[PPT] - Endpoint behavior of modulation invariant singular integrals PowerPoint Presentation

SLIDE 1

Endpoint behavior of modulation invariant singular integrals

Francesco Di Plinio

INdAM-Cofund Marie Curie Fellow at Universit` a degli Studi Roma “Tor Vergata” Institute of Scientific Computing and Applied Mathematics at Indiana University, Fellow

XXXIII Convegno Nazionale di Analisi Armonica Alba, 17-20 Giugno 2013

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Partly joint work with Ciprian Demeter and Christoph Thiele.

C. Demeter, F. Di Plinio, Endpoint bounds for the quartile operator, arXiv

preprint, to appear on Journal of Fourier Analysis and Applications

F. Di Plinio, Lacunary Fourier and Walsh-Fourier series near L1, arXiv

preprint

SLIDE 3

The Bilinear Hilbert Transform

BHT(f, g)(x) = p.v. ˆ

R

f(x − t)g(x + t)dt t ∼ ˆ

ξ>η

ˆ f(ξ)ˆ g(η)eix(ξ+η) dξdη. Multiplier singular on a line {ξ = η} ❀ modulation invariant paraproduct: BHT(Mζf, Mζg) = M2ζBHT(f, g) Same scaling as pointwise product ❀ H¨

lder-type bounds:

BHT : Lp1,q1(R) × Lp2,q2(R) → Lp3,q3(R) = ⇒ 1 p1 + 1 p2 = 1 p3 Lacey and Thiele showed BHT : Lp1(R) × Lp2(R) → Lp3(R) whenever (p1, p2, p3) H¨

lder tuple with p1, p2 > 1 and p3 > 2

3.

F. Di Plinio (Rome Tor Vergata)

Endpoint bounds 3 / 15

SLIDE 4

The Bilinear Hilbert Transform

BHT(f, g)(x) = p.v. ˆ

R

f(x − t)g(x + t)dt t ∼ ˆ

ξ>η

ˆ f(ξ)ˆ g(η)eix(ξ+η) dξdη. Multiplier singular on a line {ξ = η} ❀ modulation invariant paraproduct: BHT(Mζf, Mζg) = M2ζBHT(f, g) Same scaling as pointwise product ❀ H¨

lder-type bounds:

BHT : Lp1,q1(R) × Lp2,q2(R) → Lp3,q3(R) = ⇒ 1 p1 + 1 p2 = 1 p3 Lacey and Thiele showed BHT : Lp1(R) × Lp2(R) → Lp3(R) whenever (p1, p2, p3) H¨

lder tuple with p1, p2 > 1 and p3 > 2

3.

F. Di Plinio (Rome Tor Vergata)

Endpoint bounds 3 / 15

SLIDE 5

The Bilinear Hilbert Transform

BHT(f, g)(x) = p.v. ˆ

R

f(x − t)g(x + t)dt t ∼ ˆ

ξ>η

ˆ f(ξ)ˆ g(η)eix(ξ+η) dξdη. Multiplier singular on a line {ξ = η} ❀ modulation invariant paraproduct: BHT(Mζf, Mζg) = M2ζBHT(f, g) Same scaling as pointwise product ❀ H¨

lder-type bounds:

BHT : Lp1,q1(R) × Lp2,q2(R) → Lp3,q3(R) = ⇒ 1 p1 + 1 p2 = 1 p3 Lacey and Thiele showed BHT : Lp1(R) × Lp2(R) → Lp3(R) whenever (p1, p2, p3) H¨

lder tuple with p1, p2 > 1 and p3 > 2

3.

F. Di Plinio (Rome Tor Vergata)

Endpoint bounds 3 / 15

SLIDE 6

The Bilinear Hilbert Transform

BHT(f, g)(x) = p.v. ˆ

R

f(x − t)g(x + t)dt t ∼ ˆ

ξ>η

ˆ f(ξ)ˆ g(η)eix(ξ+η) dξdη. Multiplier singular on a line {ξ = η} ❀ modulation invariant paraproduct: BHT(Mζf, Mζg) = M2ζBHT(f, g) Same scaling as pointwise product ❀ H¨

lder-type bounds:

BHT : Lp1,q1(R) × Lp2,q2(R) → Lp3,q3(R) = ⇒ 1 p1 + 1 p2 = 1 p3 Lacey and Thiele showed BHT : Lp1(R) × Lp2(R) → Lp3(R) whenever (p1, p2, p3) H¨

lder tuple with p1, p2 > 1 and p3 > 2

3.

F. Di Plinio (Rome Tor Vergata)

Endpoint bounds 3 / 15

SLIDE 7

Fourier and Walsh models for BHT

Discretize into squares Q = ω × |ω| + ω ❀ dist(Q, {ξ = η}) ∼ |ω|
supp ˆ

f ⊂ ω, supp ˆ g ⊂ |ω| + ω ❀ supp

BHT(f, g) = supp

fg =⊂ 2|ω| + ω

Model sum

Setting s = Is × ωs, s1 = Is × (ωs + |ωs|), s2 = Is × (ωs + 2|ωs|), BHT(f, g)(x) ∼ BHT(f, g)(x) =

s∈Su

1

|Is|

f, ϕsg, ϕs1ϕs2(x)

Rmk. The models BHT fail to map into Lp3 for p3 < 2
3. Not known for

BHT. What about p3 = 2

3? L

2 3 candidate range for BHT on L1 × L2.

F. Di Plinio (Rome Tor Vergata)

Endpoint bounds 4 / 15

SLIDE 8

Fourier and Walsh models for BHT

Discretize into squares Q = ω × |ω| + ω ❀ dist(Q, {ξ = η}) ∼ |ω|
supp ˆ

f ⊂ ω, supp ˆ g ⊂ |ω| + ω ❀ supp

BHT(f, g) = supp

fg =⊂ 2|ω| + ω

Model sum

Setting s = Is × ωs, s1 = Is × (ωs + |ωs|), s2 = Is × (ωs + 2|ωs|), BHT(f, g)(x) ∼ BHT(f, g)(x) =

s∈Su

1

|Is|

f, ϕsg, ϕs1ϕs2(x)

Rmk. The models BHT fail to map into Lp3 for p3 < 2
3. Not known for

BHT. What about p3 = 2

3? L

2 3 candidate range for BHT on L1 × L2.

F. Di Plinio (Rome Tor Vergata)

Endpoint bounds 4 / 15

SLIDE 9

Fourier and Walsh models for BHT

Discretize into squares Q = ω × |ω| + ω ❀ dist(Q, {ξ = η}) ∼ |ω|
supp ˆ

f ⊂ ω, supp ˆ g ⊂ |ω| + ω ❀ supp

BHT(f, g) = supp

fg =⊂ 2|ω| + ω

Model sum

Setting s = Is × ωs, s1 = Is × (ωs + |ωs|), s2 = Is × (ωs + 2|ωs|), BHT(f, g)(x) ∼ BHT(f, g)(x) =

s∈Su

1

|Is|

f, ϕsg, ϕs1ϕs2(x)

Rmk. The models BHT fail to map into Lp3 for p3 < 2
3. Not known for

BHT. What about p3 = 2

3? L

2 3 candidate range for BHT on L1 × L2.

F. Di Plinio (Rome Tor Vergata)

Endpoint bounds 4 / 15

SLIDE 10

Fourier and Walsh models for BHT

Discretize into squares Q = ω × |ω| + ω ❀ dist(Q, {ξ = η}) ∼ |ω|
supp ˆ

f ⊂ ω, supp ˆ g ⊂ |ω| + ω ❀ supp

BHT(f, g) = supp

fg =⊂ 2|ω| + ω

Model sum

Setting s = Is × ωs, s1 = Is × (ωs + |ωs|), s2 = Is × (ωs + 2|ωs|), BHT(f, g)(x) ∼ BHT(f, g)(x) =

s∈Su

1

|Is|

f, ϕsg, ϕs1ϕs2(x)

Rmk. The models BHT fail to map into Lp3 for p3 < 2
3. Not known for

BHT. What about p3 = 2

3? L

2 3 candidate range for BHT on L1 × L2.

F. Di Plinio (Rome Tor Vergata)

Endpoint bounds 4 / 15

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RWT bounds for BHT and near-endpoint results

Conj. BHT : Lp1 × Lp2 → L

2 3 ,∞

∀1 < p1, p2 < 2,

1 p1 + 1 p2 = 3 2.

