ

(f)∥∥∥Lp′


Replace cubes by dyadic cubes:

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(∫RnTµΦf(x)p

′dx

)1/p′

≤ Cµ(∫

Rnf(x)p

′dµ(x)

)1/p′

f ≥ 0


∥∥∥TµΦ(f)∥∥∥Lp′ ≈

∥∥∥MµΦ

(f)∥∥∥Lp′



≈∥∥∥Mµ,d

Φ(f)

∥∥∥Lp′

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(∫RnTµΦf(x)p

′dx

)1/p′

≤ Cµ(∫

Rnf(x)p

′dµ(x)

)1/p′

f ≥ 0


∥∥∥TµΦ(f)∥∥∥Lp′ ≈

∥∥∥MµΦ

(f)∥∥∥Lp′



≈∥∥∥Mµ,d

Φ(f)

∥∥∥Lp′

Finally step: prove a testing type condition “a la Sawyer”

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Solution to the Fefferman-Phong problem

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After proving

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After provingCλ(V ) ≈ sup

Q

1

v(Q)

∫QI2(vχQ) vdx

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Q

1

v(Q)

∫QI2(vχQ) vdx

they proved,

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Q

1

v(Q)

∫QI2(vχQ) vdx

they proved,

Theorem (Kerman-Sawyer)There are dimensional constans c, C such that such that the least eigenvalueλ1 of satisfies

Esmall ≤ −λ1 ≤ Elargewhere

Esmall = supQ

|Q|−2/n :

1

v(Q)

∫QI2(vχQ) vdx ≥ C

and

Elarge = supQ

|Q|−2/n :

1

v(Q)

∫QI2(vχQ) vdx ≥ c

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The power bump condition

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In the work of of Fefferman and Phong the functionals

supQ

`(Q)2(

1

|Q|

∫Qvr dx

)1/r

was first considered instead of

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supQ

`(Q)2(

1

|Q|

∫Qvr dx

)1/r


supQ

1

v(Q)

∫QI2(vχQ) vdx ≈ Cλ(V )

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supQ

`(Q)2(

1

|Q|

∫Qvr dx

)1/r


supQ

1

v(Q)


Related work by Chang-Wilson-Wolff

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supQ

`(Q)2(

1

|Q|

∫Qvr dx

)1/r


supQ

1

v(Q)


Related work by Chang-Wilson-WolffReferences:

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supQ

`(Q)2(

1

|Q|

∫Qvr dx

)1/r


supQ

1

v(Q)


Related work by Chang-Wilson-WolffReferences:R. Kerman and E. Sawyer:”Weighted norm inequalities for potentials with applications to Schrodingeroperators, Fourier transforms and Carleson measures”, Bulletin of theA.M.S (1985)

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supQ

`(Q)2(

1

|Q|

∫Qvr dx

)1/r


supQ

1

v(Q)


Related work by Chang-Wilson-WolffReferences:R. Kerman and E. Sawyer:”Weighted norm inequalities for potentials with applications to Schrodingeroperators, Fourier transforms and Carleson measures”, Bulletin of theA.M.S (1985)

”The trace inequality and eigenvalue estimates for Schrodinger opera-tors”, Annales de l’Institut Fourier, 36 (1986).

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Schrodinger operators with change of sign

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Schrodinger operators: H = −∆− V +W , V,W ≥ 0

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where C is the smallest constant satisfying∫Rn|f(y)|2 v(y)dy ≤ C

∫Rn

(|∇f(y)|2 + (w + λ)|f(y)|2

)dy

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∫Rn

(|∇f(y)|2 + (w + λ)|f(y)|2

)dy

In this connection, Eric considered the following functional:

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∫Rn

(|∇f(y)|2 + (w + λ)|f(y)|2

)dy


E(Q, v,w) = minZ(Q, v),v(Q)

w(Q)

where

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∫Rn

(|∇f(y)|2 + (w + λ)|f(y)|2

)dy


E(Q, v,w) = minZ(Q, v),v(Q)

w(Q)

where

Z(Q, v) =1

v(Q)

