The A class of weights and some of its extensions Carlos P erez - - PowerPoint PPT Presentation

“The A∞ class of weights and some of its extensions” Carlos P´ erez University of the Basque Country and BCAM Probability and Analysis 2019 Banach Center for Mathematics Be ¸dlewo, May 22, 2019

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Collaborators

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Collaborators

• new results joint with

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Collaborators

• new results joint with

Javier Canto

The Cp class of weights cperez@bcamath.org

Collaborators

• new results joint with

Javier Canto

• some other new results with

The Cp class of weights cperez@bcamath.org

Collaborators

• new results joint with

Javier Canto

• some other new results with
• S. Ombrosi, E. Rela and I. Rivera-Rios

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El teorema de Muckenhoupt

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El teorema de Muckenhoupt Thm (B. Muckenhoupt (≈ 1971)) Let p ∈ (1, ∞), then M : Lp(w) − → Lp(w) if and only if

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El teorema de Muckenhoupt Thm (B. Muckenhoupt (≈ 1971)) Let p ∈ (1, ∞), then M : Lp(w) − → Lp(w) if and only if w satisfies the Ap condition

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El teorema de Muckenhoupt Thm (B. Muckenhoupt (≈ 1971)) Let p ∈ (1, ∞), then M : Lp(w) − → Lp(w) if and only if w satisfies the Ap condition [w]Ap = sup

Q

• 1

|Q|

• Q w dx

1 |Q|

• Q w

−1 p−1 dx

p−1

< ∞

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The Hunt-Muckenhoupt-Wheeden theorem (1973)

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The Hunt-Muckenhoupt-Wheeden theorem (1973) Thm Let p ∈ (1, ∞), then H : Lp(w) → Lp(w) if and only if

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The Hunt-Muckenhoupt-Wheeden theorem (1973) Thm Let p ∈ (1, ∞), then H : Lp(w) → Lp(w) if and only if w ∈ Ap

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The Hunt-Muckenhoupt-Wheeden theorem (1973) Thm Let p ∈ (1, ∞), then H : Lp(w) → Lp(w) if and only if w ∈ Ap This result was greatly improved.

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The A∞ theorem of Coifman-C. Fefferman (1974)

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The A∞ theorem of Coifman-C. Fefferman (1974) T will always be a Calder´

• n-Zygmund operator (and often T ∗).

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The A∞ theorem of Coifman-C. Fefferman (1974) T will always be a Calder´

• n-Zygmund operator (and often T ∗).

Thm Let p ∈ (0, ∞), and w ∈ A∞. Then there exists a constant c such that

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The A∞ theorem of Coifman-C. Fefferman (1974) T will always be a Calder´

• n-Zygmund operator (and often T ∗).

Thm Let p ∈ (0, ∞), and w ∈ A∞. Then there exists a constant c such that T ∗fLp(w) ≤ c MfLp(w)

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The A∞ theorem of Coifman-C. Fefferman (1974) T will always be a Calder´

• n-Zygmund operator (and often T ∗).

Thm Let p ∈ (0, ∞), and w ∈ A∞. Then there exists a constant c such that T ∗fLp(w) ≤ c MfLp(w)

• The proof is based on the ”good λ” technique of Bukholder and Gundy

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The A∞ theorem of Coifman-C. Fefferman (1974) T will always be a Calder´

• n-Zygmund operator (and often T ∗).

Thm Let p ∈ (0, ∞), and w ∈ A∞. Then there exists a constant c such that T ∗fLp(w) ≤ c MfLp(w)

• The proof is based on the ”good λ” technique of Bukholder and Gundy
• The A∞ class is defined originally as:

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The A∞ theorem of Coifman-C. Fefferman (1974) T will always be a Calder´

• n-Zygmund operator (and often T ∗).

Thm Let p ∈ (0, ∞), and w ∈ A∞. Then there exists a constant c such that T ∗fLp(w) ≤ c MfLp(w)

• The proof is based on the ”good λ” technique of Bukholder and Gundy
• The A∞ class is defined originally as:

w(E) ≤ c

• |E|

|Q|

δ

w(Q) E ⊂ Q

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The A∞ theorem of Coifman-C. Fefferman (1974) T will always be a Calder´

• n-Zygmund operator (and often T ∗).

Thm Let p ∈ (0, ∞), and w ∈ A∞. Then there exists a constant c such that T ∗fLp(w) ≤ c MfLp(w)

• The proof is based on the ”good λ” technique of Bukholder and Gundy
• The A∞ class is defined originally as:

w(E) ≤ c

• |E|

|Q|

δ

w(Q) E ⊂ Q

• The classical most important consequence:

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The A∞ theorem of Coifman-C. Fefferman (1974) T will always be a Calder´

• n-Zygmund operator (and often T ∗).

