∗
Introduction Norms inequalities for intrinsic square function Norm inequalities for intrinsic Littlewood-Paley gλ
-function
Intrinsic square functions on functions
spaces including weighted Morrey spaces
Justin Feuto
Université Félix Houphouët-Boigny Abidjan
Ecole de Recherche CIMPA ”Analyse et Probabilités”,
UFRMI, 17 au 28 mars 2014 Cocody (Côte d’Ivoire)
Justin Feuto
Intrinsic square functions on functions spaces including weig
∗
Introduction Norms inequalities for intrinsic square function Norm inequalities for intrinsic Littlewood-Paley gλ
-function
1
Introduction
2
Norms inequalities for intrinsic square function
3
Norm inequalities for intrinsic Littlewood-Paley
gλ∗ -function
4
Commutator of those operators
Justin Feuto
Intrinsic square functions on functions spaces including weig
∗
Introduction Norms inequalities for intrinsic square function Norm inequalities for intrinsic Littlewood-Paley gλ
-function
Outline
1
Introduction
2
Norms inequalities for intrinsic square function
3
Norm inequalities for intrinsic Littlewood-Paley
gλ∗ -function
4
Commutator of those operators
Justin Feuto
Intrinsic square functions on functions spaces including weig
∗
Introduction Norms inequalities for intrinsic square function Norm inequalities for intrinsic Littlewood-Paley gλ
-function
Wiener amalgam spaces
Let 0 < p, q ≤ ∞. For r > 0, put :
 1

p
p
 R − p1 |B(y , r )|
f χB(y,r ) q dy
n
R
r kf kq,p =

 ess sup n f χ
B(y,r ) q
y∈R
if
p<∞
if
p=∞
,
and
n
(Lq , Lp ) = f : Rn → C measurable : kf kq,p :=
1 kf kq,p
o
<∞
Lq ∪ Lp ⊆ (Lq , Lp ) if q ≤ p
(Lq , Lp ) ⊆ Lq ∩ Lp if p ≤ q
Justin Feuto
Intrinsic square functions on functions spaces including weig
∗
Introduction Norms inequalities for intrinsic square function Norm inequalities for intrinsic Littlewood-Paley gλ
-function
Wiener amalgam spaces
Let 0 < p, q ≤ ∞. For r > 0, put :
 1

p
p
 R − p1 |B(y , r )|
f χB(y,r ) q dy
n
R
r kf kq,p =

 ess sup n f χ
B(y,r ) q
y∈R
if
p<∞
if
p=∞
,
and
n
(Lq , Lp ) = f : Rn → C measurable : kf kq,p :=
1 kf kq,p
o
<∞
Lq ∪ Lp ⊆ (Lq , Lp ) if q ≤ p
(Lq , Lp ) ⊆ Lq ∩ Lp if p ≤ q
Justin Feuto
Intrinsic square functions on functions spaces including weig
∗
Introduction Norms inequalities for intrinsic square function Norm inequalities for intrinsic Littlewood-Paley gλ
-function
Wiener amalgam spaces
Let 0 < p, q ≤ ∞. For r > 0, put :
 1

p
p
 R − p1 |B(y , r )|
f χB(y,r ) q dy
n
R
r kf kq,p =

 ess sup n f χ
B(y,r ) q
y∈R
if
p<∞
if
p=∞
,
and
n
(Lq , Lp ) = f : Rn → C measurable : kf kq,p :=
1 kf kq,p
o
<∞
Lq ∪ Lp ⊆ (Lq , Lp ) if q ≤ p
(Lq , Lp ) ⊆ Lq ∩ Lp if p ≤ q
Justin Feuto
Intrinsic square functions on functions spaces including weig
∗
Introduction Norms inequalities for intrinsic square function Norm inequalities for intrinsic Littlewood-Paley gλ
-function
Some dilation invariant subspaces of (Lq , Lp )
d
For α > 0 and r > 0, we define δrα : f 7→ r α f (r ·).
Definition
For 1 ≤ q, p, α ≤ ∞,
q p α
q p
α
(L , L ) = f ∈ (L , L ) : kf kq,p,α := sup kδr f kq,p < ∞
r >0
kf kq,p,α

