“a tribute to eric sawyer” carlos perez´ · cv b.sc. mcgill university, mathematics 1973 ph.d....
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“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 carlosperez@us.es
cv
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Some theorems by E. Sawyer carlosperez@us.es
cv
B.Sc. McGill University, Mathematics 1973
1
Some theorems by E. Sawyer carlosperez@us.es
cv
B.Sc. McGill University, Mathematics 1973
Ph.D. McGill University, Mathematics 1977
1
Some theorems by E. Sawyer carlosperez@us.es
cv
B.Sc. McGill University, Mathematics 1973
Ph.D. McGill University, Mathematics 1977
Title of his thesis:
1
Some theorems by E. Sawyer carlosperez@us.es
cv
B.Sc. McGill University, Mathematics 1973
Ph.D. McGill University, Mathematics 1977
Title of his thesis:
“Inner Functions on Polydiscs”
1
Some theorems by E. Sawyer carlosperez@us.es
cv
B.Sc. McGill University, Mathematics 1973
Ph.D. McGill University, Mathematics 1977
Title of his thesis:
“Inner Functions on Polydiscs”
Advisor: Kohur N. Gowrisankaran
1
Some theorems by E. Sawyer carlosperez@us.es
cv
B.Sc. McGill University, Mathematics 1973
Ph.D. McGill University, Mathematics 1977
Title of his thesis:
“Inner Functions on Polydiscs”
Advisor: Kohur N. Gowrisankaran
Posdoctoral position at the University of Wisconsin, Madison
1
Some theorems by E. Sawyer carlosperez@us.es
cv
B.Sc. McGill University, Mathematics 1973
Ph.D. McGill University, Mathematics 1977
Title of his thesis:
“Inner Functions on Polydiscs”
Advisor: Kohur N. Gowrisankaran
Posdoctoral position at the University of Wisconsin, Madison
McMaster University, 1980
1
Some theorems by E. Sawyer carlosperez@us.es
cv
B.Sc. McGill University, Mathematics 1973
Ph.D. McGill University, Mathematics 1977
Title of his thesis:
“Inner Functions on Polydiscs”
Advisor: Kohur N. Gowrisankaran
Posdoctoral position at the University of Wisconsin, Madison
McMaster University, 1980
McKay Professor since 2000
1
Some theorems by E. Sawyer carlosperez@us.es
cv
B.Sc. McGill University, Mathematics 1973
Ph.D. McGill University, Mathematics 1977
Title of his thesis:
“Inner Functions on Polydiscs”
Advisor: Kohur N. Gowrisankaran
Posdoctoral position at the University of Wisconsin, Madison
McMaster University, 1980
McKay Professor since 2000
Two PhD students: Scott Rodney and Lu Qi Wang
1
Some theorems by E. Sawyer carlosperez@us.es
cv
B.Sc. McGill University, Mathematics 1973
Ph.D. McGill University, Mathematics 1977
Title of his thesis:
“Inner Functions on Polydiscs”
Advisor: Kohur N. Gowrisankaran
Posdoctoral position at the University of Wisconsin, Madison
McMaster University, 1980
McKay Professor since 2000
Two PhD students: Scott Rodney and Lu Qi Wang
Many posdoctoral researchers: Steve Hofman, Cristian Rios, Alex Iose-vich, Yong Sheng Han, Serban Costea.
1
Some theorems by E. Sawyer carlosperez@us.es
Co-authors
Andersen, Kenneth F.
2
Some theorems by E. Sawyer carlosperez@us.es
Co-authors
Andersen, Kenneth F.Arcozzi, Nicola
2
Some theorems by E. Sawyer carlosperez@us.es
Co-authors
Andersen, Kenneth F.Arcozzi, Nicola
Baernstein, Albert
2
Some theorems by E. Sawyer carlosperez@us.es
Co-authors
Andersen, Kenneth F.Arcozzi, Nicola
Baernstein, AlbertChanillo, Sagun
2
Some theorems by E. Sawyer carlosperez@us.es
Co-authors
Andersen, Kenneth F.Arcozzi, Nicola
Baernstein, AlbertChanillo, SagunChen, Wengu
2
Some theorems by E. Sawyer carlosperez@us.es
Co-authors
Andersen, Kenneth F.Arcozzi, Nicola
Baernstein, AlbertChanillo, SagunChen, Wengu
Costea, Serban
2
Some theorems by E. Sawyer carlosperez@us.es
Co-authors
Andersen, Kenneth F.Arcozzi, Nicola
Baernstein, AlbertChanillo, SagunChen, Wengu
Costea, SerbanCsima, Jozsef
2
Some theorems by E. Sawyer carlosperez@us.es
Co-authors
Andersen, Kenneth F.Arcozzi, Nicola
Baernstein, AlbertChanillo, SagunChen, Wengu
Costea, SerbanCsima, JozsefGuan, Pengfei
2
Some theorems by E. Sawyer carlosperez@us.es
Co-authors
Andersen, Kenneth F.Arcozzi, Nicola
Baernstein, AlbertChanillo, SagunChen, Wengu
Costea, SerbanCsima, JozsefGuan, Pengfei
Han, Yong Sheng
2
Some theorems by E. Sawyer carlosperez@us.es
Co-authors
Andersen, Kenneth F.Arcozzi, Nicola
Baernstein, AlbertChanillo, SagunChen, Wengu
Costea, SerbanCsima, JozsefGuan, Pengfei
Han, Yong ShengIosevich, Alexander
2
Some theorems by E. Sawyer carlosperez@us.es
Co-authors
Andersen, Kenneth F.Arcozzi, Nicola
Baernstein, AlbertChanillo, SagunChen, Wengu
Costea, SerbanCsima, JozsefGuan, Pengfei
Han, Yong ShengIosevich, Alexander
Kerman, Ronald
2
Some theorems by E. Sawyer carlosperez@us.es
Co-authors
Andersen, Kenneth F.Arcozzi, Nicola
Baernstein, AlbertChanillo, SagunChen, Wengu
Costea, SerbanCsima, JozsefGuan, Pengfei
Han, Yong ShengIosevich, Alexander
Kerman, RonaldLacey, Michael T.