Thm Let p ∈ (0, ∞), and w ∈ A∞. Then there exists a constant c such that T ∗fLp(w) ≤ c MfLp(w)

• The proof is based on the ”good λ” technique of Bukholder and Gundy
• The A∞ class is defined originally as:

w(E) ≤ c

• |E|

|Q|

δ

w(Q) E ⊂ Q

• The classical most important consequence:

Corollary Let p ∈ (1, ∞) and let w ∈ Ap. Then T ∗ : Lp(w) → Lp(w)

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Other similar results

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Other similar results Thm Let p ∈ (0, ∞) and w ∈ A∞. Then

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Other similar results Thm Let p ∈ (0, ∞) and w ∈ A∞. Then MfLp(w) ≤ c M#fLp(w)

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Other similar results Thm Let p ∈ (0, ∞) and w ∈ A∞. Then MfLp(w) ≤ c M#fLp(w)

• Recall that

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Other similar results Thm Let p ∈ (0, ∞) and w ∈ A∞. Then MfLp(w) ≤ c M#fLp(w)

• Recall that

M#f(x) = supx∈Q 1

|Q|

• Q |f − fQ|

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Other similar results Thm Let p ∈ (0, ∞) and w ∈ A∞. Then MfLp(w) ≤ c M#fLp(w)

• Recall that

M#f(x) = supx∈Q 1

|Q|

• Q |f − fQ|

Thm Let p ∈ (0, ∞) and w ∈ A∞. Also let b ∈ BMO. Then

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Other similar results Thm Let p ∈ (0, ∞) and w ∈ A∞. Then MfLp(w) ≤ c M#fLp(w)

• Recall that

M#f(x) = supx∈Q 1

|Q|

• Q |f − fQ|

Thm Let p ∈ (0, ∞) and w ∈ A∞. Also let b ∈ BMO. Then [b, T]fLp(w) ≤ c M2fLp(w)

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The A∞ class and the RHI property

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The A∞ class and the RHI property The following conditions are equivalent:

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The A∞ class and the RHI property The following conditions are equivalent:

• 1)

w ∈ A∞

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The A∞ class and the RHI property The following conditions are equivalent:

• 1)

w ∈ A∞

• 2)

w ∈ ∪p≥1Ap

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The A∞ class and the RHI property The following conditions are equivalent:

• 1)

w ∈ A∞

• 2)

w ∈ ∪p≥1Ap

• 3) w satisfies a Reverse H¨
• lder Inequality: for some δ, c > 0

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The A∞ class and the RHI property The following conditions are equivalent:

• 1)

w ∈ A∞

• 2)

w ∈ ∪p≥1Ap

• 3) w satisfies a Reverse H¨
• lder Inequality: for some δ, c > 0
• 1

|Q|

• Q w1+δ dx
• 1

1+δ

≤ c |Q|

• Q w dx

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The A∞ class and the RHI property The following conditions are equivalent:

• 1)

w ∈ A∞

• 2)

w ∈ ∪p≥1Ap

• 3) w satisfies a Reverse H¨
• lder Inequality: for some δ, c > 0
• 1

|Q|

• Q w1+δ dx
• 1

1+δ

≤ c |Q|

• Q w dx
• 4) w satisfies the following condition:

[w]A∞ := sup

Q

1 w(Q)

• Q M(wχQ) dx < ∞

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Praising the A∞ theorem: other consequences

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Praising the A∞ theorem: other consequences Thm Let 1 < p < ∞ and let w ≥ 0. Then,

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Praising the A∞ theorem: other consequences Thm Let 1 < p < ∞ and let w ≥ 0. Then, TfLp(w) ≤ c fLp(M[p]+1w)

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Praising the A∞ theorem: other consequences Thm Let 1 < p < ∞ and let w ≥ 0. Then, TfLp(w) ≤ c fLp(M[p]+1w)

• The theorem is fully sharp

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Praising the A∞ theorem: other consequences Thm Let 1 < p < ∞ and let w ≥ 0. Then, TfLp(w) ≤ c fLp(M[p]+1w)

• The theorem is fully sharp
• Key observation:

(Mµ)−λ ∈ A∞

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Quantitative versions of the A∞ thm

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Quantitative versions of the A∞ thm If 1 ≤ q < ∞, TfL1(w) [w]AqMfL1(w)

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Quantitative versions of the A∞ thm If 1 ≤ q < ∞, TfL1(w) [w]AqMfL1(w)

• There is a much better result:

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Quantitative versions of the A∞ thm If 1 ≤ q < ∞, TfL1(w) [w]AqMfL1(w)

• There is a much better result:

Thm If p ∈ (0, ∞), TfLp(w) max{1, p} [w]A∞MfLp(w)

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Quantitative versions of the A∞ thm If 1 ≤ q < ∞, TfL1(w) [w]AqMfL1(w)

• There is a much better result:

Thm If p ∈ (0, ∞), TfLp(w) max{1, p} [w]A∞MfLp(w) Recall, we are using here the following constant:

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Quantitative versions of the A∞ thm If 1 ≤ q < ∞, TfL1(w) [w]AqMfL1(w)

• There is a much better result:

Thm If p ∈ (0, ∞), TfLp(w) max{1, p} [w]A∞MfLp(w) Recall, we are using here the following constant: [w]A∞ = sup

Q

1 w(Q)

• Q M(wχQ) dx

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Key points

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Key points

• 1) The quantitative RHI

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Key points

• 1) The quantitative RHI

Thm

• T. Hyt¨
• nen and C. P.