∫QI2(vχQ) vdx

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The main result

Theorem (Sawyer)If

supQ

E(Q, v,w) <∞

and v and w satisfy some extra condition then

∫Rn|f(y)|2 v(y)dy ≤ C

∫Rn

(|∇f(y)|2 + (w + λ)|f(y)|2

)dy

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The main result

Theorem (Sawyer)If

supQ

E(Q, v,w) <∞



∫Rn

(|∇f(y)|2 + (w + λ)|f(y)|2

)dy

The proof is really complicated!!!

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The main result

Theorem (Sawyer)If

supQ

E(Q, v,w) <∞



∫Rn

(|∇f(y)|2 + (w + λ)|f(y)|2

)dy

The proof is really complicated!!!

Eric Sawyer: A weighted inequality and eigenvalue estimates for Schrodingeroperators”, Indiana Journal of Mathematics, 35, (1986)

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The model example: Muckenhoupt Ap class of weights

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The Hardy–Littlewood maximal function:

Mf(x) = supx∈Q

1

|Q|

∫Q|f(y)| dy.

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The Hardy–Littlewood maximal function:

Mf(x) = supx∈Q

1

|Q|

∫Q|f(y)| dy.

TheoremLet 1 < p <∞, then

M : Lp(w) −→ Lp(w)

if and only if w satisfies the Ap condition:

supQ

(1

|Q|

∫Qw dx

)(1

|Q|

∫Qw1−p′ dx

)p−1<∞

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Two weight theory for the maximal function

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Two weight theory for the maximal functionModel example of Eric “testing philosphy”

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∫Rn

(Mf)p udx ≤ C∫Rn|f |p vdx

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∫Rn


The natural Ap condition

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∫Rn



(Ap) supQ

(1

|Q|

∫Qu dx

)(1

|Q|

∫Qv1−p′ dx

)p−1<∞

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∫Rn



(Ap) supQ

(1

|Q|

∫Qu dx

)(1

|Q|

∫Qv1−p′ dx

)p−1<∞

is NECESSARY but NOT SUFFICIENT

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∫Rn



(Ap) supQ

(1

|Q|

∫Qu dx

)(1

|Q|

∫Qv1−p′ dx

)p−1<∞


(known from the 70’s, Muckenhoupt and Wheeden.)

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∫Rn



(Ap) supQ

(1

|Q|

∫Qu dx

)(1

|Q|

∫Qv1−p′ dx

)p−1<∞


(known from the 70’s, Muckenhoupt and Wheeden.)However:

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∫Rn



(Ap) supQ

(1

|Q|

∫Qu dx

)(1

|Q|

∫Qv1−p′ dx

)p−1<∞


(known from the 70’s, Muckenhoupt and Wheeden.)However:

(u, v) ∈ Ap ⇐⇒ M : Lp(v) −→ Lp,∞(u)

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Eric’s theorem and Eric’s condition

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Theorem [Sawyer] (1981)Let 1 < p <∞ and let (u, v) be a couple of weights, then

∫RnMf(x)p udx ≤ C

∫Rn|f(x)|pvdx

if and only if (u, v) satisfies the Sp condition: there is a constant c s.t.

(Sp)∫QM(χ

Qσ)p udx ≤ c σ(Q)

where σ = v1−p′

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∫Rn|f(x)|pvdx


(Sp)∫QM(χ



He tested this “philosophy” with many other operators.

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∫Rn|f(x)|pvdx


(Sp)∫QM(χ



He tested this “philosophy” with many other operators.

For instance: One sided maximal Hardy–Littlewood maximal function

M+f(x) = suph>0

1

h

∫ x+h

x|f(t)| dt

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First E. Sawyer’s theorem for fractional integrals

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Recall the definition of the classical fractional integral: if 0 < α < n

Iα(f)(x) =∫Rn

f(y)

|x− y|n−αdy.