p p1

1
1
1 R 
−
−

 sup Rd |B(y , r )| α q p f χB(y,r ) dy
q
r >0
≈
1
1 
−

|B(y, r )| α q f χB(y,r ) q

 supsup ess
d
if p < ∞
if p = ∞
r >0 y∈R
(Lq , Lp )α = Lα if α ∈ {p, q} , Lα ,→ (Lq , Lp )α ,→ (Lq , Lp )
Justin Feuto
Intrinsic square functions on functions spaces including weig
∗
Introduction Norms inequalities for intrinsic square function Norm inequalities for intrinsic Littlewood-Paley gλ
-function
Some dilation invariant subspaces of (Lq , Lp )
d
For α > 0 and r > 0, we define δrα : f 7→ r α f (r ·).
Definition
For 1 ≤ q, p, α ≤ ∞,
q p α
q p
α
(L , L ) = f ∈ (L , L ) : kf kq,p,α := sup kδr f kq,p < ∞
r >0
kf kq,p,α

p p1

1
1
1 R 
−
−

 sup Rd |B(y , r )| α q p f χB(y,r ) dy
q
r >0
≈
1
1 
−

|B(y, r )| α q f χB(y,r ) q

 supsup ess
d
if p < ∞
if p = ∞
r >0 y∈R
(Lq , Lp )α = Lα if α ∈ {p, q} , Lα ,→ (Lq , Lp )α ,→ (Lq , Lp )
Justin Feuto
Intrinsic square functions on functions spaces including weig
∗
Introduction Norms inequalities for intrinsic square function Norm inequalities for intrinsic Littlewood-Paley gλ
-function
Some dilation invariant subspaces of (Lq , Lp )
d
For α > 0 and r > 0, we define δrα : f 7→ r α f (r ·).
Definition
For 1 ≤ q, p, α ≤ ∞,
q p α
q p
α
(L , L ) = f ∈ (L , L ) : kf kq,p,α := sup kδr f kq,p < ∞
r >0
kf kq,p,α