Let w ∈ A∞, then

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Key points

• 1) The quantitative RHI

Thm

• T. Hyt¨
• nen and C. P.

Let w ∈ A∞, then

• 1

|Q|

• Q w1+δ
• 1

1+δ

≤ 2 |Q|

• Q w

where

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Key points

• 1) The quantitative RHI

Thm

• T. Hyt¨
• nen and C. P.

Let w ∈ A∞, then

• 1

|Q|

• Q w1+δ
• 1

1+δ

≤ 2 |Q|

• Q w

where δ = 1 cn [w]A∞

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Key points

• 1) The quantitative RHI

Thm

• T. Hyt¨
• nen and C. P.

Let w ∈ A∞, then

• 1

|Q|

• Q w1+δ
• 1

1+δ

≤ 2 |Q|

• Q w

where δ = 1 cn [w]A∞

• 2) The local exponential decay

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Key points

• 1) The quantitative RHI

Thm

• T. Hyt¨
• nen and C. P.

Let w ∈ A∞, then

• 1

|Q|

• Q w1+δ
• 1

1+δ

≤ 2 |Q|

• Q w

where δ = 1 cn [w]A∞

• 2) The local exponential decay
• y ∈ Q : |Tf(y)| > 2 t, Mf(y) ≤ t ε
• |Q|

≤ cε

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Key points

• 1) The quantitative RHI

Thm

• T. Hyt¨
• nen and C. P.

Let w ∈ A∞, then

• 1

|Q|

• Q w1+δ
• 1

1+δ

≤ 2 |Q|

• Q w

where δ = 1 cn [w]A∞

• 2) The local exponential decay
• y ∈ Q : |Tf(y)| > 2 t, Mf(y) ≤ t ε
• |Q|

≤ cε ≤ c e−c

ε

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More consequences: the A1 theory

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More consequences: the A1 theory

• w ∈ A1

if M(w) ≤ [w]A1 w

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More consequences: the A1 theory

• w ∈ A1

if M(w) ≤ [w]A1 w Thm ( C.P., A. Lerner & S. Ombrosi ≈ 2009) Let w ∈ A1. a) Let 1 < p < ∞. Then TLp(w) ≤ c pp′ [w]A1

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More consequences: the A1 theory

• w ∈ A1

if M(w) ≤ [w]A1 w Thm ( C.P., A. Lerner & S. Ombrosi ≈ 2009) Let w ∈ A1. a) Let 1 < p < ∞. Then TLp(w) ≤ c pp′ [w]A1 b) TL1(w)→L1,∞(w) ≤ c [w]A1 log(e + [w]A1)

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More consequences: the A1 theory

• w ∈ A1

if M(w) ≤ [w]A1 w Thm ( C.P., A. Lerner & S. Ombrosi ≈ 2009) Let w ∈ A1. a) Let 1 < p < ∞. Then TLp(w) ≤ c pp′ [w]A1 b) TL1(w)→L1,∞(w) ≤ c [w]A1 log(e + [w]A1)

• We thought that the correct result was linear, but it is false.

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More consequences: the A1 theory

• w ∈ A1

if M(w) ≤ [w]A1 w Thm ( C.P., A. Lerner & S. Ombrosi ≈ 2009) Let w ∈ A1. a) Let 1 < p < ∞. Then TLp(w) ≤ c pp′ [w]A1 b) TL1(w)→L1,∞(w) ≤ c [w]A1 log(e + [w]A1)

• We thought that the correct result was linear, but it is false.
• Adam Ose

¸kowski found a different interesting argument

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More consequences: the A1 theory

• w ∈ A1

if M(w) ≤ [w]A1 w Thm ( C.P., A. Lerner & S. Ombrosi ≈ 2009) Let w ∈ A1. a) Let 1 < p < ∞. Then TLp(w) ≤ c pp′ [w]A1 b) TL1(w)→L1,∞(w) ≤ c [w]A1 log(e + [w]A1)

• We thought that the correct result was linear, but it is false.
• Adam Ose

¸kowski found a different interesting argument

• Lerner-Nazarov-Ombrosi: the result is sharp.