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Recall the definition of the classical fractional integral: if 0 < α < n

Iα(f)(x) =∫Rn

f(y)

|x− y|n−αdy.

Theorem [Sawyer] (1985)

∥∥∥Iα(f)∥∥∥Lp,∞(u)

≤ c ‖f‖Lp(v)

if and only if (u, v) satisfies the following condition:

∫QIα(uχ

Q)p′σdx ≤ c u(Q)


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Second E. Sawyer’s theorem for fractional integrals

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∫Rn|Iα(f)|p udx ≤ C

∫Rn|f |p vdx

if and only if (u, v) satisfies the Sp condition:

∫QIα(σχ

Q)p udx ≤ c σ(Q) &

∫QIα(uχ



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∫Rn|Iα(f)|p udx ≤ C

∫Rn|f |p vdx

if and only if (u, v) satisfies the Sp condition:

∫QIα(σχ

Q)p udx ≤ c σ(Q) &

∫QIα(uχ



”A characterization of two weight norm inequalities for fractional andPoisson integrals”, Trans. A.M.S. (1988)

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Sawyer-Wheeden two weight bump condition

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The two weight Fefferman-Phong condition in 1992 by Sawyer and Wheedenshowed that the

Theorem [Sawyer-Wheeden] (1992)Suppose that (u, v) be a couple of weights such that for some r > 1

K = supQ

`(Q)α(

1

|Q|

∫Qur dx

)1/r ( 1

|Q|

∫Qv−r dx

)1/r<∞

Then ∫Rn|Iαf(x)|2 u(x)dx ≤ C

∫Rn|f |2v(x)dx.

E. Sawyer and R.L. Wheeden, ”Weighted inequalities for fractional in-tegrals on Euclidean and homogeneous spaces, American Journal ofMathematics, (1992)

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Tips of the proof

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Tips of the proof

Iαf(x) ≈∑Q∈D

|Q|α/n

|Q|

∫Qf(y) dy χQ(x),

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Tips of the proof

Iαf(x) ≈∑Q∈D

|Q|α/n

|Q|

∫Qf(y) dy χQ(x),

and then after “averaging”

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Tips of the proof

Iαf(x) ≈∑Q∈D

|Q|α/n

|Q|

∫Qf(y) dy χQ(x),


Iαf(x) ≈∑k,j

|Qj,k|α/n

|Qj,k|

∫Qj,k

f(y) dy χQj,k(x),

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Tips of the proof

Iαf(x) ≈∑Q∈D

|Q|α/n

|Q|

∫Qf(y) dy χQ(x),


Iαf(x) ≈∑k,j

|Qj,k|α/n

|Qj,k|

∫Qj,k

f(y) dy χQj,k(x),

where the family Qj,k is formed by special dyadic Calderon-Zyygmund cubes.

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Some comments

• This and other similar results were extended to spaces of homogeneoustype.

20


Some comments


• The case when α = 0 corresponds to the Hilbert transform is due to Neuge-bauer.

20


Some comments



• Improves: Chang-Wilson-Wolff and Fefferman-Phong

20


Some comments




• The failure of the Besicovitch covering lemma for the Heisenberg group wasobtained (also independently by Koranyi and Reimann).

20


Some comments





• A construction of a dyadic grid for spaces of homogeneous type was given(also independently by Christ, see as well David)

20


Some comments





• A construction of a dyadic grid for spaces of homogeneous type was given(also independently by Christ, see as well David)

• later with Cruz-Uribe and Martell, obtained nice results for singular integralsin this vein.

20


Mixed weak type inequalities and Eric’s conjecture

21



This result of E. Sawyer can be seen as a very sophisticated version of theclassical weak type (1,1) estimate for the Hardy-Littlewood maximal function.

21




Theorem (E. Sawyer) Let u, v ∈ A1(R), then there is a constant c suchthat, ∥∥∥M(f)

v

∥∥∥L1,∞(uv)

≤ c∫R|f(x)|udx.

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v

∥∥∥L1,∞(uv)

≤ c∫R|f(x)|udx.