2
Some theorems by E. Sawyer carlosperez@us.es
Co-authors
Andersen, Kenneth F.Arcozzi, Nicola
Baernstein, AlbertChanillo, SagunChen, Wengu
Costea, SerbanCsima, JozsefGuan, Pengfei
Han, Yong ShengIosevich, Alexander
Kerman, RonaldLacey, Michael T.
Martin-Reyes, Francisco Javier
2
Some theorems by E. Sawyer carlosperez@us.es
Co-authors
Andersen, Kenneth F.Arcozzi, Nicola
Baernstein, AlbertChanillo, SagunChen, Wengu
Costea, SerbanCsima, JozsefGuan, Pengfei
Han, Yong ShengIosevich, Alexander
Kerman, RonaldLacey, Michael T.
Martin-Reyes, Francisco JavierRios, Cristian
2
Some theorems by E. Sawyer carlosperez@us.es
Co-authors
Andersen, Kenneth F.Arcozzi, Nicola
Baernstein, AlbertChanillo, SagunChen, Wengu
Costea, SerbanCsima, JozsefGuan, Pengfei
Han, Yong ShengIosevich, Alexander
Kerman, RonaldLacey, Michael T.
Martin-Reyes, Francisco JavierRios, Cristian
Rochberg, Richard
2
Some theorems by E. Sawyer carlosperez@us.es
Co-authors
Andersen, Kenneth F.Arcozzi, Nicola
Baernstein, AlbertChanillo, SagunChen, Wengu
Costea, SerbanCsima, JozsefGuan, Pengfei
Han, Yong ShengIosevich, Alexander
Kerman, RonaldLacey, Michael T.
Martin-Reyes, Francisco JavierRios, Cristian
Rochberg, RichardSeeger, Andreas
2
Some theorems by E. Sawyer carlosperez@us.es
Co-authors
Andersen, Kenneth F.Arcozzi, Nicola
Baernstein, AlbertChanillo, SagunChen, Wengu
Costea, SerbanCsima, JozsefGuan, Pengfei
Han, Yong ShengIosevich, Alexander
Kerman, RonaldLacey, Michael T.
Martin-Reyes, Francisco JavierRios, Cristian
Rochberg, RichardSeeger, Andreas
Uriarte-Tuero, Ignacio
2
Some theorems by E. Sawyer carlosperez@us.es
Co-authors
Andersen, Kenneth F.Arcozzi, Nicola
Baernstein, AlbertChanillo, SagunChen, Wengu
Costea, SerbanCsima, JozsefGuan, Pengfei
Han, Yong ShengIosevich, Alexander
Kerman, RonaldLacey, Michael T.
Martin-Reyes, Francisco JavierRios, Cristian
Rochberg, RichardSeeger, Andreas
Uriarte-Tuero, IgnacioWheeden, Richard L.
2
Some theorems by E. Sawyer carlosperez@us.es
Co-authors
Andersen, Kenneth F.Arcozzi, Nicola
Baernstein, AlbertChanillo, SagunChen, Wengu
Costea, SerbanCsima, JozsefGuan, Pengfei
Han, Yong ShengIosevich, Alexander
Kerman, RonaldLacey, Michael T.
Martin-Reyes, Francisco JavierRios, Cristian
Rochberg, RichardSeeger, Andreas
Uriarte-Tuero, IgnacioWheeden, Richard L.
Wick, Brett D.
2
Some theorems by E. Sawyer carlosperez@us.es
Co-authors
Andersen, Kenneth F.Arcozzi, Nicola
Baernstein, AlbertChanillo, SagunChen, Wengu
Costea, SerbanCsima, JozsefGuan, Pengfei
Han, Yong ShengIosevich, Alexander
Kerman, RonaldLacey, Michael T.
Martin-Reyes, Francisco JavierRios, Cristian
Rochberg, RichardSeeger, Andreas
Uriarte-Tuero, IgnacioWheeden, Richard L.
Wick, Brett D.Zhao, Shi Ying
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Some theorems by E. Sawyer carlosperez@us.es
More informationNumber of publications: 70
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Some theorems by E. Sawyer carlosperez@us.es
More informationNumber of publications: 70
Areas of research:
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Some theorems by E. Sawyer carlosperez@us.es
More informationNumber of publications: 70
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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Some theorems by E. Sawyer carlosperez@us.es
Schrodinger operators and the theory of weights
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Some theorems by E. Sawyer carlosperez@us.es
Schrodinger operators and the theory of weightsFefferman-Phong problem
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Some theorems by E. Sawyer carlosperez@us.es
Schrodinger operators and the theory of weightsFefferman-Phong problem
Let
−∆− V
be a Schrodinger operator with potential V .
4
Some theorems by E. Sawyer carlosperez@us.es
Schrodinger operators and the theory of weightsFefferman-Phong problem
Let
−∆− V
be a Schrodinger operator with potential V .
Consider the first eigenvalue defined by its quadratic, called the Rayleigh quo-tient
λ1 = infg∈C∞o
〈Hg, g〉〈g, g〉
4
Some theorems by E. Sawyer carlosperez@us.es
Schrodinger operators and the theory of weightsFefferman-Phong problem
Let
−∆− V
be a Schrodinger operator with potential V .
Consider the first eigenvalue defined by its quadratic, called the Rayleigh quo-tient
λ1 = infg∈C∞o
〈Hg, g〉〈g, g〉
By a formula due to of R. Kerman and E. Sawyer
4
Some theorems by E. Sawyer carlosperez@us.es
Schrodinger operators and the theory of weightsFefferman-Phong problem
Let
−∆− V
be a Schrodinger operator with potential V .
Consider the first eigenvalue defined by its quadratic, called the Rayleigh quo-tient
λ1 = infg∈C∞o
〈Hg, g〉〈g, g〉
By a formula due to of R. Kerman and E. Sawyer
−λ1 = infλ ≥ 0 : Cλ(V ) ≤ 1,
where Cλ(V ) is the smallest constant satisfying
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Some theorems by E. Sawyer carlosperez@us.es
Schrodinger operators and the theory of weightsFefferman-Phong problem
Let
−∆− V
be a Schrodinger operator with potential V .