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More praises:

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More praises:

• 1) Vector-valued extensions

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More praises:

• 1) Vector-valued extensions

Thm Let p, q ∈ (0, ∞) and w ∈ A∞. Then

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More praises:

• 1) Vector-valued extensions

Thm Let p, q ∈ (0, ∞) and w ∈ A∞. Then

• j

(Tfj)q

1

q

• Lp(w)

≤ C

• j

(Mfj)q

1

q

• Lp(w)

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More praises:

• 1) Vector-valued extensions

Thm Let p, q ∈ (0, ∞) and w ∈ A∞. Then

• j

(Tfj)q

1

q

• Lp(w)

≤ C

• j

(Mfj)q

1

q

• Lp(w)

and

• j

(Tfj)q

1

q

• Lp,∞(w)

≤ C

• j

(Mfj)q

1

q

• Lp,∞(w)

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More praises:

• 1) Vector-valued extensions

Thm Let p, q ∈ (0, ∞) and w ∈ A∞. Then

• j

(Tfj)q

1

q

• Lp(w)

≤ C

• j

(Mfj)q

1

q

• Lp(w)

and

• j

(Tfj)q

1

q

• Lp,∞(w)

≤ C

• j

(Mfj)q

1

q

• Lp,∞(w)
• 2) Sawyer’s problem where one of the key results is

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More praises:

• 1) Vector-valued extensions

Thm Let p, q ∈ (0, ∞) and w ∈ A∞. Then

• j

(Tfj)q

1

q

• Lp(w)

≤ C

• j

(Mfj)q

1

q

• Lp(w)

and

• j

(Tfj)q

1

q

• Lp,∞(w)

≤ C

• j

(Mfj)q

1

q

• Lp,∞(w)
• 2) Sawyer’s problem where one of the key results is

Thm Let u ∈ A1(Rn) and v ∈ A∞(Rn). Then

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More praises:

• 1) Vector-valued extensions

Thm Let p, q ∈ (0, ∞) and w ∈ A∞. Then

• j

(Tfj)q

1

q

• Lp(w)

≤ C

• j

(Mfj)q

1

q

• Lp(w)

and

• j

(Tfj)q

1

q

• Lp,∞(w)

≤ C

• j

(Mfj)q

1

q

• Lp,∞(w)
• 2) Sawyer’s problem where one of the key results is

Thm Let u ∈ A1(Rn) and v ∈ A∞(Rn). Then

• T ∗(fv)

v

• L1,∞(uv) ≤ c
• M(fv)

v

• L1,∞(uv)

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More praises:

• 1) Vector-valued extensions

Thm Let p, q ∈ (0, ∞) and w ∈ A∞. Then

• j

(Tfj)q

1

q

• Lp(w)

≤ C

• j

(Mfj)q

1

q

• Lp(w)

and

• j

(Tfj)q

1

q

• Lp,∞(w)

≤ C

• j

(Mfj)q

1

q

• Lp,∞(w)
• 2) Sawyer’s problem where one of the key results is

Thm Let u ∈ A1(Rn) and v ∈ A∞(Rn). Then

• T ∗(fv)

v

• L1,∞(uv) ≤ c
• M(fv)

v

• L1,∞(uv)
• (work with D. Cruz-Uribe, JM Martell).

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The Cp condition

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The Cp condition Recall the A∞ theorem

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The Cp condition Recall the A∞ theorem T ∗fLp(w) ≤ c MfLp(w)

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The Cp condition Recall the A∞ theorem T ∗fLp(w) ≤ c MfLp(w)

• Key observation:

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The Cp condition Recall the A∞ theorem T ∗fLp(w) ≤ c MfLp(w)

• Key observation: If p > 1, Muckenhoupt proved that then w ∈ Cp :

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The Cp condition Recall the A∞ theorem T ∗fLp(w) ≤ c MfLp(w)

• Key observation: If p > 1, Muckenhoupt proved that then w ∈ Cp :

Definition w is in the Cp class if there are constants c, δ > 0 such that

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The Cp condition Recall the A∞ theorem T ∗fLp(w) ≤ c MfLp(w)

• Key observation: If p > 1, Muckenhoupt proved that then w ∈ Cp :

Definition w is in the Cp class if there are constants c, δ > 0 such that w(E) ≤ C

• |E|

|Q|

δ

Rn(MχQ(x))pw(x)dx

E ⊂ Q

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The Cp condition Recall the A∞ theorem T ∗fLp(w) ≤ c MfLp(w)

• Key observation: If p > 1, Muckenhoupt proved that then w ∈ Cp :

Definition w is in the Cp class if there are constants c, δ > 0 such that w(E) ≤ C

• |E|

|Q|

δ

Rn(MχQ(x))pw(x)dx

E ⊂ Q

• Compare with the A∞ condition:

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The Cp condition Recall the A∞ theorem T ∗fLp(w) ≤ c MfLp(w)

• Key observation: If p > 1, Muckenhoupt proved that then w ∈ Cp :

Definition w is in the Cp class if there are constants c, δ > 0 such that w(E) ≤ C

• |E|

|Q|

δ

Rn(MχQ(x))pw(x)dx

E ⊂ Q

• Compare with the A∞ condition:

w(E) ≤ c

• |E|

|Q|

δ

w(Q) E ⊂ Q

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The Cp condition Recall the A∞ theorem T ∗fLp(w) ≤ c MfLp(w)

• Key observation: If p > 1, Muckenhoupt proved that then w ∈ Cp :

Definition w is in the Cp class if there are constants c, δ > 0 such that w(E) ≤ C

• |E|

|Q|

δ

Rn(MχQ(x))pw(x)dx

E ⊂ Q

• Compare with the A∞ condition:

w(E) ≤ c

• |E|

|Q|

δ

w(Q) E ⊂ Q

• Hence:

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The Cp condition Recall the A∞ theorem T ∗fLp(w) ≤ c MfLp(w)

• Key observation: If p > 1, Muckenhoupt proved that then w ∈ Cp :

Definition w is in the Cp class if there are constants c, δ > 0 such that w(E) ≤ C

• |E|

|Q|

δ

Rn(MχQ(x))pw(x)dx

E ⊂ Q

• Compare with the A∞ condition:

w(E) ≤ c

• |E|

|Q|

δ

w(Q) E ⊂ Q

• Hence:

A∞ ⊂ Cp

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The Cp condition Recall the A∞ theorem T ∗fLp(w) ≤ c MfLp(w)

• Key observation: If p > 1, Muckenhoupt proved that then w ∈ Cp :

Definition w is in the Cp class if there are constants c, δ > 0 such that w(E) ≤ C

• |E|

|Q|

δ

Rn(MχQ(x))pw(x)dx

E ⊂ Q

• Compare with the A∞ condition:

w(E) ≤ c

• |E|

|Q|

δ

w(Q) E ⊂ Q

• Hence:

A∞ ⊂ Cp

• Open problem, Is the Cp condition sufficient?

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The Cp theorems

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The Cp theorems Thm (E. Sawyer, 1984) If p ∈ (1, ∞) and w ∈ Cp+ǫ

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The Cp theorems Thm (E. Sawyer, 1984) If p ∈ (1, ∞) and w ∈ Cp+ǫ TfLp(w) ≤ c MfLp(w)

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The Cp theorems Thm (E. Sawyer, 1984) If p ∈ (1, ∞) and w ∈ Cp+ǫ TfLp(w) ≤ c MfLp(w)

• The proof is a sophisticated version of Coifman-Fefferman’s A∞’s proof

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The Cp theorems Thm (E. Sawyer, 1984) If p ∈ (1, ∞) and w ∈ Cp+ǫ TfLp(w) ≤ c MfLp(w)

• The proof is a sophisticated version of Coifman-Fefferman’s A∞’s proof
• There is another interesting related result

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The Cp theorems Thm (E. Sawyer, 1984) If p ∈ (1, ∞) and w ∈ Cp+ǫ TfLp(w) ≤ c MfLp(w)

• The proof is a sophisticated version of Coifman-Fefferman’s A∞’s proof
• There is another interesting related result

Thm (K. Yabuta, 1990) If p ∈ (1, ∞) and w ∈ Cp+ǫ

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The Cp theorems Thm (E. Sawyer, 1984) If p ∈ (1, ∞) and w ∈ Cp+ǫ TfLp(w) ≤ c MfLp(w)

• The proof is a sophisticated version of Coifman-Fefferman’s A∞’s proof
• There is another interesting related result

Thm (K. Yabuta, 1990) If p ∈ (1, ∞) and w ∈ Cp+ǫ MfLp(w) ≤ c M#fLp(w)

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The Cp theorems Thm (E. Sawyer, 1984) If p ∈ (1, ∞) and w ∈ Cp+ǫ TfLp(w) ≤ c MfLp(w)

• The proof is a sophisticated version of Coifman-Fefferman’s A∞’s proof
• There is another interesting related result

Thm (K. Yabuta, 1990) If p ∈ (1, ∞) and w ∈ Cp+ǫ MfLp(w) ≤ c M#fLp(w)

• Recall that

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The Cp theorems Thm (E. Sawyer, 1984) If p ∈ (1, ∞) and w ∈ Cp+ǫ TfLp(w) ≤ c MfLp(w)

• The proof is a sophisticated version of Coifman-Fefferman’s A∞’s proof
• There is another interesting related result

Thm (K. Yabuta, 1990) If p ∈ (1, ∞) and w ∈ Cp+ǫ MfLp(w) ≤ c M#fLp(w)