E. Sawyer, “A weighted weak type inequality for the maximal function”,Proceedings of the A.M.S., 93(1985).

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v

∥∥∥L1,∞(uv)

≤ c∫R|f(x)|udx.

E. Sawyer, “A weighted weak type inequality for the maximal function”,Proceedings of the A.M.S., 93(1985).• Of course, the case v = u = 1 is the classical unweighted result.

21





v

∥∥∥L1,∞(uv)

≤ c∫R|f(x)|udx.

E. Sawyer, “A weighted weak type inequality for the maximal function”,Proceedings of the A.M.S., 93(1985).• Of course, the case v = u = 1 is the classical unweighted result.• The case v = 1 is the classical Fefferman-Stein’s inequality and the caseu = 1 is also immediate:

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v

∥∥∥L1,∞(uv)

≤ c∫R|f(x)|udx.


M(f)

v≤ CMv(

f

v)

21





v

∥∥∥L1,∞(uv)

≤ c∫R|f(x)|udx.


M(f)

v≤ CMv(

f

v)

• The problem in general is that the product uv can be very singular...

21





v

∥∥∥L1,∞(uv)

≤ c∫R|f(x)|udx.


M(f)

v≤ CMv(

f

v)

• The problem in general is that the product uv can be very singular...• To assume that u ∈ A1 is natural because of the case v = 1.

21


Why Eric look for such a sophisticated a result?

22



To give a different proof of Muckenhoupt’s Ap theorem assuming that theweights are already factored:

w = uv1−p u, v ∈ A1.

22




w = uv1−p u, v ∈ A1.

Indeed, let

22




w = uv1−p u, v ∈ A1.

Indeed, let

Sv(f) =M(f v)

v

22




w = uv1−p u, v ∈ A1.

Indeed, let

Sv(f) =M(f v)

v

Then, Eric’s result says that Sv is of weak type (1,1) w. r. to the measureuv:

Sv : L1(uv)→ L1,∞(uv)

22




w = uv1−p u, v ∈ A1.

Indeed, let

Sv(f) =M(f v)

v


Sv : L1(uv)→ L1,∞(uv)

But observe that Sv is also bounded on L∞, (v ∈ A1 ) and hence onL∞(uv).

22




w = uv1−p u, v ∈ A1.

Indeed, let

Sv(f) =M(f v)

v


Sv : L1(uv)→ L1,∞(uv)


Then by the Marcinkiewicz interpolation theorem:

22




w = uv1−p u, v ∈ A1.

Indeed, let

Sv(f) =M(f v)

v


Sv : L1(uv)→ L1,∞(uv)


Then by the Marcinkiewicz interpolation theorem:

M : Lp(uv1−p)→ Lp(uv1−p)

22


Two conjectures by Eric

23



• First conjecture: to replace M by the Hilbert transform H , namely

23




∥∥∥H(fv)

v

∥∥∥L1,∞(uv)

≤ c∫R|f(x)|uv dx

23




∥∥∥H(fv)

v

∥∥∥L1,∞(uv)


• Second conjecture: to replace M in one dimension by M in Rn, namely

23




∥∥∥H(fv)

v

∥∥∥L1,∞(uv)


• Second conjecture: to replace M in one dimension by M in Rn, namely

∥∥∥M(fv)

v

∥∥∥L1,∞(uv)

≤ c∫Rn|f(x)|uv dx

23


Results

24


Results

Joint work with J. M. Martell and D. Cruz-Uribe

24


Results

Joint work with J. M. Martell and D. Cruz-UribeT is any Calderon-Zygmund .

24


Results


Theorem If u ∈ A1(Rn), and v ∈ A1(Rn), then there is a constant c suchthat ∥∥∥M(fv)

v

∥∥∥L1,∞(uv)


and ∥∥∥T (fv)

v

∥∥∥L1,∞(uv)


24


Results



v

∥∥∥L1,∞(uv)


and ∥∥∥T (fv)

v

∥∥∥L1,∞(uv)


and also

24


Results



v

∥∥∥L1,∞(uv)


and ∥∥∥T (fv)

v

∥∥∥L1,∞(uv)


and also

Theorem If u ∈ A1(Rn), and v ∈ A∞(u), then the same conclusion holds.