Appropriate substitutes near L1 × L2. LT result follows via (generalized) RWT interpolation of (RWE) |BHT(f1, f2), f3| |F1|

1 p1 |F2| 1 p2 |G3|1− 1 p3 ,

|fj| ≤ 1Fj where F3 ⊂ G3 major set, in the open range 1 < p1, p2 ≤ ∞, p3 > 2

3

Estimate (RWE) for p3 = 2

3 (weaker than Conj.) is OPEN

Theorem (Bilyk-Grafakos06)

Log-bumped version near (1, 2, 2

3): for |F1| ≤| F2|

(BG) |BHT(f1, f2), f3| |F1||F2|

1 2 |F3|− 1 2 log

e +

|F3|2 |F1||F2|

2.

F. Di Plinio (Rome Tor Vergata)

Endpoint bounds 5 / 15

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RWT bounds for BHT and near-endpoint results

Conj. BHT : Lp1 × Lp2 → L

2 3 ,∞

∀1 < p1, p2 < 2,

1 p1 + 1 p2 = 3 2.

Appropriate substitutes near L1 × L2. LT result follows via (generalized) RWT interpolation of (RWE) |BHT(f1, f2), f3| |F1|

1 p1 |F2| 1 p2 |G3|1− 1 p3 ,

|fj| ≤ 1Fj where F3 ⊂ G3 major set, in the open range 1 < p1, p2 ≤ ∞, p3 > 2

3

Estimate (RWE) for p3 = 2

3 (weaker than Conj.) is OPEN

Theorem (Bilyk-Grafakos06)

Log-bumped version near (1, 2, 2

3): for |F1| ≤| F2|

(BG) |BHT(f1, f2), f3| |F1||F2|

1 2 |F3|− 1 2 log

e +

|F3|2 |F1||F2|

2.

F. Di Plinio (Rome Tor Vergata)

Endpoint bounds 5 / 15

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RWT bounds for BHT and near-endpoint results

Conj. BHT : Lp1 × Lp2 → L

2 3 ,∞

∀1 < p1, p2 < 2,

1 p1 + 1 p2 = 3 2.

Appropriate substitutes near L1 × L2. LT result follows via (generalized) RWT interpolation of (RWE) |BHT(f1, f2), f3| |F1|

1 p1 |F2| 1 p2 |G3|1− 1 p3 ,

|fj| ≤ 1Fj where F3 ⊂ G3 major set, in the open range 1 < p1, p2 ≤ ∞, p3 > 2

3

Estimate (RWE) for p3 = 2

3 (weaker than Conj.) is OPEN

Theorem (Bilyk-Grafakos06)

Log-bumped version near (1, 2, 2

3): for |F1| ≤| F2|

(BG) |BHT(f1, f2), f3| |F1||F2|

1 2 |F3|− 1 2 log

e +

|F3|2 |F1||F2|

2.

F. Di Plinio (Rome Tor Vergata)

Endpoint bounds 5 / 15

SLIDE 14

RWT bounds for BHT and near-endpoint results

Conj. BHT : Lp1 × Lp2 → L

2 3 ,∞

∀1 < p1, p2 < 2,

1 p1 + 1 p2 = 3 2.

Appropriate substitutes near L1 × L2. LT result follows via (generalized) RWT interpolation of (RWE) |BHT(f1, f2), f3| |F1|

1 p1 |F2| 1 p2 |G3|1− 1 p3 ,

|fj| ≤ 1Fj where F3 ⊂ G3 major set, in the open range 1 < p1, p2 ≤ ∞, p3 > 2

3

Estimate (RWE) for p3 = 2

3 (weaker than Conj.) is OPEN

Theorem (Bilyk-Grafakos06)

Log-bumped version near (1, 2, 2

3): for |F1| ≤| F2|

(BG) |BHT(f1, f2), f3| |F1||F2|

1 2 |F3|− 1 2 log

e +

|F3|2 |F1||F2|

2.

F. Di Plinio (Rome Tor Vergata)

Endpoint bounds 5 / 15

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RWT bounds for BHT and near-endpoint results

Theorems (Carro-Grafakos-Martell-Soria09)

Proxy for L

2 3 ,∞: weighted Lorentz quasi-Banach space

L

2 3 ,∞

f

L

2 3 ,∞ = sup

t>0 t

3 2

log(e+t)f∗(t).

Using (RWE) and bilinear extrapolation (CGMS1) BHT : L1, 2

3 (log L) 4 3 × L2, 2 3 (log L) 4 3 →

L

2 3 ,∞

Using (RWE) and (ε, δ)-atomic approximability of BHT (CGMS2) BHT : L(log L)2(log2 L)

1 2 (log3 L) 1 2 +η ×L2, 2 3 (log L) 4 3 →

L

2 3 ,∞

F. Di Plinio (Rome Tor Vergata)

Endpoint bounds 6 / 15

SLIDE 16

RWT bounds for BHT and near-endpoint results

Theorems (Carro-Grafakos-Martell-Soria09)

Proxy for L

2 3 ,∞: weighted Lorentz quasi-Banach space

L

2 3 ,∞

f

L

2 3 ,∞ = sup

t>0 t

3 2

log(e+t)f∗(t).

Using (RWE) and bilinear extrapolation (CGMS1) BHT : L1, 2

3 (log L) 4 3 × L2, 2 3 (log L) 4 3 →

L

2 3 ,∞

Using (RWE) and (ε, δ)-atomic approximability of BHT (CGMS2) BHT : L(log L)2(log2 L)

1 2 (log3 L) 1 2 +η ×L2, 2 3 (log L) 4 3 →

L

2 3 ,∞

F. Di Plinio (Rome Tor Vergata)

Endpoint bounds 6 / 15

SLIDE 17

RWT bounds for BHT and near-endpoint results

Theorems (Carro-Grafakos-Martell-Soria09)

Proxy for L

2 3 ,∞: weighted Lorentz quasi-Banach space

L

2 3 ,∞

f

L

2 3 ,∞ = sup

t>0 t

3 2

log(e+t)f∗(t).

Using (RWE) and bilinear extrapolation (CGMS1) BHT : L1, 2

3 (log L) 4 3 × L2, 2 3 (log L) 4 3 →

L

2 3 ,∞

Using (RWE) and (ε, δ)-atomic approximability of BHT (CGMS2) BHT : L(log L)2(log2 L)

1 2 (log3 L) 1 2 +η ×L2, 2 3 (log L) 4 3 →

L

2 3 ,∞

F. Di Plinio (Rome Tor Vergata)

Endpoint bounds 6 / 15

SLIDE 18

RWT bounds for BHT and near-endpoint results

Theorems (Carro-Grafakos-Martell-Soria09)

Proxy for L

2 3 ,∞: weighted Lorentz quasi-Banach space

L

2 3 ,∞

f

L

2 3 ,∞ = sup

t>0 t

3 2

log(e+t)f∗(t).

Using (RWE) and bilinear extrapolation (CGMS1) BHT : L1, 2

3 (log L) 4 3 × L2, 2 3 (log L) 4 3 →

L

2 3 ,∞

Using (RWE) and (ε, δ)-atomic approximability of BHT (CGMS2) BHT : L(log L)2(log2 L)

1 2 (log3 L) 1 2 +η ×L2, 2 3 (log L) 4 3 →

L

2 3 ,∞

F. Di Plinio (Rome Tor Vergata)

Endpoint bounds 6 / 15

SLIDE 19

The quartile operator

Walsh basis {Wn : n ∈ N}: orthonormal basis of L2(T) defined as Wn(x) =

k∈N
sign sin(2k2πx)

εk(n), εk(n) := ⌊2−kn⌋ mod 2. · A quartile s = Is × ωs is a dyadic rectangle of area 4: |ωs| = 4|Is|−1 · ωs = union of its dyadic grandchildren ωsj, j = 1, . . . , 4 · denote by sj the tiles Is × ωsj, · to each tile t = It × ωt, associate a Walsh wave packet by wt(x) = |It|−1/2Wnt x − inf It |It|

,

nt := |It| inf ωt. · perfect localization: t ∩ t′ = ∅ = ⇒ wt, wt′ = 0. The quartile operator: Q(f1, f2)(x) =

quartiles

1

|Is|

f1, ws1f2, ws2ws3(x)

F. Di Plinio (Rome Tor Vergata)

Endpoint bounds 7 / 15

SLIDE 20

The quartile operator

Walsh basis {Wn : n ∈ N}: orthonormal basis of L2(T) defined as Wn(x) =

k∈N
sign sin(2k2πx)