p p1

1
1
1 R 
−
−

 sup Rd |B(y , r )| α q p f χB(y,r ) dy
q
r >0
≈
1
1 
−

|B(y, r )| α q f χB(y,r ) q

 supsup ess
d
if p < ∞
if p = ∞
r >0 y∈R
(Lq , Lp )α = Lα if α ∈ {p, q} , Lα ,→ (Lq , Lp )α ,→ (Lq , Lp )
Justin Feuto
Intrinsic square functions on functions spaces including weig
∗
Introduction Norms inequalities for intrinsic square function Norm inequalities for intrinsic Littlewood-Paley gλ
-function
Norm inequalities of some classical operators
Maximal operator defined by
Mf (x) = sup |B(x, r )|
r >0
−1
Z
|f (y)| dy
B(x,r )
are bounded on (Lq , Lp )α
The Hilbert operator (n = 1) is bounded on (Lq , Lp )α .
Some CZO are bounded on (Lq , Lp )α
For < γ < n, the Riesz potential Iγ f (x) ≈
R
f (y)
Rn |x−y|n−γ dy is
∗
bounded from (Lq , Lp )α to (Lqe , Lep )α for
1 < q ≤ α ≤ p ≤ ∞ and 0 < dγ < α1 , where
1
= q1 − αq dγ and p̃1 = p1 − αp dγ .
q̃
Justin Feuto
1
α∗
=
1
α
− dγ . ,
Intrinsic square functions on functions spaces including weig
∗
Introduction Norms inequalities for intrinsic square function Norm inequalities for intrinsic Littlewood-Paley gλ
-function
Norm inequalities of some classical operators
Maximal operator defined by
Mf (x) = sup |B(x, r )|
r >0
−1
Z
|f (y)| dy
B(x,r )
are bounded on (Lq , Lp )α
The Hilbert operator (n = 1) is bounded on (Lq , Lp )α .
Some CZO are bounded on (Lq , Lp )α
For < γ < n, the Riesz potential Iγ f (x) ≈
R
f (y)
Rn |x−y|n−γ dy is
∗
bounded from (Lq , Lp )α to (Lqe , Lep )α for
1 < q ≤ α ≤ p ≤ ∞ and 0 < dγ < α1 , where
1
= q1 − αq dγ and p̃1 = p1 − αp dγ .
q̃
Justin Feuto
1
α∗
=
1
α
− dγ . ,
Intrinsic square functions on functions spaces including weig
∗
Introduction Norms inequalities for intrinsic square function Norm inequalities for intrinsic Littlewood-Paley gλ
-function
Norm inequalities of some classical operators
Maximal operator defined by
Mf (x) = sup |B(x, r )|
r >0
−1
Z
|f (y)| dy
B(x,r )
are bounded on (Lq , Lp )α
The Hilbert operator (n = 1) is bounded on (Lq , Lp )α .
Some CZO are bounded on (Lq , Lp )α
For < γ < n, the Riesz potential Iγ f (x) ≈
R
f (y)
Rn |x−y|n−γ dy is
∗
bounded from (Lq , Lp )α to (Lqe , Lep )α for
1 < q ≤ α ≤ p ≤ ∞ and 0 < dγ < α1 , where
1
= q1 − αq dγ and p̃1 = p1 − αp dγ .
q̃
Justin Feuto
1
α∗
=
1
α
− dγ . ,
Intrinsic square functions on functions spaces including weig
∗
Introduction Norms inequalities for intrinsic square function Norm inequalities for intrinsic Littlewood-Paley gλ
-function
Norm inequalities of some classical operators
Maximal operator defined by
Mf (x) = sup |B(x, r )|
r >0
−1
Z
|f (y)| dy
B(x,r )
are bounded on (Lq , Lp )α
The Hilbert operator (n = 1) is bounded on (Lq , Lp )α .
Some CZO are bounded on (Lq , Lp )α
For < γ < n, the Riesz potential Iγ f (x) ≈
R
f (y)
Rn |x−y|n−γ dy is
∗
bounded from (Lq , Lp )α to (Lqe , Lep )α for
1 < q ≤ α ≤ p ≤ ∞ and 0 < dγ < α1 , where
1
= q1 − αq dγ and p̃1 = p1 − αp dγ .
q̃
Justin Feuto
1
α∗
=
1
α
− dγ . ,
Intrinsic square functions on functions spaces including weig
∗
Introduction Norms inequalities for intrinsic square function Norm inequalities for intrinsic Littlewood-Paley gλ
-function
Weighted (Lq , Lp )α spaces
Let 1 ≤ q ≤ α ≤ p ≤ ∞ and w be a weight. For f : Rn → C,