Consider the first eigenvalue defined by its quadratic, called the Rayleigh quo-tient
λ1 = infg∈C∞o
〈Hg, g〉〈g, g〉
By a formula due to of R. Kerman and E. Sawyer
−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
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Some theorems by E. Sawyer carlosperez@us.es
The trace inequality I
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Some theorems by E. Sawyer carlosperez@us.es
The trace inequality IBy taking the Fourier transform, is easy to see that this is equivalent to,
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Some theorems by E. Sawyer carlosperez@us.es
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
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Some theorems by E. Sawyer carlosperez@us.es
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
where Jα,λ, α, λ > 0 is the Bessel potential defined by
Jα,λf = Kα,λ ? f.
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Some theorems by E. Sawyer carlosperez@us.es
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
where Jα,λ, α, λ > 0 is the Bessel potential defined by
Jα,λf = Kα,λ ? f.
This is a what is called a trace inequality.
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Some theorems by E. Sawyer carlosperez@us.es
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
where Jα,λ, α, λ > 0 is the Bessel potential defined by
Jα,λf = Kα,λ ? f.
This is a what is called a trace inequality.
In other words Cλ(V ) = ‖J1,λ‖L2→L2(V )
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Some theorems by E. Sawyer carlosperez@us.es
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
where Jα,λ, α, λ > 0 is the Bessel potential defined by
Jα,λf = Kα,λ ? f.
This is a what is called a trace inequality.
In other words Cλ(V ) = ‖J1,λ‖L2→L2(V )
The case λ = 0 is specially interesting and we have
5
Some theorems by E. Sawyer carlosperez@us.es
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
where Jα,λ, α, λ > 0 is the Bessel potential defined by
Jα,λf = Kα,λ ? f.
This is a what is called a trace inequality.
In other words Cλ(V ) = ‖J1,λ‖L2→L2(V )
The case λ = 0 is specially interesting and we have
C0(V ) = ‖I1‖L2→L2(V )
where now Iα , 0 < α < n , is the Riesz potential operator defined by
5
Some theorems by E. Sawyer carlosperez@us.es
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
where Jα,λ, α, λ > 0 is the Bessel potential defined by
Jα,λf = Kα,λ ? f.
This is a what is called a trace inequality.
In other words Cλ(V ) = ‖J1,λ‖L2→L2(V )
The case λ = 0 is specially interesting and we have
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.
5
Some theorems by E. Sawyer carlosperez@us.es
The trace inequality II
6
Some theorems by E. Sawyer carlosperez@us.es
The trace inequality II
Then, is interesting to find conditions on the positive measure dµ such that
6
Some theorems by E. Sawyer carlosperez@us.es
The trace inequality II
Then, is interesting to find conditions on the positive measure dµ such that
∫RnTΦf(x)p dµ(x) ≤ c
∫Rnf(x)p dx
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Some theorems by E. Sawyer carlosperez@us.es
The trace inequality II
Then, is interesting to find conditions on the positive measure dµ such that
∫RnTΦf(x)p dµ(x) ≤ c
∫Rnf(x)p dx
Kerman and Sawyer related these operators with their dual ones, namely
6
Some theorems by E. Sawyer carlosperez@us.es
The trace inequality II
Then, is interesting to find conditions on the positive measure dµ such that
∫RnTΦf(x)p dµ(x) ≤ c
∫Rnf(x)p dx
Kerman and Sawyer related these operators with their dual ones, namely
TµΦ(f)(x) =
∫Rn
Φ(x− y)f(y) dµ(y)
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Some theorems by E. Sawyer carlosperez@us.es
The trace inequality II
Then, is interesting to find conditions on the positive measure dµ such that
∫RnTΦf(x)p dµ(x) ≤ c
∫Rnf(x)p dx
Kerman and Sawyer related these operators with their dual ones, namely
TµΦ(f)(x) =
∫Rn
Φ(x− y)f(y) dµ(y)
and its associated maximal type fractional operators
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Some theorems by E. Sawyer carlosperez@us.es
The trace inequality II
Then, is interesting to find conditions on the positive measure dµ such that
∫RnTΦf(x)p dµ(x) ≤ c
∫Rnf(x)p dx
Kerman and Sawyer related these operators with their dual ones, namely
TµΦ(f)(x) =
∫Rn
Φ(x− y)f(y) dµ(y)
and its associated maximal type fractional operators
MµΦ
(f)(x) = supx∈Q
Φ(`(Q))
|Q|
∫Qf(y) dµ(y).
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Some theorems by E. Sawyer carlosperez@us.es
Kerman-Sawyer’s theorem for the trace inequality
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Some theorems by E. Sawyer carlosperez@us.es
Kerman-Sawyer’s theorem for the trace inequality
Φ is a decreasing function and µ a positive measure on Rn.