• Recall that

M#f(x) = supx∈Q 1

|Q|

• Q |f − fQ|

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Recent extensions and improvements I: (with E. Cejas, I. Rivera-Rios & K. Li)

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Recent extensions and improvements I: (with E. Cejas, I. Rivera-Rios & K. Li) Thm Let p ∈ (0, ∞) and w ∈ Cmax{1,p}+ǫ

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Recent extensions and improvements I: (with E. Cejas, I. Rivera-Rios & K. Li) Thm Let p ∈ (0, ∞) and w ∈ Cmax{1,p}+ǫ TfLp(w) ≤ cT,p,ǫMfLp(w)

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Recent extensions and improvements I: (with E. Cejas, I. Rivera-Rios & K. Li) Thm Let p ∈ (0, ∞) and w ∈ Cmax{1,p}+ǫ TfLp(w) ≤ cT,p,ǫMfLp(w)

• The case of multilinear Calder´
• n-Zygmund operators we obtained results

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Recent extensions and improvements I: (with E. Cejas, I. Rivera-Rios & K. Li) Thm Let p ∈ (0, ∞) and w ∈ Cmax{1,p}+ǫ TfLp(w) ≤ cT,p,ǫMfLp(w)

• The case of multilinear Calder´
• n-Zygmund operators we obtained results

Thm Let p ∈ (0, ∞) and w ∈ Cmax{1,mp}+ǫ

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Recent extensions and improvements I: (with E. Cejas, I. Rivera-Rios & K. Li) Thm Let p ∈ (0, ∞) and w ∈ Cmax{1,p}+ǫ TfLp(w) ≤ cT,p,ǫMfLp(w)

• The case of multilinear Calder´
• n-Zygmund operators we obtained results

Thm Let p ∈ (0, ∞) and w ∈ Cmax{1,mp}+ǫ T( f)Lp(w) ≤ c M( f)Lp(w)

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Recent extensions and improvements I: (with E. Cejas, I. Rivera-Rios & K. Li) Thm Let p ∈ (0, ∞) and w ∈ Cmax{1,p}+ǫ TfLp(w) ≤ cT,p,ǫMfLp(w)

• The case of multilinear Calder´
• n-Zygmund operators we obtained results

Thm Let p ∈ (0, ∞) and w ∈ Cmax{1,mp}+ǫ T( f)Lp(w) ≤ c M( f)Lp(w)

• Key point: the following pointwise inequality

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Recent extensions and improvements I: (with E. Cejas, I. Rivera-Rios & K. Li) Thm Let p ∈ (0, ∞) and w ∈ Cmax{1,p}+ǫ TfLp(w) ≤ cT,p,ǫMfLp(w)

• The case of multilinear Calder´
• n-Zygmund operators we obtained results

Thm Let p ∈ (0, ∞) and w ∈ Cmax{1,mp}+ǫ T( f)Lp(w) ≤ c M( f)Lp(w)

• Key point: the following pointwise inequality

M#

δ (T(

f ))(x) ≤ c M( f )(x), 0 < δ < 1 m

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Improvements: joint work with Javier Canto

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Improvements: joint work with Javier Canto Thm If p ∈ (1, ∞) and w ∈ Cp+ǫ

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Improvements: joint work with Javier Canto Thm If p ∈ (1, ∞) and w ∈ Cp+ǫ TfLp(w) ([w]Cp+ǫ + 1) log(e + [w]Cp+ǫ) MfLp(w)

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Improvements: joint work with Javier Canto Thm If p ∈ (1, ∞) and w ∈ Cp+ǫ TfLp(w) ([w]Cp+ǫ + 1) log(e + [w]Cp+ǫ) MfLp(w)

• We need to define the constant [w]Cp

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Improvements: joint work with Javier Canto Thm If p ∈ (1, ∞) and w ∈ Cp+ǫ TfLp(w) ([w]Cp+ǫ + 1) log(e + [w]Cp+ǫ) MfLp(w)

• We need to define the constant [w]Cp
• The log appears as a consequence of the non-local nature of the condition

Cp, but we conjecture that it should be linear:

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Improvements: joint work with Javier Canto Thm If p ∈ (1, ∞) and w ∈ Cp+ǫ TfLp(w) ([w]Cp+ǫ + 1) log(e + [w]Cp+ǫ) MfLp(w)

• We need to define the constant [w]Cp
• The log appears as a consequence of the non-local nature of the condition

Cp, but we conjecture that it should be linear: Conjecture

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Improvements: joint work with Javier Canto Thm If p ∈ (1, ∞) and w ∈ Cp+ǫ TfLp(w) ([w]Cp+ǫ + 1) log(e + [w]Cp+ǫ) MfLp(w)

• We need to define the constant [w]Cp
• The log appears as a consequence of the non-local nature of the condition