1
R
R
kf kqw := Rn |f (x)|q w(x)dx q , w(B) = B w(x)dx
1
p
p
1
R − q1 − p1 α
dy
w(B(y,
r
))
f
χ
n
r kf kqw ,p,α :=
B(y,r
)
R
q
w
kf kqw ,p,α := supr >0 r kf kqw ,p,α
and the usual modification when p = ∞.
Definition
n
o
(Lqw , Lp )α (Rn ) = f : Rn → C measurable : kf kqw ,p,α < ∞
1
2
For w ≡ 1, we obtain (Lq , Lp )α (Rn )
For q < α and p = ∞, (Lqw , L∞ )α (Rn ) is the weighted
1
1
n
Morrey spaces Lq,κ
w (R ), with κ = q − α
Justin Feuto
Intrinsic square functions on functions spaces including weig
∗
Introduction Norms inequalities for intrinsic square function Norm inequalities for intrinsic Littlewood-Paley gλ
-function
Outline
1
Introduction
2
Norms inequalities for intrinsic square function
3
Norm inequalities for intrinsic Littlewood-Paley
gλ∗ -function
4
Commutator of those operators
Justin Feuto
Intrinsic square functions on functions spaces including weig
∗
Introduction Norms inequalities for intrinsic square function Norm inequalities for intrinsic Littlewood-Paley gλ
-function
Some maximal function
Let 0 < γ ≤ 1 , B = {x ∈ Rn : |x| ≤ 1} and ϕ : Rn → C.
1
The function ϕ ∈ Cγ iff
suppϕ
⊂B
R
ϕ(x)dx
=0
Rn
γ
|ϕ(x) − ϕ(x 0 )| ≤ |x − x 0 | , ∀x, x 0 ∈ Rn
2
For f ∈ L1loc put
fγ∗ (y , t) = sup |f ∗ ϕt (y )| , (y, t) ∈ Rn+1
+ ,
ϕ∈Cγ
where Rn+1
= Rn × (0, ∞) and ϕt (x) = t −n ϕ(t −1 x).
+
Justin Feuto
Intrinsic square functions on functions spaces including weig
∗
Introduction Norms inequalities for intrinsic square function Norm inequalities for intrinsic Littlewood-Paley gλ
-function
Some maximal function
Let 0 < γ ≤ 1 , B = {x ∈ Rn : |x| ≤ 1} and ϕ : Rn → C.
1
The function ϕ ∈ Cγ iff
suppϕ
⊂B
R
ϕ(x)dx
=0
Rn
γ
|ϕ(x) − ϕ(x 0 )| ≤ |x − x 0 | , ∀x, x 0 ∈ Rn
2
For f ∈ L1loc put
fγ∗ (y , t) = sup |f ∗ ϕt (y )| , (y, t) ∈ Rn+1
+ ,
ϕ∈Cγ
where Rn+1
= Rn × (0, ∞) and ϕt (x) = t −n ϕ(t −1 x).
+
Justin Feuto
Intrinsic square functions on functions spaces including weig
∗
Introduction Norms inequalities for intrinsic square function Norm inequalities for intrinsic Littlewood-Paley gλ
-function
Some maximal function
Let 0 < γ ≤ 1 , B = {x ∈ Rn : |x| ≤ 1} and ϕ : Rn → C.
1
The function ϕ ∈ Cγ iff
suppϕ
⊂B
R
ϕ(x)dx
=0
Rn
γ
|ϕ(x) − ϕ(x 0 )| ≤ |x − x 0 | , ∀x, x 0 ∈ Rn
2
For f ∈ L1loc put
fγ∗ (y , t) = sup |f ∗ ϕt (y )| , (y, t) ∈ Rn+1
+ ,
ϕ∈Cγ
where Rn+1
= Rn × (0, ∞) and ϕt (x) = t −n ϕ(t −1 x).
+
Justin Feuto
Intrinsic square functions on functions spaces including weig
∗
Introduction Norms inequalities for intrinsic square function Norm inequalities for intrinsic Littlewood-Paley gλ
-function
Some maximal function
Let 0 < γ ≤ 1 , B = {x ∈ Rn : |x| ≤ 1} and ϕ : Rn → C.
1
The function ϕ ∈ Cγ iff
suppϕ
⊂B
R
ϕ(x)dx
=0
Rn
γ
|ϕ(x) − ϕ(x 0 )| ≤ |x − x 0 | , ∀x, x 0 ∈ Rn
2
For f ∈ L1loc put
fγ∗ (y , t) = sup |f ∗ ϕt (y )| , (y, t) ∈ Rn+1
+ ,
ϕ∈Cγ
where Rn+1
= Rn × (0, ∞) and ϕt (x) = t −n ϕ(t −1 x).
+
Justin Feuto
Intrinsic square functions on functions spaces including weig
∗
Introduction Norms inequalities for intrinsic square function Norm inequalities for intrinsic Littlewood-Paley gλ
-function
Some maximal function
Let 0 < γ ≤ 1 , B = {x ∈ Rn : |x| ≤ 1} and ϕ : Rn → C.
1
The function ϕ ∈ Cγ iff
suppϕ
⊂B
R
ϕ(x)dx
=0