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Some theorems by E. Sawyer carlosperez@us.es
Kerman-Sawyer’s theorem for the trace inequality
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 theorems by E. Sawyer carlosperez@us.es
Some comments about of the proof
8
Some theorems by E. Sawyer carlosperez@us.es
Some comments about of the proofUse of duality:
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Some theorems by E. Sawyer carlosperez@us.es
Some comments about of the proofUse of duality:
(∫RnTµΦf(x)p
′dx
)1/p′
≤ Cµ(∫
Rnf(x)p
′dµ(x)
)1/p′
f ≥ 0
8
Some theorems by E. Sawyer carlosperez@us.es
Some comments about of the proofUse of duality:
(∫RnTµΦf(x)p
′dx
)1/p′
≤ Cµ(∫
Rnf(x)p
′dµ(x)
)1/p′
f ≥ 0
Control by the maximal function:
8
Some theorems by E. Sawyer carlosperez@us.es
Some comments about of the proofUse of duality:
(∫RnTµΦf(x)p
′dx
)1/p′
≤ Cµ(∫
Rnf(x)p
′dµ(x)
)1/p′
f ≥ 0
Control by the maximal function:
∥∥∥TµΦ(f)∥∥∥Lp′ ≈
∥∥∥MµΦ
(f)∥∥∥Lp′
8
Some theorems by E. Sawyer carlosperez@us.es
Some comments about of the proofUse of duality:
(∫RnTµΦf(x)p
′dx
)1/p′
≤ Cµ(∫
Rnf(x)p
′dµ(x)
)1/p′
f ≥ 0
Control by the maximal function:
∥∥∥TµΦ(f)∥∥∥Lp′ ≈
∥∥∥MµΦ
(f)∥∥∥Lp′
Proof by good-λ as in the work of Muckenhoupt-Wheeden
8
Some theorems by E. Sawyer carlosperez@us.es
Some comments about of the proofUse of duality:
(∫RnTµΦf(x)p
′dx
)1/p′
≤ Cµ(∫
Rnf(x)p
′dµ(x)
)1/p′
f ≥ 0
Control by the maximal function:
∥∥∥TµΦ(f)∥∥∥Lp′ ≈
∥∥∥MµΦ
(f)∥∥∥Lp′
Proof by good-λ as in the work of Muckenhoupt-Wheeden
Replace cubes by dyadic cubes:
8
Some theorems by E. Sawyer carlosperez@us.es
Some comments about of the proofUse of duality:
(∫RnTµΦf(x)p
′dx
)1/p′
≤ Cµ(∫
Rnf(x)p
′dµ(x)
)1/p′
f ≥ 0
Control by the maximal function:
∥∥∥TµΦ(f)∥∥∥Lp′ ≈
∥∥∥MµΦ
(f)∥∥∥Lp′
Proof by good-λ as in the work of Muckenhoupt-Wheeden
Replace cubes by dyadic cubes:
≈∥∥∥Mµ,d
Φ(f)
∥∥∥Lp′
8
Some theorems by E. Sawyer carlosperez@us.es
Some comments about of the proofUse of duality:
(∫RnTµΦf(x)p
′dx
)1/p′
≤ Cµ(∫
Rnf(x)p
′dµ(x)
)1/p′
f ≥ 0
Control by the maximal function:
∥∥∥TµΦ(f)∥∥∥Lp′ ≈
∥∥∥MµΦ
(f)∥∥∥Lp′
Proof by good-λ as in the work of Muckenhoupt-Wheeden
Replace cubes by dyadic cubes:
≈∥∥∥Mµ,d
Φ(f)
∥∥∥Lp′
Finally step: prove a testing type condition “a la Sawyer”
8
Some theorems by E. Sawyer carlosperez@us.es
Solution to the Fefferman-Phong problem
9
Some theorems by E. Sawyer carlosperez@us.es
Solution to the Fefferman-Phong problem
After proving
9
Some theorems by E. Sawyer carlosperez@us.es
Solution to the Fefferman-Phong problem
After provingCλ(V ) ≈ sup
Q
1
v(Q)
∫QI2(vχQ) vdx
9
Some theorems by E. Sawyer carlosperez@us.es
Solution to the Fefferman-Phong problem
After provingCλ(V ) ≈ sup
Q
1
v(Q)
∫QI2(vχQ) vdx
they proved,
9
Some theorems by E. Sawyer carlosperez@us.es
Solution to the Fefferman-Phong problem
After provingCλ(V ) ≈ sup
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
9
Some theorems by E. Sawyer carlosperez@us.es
The power bump condition
10
Some theorems by E. Sawyer carlosperez@us.es
The power bump condition
In the work of of Fefferman and Phong the functionals
supQ
`(Q)2(
1
|Q|
∫Qvr dx
)1/r
was first considered instead of
10
Some theorems by E. Sawyer carlosperez@us.es
The power bump condition
In the work of of Fefferman and Phong the functionals
supQ
`(Q)2(
1
|Q|
∫Qvr dx
)1/r
was first considered instead of
supQ
1
v(Q)
∫QI2(vχQ) vdx ≈ Cλ(V )
10
Some theorems by E. Sawyer carlosperez@us.es
The power bump condition
In the work of of Fefferman and Phong the functionals
supQ
`(Q)2(
1
|Q|
∫Qvr dx
)1/r
was first considered instead of
supQ
1
v(Q)
∫QI2(vχQ) vdx ≈ Cλ(V )
Related work by Chang-Wilson-Wolff
10
Some theorems by E. Sawyer carlosperez@us.es
The power bump condition
In the work of of Fefferman and Phong the functionals
supQ
`(Q)2(
1
|Q|
∫Qvr dx
)1/r
was first considered instead of
supQ
1
v(Q)
∫QI2(vχQ) vdx ≈ Cλ(V )
Related work by Chang-Wilson-WolffReferences:
10
Some theorems by E. Sawyer carlosperez@us.es
The power bump condition
In the work of of Fefferman and Phong the functionals
supQ
`(Q)2(
1
|Q|
∫Qvr dx
)1/r
was first considered instead of
supQ
1
v(Q)
∫QI2(vχQ) vdx ≈ Cλ(V )
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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Some theorems by E. Sawyer carlosperez@us.es
The power bump condition
In the work of of Fefferman and Phong the functionals
supQ
`(Q)2(
1
|Q|
∫Qvr dx
)1/r
was first considered instead of
supQ
1
v(Q)
∫QI2(vχQ) vdx ≈ Cλ(V )
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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Some theorems by E. Sawyer carlosperez@us.es
Schrodinger operators with change of sign
11
Some theorems by E. Sawyer carlosperez@us.es
Schrodinger operators with change of sign
Schrodinger operators: H = −∆− V +W , V,W ≥ 0
11
Some theorems by E. Sawyer carlosperez@us.es
Schrodinger operators with change of sign
Schrodinger operators: H = −∆− V +W , V,W ≥ 0
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
11
Some theorems by E. Sawyer carlosperez@us.es
Schrodinger operators with change of sign
Schrodinger operators: H = −∆− V +W , V,W ≥ 0
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
In this connection, Eric considered the following functional:
11
Some theorems by E. Sawyer carlosperez@us.es
Schrodinger operators with change of sign
Schrodinger operators: H = −∆− V +W , V,W ≥ 0
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
In this connection, Eric considered the following functional:
E(Q, v,w) = minZ(Q, v),v(Q)
w(Q)
where
11
Some theorems by E. Sawyer carlosperez@us.es
Schrodinger operators with change of sign
Schrodinger operators: H = −∆− V +W , V,W ≥ 0
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
In this connection, Eric considered the following functional:
E(Q, v,w) = minZ(Q, v),v(Q)
w(Q)
where
Z(Q, v) =1
v(Q)
∫QI2(vχQ) vdx
11
Some theorems by E. Sawyer carlosperez@us.es
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
12
Some theorems by E. Sawyer carlosperez@us.es
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
The proof is really complicated!!!