εk(n), εk(n) := ⌊2−kn⌋ mod 2. · A quartile s = Is × ωs is a dyadic rectangle of area 4: |ωs| = 4|Is|−1 · ωs = union of its dyadic grandchildren ωsj, j = 1, . . . , 4 · denote by sj the tiles Is × ωsj, · to each tile t = It × ωt, associate a Walsh wave packet by wt(x) = |It|−1/2Wnt x − inf It |It|

,

nt := |It| inf ωt. · perfect localization: t ∩ t′ = ∅ = ⇒ wt, wt′ = 0. The quartile operator: Q(f1, f2)(x) =

quartiles

1

|Is|

f1, ws1f2, ws2ws3(x)

F. Di Plinio (Rome Tor Vergata)

Endpoint bounds 7 / 15

SLIDE 21

The quartile operator

Walsh basis {Wn : n ∈ N}: orthonormal basis of L2(T) defined as Wn(x) =

k∈N
sign sin(2k2πx)

εk(n), εk(n) := ⌊2−kn⌋ mod 2. · A quartile s = Is × ωs is a dyadic rectangle of area 4: |ωs| = 4|Is|−1 · ωs = union of its dyadic grandchildren ωsj, j = 1, . . . , 4 · denote by sj the tiles Is × ωsj, · to each tile t = It × ωt, associate a Walsh wave packet by wt(x) = |It|−1/2Wnt x − inf It |It|

,

nt := |It| inf ωt. · perfect localization: t ∩ t′ = ∅ = ⇒ wt, wt′ = 0. The quartile operator: Q(f1, f2)(x) =

quartiles

1

|Is|

f1, ws1f2, ws2ws3(x)

F. Di Plinio (Rome Tor Vergata)

Endpoint bounds 7 / 15

SLIDE 22

The quartile operator

Walsh basis {Wn : n ∈ N}: orthonormal basis of L2(T) defined as Wn(x) =

k∈N
sign sin(2k2πx)

εk(n), εk(n) := ⌊2−kn⌋ mod 2. · A quartile s = Is × ωs is a dyadic rectangle of area 4: |ωs| = 4|Is|−1 · ωs = union of its dyadic grandchildren ωsj, j = 1, . . . , 4 · denote by sj the tiles Is × ωsj, · to each tile t = It × ωt, associate a Walsh wave packet by wt(x) = |It|−1/2Wnt x − inf It |It|

,

nt := |It| inf ωt. · perfect localization: t ∩ t′ = ∅ = ⇒ wt, wt′ = 0. The quartile operator: Q(f1, f2)(x) =

quartiles

1

|Is|

f1, ws1f2, ws2ws3(x)

F. Di Plinio (Rome Tor Vergata)

Endpoint bounds 7 / 15

SLIDE 23

The quartile operator

Walsh basis {Wn : n ∈ N}: orthonormal basis of L2(T) defined as Wn(x) =

k∈N
sign sin(2k2πx)

εk(n), εk(n) := ⌊2−kn⌋ mod 2. · A quartile s = Is × ωs is a dyadic rectangle of area 4: |ωs| = 4|Is|−1 · ωs = union of its dyadic grandchildren ωsj, j = 1, . . . , 4 · denote by sj the tiles Is × ωsj, · to each tile t = It × ωt, associate a Walsh wave packet by wt(x) = |It|−1/2Wnt x − inf It |It|

,

nt := |It| inf ωt. · perfect localization: t ∩ t′ = ∅ = ⇒ wt, wt′ = 0. The quartile operator: Q(f1, f2)(x) =

quartiles

1

|Is|

f1, ws1f2, ws2ws3(x)

F. Di Plinio (Rome Tor Vergata)

Endpoint bounds 7 / 15

SLIDE 24

The quartile operator

Walsh basis {Wn : n ∈ N}: orthonormal basis of L2(T) defined as Wn(x) =

k∈N
sign sin(2k2πx)

εk(n), εk(n) := ⌊2−kn⌋ mod 2. · A quartile s = Is × ωs is a dyadic rectangle of area 4: |ωs| = 4|Is|−1 · ωs = union of its dyadic grandchildren ωsj, j = 1, . . . , 4 · denote by sj the tiles Is × ωsj, · to each tile t = It × ωt, associate a Walsh wave packet by wt(x) = |It|−1/2Wnt x − inf It |It|

,

nt := |It| inf ωt. · perfect localization: t ∩ t′ = ∅ = ⇒ wt, wt′ = 0. The quartile operator: Q(f1, f2)(x) =

quartiles

1

|Is|

f1, ws1f2, ws2ws3(x)

F. Di Plinio (Rome Tor Vergata)

Endpoint bounds 7 / 15

SLIDE 25

The quartile operator

Walsh basis {Wn : n ∈ N}: orthonormal basis of L2(T) defined as Wn(x) =

k∈N
sign sin(2k2πx)

εk(n), εk(n) := ⌊2−kn⌋ mod 2. · A quartile s = Is × ωs is a dyadic rectangle of area 4: |ωs| = 4|Is|−1 · ωs = union of its dyadic grandchildren ωsj, j = 1, . . . , 4 · denote by sj the tiles Is × ωsj, · to each tile t = It × ωt, associate a Walsh wave packet by wt(x) = |It|−1/2Wnt x − inf It |It|

,

nt := |It| inf ωt. · perfect localization: t ∩ t′ = ∅ = ⇒ wt, wt′ = 0. The quartile operator: Q(f1, f2)(x) =

quartiles

1

|Is|

f1, ws1f2, ws2ws3(x)

F. Di Plinio (Rome Tor Vergata)

Endpoint bounds 7 / 15

SLIDE 26

Weak-type endpoint bounds for the quartile operator

Theorems (Demeter-D2012). Let 1 < p1, p2 < 2, 1

p1 + 1 p2 = 3

2. Then

(WRW) Q(f1, f2) 2

3 ,∞ f1p1|F2| 1 p2 ,

|f2| ≤ 1F2. Substitute near L1 × L2: for all 1 < p < 2, (E12) Q(f1, f2)r,∞ ≤ C(p′)f1pf22,

1 r = 1 p + 1 2

Linear extrapolation and (WRW) yield Q : Lp1 × Lp2, 2

3 → L 2 3 ,∞.

The constant O(p′) in (E12) allows for (Log) Q(f1, f2)

L

2 3 ,∞ f11 log

f1∞

f11

f22

which in turn implies Q : L log L(log2 L)

1 2 (log3 L) 1 2 +η × L2 →

L

2 3 ,∞.

F. Di Plinio (Rome Tor Vergata)

Endpoint bounds 8 / 15

SLIDE 27

Weak-type endpoint bounds for the quartile operator

Theorems (Demeter-D2012). Let 1 < p1, p2 < 2, 1

p1 + 1 p2 = 3

2. Then

(WRW) Q(f1, f2) 2

3 ,∞ f1p1|F2| 1 p2 ,

|f2| ≤ 1F2. Substitute near L1 × L2: for all 1 < p < 2, (E12) Q(f1, f2)r,∞ ≤ C(p′)f1pf22,

1 r = 1 p + 1 2

Linear extrapolation and (WRW) yield Q : Lp1 × Lp2, 2

3 → L 2 3 ,∞.

The constant O(p′) in (E12) allows for (Log) Q(f1, f2)

L

2 3 ,∞ f11 log

f1∞

f11

f22

which in turn implies Q : L log L(log2 L)

1 2 (log3 L) 1 2 +η × L2 →

L

2 3 ,∞.

F. Di Plinio (Rome Tor Vergata)

Endpoint bounds 8 / 15

SLIDE 28

Weak-type endpoint bounds for the quartile operator

Theorems (Demeter-D2012). Let 1 < p1, p2 < 2, 1

p1 + 1 p2 = 3

2. Then

(WRW) Q(f1, f2) 2

3 ,∞ f1p1|F2| 1 p2 ,

|f2| ≤ 1F2. Substitute near L1 × L2: for all 1 < p < 2, (E12) Q(f1, f2)r,∞ ≤ C(p′)f1pf22,

1 r = 1 p + 1 2

Linear extrapolation and (WRW) yield Q : Lp1 × Lp2, 2

3 → L 2 3 ,∞.

The constant O(p′) in (E12) allows for (Log) Q(f1, f2)

L

2 3 ,∞ f11 log

f1∞

f11

f22

which in turn implies Q : L log L(log2 L)

1 2 (log3 L) 1 2 +η × L2 →

L

2 3 ,∞.