Rn
γ
|ϕ(x) − ϕ(x 0 )| ≤ |x − x 0 | , ∀x, x 0 ∈ Rn
2
For f ∈ L1loc put
fγ∗ (y , t) = sup |f ∗ ϕt (y )| , (y, t) ∈ Rn+1
+ ,
ϕ∈Cγ
where Rn+1
= Rn × (0, ∞) and ϕt (x) = t −n ϕ(t −1 x).
+
Justin Feuto
Intrinsic square functions on functions spaces including weig
∗
Introduction Norms inequalities for intrinsic square function Norm inequalities for intrinsic Littlewood-Paley gλ
-function
Intrinsic square function
Definition
We defined Sγ by
"Z
Sγ (f )(x) =
Γ(x)
dydt
fγ∗ (y , t)2 n+1
t
#1
2
,
n
o
n+1
where for x ∈ Rn , Γ(x) = (y, t) ∈ R+
: |x − y | < t .
Justin Feuto
Intrinsic square functions on functions spaces including weig
∗
Introduction Norms inequalities for intrinsic square function Norm inequalities for intrinsic Littlewood-Paley gλ
-function
Norm inequality for Sγ
Theorem [Wilson]
Let w ∈ Aq , 1 < q < ∞. Then
kSγ f kqw 4 kf kqw .
Theorem [Wang]
Let 1 < q < ∞, 0 < κ < 1 and w ∈ Aq . Then
kSγ f kLq,κ 4 kf kLq,κ
.
w
w
Theorem [F.]
Let 0 < γ ≤ 1, 1 < q ≤ α < p ≤ ∞ and w ∈ Aq . The operators
Sγ are bounded in (Lqw , Lp )α (Rn ),i.e.,
kSγ f kqw ,p,α 4 kf kqw ,p,α
Justin Feuto
Intrinsic square functions on functions spaces including weig
∗
Introduction Norms inequalities for intrinsic square function Norm inequalities for intrinsic Littlewood-Paley gλ
-function
Norm inequality for Sγ
Theorem [Wilson]
Let w ∈ Aq , 1 < q < ∞. Then
kSγ f kqw 4 kf kqw .
Theorem [Wang]
Let 1 < q < ∞, 0 < κ < 1 and w ∈ Aq . Then
kSγ f kLq,κ 4 kf kLq,κ
.
w
w
Theorem [F.]
Let 0 < γ ≤ 1, 1 < q ≤ α < p ≤ ∞ and w ∈ Aq . The operators
Sγ are bounded in (Lqw , Lp )α (Rn ),i.e.,
kSγ f kqw ,p,α 4 kf kqw ,p,α
Justin Feuto
Intrinsic square functions on functions spaces including weig
∗
Introduction Norms inequalities for intrinsic square function Norm inequalities for intrinsic Littlewood-Paley gλ
-function
Norm inequality for Sγ
Theorem [Wilson]
Let w ∈ Aq , 1 < q < ∞. Then
kSγ f kqw 4 kf kqw .
Theorem [Wang]
Let 1 < q < ∞, 0 < κ < 1 and w ∈ Aq . Then
kSγ f kLq,κ 4 kf kLq,κ
.
w
w
Theorem [F.]
Let 0 < γ ≤ 1, 1 < q ≤ α < p ≤ ∞ and w ∈ Aq . The operators
Sγ are bounded in (Lqw , Lp )α (Rn ),i.e.,
kSγ f kqw ,p,α 4 kf kqw ,p,α
Justin Feuto
Intrinsic square functions on functions spaces including weig
∗
Introduction Norms inequalities for intrinsic square function Norm inequalities for intrinsic Littlewood-Paley gλ
-function
Intermediate result
Proposition [F.]
Let 1 ≤ s ≤ q < ∞, w ∈ Aq/s and T : Lqloc (w) → Lqloc (w) a sub
linear operator which satisfies the following property : for all
balls B ⊂ Rn
1
s
R
P∞
s
1
T (f χ(2B)c )(x) 4 k=1 k 2k+1 B 2k +1 B |f (z)| dz
a.e. on B.
|
|
Then
1
2
if q > 1 and T is bounded on Lq (w), then it is also
bounded on (Lq (w), Lp )α , for q ≤ α < p ≤ ∞,
if for all λ > 0
w({x ∈ Rn : |T f (x)| > λ} ≤ C λ1
R
Rn
|f (y )| w(y )dy ,
then for 1 ≤ α < p ≤ ∞, T is bounded from (L1 (w), Lp )α to
(L1,∞ (w), Lp )α .
Justin Feuto
Intrinsic square functions on functions spaces including weig
∗
Introduction Norms inequalities for intrinsic square function Norm inequalities for intrinsic Littlewood-Paley gλ
-function
Intermediate result
Proposition [F.]
Let 1 ≤ s ≤ q < ∞, w ∈ Aq/s and T : Lqloc (w) → Lqloc (w) a sub