12
Some theorems by E. Sawyer carlosperez@us.es
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
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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Some theorems by E. Sawyer carlosperez@us.es
The model example: Muckenhoupt Ap class of weights
13
Some theorems by E. Sawyer carlosperez@us.es
The model example: Muckenhoupt Ap class of weights
The Hardy–Littlewood maximal function:
Mf(x) = supx∈Q
1
|Q|
∫Q|f(y)| dy.
13
Some theorems by E. Sawyer carlosperez@us.es
The model example: Muckenhoupt Ap class of weights
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<∞
13
Some theorems by E. Sawyer carlosperez@us.es
Two weight theory for the maximal function
14
Some theorems by E. Sawyer carlosperez@us.es
Two weight theory for the maximal functionModel example of Eric “testing philosphy”
14
Some theorems by E. Sawyer carlosperez@us.es
Two weight theory for the maximal functionModel example of Eric “testing philosphy”
∫Rn
(Mf)p udx ≤ C∫Rn|f |p vdx
14
Some theorems by E. Sawyer carlosperez@us.es
Two weight theory for the maximal functionModel example of Eric “testing philosphy”
∫Rn
(Mf)p udx ≤ C∫Rn|f |p vdx
The natural Ap condition
14
Some theorems by E. Sawyer carlosperez@us.es
Two weight theory for the maximal functionModel example of Eric “testing philosphy”
∫Rn
(Mf)p udx ≤ C∫Rn|f |p vdx
The natural Ap condition
(Ap) supQ
(1
|Q|
∫Qu dx
)(1
|Q|
∫Qv1−p′ dx
)p−1<∞
14
Some theorems by E. Sawyer carlosperez@us.es
Two weight theory for the maximal functionModel example of Eric “testing philosphy”
∫Rn
(Mf)p udx ≤ C∫Rn|f |p vdx
The natural Ap condition
(Ap) supQ
(1
|Q|
∫Qu dx
)(1
|Q|
∫Qv1−p′ dx
)p−1<∞
is NECESSARY but NOT SUFFICIENT
14
Some theorems by E. Sawyer carlosperez@us.es
Two weight theory for the maximal functionModel example of Eric “testing philosphy”
∫Rn
(Mf)p udx ≤ C∫Rn|f |p vdx
The natural Ap condition
(Ap) supQ
(1
|Q|
∫Qu dx
)(1
|Q|
∫Qv1−p′ dx
)p−1<∞
is NECESSARY but NOT SUFFICIENT
(known from the 70’s, Muckenhoupt and Wheeden.)
14
Some theorems by E. Sawyer carlosperez@us.es
Two weight theory for the maximal functionModel example of Eric “testing philosphy”
∫Rn
(Mf)p udx ≤ C∫Rn|f |p vdx
The natural Ap condition
(Ap) supQ
(1
|Q|
∫Qu dx
)(1
|Q|
∫Qv1−p′ dx
)p−1<∞
is NECESSARY but NOT SUFFICIENT
(known from the 70’s, Muckenhoupt and Wheeden.)However:
14
Some theorems by E. Sawyer carlosperez@us.es
Two weight theory for the maximal functionModel example of Eric “testing philosphy”
∫Rn
(Mf)p udx ≤ C∫Rn|f |p vdx
The natural Ap condition
(Ap) supQ
(1
|Q|
∫Qu dx
)(1
|Q|
∫Qv1−p′ dx
)p−1<∞
is NECESSARY but NOT SUFFICIENT
(known from the 70’s, Muckenhoupt and Wheeden.)However:
(u, v) ∈ Ap ⇐⇒ M : Lp(v) −→ Lp,∞(u)
14
Some theorems by E. Sawyer carlosperez@us.es
Eric’s theorem and Eric’s condition
15
Some theorems by E. Sawyer carlosperez@us.es
Eric’s theorem and Eric’s condition
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′
15
Some theorems by E. Sawyer carlosperez@us.es
Eric’s theorem and Eric’s condition
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′
He tested this “philosophy” with many other operators.
15
Some theorems by E. Sawyer carlosperez@us.es
Eric’s theorem and Eric’s condition
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′
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
15
Some theorems by E. Sawyer carlosperez@us.es
First E. Sawyer’s theorem for fractional integrals
16
Some theorems by E. Sawyer carlosperez@us.es
First E. Sawyer’s theorem for fractional integrals
Recall the definition of the classical fractional integral: if 0 < α < n
Iα(f)(x) =∫Rn
f(y)
|x− y|n−αdy.
16
Some theorems by E. Sawyer carlosperez@us.es
First E. Sawyer’s theorem for fractional integrals
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)
where σ = v1−p′
16
Some theorems by E. Sawyer carlosperez@us.es
Second E. Sawyer’s theorem for fractional integrals
17
Some theorems by E. Sawyer carlosperez@us.es
Second E. Sawyer’s theorem for fractional integrals
Theorem [Sawyer] (1988)
∫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
Q)p′σdx ≤ c u(Q)
where σ = v1−p′
17
Some theorems by E. Sawyer carlosperez@us.es
Second E. Sawyer’s theorem for fractional integrals
Theorem [Sawyer] (1988)
∫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
Q)p′σdx ≤ c u(Q)
where σ = v1−p′
”A characterization of two weight norm inequalities for fractional andPoisson integrals”, Trans. A.M.S. (1988)
17
Some theorems by E. Sawyer carlosperez@us.es
Sawyer-Wheeden two weight bump condition
18
Some theorems by E. Sawyer carlosperez@us.es
Sawyer-Wheeden two weight bump condition
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.