24


Results



v

∥∥∥L1,∞(uv)


and ∥∥∥T (fv)

v

∥∥∥L1,∞(uv)


and also


Example: u = 1 and v ∈ A∞

24


Results



v

∥∥∥L1,∞(uv)


and ∥∥∥T (fv)

v

∥∥∥L1,∞(uv)


and also


Example: u = 1 and v ∈ A∞Some special cases were considered in the 70’s by Muckenhoupt-Wheedenfor the Hilbert Transform.

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Key Lemma

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Key Lemma

Our approach is completely different

25


Key Lemma


We found a sort of “mechanism” (a very general extrapolation type theorem) toreduce everything to prove the corresponding estimate for the dyadic maximaloperator.

25


Key Lemma



Lemma Let u ∈ A1(Rn) and v ∈ A∞(Rn), then∥∥∥T (fv)

v

∥∥∥L1,∞(uv)

≤ c∥∥∥Md(fv)

v

∥∥∥L1,∞(uv)

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Key Lemma



Lemma Let u ∈ A1(Rn) and v ∈ A∞(Rn), then∥∥∥T (fv)

v

∥∥∥L1,∞(uv)

≤ c∥∥∥Md(fv)

v

∥∥∥L1,∞(uv)

In fact this results holds for non A∞ weights on v.

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The final key inequality and Sawyer’s conjecture

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The final key inequality and Sawyer’s conjectureThen the proof of the Theorem is reduced to prove a Sawyer´s type esti-mate for the dyadic maximal operator but in Rn:

26



Theorem If u ∈ A1(Rn), and v ∈ A1(Rn), then there is a constant c suchthat ∥∥∥Md(fv)

v

∥∥∥L1,∞(uv)


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v

∥∥∥L1,∞(uv)


We were very lucky because the proof is basically the same as in theEric’s paper, we have reduced everything to the dyadic and is a bit sim-pler but

26




v

∥∥∥L1,∞(uv)


We were very lucky because the proof is basically the same as in theEric’s paper, we have reduced everything to the dyadic and is a bit sim-pler butWe DON’T UNDERSTAND IT

26




v

∥∥∥L1,∞(uv)


We were very lucky because the proof is basically the same as in theEric’s paper, we have reduced everything to the dyadic and is a bit sim-pler butWe DON’T UNDERSTAND ITWe are not happy neither with the proof nor with the result.

26




v

∥∥∥L1,∞(uv)


We were very lucky because the proof is basically the same as in theEric’s paper, we have reduced everything to the dyadic and is a bit sim-pler butWe DON’T UNDERSTAND ITWe are not happy neither with the proof nor with the result.

Sawyer’s conjecture: The result should hold in the following case

u ∈ A1(Rn) & v ∈ A∞(Rn)

which corresponds to the most singular case.

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Proof: the two basic Ingredients

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• The Coifman-Fefferman’s estimate: if 0 < p <∞ and w ∈ A∞

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Let T be any Calderon-Zygmund Operator and let 0 < p <∞ and w ∈ A∞.Then

‖Tf‖Lp(w)

≤ c ‖Mf‖Lp(w)

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‖Tf‖Lp(w)

≤ c ‖Mf‖Lp(w)

• and the use of the so called Rubio de Francia’s algorithm for a non standardoperator, namely Eric’s operator:

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‖Tf‖Lp(w)

≤ c ‖Mf‖Lp(w)


Rh(x) =∞∑k=0

Skvh(x)

2k ‖Sv‖k.

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‖Tf‖Lp(w)

≤ c ‖Mf‖Lp(w)


Rh(x) =∞∑k=0

Skvh(x)

2k ‖Sv‖k.

where

Svf(x) =M(fv)(x)

v(x)

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Singular Integrals and the Cp class of weights

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Motivation: to understand the following version of the theorem of Coifman-Fefferman.