F. Di Plinio (Rome Tor Vergata)

Endpoint bounds 8 / 15

SLIDE 29

Weak-type endpoint bounds for the quartile operator

Theorems (Demeter-D2012). Let 1 < p1, p2 < 2, 1

p1 + 1 p2 = 3

2. Then

(WRW) Q(f1, f2) 2

3 ,∞ f1p1|F2| 1 p2 ,

|f2| ≤ 1F2. Substitute near L1 × L2: for all 1 < p < 2, (E12) Q(f1, f2)r,∞ ≤ C(p′)f1pf22,

1 r = 1 p + 1 2

Linear extrapolation and (WRW) yield Q : Lp1 × Lp2, 2

3 → L 2 3 ,∞.

The constant O(p′) in (E12) allows for (Log) Q(f1, f2)

L

2 3 ,∞ f11 log

f1∞

f11

f22

which in turn implies Q : L log L(log2 L)

1 2 (log3 L) 1 2 +η × L2 →

L

2 3 ,∞.

F. Di Plinio (Rome Tor Vergata)

Endpoint bounds 8 / 15

SLIDE 30

Weak-type endpoint bounds for the quartile operator

Theorems (Demeter-D2012). Let 1 < p1, p2 < 2, 1

p1 + 1 p2 = 3

2. Then

(WRW) Q(f1, f2) 2

3 ,∞ f1p1|F2| 1 p2 ,

|f2| ≤ 1F2. Substitute near L1 × L2: for all 1 < p < 2, (E12) Q(f1, f2)r,∞ ≤ C(p′)f1pf22,

1 r = 1 p + 1 2

Linear extrapolation and (WRW) yield Q : Lp1 × Lp2, 2

3 → L 2 3 ,∞.

The constant O(p′) in (E12) allows for (Log) Q(f1, f2)

L

2 3 ,∞ f11 log

f1∞

f11

f22

which in turn implies Q : L log L(log2 L)

1 2 (log3 L) 1 2 +η × L2 →

L

2 3 ,∞.

F. Di Plinio (Rome Tor Vergata)

Endpoint bounds 8 / 15

SLIDE 31

Weak-type endpoint bounds for the quartile operator

Theorems (Demeter-D2012). Let 1 < p1, p2 < 2, 1

p1 + 1 p2 = 3

2. Then

(WRW) Q(f1, f2) 2

3 ,∞ f1p1|F2| 1 p2 ,

|f2| ≤ 1F2. Substitute near L1 × L2: for all 1 < p < 2, (E12) Q(f1, f2)r,∞ ≤ C(p′)f1pf22,

1 r = 1 p + 1 2

Linear extrapolation and (WRW) yield Q : Lp1 × Lp2, 2

3 → L 2 3 ,∞.

The constant O(p′) in (E12) allows for (Log) Q(f1, f2)

L

2 3 ,∞ f11 log

f1∞

f11

f22

which in turn implies Q : L log L(log2 L)

1 2 (log3 L) 1 2 +η × L2 →

L

2 3 ,∞.

F. Di Plinio (Rome Tor Vergata)

Endpoint bounds 8 / 15

SLIDE 32

A multifrequency Calderon-Zygmund decomposition

Lemma

Let 1 < p < 2, an interval I ⊂ R, fI : I → C,

1

|I|

ˆ

I

|f(x)|p dx 1

p = λ,

and a collection of frequencies ξ1, . . . , ξM ∈ Rξ be given. Then fI = gI + bI, with

gI, bI supported on 3I;
gI2 λM

1 2 − 1 p′ |I| 1 2 ,

bI1 λM

1 2 − 1 p′ |I|,

ˆ

R

bI(x)e2πiξj dx = 0 ∀j = 1, . . . , M. The proof is based on the inequality (due to Borwein-Erdelyi) vL∞(−1,1) ≤ M

1 2 vL2(−3,3)

∀v ∈ span

e2πixξjM

j=1 ֒

→ L2(−3, 3)

F. Di Plinio (Rome Tor Vergata)

Endpoint bounds 9 / 15

SLIDE 33

A multifrequency Calderon-Zygmund decomposition

Lemma

Let 1 < p < 2, an interval I ⊂ R, fI : I → C,

1

|I|

ˆ

I

|f(x)|p dx 1

p = λ,

and a collection of frequencies ξ1, . . . , ξM ∈ Rξ be given. Then fI = gI + bI, with

gI, bI supported on 3I;
gI2 λM

1 2 − 1 p′ |I| 1 2 ,

bI1 λM

1 2 − 1 p′ |I|,

ˆ

R

bI(x)e2πiξj dx = 0 ∀j = 1, . . . , M. The proof is based on the inequality (due to Borwein-Erdelyi) vL∞(−1,1) ≤ M

1 2 vL2(−3,3)

∀v ∈ span

e2πixξjM

j=1 ֒

→ L2(−3, 3)

F. Di Plinio (Rome Tor Vergata)

Endpoint bounds 9 / 15

SLIDE 34

A multifrequency Calderon-Zygmund decomposition

Lemma

Let 1 < p < 2, an interval I ⊂ R, fI : I → C,

1

|I|

ˆ

I

|f(x)|p dx 1

p = λ,

and a collection of frequencies ξ1, . . . , ξM ∈ Rξ be given. Then fI = gI + bI, with

gI, bI supported on 3I;
gI2 λM

1 2 − 1 p′ |I| 1 2 ,

bI1 λM

1 2 − 1 p′ |I|,

ˆ

R

bI(x)e2πiξj dx = 0 ∀j = 1, . . . , M. The proof is based on the inequality (due to Borwein-Erdelyi) vL∞(−1,1) ≤ M

1 2 vL2(−3,3)

∀v ∈ span

e2πixξjM

j=1 ֒

→ L2(−3, 3)

F. Di Plinio (Rome Tor Vergata)

Endpoint bounds 9 / 15

SLIDE 35

A multifrequency Calderon-Zygmund decomposition

Lemma

Let 1 < p < 2, an interval I ⊂ R, fI : I → C,

1

|I|

ˆ

I

|f(x)|p dx 1

p = λ,

and a collection of frequencies ξ1, . . . , ξM ∈ Rξ be given. Then fI = gI + bI, with

gI, bI supported on 3I;
gI2 λM

1 2 − 1 p′ |I| 1 2 ,

bI1 λM

1 2 − 1 p′ |I|,

ˆ

R

bI(x)e2πiξj dx = 0 ∀j = 1, . . . , M. The proof is based on the inequality (due to Borwein-Erdelyi) vL∞(−1,1) ≤ M

1 2 vL2(−3,3)

∀v ∈ span

e2πixξjM

j=1 ֒

→ L2(−3, 3)

F. Di Plinio (Rome Tor Vergata)

Endpoint bounds 9 / 15

SLIDE 36

A multifrequency Calderon-Zygmund decomposition

Lemma

Let 1 < p < 2, an interval I ⊂ R, fI : I → C,

1

|I|

ˆ

I

|f(x)|p dx 1

p = λ,

and a collection of frequencies ξ1, . . . , ξM ∈ Rξ be given. Then fI = gI + bI, with

gI, bI supported on 3I;
gI2 λM

1 2 − 1 p′ |I| 1 2 ,

bI1 λM

1 2 − 1 p′ |I|,

ˆ

R

bI(x)e2πiξj dx = 0 ∀j = 1, . . . , M. The proof is based on the inequality (due to Borwein-Erdelyi) vL∞(−1,1) ≤ M

1 2 vL2(−3,3)

∀v ∈ span

e2πixξjM

j=1 ֒

→ L2(−3, 3)

F. Di Plinio (Rome Tor Vergata)

Endpoint bounds 9 / 15

SLIDE 37

A multifrequency Calderon-Zygmund decomposition

Lemma

Let 1 < p < 2, an interval I ⊂ R, fI : I → C,

1

|I|

ˆ

I

|f(x)|p dx 1

p = λ,

and a collection of frequencies ξ1, . . . , ξM ∈ Rξ be given. Then fI = gI + bI, with

gI, bI supported on 3I;
gI2 λM

1 2 − 1 p′ |I| 1 2 ,

bI1 λM

1 2 − 1 p′ |I|,

ˆ

R

bI(x)e2πiξj dx = 0 ∀j = 1, . . . , M. The proof is based on the inequality (due to Borwein-Erdelyi) vL∞(−1,1) ≤ M

1 2 vL2(−3,3)

∀v ∈ span

e2πixξjM

j=1 ֒

→ L2(−3, 3)