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Some theorems by E. Sawyer carlosperez@us.es
Sawyer-Wheeden two weight bump condition
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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Some theorems by E. Sawyer carlosperez@us.es
Tips of the proof
19
Some theorems by E. Sawyer carlosperez@us.es
Tips of the proof
Iαf(x) ≈∑Q∈D
|Q|α/n
|Q|
∫Qf(y) dy χQ(x),
19
Some theorems by E. Sawyer carlosperez@us.es
Tips of the proof
Iαf(x) ≈∑Q∈D
|Q|α/n
|Q|
∫Qf(y) dy χQ(x),
and then after “averaging”
19
Some theorems by E. Sawyer carlosperez@us.es
Tips of the proof
Iαf(x) ≈∑Q∈D
|Q|α/n
|Q|
∫Qf(y) dy χQ(x),
and then after “averaging”
Iαf(x) ≈∑k,j
|Qj,k|α/n
|Qj,k|
∫Qj,k
f(y) dy χQj,k(x),
19
Some theorems by E. Sawyer carlosperez@us.es
Tips of the proof
Iαf(x) ≈∑Q∈D
|Q|α/n
|Q|
∫Qf(y) dy χQ(x),
and then after “averaging”
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.
19
Some theorems by E. Sawyer carlosperez@us.es
Some comments
• This and other similar results were extended to spaces of homogeneoustype.
20
Some theorems by E. Sawyer carlosperez@us.es
Some comments
• This and other similar results were extended to spaces of homogeneoustype.
• The case when α = 0 corresponds to the Hilbert transform is due to Neuge-bauer.
20
Some theorems by E. Sawyer carlosperez@us.es
Some comments
• This and other similar results were extended to spaces of homogeneoustype.
• The case when α = 0 corresponds to the Hilbert transform is due to Neuge-bauer.
• Improves: Chang-Wilson-Wolff and Fefferman-Phong
20
Some theorems by E. Sawyer carlosperez@us.es
Some comments
• This and other similar results were extended to spaces of homogeneoustype.
• The case when α = 0 corresponds to the Hilbert transform is due to Neuge-bauer.
• Improves: Chang-Wilson-Wolff and Fefferman-Phong
• The failure of the Besicovitch covering lemma for the Heisenberg group wasobtained (also independently by Koranyi and Reimann).
20
Some theorems by E. Sawyer carlosperez@us.es
Some comments
• This and other similar results were extended to spaces of homogeneoustype.
• The case when α = 0 corresponds to the Hilbert transform is due to Neuge-bauer.
• Improves: Chang-Wilson-Wolff and Fefferman-Phong
• The failure of the Besicovitch covering lemma for the Heisenberg group wasobtained (also independently by Koranyi and Reimann).
• A construction of a dyadic grid for spaces of homogeneous type was given(also independently by Christ, see as well David)
20
Some theorems by E. Sawyer carlosperez@us.es
Some comments
• This and other similar results were extended to spaces of homogeneoustype.
• The case when α = 0 corresponds to the Hilbert transform is due to Neuge-bauer.
• Improves: Chang-Wilson-Wolff and Fefferman-Phong
• The failure of the Besicovitch covering lemma for the Heisenberg group wasobtained (also independently by Koranyi and Reimann).
• 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
Some theorems by E. Sawyer carlosperez@us.es
Mixed weak type inequalities and Eric’s conjecture
21
Some theorems by E. Sawyer carlosperez@us.es
Mixed weak type inequalities and Eric’s conjecture
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
Some theorems by E. Sawyer carlosperez@us.es
Mixed weak type inequalities and Eric’s conjecture
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.
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.
21
Some theorems by E. Sawyer carlosperez@us.es
Mixed weak type inequalities and Eric’s conjecture
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.
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.
E. Sawyer, “A weighted weak type inequality for the maximal function”,Proceedings of the A.M.S., 93(1985).
21
Some theorems by E. Sawyer carlosperez@us.es
Mixed weak type inequalities and Eric’s conjecture
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.
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.
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
Some theorems by E. Sawyer carlosperez@us.es
Mixed weak type inequalities and Eric’s conjecture
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.
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.
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:
21
Some theorems by E. Sawyer carlosperez@us.es
Mixed weak type inequalities and Eric’s conjecture
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.
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.
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:
M(f)
v≤ CMv(
f
v)
21
Some theorems by E. Sawyer carlosperez@us.es
Mixed weak type inequalities and Eric’s conjecture
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.
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.
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:
M(f)
v≤ CMv(
f
v)
• The problem in general is that the product uv can be very singular...
21
Some theorems by E. Sawyer carlosperez@us.es
Mixed weak type inequalities and Eric’s conjecture
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.
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.
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:
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
Some theorems by E. Sawyer carlosperez@us.es
Why Eric look for such a sophisticated a result?
22
Some theorems by E. Sawyer carlosperez@us.es
Why Eric look for such a sophisticated a result?
To give a different proof of Muckenhoupt’s Ap theorem assuming that theweights are already factored:
w = uv1−p u, v ∈ A1.
22
Some theorems by E. Sawyer carlosperez@us.es
Why Eric look for such a sophisticated a result?
To give a different proof of Muckenhoupt’s Ap theorem assuming that theweights are already factored:
w = uv1−p u, v ∈ A1.
Indeed, let
22
Some theorems by E. Sawyer carlosperez@us.es
Why Eric look for such a sophisticated a result?
To give a different proof of Muckenhoupt’s Ap theorem assuming that theweights are already factored:
w = uv1−p u, v ∈ A1.
Indeed, let
Sv(f) =M(f v)
v
22
Some theorems by E. Sawyer carlosperez@us.es
Why Eric look for such a sophisticated a result?
To give a different proof of Muckenhoupt’s Ap theorem assuming that theweights are already factored:
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
Some theorems by E. Sawyer carlosperez@us.es
Why Eric look for such a sophisticated a result?
To give a different proof of Muckenhoupt’s Ap theorem assuming that theweights are already factored:
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)
But observe that Sv is also bounded on L∞, (v ∈ A1 ) and hence onL∞(uv).
22
Some theorems by E. Sawyer carlosperez@us.es
Why Eric look for such a sophisticated a result?
To give a different proof of Muckenhoupt’s Ap theorem assuming that theweights are already factored:
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)
But observe that Sv is also bounded on L∞, (v ∈ A1 ) and hence onL∞(uv).
Then by the Marcinkiewicz interpolation theorem:
22
Some theorems by E. Sawyer carlosperez@us.es
Why Eric look for such a sophisticated a result?
To give a different proof of Muckenhoupt’s Ap theorem assuming that theweights are already factored:
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)
But observe that Sv is also bounded on L∞, (v ∈ A1 ) and hence onL∞(uv).