Cp, but we conjecture that it should be linear: Conjecture TfLp(w) ([w]Cp+ǫ + 1)MfLp(w)

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Improvements: joint work with Javier Canto Thm If p ∈ (1, ∞) and w ∈ Cp+ǫ TfLp(w) ([w]Cp+ǫ + 1) log(e + [w]Cp+ǫ) MfLp(w)

• We need to define the constant [w]Cp
• The log appears as a consequence of the non-local nature of the condition

Cp, but we conjecture that it should be linear: Conjecture TfLp(w) ([w]Cp+ǫ + 1)MfLp(w) Thm If p ∈ (1, ∞) and w ∈ Cp+ǫ

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Improvements: joint work with Javier Canto Thm If p ∈ (1, ∞) and w ∈ Cp+ǫ TfLp(w) ([w]Cp+ǫ + 1) log(e + [w]Cp+ǫ) MfLp(w)

• We need to define the constant [w]Cp
• The log appears as a consequence of the non-local nature of the condition

Cp, but we conjecture that it should be linear: Conjecture TfLp(w) ([w]Cp+ǫ + 1)MfLp(w) Thm If p ∈ (1, ∞) and w ∈ Cp+ǫ MfLp(w) p2 ǫ ([w]Cp+ǫ + 1) log(e + [w]Cp+ǫ) M#fLp(w)

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Properties

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Properties Recall the definition of the Cp class:

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Properties Recall the definition of the Cp class: w(E) ≤ C

• |E|

|Q|

δ

Rn(MχQ(x))pw(x)dx

E ⊂ Q

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Properties Recall the definition of the Cp class: w(E) ≤ C

• |E|

|Q|

δ

Rn(MχQ(x))pw(x)dx

E ⊂ Q

• The definition of Cp implies that for appropriate constants c and δ

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Properties Recall the definition of the Cp class: w(E) ≤ C

• |E|

|Q|

δ

Rn(MχQ(x))pw(x)dx

E ⊂ Q

• The definition of Cp implies that for appropriate constants c and δ
• 1

|Q|

• Q w1+δ
• 1

1+δ

≤ c |Q|

• Rn(MχQ)pw

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Properties Recall the definition of the Cp class: w(E) ≤ C

• |E|

|Q|

δ

Rn(MχQ(x))pw(x)dx

E ⊂ Q

• The definition of Cp implies that for appropriate constants c and δ
• 1

|Q|

• Q w1+δ
• 1

1+δ

≤ c |Q|

• Rn(MχQ)pw

However these constants are not so convenient or precise

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Key point I: A quantitative RHI for Cp weights

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Key point I: A quantitative RHI for Cp weights To improve the RHI result we need to define an appropriate weight constant.

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Key point I: A quantitative RHI for Cp weights To improve the RHI result we need to define an appropriate weight constant. Definition [w]Cp := sup

Q

1

• Rn(MχQ)pw
• Q M(χQw)

E ⊂ Q

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Key point I: A quantitative RHI for Cp weights To improve the RHI result we need to define an appropriate weight constant. Definition [w]Cp := sup

Q

1

• Rn(MχQ)pw
• Q M(χQw)

E ⊂ Q

• If 0 < q ≤ p

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Key point I: A quantitative RHI for Cp weights To improve the RHI result we need to define an appropriate weight constant. Definition [w]Cp := sup

Q

1

• Rn(MχQ)pw
• Q M(χQw)

E ⊂ Q

• If 0 < q ≤ p

[w]Cq ≤ [w]Cp ≤ [w]A∞

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Key point I: A quantitative RHI for Cp weights To improve the RHI result we need to define an appropriate weight constant. Definition [w]Cp := sup

Q

1

• Rn(MχQ)pw
• Q M(χQw)

E ⊂ Q

• If 0 < q ≤ p

[w]Cq ≤ [w]Cp ≤ [w]A∞

• The quantitative optimal result is the following:

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Key point I: A quantitative RHI for Cp weights To improve the RHI result we need to define an appropriate weight constant. Definition [w]Cp := sup

Q

1

• Rn(MχQ)pw
• Q M(χQw)

E ⊂ Q

• If 0 < q ≤ p

[w]Cq ≤ [w]Cp ≤ [w]A∞

• The quantitative optimal result is the following:

Thm Let p ∈ (1, ∞) and let w ∈ Cp. Then

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Key point I: A quantitative RHI for Cp weights To improve the RHI result we need to define an appropriate weight constant. Definition [w]Cp := sup

Q

1

• Rn(MχQ)pw
• Q M(χQw)

E ⊂ Q

• If 0 < q ≤ p

[w]Cq ≤ [w]Cp ≤ [w]A∞

• The quantitative optimal result is the following:

Thm Let p ∈ (1, ∞) and let w ∈ Cp. Then

• 1

|Q|

• Q w1+δ
• 1

1+δ

≤ 4 |Q|

• Rn(MχQ)pw

The Cp class of weights cperez@bcamath.org

Key point I: A quantitative RHI for Cp weights To improve the RHI result we need to define an appropriate weight constant. Definition [w]Cp := sup

Q

1

• Rn(MχQ)pw
• Q M(χQw)

E ⊂ Q

• If 0 < q ≤ p

[w]Cq ≤ [w]Cp ≤ [w]A∞

• The quantitative optimal result is the following:

Thm Let p ∈ (1, ∞) and let w ∈ Cp. Then

• 1

|Q|

• Q w1+δ
• 1

1+δ

≤ 4 |Q|

• Rn(MχQ)pw

where δ = 1 cn,p max{[w]Cp, 1}

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Key point II: An extension of the John-Nirenberg’s theorem Thm Let 1 ≤ p < ∞, then

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Key point II: An extension of the John-Nirenberg’s theorem Thm Let 1 ≤ p < ∞, then

• 1

|Q|

• Q

MQ(f − fQ)(x)

M#f(x)

p

dx

1

p

≤ cn p

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Key point II: An extension of the John-Nirenberg’s theorem Thm Let 1 ≤ p < ∞, then

• 1

|Q|

• Q

MQ(f − fQ)(x)

M#f(x)

p

dx

1

p

≤ cn p

• As a consequence we have the local exponential decay

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Key point II: An extension of the John-Nirenberg’s theorem Thm Let 1 ≤ p < ∞, then

• 1

|Q|

• Q

MQ(f − fQ)(x)

M#f(x)

p

dx

1

p

≤ cn p

• As a consequence we have the local exponential decay
• y∈Q:MQ(f−fQ)(x)>t, M#f(x)≤t ε
• |Q|

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Key point II: An extension of the John-Nirenberg’s theorem Thm Let 1 ≤ p < ∞, then

• 1

|Q|

• Q

MQ(f − fQ)(x)

M#f(x)

p

dx

1

p

≤ cn p

• As a consequence we have the local exponential decay
• y∈Q:MQ(f−fQ)(x)>t, M#f(x)≤t ε
• |Q|

≤ c e−c

ε

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Two characterizations

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Two characterizations Let f ∈ BMO and let w ∈ A∞, then one can show that

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Two characterizations Let f ∈ BMO and let w ∈ A∞, then one can show that sup

Q

1 w(Q)

• Q |f − fQ| wdx ≤ cn[w]A∞ fBMO,

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Two characterizations Let f ∈ BMO and let w ∈ A∞, then one can show that sup

Q

1 w(Q)

• Q |f − fQ| wdx ≤ cn[w]A∞ fBMO,
• Is this is true for other weights? (doubling, for instance)

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Two characterizations Let f ∈ BMO and let w ∈ A∞, then one can show that sup

Q

1 w(Q)

• Q |f − fQ| wdx ≤ cn[w]A∞ fBMO,
• Is this is true for other weights? (doubling, for instance)
• Then answer is in the negative:

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Two characterizations Let f ∈ BMO and let w ∈ A∞, then one can show that sup

Q

1 w(Q)

• Q |f − fQ| wdx ≤ cn[w]A∞ fBMO,
• Is this is true for other weights? (doubling, for instance)
• Then answer is in the negative:

Thm (another characterization of A∞) [w]A∞ ≈ sup

f:fBMO=1

sup

Q

1 w(Q)

• Q |f(x) − fQ| wdx

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Two characterizations Let f ∈ BMO and let w ∈ A∞, then one can show that sup

Q

1 w(Q)

• Q |f − fQ| wdx ≤ cn[w]A∞ fBMO,
• Is this is true for other weights? (doubling, for instance)
• Then answer is in the negative:

Thm (another characterization of A∞) [w]A∞ ≈ sup

f:fBMO=1

sup

Q

1 w(Q)

• Q |f(x) − fQ| wdx
• Similarly for the Cp class:

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Two characterizations Let f ∈ BMO and let w ∈ A∞, then one can show that sup

Q

1 w(Q)

• Q |f − fQ| wdx ≤ cn[w]A∞ fBMO,
• Is this is true for other weights? (doubling, for instance)
• Then answer is in the negative:

Thm (another characterization of A∞) [w]A∞ ≈ sup

f:fBMO=1

sup

Q

1 w(Q)

• Q |f(x) − fQ| wdx
• Similarly for the Cp class:

Thm [w]Cp ≈ sup

f:fBMO=1

sup

Q

1

• Rn M(χQ)pw
• Q |f(x) − fQ| wdx

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The Cp class of weights cperez@bcamath.org

The Cp class of weights cperez@bcamath.org

DZIEKUJE BARDZO

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DZIEKUJE BARDZO THANK YOU VERY MUCH