F. Di Plinio (Rome Tor Vergata)

Endpoint bounds 9 / 15

SLIDE 38

A multifrequency Calderon-Zygmund decomposition

Lemma

Let 1 < p < 2, an interval I ⊂ R, fI : I → C,

1

|I|

ˆ

I

|f(x)|p dx 1

p = λ,

and a collection of frequencies ξ1, . . . , ξM ∈ Rξ be given. Then fI = gI + bI, with

gI, bI supported on 3I;
gI2 λM

1 2 − 1 p′ |I| 1 2 ,

bI1 λM

1 2 − 1 p′ |I|,

ˆ

R

bI(x)e2πiξj dx = 0 ∀j = 1, . . . , M. The proof is based on the inequality (due to Borwein-Erdelyi) vL∞(−1,1) ≤ M

1 2 vL2(−3,3)

∀v ∈ span

e2πixξjM

j=1 ֒

→ L2(−3, 3)

F. Di Plinio (Rome Tor Vergata)

Endpoint bounds 9 / 15

SLIDE 39

Proof of (E12)

By scaling, proving (E12) same as: for all f1p = f22 = |F3| = 1,

ΛS(f1, f2, 1G3) :=
s∈S

1

|Is|

f1, ws1f2, ws2ws3, 1G3

p′

where, setting E := {Mpf1 1}, G3 := F3\ is a major subset of F3.

Trees and size

a tree T: collection of quartiles with Is ⊂ IT, ωs ∋ ξT for all s ∈ T
if ΠT,jf is the projection onto span{wsj : s ∈ T} ֒

→ L2(R), sizej(f, T) := ΠT,jfBMO(R).

if F is a union of trees T with size3(h3, T) ≤ σ,

ΛF(h1, h2, h3) σh12h22; (essentially QF : L2 × BMO → L2).

F. Di Plinio (Rome Tor Vergata)

Endpoint bounds 10 / 15

SLIDE 40

Proof of (E12)

By scaling, proving (E12) same as: for all f1p = f22 = |F3| = 1,

ΛS(f1, f2, 1G3) :=
s∈S

1

|Is|

f1, ws1f2, ws2ws3, 1G3

p′

where, setting E := {Mpf1 1}, G3 := F3\ is a major subset of F3.

Trees and size

a tree T: collection of quartiles with Is ⊂ IT, ωs ∋ ξT for all s ∈ T
if ΠT,jf is the projection onto span{wsj : s ∈ T} ֒

→ L2(R), sizej(f, T) := ΠT,jfBMO(R).

if F is a union of trees T with size3(h3, T) ≤ σ,

ΛF(h1, h2, h3) σh12h22; (essentially QF : L2 × BMO → L2).

F. Di Plinio (Rome Tor Vergata)

Endpoint bounds 10 / 15

SLIDE 41

Proof of (E12)

By scaling, proving (E12) same as: for all f1p = f22 = |F3| = 1,

ΛS(f1, f2, 1G3) :=
s∈S

1

|Is|

f1, ws1f2, ws2ws3, 1G3

p′

where, setting E := {Mpf1 1}, G3 := F3\ is a major subset of F3.

Trees and size

a tree T: collection of quartiles with Is ⊂ IT, ωs ∋ ξT for all s ∈ T
if ΠT,jf is the projection onto span{wsj : s ∈ T} ֒

→ L2(R), sizej(f, T) := ΠT,jfBMO(R).

if F is a union of trees T with size3(h3, T) ≤ σ,

ΛF(h1, h2, h3) σh12h22; (essentially QF : L2 × BMO → L2).

F. Di Plinio (Rome Tor Vergata)

Endpoint bounds 10 / 15

SLIDE 42

Proof of (E12)

By scaling, proving (E12) same as: for all f1p = f22 = |F3| = 1,

ΛS(f1, f2, 1G3) :=
s∈S

1

|Is|

f1, ws1f2, ws2ws3, 1G3

p′

where, setting E := {Mpf1 1}, G3 := F3\ is a major subset of F3.

Trees and size

a tree T: collection of quartiles with Is ⊂ IT, ωs ∋ ξT for all s ∈ T
if ΠT,jf is the projection onto span{wsj : s ∈ T} ֒

→ L2(R), sizej(f, T) := ΠT,jfBMO(R).

if F is a union of trees T with size3(h3, T) ≤ σ,

ΛF(h1, h2, h3) σh12h22; (essentially QF : L2 × BMO → L2).

F. Di Plinio (Rome Tor Vergata)

Endpoint bounds 10 / 15

SLIDE 43

Proof of (E12)

By scaling, proving (E12) same as: for all f1p = f22 = |F3| = 1,

ΛS(f1, f2, 1G3) :=
s∈S

1

|Is|

f1, ws1f2, ws2ws3, 1G3

p′

where, setting E := {Mpf1 1}, G3 := F3\ is a major subset of F3.

Trees and size

a tree T: collection of quartiles with Is ⊂ IT, ωs ∋ ξT for all s ∈ T
if ΠT,jf is the projection onto span{wsj : s ∈ T} ֒

→ L2(R), sizej(f, T) := ΠT,jfBMO(R).

if F is a union of trees T with size3(h3, T) ≤ σ,

ΛF(h1, h2, h3) σh12h22; (essentially QF : L2 × BMO → L2).

F. Di Plinio (Rome Tor Vergata)

Endpoint bounds 10 / 15

SLIDE 44

Proof of (E12)

By scaling, proving (E12) same as: for all f1p = f22 = |F3| = 1,

ΛS(f1, f2, 1G3) :=
s∈S

1

|Is|

f1, ws1f2, ws2ws3, 1G3

p′

where, setting E := {Mpf1 1}, G3 := F3\ is a major subset of F3.

Trees and size

a tree T: collection of quartiles with Is ⊂ IT, ωs ∋ ξT for all s ∈ T
if ΠT,jf is the projection onto span{wsj : s ∈ T} ֒

→ L2(R), sizej(f, T) := ΠT,jfBMO(R).

if F is a union of trees T with size3(h3, T) ≤ σ,

ΛF(h1, h2, h3) σh12h22; (essentially QF : L2 × BMO → L2).

F. Di Plinio (Rome Tor Vergata)

Endpoint bounds 10 / 15

SLIDE 45

The multi-frequency projection lemma in action

· Decompose S =

n≥0

Fn, Fn union of 22n trees T with size3(1G3, T) ≤ 2−n. · for each I m.d.i. of {Mpf 1}, one has

1 |I|

´

I |f|p ∼ 1.

· Use lemma to decompose f1I wrt the M ∼ 22n frequencies ξT, T ∈ Fn : − gI2 ≤ 2n2− 2

p′ n|I| 1 2 so that

g :=

I gI

2

I gI2 2

1

2 2n2− 2 p′ n;

− |bI, ws1| small for all s ∈ T ∈ Fn due to ws1 being frequency localized near ξT and bI having mean zero wrt ξT Using the forest estimate on the good part ΛF(g, f2, 1G3)

sup

T∈Fn

size3(1G3, T)

g2f22 2− 2

p′ n,

which sums to ∼ p′ over n ≥ 0. Done!

Rem. In the Walsh case, bI, ws1 = 0, so the bad part has no contribution.
F. Di Plinio (Rome Tor Vergata)

Endpoint bounds 11 / 15

SLIDE 46

The multi-frequency projection lemma in action

· Decompose S =

n≥0

Fn, Fn union of 22n trees T with size3(1G3, T) ≤ 2−n. · for each I m.d.i. of {Mpf 1}, one has

1 |I|

´

I |f|p ∼ 1.

· Use lemma to decompose f1I wrt the M ∼ 22n frequencies ξT, T ∈ Fn : − gI2 ≤ 2n2− 2

p′ n|I| 1 2 so that

g :=

I gI

2

I gI2 2

1

2 2n2− 2 p′ n;

− |bI, ws1| small for all s ∈ T ∈ Fn due to ws1 being frequency localized near ξT and bI having mean zero wrt ξT Using the forest estimate on the good part ΛF(g, f2, 1G3)

sup

T∈Fn

size3(1G3, T)

g2f22 2− 2

p′ n,

which sums to ∼ p′ over n ≥ 0. Done!

Rem. In the Walsh case, bI, ws1 = 0, so the bad part has no contribution.
F. Di Plinio (Rome Tor Vergata)

Endpoint bounds 11 / 15

SLIDE 47

The multi-frequency projection lemma in action

· Decompose S =

n≥0

Fn, Fn union of 22n trees T with size3(1G3, T) ≤ 2−n. · for each I m.d.i. of {Mpf 1}, one has

1 |I|

´

I |f|p ∼ 1.