Then by the Marcinkiewicz interpolation theorem:
M : Lp(uv1−p)→ Lp(uv1−p)
22
Some theorems by E. Sawyer carlosperez@us.es
Two conjectures by Eric
23
Some theorems by E. Sawyer carlosperez@us.es
Two conjectures by Eric
• First conjecture: to replace M by the Hilbert transform H , namely
23
Some theorems by E. Sawyer carlosperez@us.es
Two conjectures by Eric
• First conjecture: to replace M by the Hilbert transform H , namely
∥∥∥H(fv)
v
∥∥∥L1,∞(uv)
≤ c∫R|f(x)|uv dx
23
Some theorems by E. Sawyer carlosperez@us.es
Two conjectures by Eric
• First conjecture: to replace M by the Hilbert transform H , namely
∥∥∥H(fv)
v
∥∥∥L1,∞(uv)
≤ c∫R|f(x)|uv dx
• Second conjecture: to replace M in one dimension by M in Rn, namely
23
Some theorems by E. Sawyer carlosperez@us.es
Two conjectures by Eric
• First conjecture: to replace M by the Hilbert transform H , namely
∥∥∥H(fv)
v
∥∥∥L1,∞(uv)
≤ c∫R|f(x)|uv dx
• 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
Some theorems by E. Sawyer carlosperez@us.es
Results
24
Some theorems by E. Sawyer carlosperez@us.es
Results
Joint work with J. M. Martell and D. Cruz-Uribe
24
Some theorems by E. Sawyer carlosperez@us.es
Results
Joint work with J. M. Martell and D. Cruz-UribeT is any Calderon-Zygmund .
24
Some theorems by E. Sawyer carlosperez@us.es
Results
Joint work with J. M. Martell and D. Cruz-UribeT is any Calderon-Zygmund .
Theorem If u ∈ A1(Rn), and v ∈ A1(Rn), then there is a constant c suchthat ∥∥∥M(fv)
v
∥∥∥L1,∞(uv)
≤ c∫Rn|f(x)|uv dx
and ∥∥∥T (fv)
v
∥∥∥L1,∞(uv)
≤ c∫Rn|f(x)|uv dx
24
Some theorems by E. Sawyer carlosperez@us.es
Results
Joint work with J. M. Martell and D. Cruz-UribeT is any Calderon-Zygmund .
Theorem If u ∈ A1(Rn), and v ∈ A1(Rn), then there is a constant c suchthat ∥∥∥M(fv)
v
∥∥∥L1,∞(uv)
≤ c∫Rn|f(x)|uv dx
and ∥∥∥T (fv)
v
∥∥∥L1,∞(uv)
≤ c∫Rn|f(x)|uv dx
and also
24
Some theorems by E. Sawyer carlosperez@us.es
Results
Joint work with J. M. Martell and D. Cruz-UribeT is any Calderon-Zygmund .
Theorem If u ∈ A1(Rn), and v ∈ A1(Rn), then there is a constant c suchthat ∥∥∥M(fv)
v
∥∥∥L1,∞(uv)
≤ c∫Rn|f(x)|uv dx
and ∥∥∥T (fv)
v
∥∥∥L1,∞(uv)
≤ c∫Rn|f(x)|uv dx
and also
Theorem If u ∈ A1(Rn), and v ∈ A∞(u), then the same conclusion holds.
24
Some theorems by E. Sawyer carlosperez@us.es
Results
Joint work with J. M. Martell and D. Cruz-UribeT is any Calderon-Zygmund .
Theorem If u ∈ A1(Rn), and v ∈ A1(Rn), then there is a constant c suchthat ∥∥∥M(fv)
v
∥∥∥L1,∞(uv)
≤ c∫Rn|f(x)|uv dx
and ∥∥∥T (fv)
v
∥∥∥L1,∞(uv)
≤ c∫Rn|f(x)|uv dx
and also
Theorem If u ∈ A1(Rn), and v ∈ A∞(u), then the same conclusion holds.
Example: u = 1 and v ∈ A∞
24
Some theorems by E. Sawyer carlosperez@us.es
Results
Joint work with J. M. Martell and D. Cruz-UribeT is any Calderon-Zygmund .
Theorem If u ∈ A1(Rn), and v ∈ A1(Rn), then there is a constant c suchthat ∥∥∥M(fv)
v
∥∥∥L1,∞(uv)
≤ c∫Rn|f(x)|uv dx
and ∥∥∥T (fv)
v
∥∥∥L1,∞(uv)
≤ c∫Rn|f(x)|uv dx
and also
Theorem If u ∈ A1(Rn), and v ∈ A∞(u), then the same conclusion holds.
Example: u = 1 and v ∈ A∞Some special cases were considered in the 70’s by Muckenhoupt-Wheedenfor the Hilbert Transform.
24
Some theorems by E. Sawyer carlosperez@us.es
Key Lemma
25
Some theorems by E. Sawyer carlosperez@us.es
Key Lemma
Our approach is completely different
25
Some theorems by E. Sawyer carlosperez@us.es
Key Lemma
Our approach is completely different
We found a sort of “mechanism” (a very general extrapolation type theorem) toreduce everything to prove the corresponding estimate for the dyadic maximaloperator.
25
Some theorems by E. Sawyer carlosperez@us.es
Key Lemma
Our approach is completely different
We found a sort of “mechanism” (a very general extrapolation type theorem) toreduce everything to prove the corresponding estimate for the dyadic maximaloperator.
Lemma Let u ∈ A1(Rn) and v ∈ A∞(Rn), then∥∥∥T (fv)
v
∥∥∥L1,∞(uv)
≤ c∥∥∥Md(fv)
v
∥∥∥L1,∞(uv)
25
Some theorems by E. Sawyer carlosperez@us.es
Key Lemma
Our approach is completely different
We found a sort of “mechanism” (a very general extrapolation type theorem) toreduce everything to prove the corresponding estimate for the dyadic maximaloperator.
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.