28




Let H be the Hilbert transform let 1 < p <∞ and w ∈ A∞.Then ∫

R|Hf |pw dx ≤ C

∫R

(Mf)pw dx

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R|Hf |pw dx ≤ C

∫R

(Mf)pw dx

This question was raised by Muckenhoupt in the 70’s. He showed that if theweight satisfies this estimate then:

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R|Hf |pw dx ≤ C

∫R

(Mf)pw dx

This question was raised by Muckenhoupt in the 70’s. He showed that if theweight satisfies this estimate then:

ω(E) ≤ C(|E||Q|

)ε ∫R

(M(χQ))pw dx

for any cube Q and for any set E ⊂ Q.

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The Cp condition: necessity

29



Rj will denote the Riesz transforms.

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Rj will denote the Riesz transforms.

Theorem (Eric)Let 1 < p <∞ . If the weight w satisfies

∫Rn|Rjf |pw dx ≤ C

∫Rn

(Mf)pw dx, 1 ≤ j ≤ n, f ∈ L∞c

Then w satifies the Cp condition:

ω(E) ≤ C(|E||Q|

)ε ∫Rn

(M(χQ))pw dx

for any cube Q and for any set E ⊂ Q.

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The Cp condition: sufficiency

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The Cp condition: sufficiency

Theorem (Eric)Let 1 < p <∞ and let ε > 0. Let T be a Singular Integral, then if weight wsatisfies the Cp+δ condition:

ω(E) ≤ C(|E||Q|

)ε ∫Rn

(M(χQ))p+δ w dx

for any cube Q and for any set E ⊂ Q.Then ∫

Rn|Tf |pw dx ≤ C

∫Rn

(Mf)pw dx.

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The Monge-Ampere Equation

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uxxuyy − u2xy = k(x, y)

near the origin with 0 ≤ k(x, y) ∈ C∞(R2).

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near the origin with 0 ≤ k(x, y) ∈ C∞(R2).Then under minimal assumption on k and a minimal initial assumption on uthen u ∈ C∞ near the origin.

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near the origin with 0 ≤ k(x, y) ∈ C∞(R2).Then under minimal assumption on k and a minimal initial assumption on uthen u ∈ C∞ near the origin.This theorem is due to P. Guan.

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near the origin with 0 ≤ k(x, y) ∈ C∞(R2).Then under minimal assumption on k and a minimal initial assumption on uthen u ∈ C∞ near the origin.This theorem is due to P. Guan.Improved in joint work with Eric (TAMS)

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near the origin with 0 ≤ k(x, y) ∈ C∞(R2).Then under minimal assumption on k and a minimal initial assumption on uthen u ∈ C∞ near the origin.This theorem is due to P. Guan.Improved in joint work with Eric (TAMS)

Theorem [Rios, Sawyer and Wheeden] (2008)Let n ≥ 3 and suppose k ≈ |x|2m can be written as a sum of squares ofsmooth functions in Ω ⊂ Rn . If u is a C2 convex solution u to the subellipticMonge-Ampere equation

detD2u(x) = k(x, u,Du) x ∈ Ω,

then u is smooth if the elementary (n − 1)st symmetric curvature kn−1 of uis positive (the case m ≥ 2 uses an additional nondegeneracy condition onthe sum of squares).

Advances in Math 2008

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Comments

32


Comments

As a conseqence they obtain the following geometric result: a C2 convex func-tion u whose graph has smooth Gaussian curvature k ≈ |x|2 is itself smoothif and only if the subGaussian curvature kn−1 of u is positive in Ω.

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future

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future

I am sure that Eric is going to produce more mathematics for us!!!

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future

I am sure that Eric is going to produce more mathematics for us!!!

THANK YOU VERYMUCH

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“a tribute to eric sawyer” carlos perez´ · cv b.sc. mcgill university, mathematics 1973 ph.d....

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