· Use lemma to decompose f1I wrt the M ∼ 22n frequencies ξT, T ∈ Fn : − gI2 ≤ 2n2− 2

p′ n|I| 1 2 so that

g :=

I gI

2

I gI2 2

1

2 2n2− 2 p′ n;

− |bI, ws1| small for all s ∈ T ∈ Fn due to ws1 being frequency localized near ξT and bI having mean zero wrt ξT Using the forest estimate on the good part ΛF(g, f2, 1G3)

sup

T∈Fn

size3(1G3, T)

g2f22 2− 2

p′ n,

which sums to ∼ p′ over n ≥ 0. Done!

Rem. In the Walsh case, bI, ws1 = 0, so the bad part has no contribution.
F. Di Plinio (Rome Tor Vergata)

Endpoint bounds 11 / 15

SLIDE 48

The multi-frequency projection lemma in action

· Decompose S =

n≥0

Fn, Fn union of 22n trees T with size3(1G3, T) ≤ 2−n. · for each I m.d.i. of {Mpf 1}, one has

1 |I|

´

I |f|p ∼ 1.

· Use lemma to decompose f1I wrt the M ∼ 22n frequencies ξT, T ∈ Fn : − gI2 ≤ 2n2− 2

p′ n|I| 1 2 so that

g :=

I gI

2

I gI2 2

1

2 2n2− 2 p′ n;

− |bI, ws1| small for all s ∈ T ∈ Fn due to ws1 being frequency localized near ξT and bI having mean zero wrt ξT Using the forest estimate on the good part ΛF(g, f2, 1G3)

sup

T∈Fn

size3(1G3, T)

g2f22 2− 2

p′ n,

which sums to ∼ p′ over n ≥ 0. Done!

Rem. In the Walsh case, bI, ws1 = 0, so the bad part has no contribution.
F. Di Plinio (Rome Tor Vergata)

Endpoint bounds 11 / 15

SLIDE 49

The multi-frequency projection lemma in action

· Decompose S =

n≥0

Fn, Fn union of 22n trees T with size3(1G3, T) ≤ 2−n. · for each I m.d.i. of {Mpf 1}, one has

1 |I|

´

I |f|p ∼ 1.

· Use lemma to decompose f1I wrt the M ∼ 22n frequencies ξT, T ∈ Fn : − gI2 ≤ 2n2− 2

p′ n|I| 1 2 so that

g :=

I gI

2

I gI2 2

1

2 2n2− 2 p′ n;

− |bI, ws1| small for all s ∈ T ∈ Fn due to ws1 being frequency localized near ξT and bI having mean zero wrt ξT Using the forest estimate on the good part ΛF(g, f2, 1G3)

sup

T∈Fn

size3(1G3, T)

g2f22 2− 2

p′ n,

which sums to ∼ p′ over n ≥ 0. Done!

Rem. In the Walsh case, bI, ws1 = 0, so the bad part has no contribution.
F. Di Plinio (Rome Tor Vergata)

Endpoint bounds 11 / 15

SLIDE 50

The multi-frequency projection lemma in action

· Decompose S =

n≥0

Fn, Fn union of 22n trees T with size3(1G3, T) ≤ 2−n. · for each I m.d.i. of {Mpf 1}, one has

1 |I|

´

I |f|p ∼ 1.

· Use lemma to decompose f1I wrt the M ∼ 22n frequencies ξT, T ∈ Fn : − gI2 ≤ 2n2− 2

p′ n|I| 1 2 so that

g :=

I gI

2

I gI2 2

1

2 2n2− 2 p′ n;

− |bI, ws1| small for all s ∈ T ∈ Fn due to ws1 being frequency localized near ξT and bI having mean zero wrt ξT Using the forest estimate on the good part ΛF(g, f2, 1G3)

sup

T∈Fn

size3(1G3, T)

g2f22 2− 2

p′ n,

which sums to ∼ p′ over n ≥ 0. Done!

Rem. In the Walsh case, bI, ws1 = 0, so the bad part has no contribution.
F. Di Plinio (Rome Tor Vergata)

Endpoint bounds 11 / 15

SLIDE 51

The multi-frequency projection lemma in action

· Decompose S =

n≥0

Fn, Fn union of 22n trees T with size3(1G3, T) ≤ 2−n. · for each I m.d.i. of {Mpf 1}, one has

1 |I|

´

I |f|p ∼ 1.

· Use lemma to decompose f1I wrt the M ∼ 22n frequencies ξT, T ∈ Fn : − gI2 ≤ 2n2− 2

p′ n|I| 1 2 so that

g :=

I gI

2

I gI2 2

1

2 2n2− 2 p′ n;

− |bI, ws1| small for all s ∈ T ∈ Fn due to ws1 being frequency localized near ξT and bI having mean zero wrt ξT Using the forest estimate on the good part ΛF(g, f2, 1G3)

sup

T∈Fn

size3(1G3, T)

g2f22 2− 2

p′ n,

which sums to ∼ p′ over n ≥ 0. Done!

Rem. In the Walsh case, bI, ws1 = 0, so the bad part has no contribution.
F. Di Plinio (Rome Tor Vergata)

Endpoint bounds 11 / 15

SLIDE 52

The multi-frequency projection lemma in action

· Decompose S =

n≥0

Fn, Fn union of 22n trees T with size3(1G3, T) ≤ 2−n. · for each I m.d.i. of {Mpf 1}, one has

1 |I|

´

I |f|p ∼ 1.

· Use lemma to decompose f1I wrt the M ∼ 22n frequencies ξT, T ∈ Fn : − gI2 ≤ 2n2− 2

p′ n|I| 1 2 so that

g :=

I gI

2

I gI2 2

1

2 2n2− 2 p′ n;

− |bI, ws1| small for all s ∈ T ∈ Fn due to ws1 being frequency localized near ξT and bI having mean zero wrt ξT Using the forest estimate on the good part ΛF(g, f2, 1G3)

sup

T∈Fn

size3(1G3, T)

g2f22 2− 2

p′ n,

which sums to ∼ p′ over n ≥ 0. Done!

Rem. In the Walsh case, bI, ws1 = 0, so the bad part has no contribution.
F. Di Plinio (Rome Tor Vergata)

Endpoint bounds 11 / 15

SLIDE 53

Endpoint bounds for the Walsh-Carleson operator, I

Walsh-Fourier series: Wnf(x) =

n

k=0

f, WkWk(x), x ∈ T, n ∈ N. Carleson-Hunt estimate: see deReyna00, SjolinSoria03 for ref. (CH)

sup

n |Wn1F |

p,∞ (p′)|F|

1 p ,

1 < p ≤ 2.

Theorem (D?13, unpublished)

For 1 < p < 2,

sup

n |Wnf|

p,∞ (p′)fp.

Theorem (D13)

Suppose {nj} is a θ-lacunary sequence. Then (WpLac)

sup

j

|Wnjf|

p,∞ θ log(p′)fp,

1 < p < 2.

F. Di Plinio (Rome Tor Vergata)

Endpoint bounds 12 / 15

SLIDE 54

Endpoint bounds for the Walsh-Carleson operator, I

Walsh-Fourier series: Wnf(x) =

n

k=0

f, WkWk(x), x ∈ T, n ∈ N. Carleson-Hunt estimate: see deReyna00, SjolinSoria03 for ref. (CH)

sup

n |Wn1F |

p,∞ (p′)|F|

1 p ,

1 < p ≤ 2.

Theorem (D?13, unpublished)

For 1 < p < 2,

sup

n |Wnf|

p,∞ (p′)fp.

Theorem (D13)

Suppose {nj} is a θ-lacunary sequence. Then (WpLac)

sup

j

|Wnjf|

p,∞ θ log(p′)fp,

1 < p < 2.

F. Di Plinio (Rome Tor Vergata)

Endpoint bounds 12 / 15

SLIDE 55

Endpoint bounds for the Walsh-Carleson operator, I

Walsh-Fourier series: Wnf(x) =

n

k=0

f, WkWk(x), x ∈ T, n ∈ N. Carleson-Hunt estimate: see deReyna00, SjolinSoria03 for ref. (CH)

sup

n |Wn1F |

p,∞ (p′)|F|

1 p ,

1 < p ≤ 2.

Theorem (D?13, unpublished)

For 1 < p < 2,

sup

n |Wnf|

p,∞ (p′)fp.