25
Some theorems by E. Sawyer carlosperez@us.es
The final key inequality and Sawyer’s conjecture
26
Some theorems by E. Sawyer carlosperez@us.es
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
Some theorems by E. Sawyer carlosperez@us.es
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:
Theorem If u ∈ A1(Rn), and v ∈ A1(Rn), then there is a constant c suchthat ∥∥∥Md(fv)
v
∥∥∥L1,∞(uv)
≤ c∫Rn|f(x)|uv dx
26
Some theorems by E. Sawyer carlosperez@us.es
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:
Theorem If u ∈ A1(Rn), and v ∈ A1(Rn), then there is a constant c suchthat ∥∥∥Md(fv)
v
∥∥∥L1,∞(uv)
≤ c∫Rn|f(x)|uv dx
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
Some theorems by E. Sawyer carlosperez@us.es
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:
Theorem If u ∈ A1(Rn), and v ∈ A1(Rn), then there is a constant c suchthat ∥∥∥Md(fv)
v
∥∥∥L1,∞(uv)
≤ c∫Rn|f(x)|uv dx
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
Some theorems by E. Sawyer carlosperez@us.es
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:
Theorem If u ∈ A1(Rn), and v ∈ A1(Rn), then there is a constant c suchthat ∥∥∥Md(fv)
v
∥∥∥L1,∞(uv)
≤ c∫Rn|f(x)|uv dx
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
Some theorems by E. Sawyer carlosperez@us.es
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:
Theorem If u ∈ A1(Rn), and v ∈ A1(Rn), then there is a constant c suchthat ∥∥∥Md(fv)
v
∥∥∥L1,∞(uv)
≤ c∫Rn|f(x)|uv dx
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.
26
Some theorems by E. Sawyer carlosperez@us.es
Proof: the two basic Ingredients
27
Some theorems by E. Sawyer carlosperez@us.es
Proof: the two basic Ingredients
• The Coifman-Fefferman’s estimate: if 0 < p <∞ and w ∈ A∞
27
Some theorems by E. Sawyer carlosperez@us.es
Proof: the two basic Ingredients
• The Coifman-Fefferman’s estimate: if 0 < p <∞ and w ∈ A∞
Let T be any Calderon-Zygmund Operator and let 0 < p <∞ and w ∈ A∞.Then
‖Tf‖Lp(w)
≤ c ‖Mf‖Lp(w)
27
Some theorems by E. Sawyer carlosperez@us.es
Proof: the two basic Ingredients
• The Coifman-Fefferman’s estimate: if 0 < p <∞ and w ∈ A∞
Let T be any Calderon-Zygmund Operator and let 0 < p <∞ and w ∈ A∞.Then
‖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:
27
Some theorems by E. Sawyer carlosperez@us.es
Proof: the two basic Ingredients
• The Coifman-Fefferman’s estimate: if 0 < p <∞ and w ∈ A∞
Let T be any Calderon-Zygmund Operator and let 0 < p <∞ and w ∈ A∞.Then
‖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:
Rh(x) =∞∑k=0
Skvh(x)
2k ‖Sv‖k.
27
Some theorems by E. Sawyer carlosperez@us.es
Proof: the two basic Ingredients
• The Coifman-Fefferman’s estimate: if 0 < p <∞ and w ∈ A∞
Let T be any Calderon-Zygmund Operator and let 0 < p <∞ and w ∈ A∞.Then
‖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:
Rh(x) =∞∑k=0
Skvh(x)
2k ‖Sv‖k.
where
Svf(x) =M(fv)(x)
v(x)
27
Some theorems by E. Sawyer carlosperez@us.es
Singular Integrals and the Cp class of weights
28
Some theorems by E. Sawyer carlosperez@us.es
Singular Integrals and the Cp class of weights
Motivation: to understand the following version of the theorem of Coifman-Fefferman.
28
Some theorems by E. Sawyer carlosperez@us.es
Singular Integrals and the Cp class of weights
Motivation: to understand the following version of the theorem of Coifman-Fefferman.
Let H be the Hilbert transform let 1 < p <∞ and w ∈ A∞.Then ∫
R|Hf |pw dx ≤ C
∫R
(Mf)pw dx
28
Some theorems by E. Sawyer carlosperez@us.es
Singular Integrals and the Cp class of weights
Motivation: to understand the following version of the theorem of Coifman-Fefferman.
Let H be the Hilbert transform let 1 < p <∞ and w ∈ A∞.Then ∫
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:
28
Some theorems by E. Sawyer carlosperez@us.es
Singular Integrals and the Cp class of weights
Motivation: to understand the following version of the theorem of Coifman-Fefferman.
Let H be the Hilbert transform let 1 < p <∞ and w ∈ A∞.Then ∫
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.
28
Some theorems by E. Sawyer carlosperez@us.es
The Cp condition: necessity
29
Some theorems by E. Sawyer carlosperez@us.es
The Cp condition: necessity
Rj will denote the Riesz transforms.
29
Some theorems by E. Sawyer carlosperez@us.es
The Cp condition: necessity
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.
29
Some theorems by E. Sawyer carlosperez@us.es
The Cp condition: sufficiency
30
Some theorems by E. Sawyer carlosperez@us.es
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.
30
Some theorems by E. Sawyer carlosperez@us.es
The Monge-Ampere Equation
31
Some theorems by E. Sawyer carlosperez@us.es
The Monge-Ampere Equation
uxxuyy − u2xy = k(x, y)
near the origin with 0 ≤ k(x, y) ∈ C∞(R2).
31
Some theorems by E. Sawyer carlosperez@us.es
The Monge-Ampere Equation
uxxuyy − u2xy = k(x, y)
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.
31
Some theorems by E. Sawyer carlosperez@us.es
The Monge-Ampere Equation
uxxuyy − u2xy = k(x, y)
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.
31
Some theorems by E. Sawyer carlosperez@us.es
The Monge-Ampere Equation
uxxuyy − u2xy = k(x, y)
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)
31
Some theorems by E. Sawyer carlosperez@us.es
The Monge-Ampere Equation
uxxuyy − u2xy = k(x, y)
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
31
Some theorems by E. Sawyer carlosperez@us.es
Comments
32
Some theorems by E. Sawyer carlosperez@us.es
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.
32
Some theorems by E. Sawyer carlosperez@us.es
future
33
Some theorems by E. Sawyer carlosperez@us.es
future
I am sure that Eric is going to produce more mathematics for us!!!
33
Some theorems by E. Sawyer carlosperez@us.es
future
I am sure that Eric is going to produce more mathematics for us!!!
THANK YOU VERYMUCH
33
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