Theorem (D13)

Suppose {nj} is a θ-lacunary sequence. Then (WpLac)

sup

j

|Wnjf|

p,∞ θ log(p′)fp,

1 < p < 2.

F. Di Plinio (Rome Tor Vergata)

Endpoint bounds 12 / 15

SLIDE 56

Endpoint bounds for the Walsh-Carleson operator, I

Walsh-Fourier series: Wnf(x) =

n

k=0

f, WkWk(x), x ∈ T, n ∈ N. Carleson-Hunt estimate: see deReyna00, SjolinSoria03 for ref. (CH)

sup

n |Wn1F |

p,∞ (p′)|F|

1 p ,

1 < p ≤ 2.

Theorem (D?13, unpublished)

For 1 < p < 2,

sup

n |Wnf|

p,∞ (p′)fp.

Theorem (D13)

Suppose {nj} is a θ-lacunary sequence. Then (WpLac)

sup

j

|Wnjf|

p,∞ θ log(p′)fp,

1 < p < 2.

F. Di Plinio (Rome Tor Vergata)

Endpoint bounds 12 / 15

SLIDE 57

Endpoint bounds and lacunary a.e. convergence

The proof of (WpLac) relies on (A) lacunary version of MF Lemma: g2

2 ≤ p′fp p

This is a form of Chang-Wilson-Wolff inequality: for {nj}θ-lacunary

j aje2πinjx
Lp′(T) √p′

j |aj|2 1

2

(B) W⋆ := supj |Wnj| : L∞ → ExpL1. Corollaries: (1) W⋆f1,∞ f1 log log f∞

f1

(Do-Lacey2012, via RWT + Antonov)

(2) W⋆ : L log2 L log4 L → L1,∞ ❀ a.e. convergence of Wnjf for f ∈ L log2 L log4 L. Remarks: · (2) compares to W⋆ : L log L log3 L → L1,∞ for the full sequence (Antonov1996,deReyna2000,Lie2012,Sjolin-Soria2003 for Walsh case). Both sharp apart from log4, log3 terms (Kalton log-cvx of L1,∞). · (1) in Fourier case due to Lie2012

F. Di Plinio (Rome Tor Vergata)

Endpoint bounds 13 / 15

SLIDE 58

Endpoint bounds and lacunary a.e. convergence

The proof of (WpLac) relies on (A) lacunary version of MF Lemma: g2

2 ≤ p′fp p

This is a form of Chang-Wilson-Wolff inequality: for {nj}θ-lacunary

j aje2πinjx
Lp′(T) √p′

j |aj|2 1

2

(B) W⋆ := supj |Wnj| : L∞ → ExpL1. Corollaries: (1) W⋆f1,∞ f1 log log f∞

f1

(Do-Lacey2012, via RWT + Antonov)

(2) W⋆ : L log2 L log4 L → L1,∞ ❀ a.e. convergence of Wnjf for f ∈ L log2 L log4 L. Remarks: · (2) compares to W⋆ : L log L log3 L → L1,∞ for the full sequence (Antonov1996,deReyna2000,Lie2012,Sjolin-Soria2003 for Walsh case). Both sharp apart from log4, log3 terms (Kalton log-cvx of L1,∞). · (1) in Fourier case due to Lie2012

F. Di Plinio (Rome Tor Vergata)

Endpoint bounds 13 / 15

SLIDE 59

Endpoint bounds and lacunary a.e. convergence

The proof of (WpLac) relies on (A) lacunary version of MF Lemma: g2

2 ≤ p′fp p

This is a form of Chang-Wilson-Wolff inequality: for {nj}θ-lacunary

j aje2πinjx
Lp′(T) √p′

j |aj|2 1

2

(B) W⋆ := supj |Wnj| : L∞ → ExpL1. Corollaries: (1) W⋆f1,∞ f1 log log f∞

f1

(Do-Lacey2012, via RWT + Antonov)

(2) W⋆ : L log2 L log4 L → L1,∞ ❀ a.e. convergence of Wnjf for f ∈ L log2 L log4 L. Remarks: · (2) compares to W⋆ : L log L log3 L → L1,∞ for the full sequence (Antonov1996,deReyna2000,Lie2012,Sjolin-Soria2003 for Walsh case). Both sharp apart from log4, log3 terms (Kalton log-cvx of L1,∞). · (1) in Fourier case due to Lie2012

F. Di Plinio (Rome Tor Vergata)

Endpoint bounds 13 / 15

SLIDE 60

Endpoint bounds and lacunary a.e. convergence

The proof of (WpLac) relies on (A) lacunary version of MF Lemma: g2

2 ≤ p′fp p

This is a form of Chang-Wilson-Wolff inequality: for {nj}θ-lacunary

j aje2πinjx
Lp′(T) √p′

j |aj|2 1

2

(B) W⋆ := supj |Wnj| : L∞ → ExpL1. Corollaries: (1) W⋆f1,∞ f1 log log f∞

f1

(Do-Lacey2012, via RWT + Antonov)

(2) W⋆ : L log2 L log4 L → L1,∞ ❀ a.e. convergence of Wnjf for f ∈ L log2 L log4 L. Remarks: · (2) compares to W⋆ : L log L log3 L → L1,∞ for the full sequence (Antonov1996,deReyna2000,Lie2012,Sjolin-Soria2003 for Walsh case). Both sharp apart from log4, log3 terms (Kalton log-cvx of L1,∞). · (1) in Fourier case due to Lie2012

F. Di Plinio (Rome Tor Vergata)

Endpoint bounds 13 / 15

SLIDE 61

Endpoint bounds and lacunary a.e. convergence

The proof of (WpLac) relies on (A) lacunary version of MF Lemma: g2

2 ≤ p′fp p

This is a form of Chang-Wilson-Wolff inequality: for {nj}θ-lacunary

j aje2πinjx
Lp′(T) √p′

j |aj|2 1

2

(B) W⋆ := supj |Wnj| : L∞ → ExpL1. Corollaries: (1) W⋆f1,∞ f1 log log f∞

f1

(Do-Lacey2012, via RWT + Antonov)

(2) W⋆ : L log2 L log4 L → L1,∞ ❀ a.e. convergence of Wnjf for f ∈ L log2 L log4 L. Remarks: · (2) compares to W⋆ : L log L log3 L → L1,∞ for the full sequence (Antonov1996,deReyna2000,Lie2012,Sjolin-Soria2003 for Walsh case). Both sharp apart from log4, log3 terms (Kalton log-cvx of L1,∞). · (1) in Fourier case due to Lie2012

F. Di Plinio (Rome Tor Vergata)

Endpoint bounds 13 / 15

SLIDE 62

Endpoint bounds and lacunary a.e. convergence

The proof of (WpLac) relies on (A) lacunary version of MF Lemma: g2

2 ≤ p′fp p

This is a form of Chang-Wilson-Wolff inequality: for {nj}θ-lacunary

j aje2πinjx
Lp′(T) √p′

j |aj|2 1

2

(B) W⋆ := supj |Wnj| : L∞ → ExpL1. Corollaries: (1) W⋆f1,∞ f1 log log f∞

f1

(Do-Lacey2012, via RWT + Antonov)

(2) W⋆ : L log2 L log4 L → L1,∞ ❀ a.e. convergence of Wnjf for f ∈ L log2 L log4 L. Remarks: · (2) compares to W⋆ : L log L log3 L → L1,∞ for the full sequence (Antonov1996,deReyna2000,Lie2012,Sjolin-Soria2003 for Walsh case). Both sharp apart from log4, log3 terms (Kalton log-cvx of L1,∞). · (1) in Fourier case due to Lie2012

F. Di Plinio (Rome Tor Vergata)

Endpoint bounds 13 / 15

SLIDE 63

Endpoint bounds and lacunary a.e. convergence

The proof of (WpLac) relies on (A) lacunary version of MF Lemma: g2

2 ≤ p′fp p

This is a form of Chang-Wilson-Wolff inequality: for {nj}θ-lacunary

j aje2πinjx
Lp′(T) √p′

j |aj|2 1

2

(B) W⋆ := supj |Wnj| : L∞ → ExpL1. Corollaries: (1) W⋆f1,∞ f1 log log f∞

f1

(Do-Lacey2012, via RWT + Antonov)

(2) W⋆ : L log2 L log4 L → L1,∞ ❀ a.e. convergence of Wnjf for f ∈ L log2 L log4 L. Remarks: · (2) compares to W⋆ : L log L log3 L → L1,∞ for the full sequence (Antonov1996,deReyna2000,Lie2012,Sjolin-Soria2003 for Walsh case). Both sharp apart from log4, log3 terms (Kalton log-cvx of L1,∞). · (1) in Fourier case due to Lie2012

F. Di Plinio (Rome Tor Vergata)

Endpoint bounds 13 / 15

SLIDE 64

Endpoint bounds and lacunary a.e. convergence

The proof of (WpLac) relies on (A) lacunary version of MF Lemma: g2

2 ≤ p′fp p

This is a form of Chang-Wilson-Wolff inequality: for {nj}θ-lacunary

j aje2πinjx
Lp′(T) √p′

j |aj|2 1

2

(B) W⋆ := supj |Wnj| : L∞ → ExpL1. Corollaries: (1) W⋆f1,∞ f1 log log f∞

f1

(Do-Lacey2012, via RWT + Antonov)

(2) W⋆ : L log2 L log4 L → L1,∞ ❀ a.e. convergence of Wnjf for f ∈ L log2 L log4 L. Remarks: · (2) compares to W⋆ : L log L log3 L → L1,∞ for the full sequence (Antonov1996,deReyna2000,Lie2012,Sjolin-Soria2003 for Walsh case). Both sharp apart from log4, log3 terms (Kalton log-cvx of L1,∞). · (1) in Fourier case due to Lie2012

F. Di Plinio (Rome Tor Vergata)

Endpoint bounds 13 / 15

SLIDE 65

Endpoint bounds and lacunary a.e. convergence

The proof of (WpLac) relies on (A) lacunary version of MF Lemma: g2

2 ≤ p′fp p

This is a form of Chang-Wilson-Wolff inequality: for {nj}θ-lacunary

j aje2πinjx
Lp′(T) √p′

j |aj|2 1

2

(B) W⋆ := supj |Wnj| : L∞ → ExpL1. Corollaries: (1) W⋆f1,∞ f1 log log f∞

f1

(Do-Lacey2012, via RWT + Antonov)

(2) W⋆ : L log2 L log4 L → L1,∞ ❀ a.e. convergence of Wnjf for f ∈ L log2 L log4 L. Remarks: · (2) compares to W⋆ : L log L log3 L → L1,∞ for the full sequence (Antonov1996,deReyna2000,Lie2012,Sjolin-Soria2003 for Walsh case). Both sharp apart from log4, log3 terms (Kalton log-cvx of L1,∞). · (1) in Fourier case due to Lie2012

F. Di Plinio (Rome Tor Vergata)

Endpoint bounds 13 / 15

SLIDE 66

Ongoing research and open problems

Bilinear Hilbert transform

· remove |f2| ≤ 1F2 in (WRW) ❀ Q : Lp1 × Lp2 → L

2 3 ,∞

· Fourier case: handling bI (mean zero wrt N freq ❀ decay)

Walsh and Fourier Carleson near L1

· sharpness of constants in (WpLac) · strong bounds: (WpLac) implies W⋆ : L log L log2 L → L1. Sharp bound would not have the log2 term. Tempting because W⋆f ∼ sup

T

|H⋆ ◦ ModξT(ΠTf)| where H⋆ max. Hilbert transform, ΠTf live in disjoint time-frequency regions. · removal of log4, log3 in a.e. convergence results (Konyagin’s conjectures, ICM 2006).

F. Di Plinio (Rome Tor Vergata)

Endpoint bounds 14 / 15

SLIDE 67

Ongoing research and open problems

Bilinear Hilbert transform

· remove |f2| ≤ 1F2 in (WRW) ❀ Q : Lp1 × Lp2 → L

2 3 ,∞

· Fourier case: handling bI (mean zero wrt N freq ❀ decay)

Walsh and Fourier Carleson near L1

· sharpness of constants in (WpLac) · strong bounds: (WpLac) implies W⋆ : L log L log2 L → L1. Sharp bound would not have the log2 term. Tempting because W⋆f ∼ sup

T

|H⋆ ◦ ModξT(ΠTf)| where H⋆ max. Hilbert transform, ΠTf live in disjoint time-frequency regions. · removal of log4, log3 in a.e. convergence results (Konyagin’s conjectures, ICM 2006).

F. Di Plinio (Rome Tor Vergata)

Endpoint bounds 14 / 15

SLIDE 68

Ongoing research and open problems

Bilinear Hilbert transform

· remove |f2| ≤ 1F2 in (WRW) ❀ Q : Lp1 × Lp2 → L

2 3 ,∞

· Fourier case: handling bI (mean zero wrt N freq ❀ decay)

Walsh and Fourier Carleson near L1

· sharpness of constants in (WpLac) · strong bounds: (WpLac) implies W⋆ : L log L log2 L → L1. Sharp bound would not have the log2 term. Tempting because W⋆f ∼ sup

T

|H⋆ ◦ ModξT(ΠTf)| where H⋆ max. Hilbert transform, ΠTf live in disjoint time-frequency regions. · removal of log4, log3 in a.e. convergence results (Konyagin’s conjectures, ICM 2006).

F. Di Plinio (Rome Tor Vergata)

Endpoint bounds 14 / 15

SLIDE 69

Ongoing research and open problems

Bilinear Hilbert transform

· remove |f2| ≤ 1F2 in (WRW) ❀ Q : Lp1 × Lp2 → L

2 3 ,∞

· Fourier case: handling bI (mean zero wrt N freq ❀ decay)

Walsh and Fourier Carleson near L1

· sharpness of constants in (WpLac) · strong bounds: (WpLac) implies W⋆ : L log L log2 L → L1. Sharp bound would not have the log2 term. Tempting because W⋆f ∼ sup

T

|H⋆ ◦ ModξT(ΠTf)| where H⋆ max. Hilbert transform, ΠTf live in disjoint time-frequency regions. · removal of log4, log3 in a.e. convergence results (Konyagin’s conjectures, ICM 2006).

F. Di Plinio (Rome Tor Vergata)

Endpoint bounds 14 / 15

SLIDE 70

Ongoing research and open problems

Bilinear Hilbert transform

· remove |f2| ≤ 1F2 in (WRW) ❀ Q : Lp1 × Lp2 → L

2 3 ,∞

· Fourier case: handling bI (mean zero wrt N freq ❀ decay)

Walsh and Fourier Carleson near L1

· sharpness of constants in (WpLac) · strong bounds: (WpLac) implies W⋆ : L log L log2 L → L1. Sharp bound would not have the log2 term. Tempting because W⋆f ∼ sup

T

|H⋆ ◦ ModξT(ΠTf)| where H⋆ max. Hilbert transform, ΠTf live in disjoint time-frequency regions. · removal of log4, log3 in a.e. convergence results (Konyagin’s conjectures, ICM 2006).

F. Di Plinio (Rome Tor Vergata)

Endpoint bounds 14 / 15

SLIDE 71

Ongoing research and open problems

Bilinear Hilbert transform

· remove |f2| ≤ 1F2 in (WRW) ❀ Q : Lp1 × Lp2 → L

2 3 ,∞

· Fourier case: handling bI (mean zero wrt N freq ❀ decay)

Walsh and Fourier Carleson near L1

· sharpness of constants in (WpLac) · strong bounds: (WpLac) implies W⋆ : L log L log2 L → L1. Sharp bound would not have the log2 term. Tempting because W⋆f ∼ sup

T

|H⋆ ◦ ModξT(ΠTf)| where H⋆ max. Hilbert transform, ΠTf live in disjoint time-frequency regions. · removal of log4, log3 in a.e. convergence results (Konyagin’s conjectures, ICM 2006).

F. Di Plinio (Rome Tor Vergata)

Endpoint bounds 14 / 15

SLIDE 72

Ongoing research and open problems

Bilinear Hilbert transform

· remove |f2| ≤ 1F2 in (WRW) ❀ Q : Lp1 × Lp2 → L

2 3 ,∞

· Fourier case: handling bI (mean zero wrt N freq ❀ decay)

Walsh and Fourier Carleson near L1

· sharpness of constants in (WpLac) · strong bounds: (WpLac) implies W⋆ : L log L log2 L → L1. Sharp bound would not have the log2 term. Tempting because W⋆f ∼ sup

T

|H⋆ ◦ ModξT(ΠTf)| where H⋆ max. Hilbert transform, ΠTf live in disjoint time-frequency regions. · removal of log4, log3 in a.e. convergence results (Konyagin’s conjectures, ICM 2006).

F. Di Plinio (Rome Tor Vergata)

Endpoint bounds 14 / 15

SLIDE 73