linear operator which satisfies the following property : for all
balls B ⊂ Rn
1
s
R
P∞
s
1
T (f χ(2B)c )(x) 4 k=1 k 2k+1 B 2k +1 B |f (z)| dz
a.e. on B.
|
|
Then
1
2
if q > 1 and T is bounded on Lq (w), then it is also
bounded on (Lq (w), Lp )α , for q ≤ α < p ≤ ∞,
if for all λ > 0
w({x ∈ Rn : |T f (x)| > λ} ≤ C λ1
R
Rn
|f (y )| w(y )dy ,
then for 1 ≤ α < p ≤ ∞, T is bounded from (L1 (w), Lp )α to
(L1,∞ (w), Lp )α .
Justin Feuto
Intrinsic square functions on functions spaces including weig
∗
Introduction Norms inequalities for intrinsic square function Norm inequalities for intrinsic Littlewood-Paley gλ
-function
Intermediate result
Proposition [F.]
Let 1 ≤ s ≤ q < ∞, w ∈ Aq/s and T : Lqloc (w) → Lqloc (w) a sub
linear operator which satisfies the following property : for all
balls B ⊂ Rn
1
s
R
P∞
s
1
T (f χ(2B)c )(x) 4 k=1 k 2k+1 B 2k +1 B |f (z)| dz
a.e. on B.
|
|
Then
1
2
if q > 1 and T is bounded on Lq (w), then it is also
bounded on (Lq (w), Lp )α , for q ≤ α < p ≤ ∞,
if for all λ > 0
w({x ∈ Rn : |T f (x)| > λ} ≤ C λ1
R
Rn
|f (y )| w(y )dy ,
then for 1 ≤ α < p ≤ ∞, T is bounded from (L1 (w), Lp )α to
(L1,∞ (w), Lp )α .
Justin Feuto
Intrinsic square functions on functions spaces including weig
∗
Introduction Norms inequalities for intrinsic square function Norm inequalities for intrinsic Littlewood-Paley gλ
-function
Sketch of the proof
Fix y ∈ Rn and r > 0. For a.e. x ∈ B = B(y , r ),
P
k
|T f (x)| 4 |T (f χ2B )(x)| + ∞
1 kf χ2k +1 B kLq (w)
k=1
w(2k+1 B) q
q > 1 take the Lq (w)-norm on B
kT f χB kLq (w) 4 kf χ2B kLq (w)
1 .
P∞
q
w(B)
+
k
kf
χ
k
k
+1
q
2
B L (w) w(2k +1 B)
k=1
q = 1. For λ > 0, we have
λw({x ∈ B : |T f (x)| > λ})
∞
X
kw(B)
4 kf χ2B kL1 (w) +
kf χ2k +1 B )kL1 (w)
w(2k+1 B)
k=1
Justin Feuto
Intrinsic square functions on functions spaces including weig
∗
Introduction Norms inequalities for intrinsic square function Norm inequalities for intrinsic Littlewood-Paley gλ
-function
Sketch of the proof
Fix y ∈ Rn and r > 0. For a.e. x ∈ B = B(y , r ),
P
k
|T f (x)| 4 |T (f χ2B )(x)| + ∞
1 kf χ2k +1 B kLq (w)
k=1
w(2k+1 B) q
q > 1 take the Lq (w)-norm on B
kT f χB kLq (w) 4 kf χ2B kLq (w)
1 .
P∞
q
w(B)
+
k
kf
χ
k
k
+1
q
2
B L (w) w(2k +1 B)
k=1
q = 1. For λ > 0, we have
λw({x ∈ B : |T f (x)| > λ})
∞
X
kw(B)
4 kf χ2B kL1 (w) +
kf χ2k +1 B )kL1 (w)
w(2k+1 B)
k=1
Justin Feuto
Intrinsic square functions on functions spaces including weig
∗
Introduction Norms inequalities for intrinsic square function Norm inequalities for intrinsic Littlewood-Paley gλ
-function
Proof of the theorem
Fix B = B(y, r ) a ball and let f2 = f χ(2B)c . For t > 0, we have
R
|f2 ∗ ϕt (u)| 4 t −n (2B)c ∩B̃(u,t) |f (z)| dz, ϕ ∈ Cγ , u ∈ Rn .
Let x ∈ B, we have

Z

|Sγ (f2 )(x)| 4
Γ(x)
Minkowski
4
∞ Z
X
t
−n
!2
Z
|f (z)| dz
(2B)c ∩B̃(u,t)
"Z
∞
0
Justin Feuto
dudt 
t n+1
B(x,t)
2
!
Z
|f (z)|
k+1 B\2k B
k=1 2
1
χB̃(z,t) (u)du
dt
t 3n+1
#1
2
dz
Intrinsic square functions on functions spaces including weig
∗
Introduction Norms inequalities for intrinsic square function Norm inequalities for intrinsic Littlewood-Paley gλ
-function
Proof of the theorem
|Sγ (f2 )(x)| 4
4
4
P∞ R
k=1
2k +1 B\2k B
P∞ R
|f (z)|
R
∞
2k −2 r
R
|f (z)|
k
k=1 k +1
P∞ 2 1 B\2R B
k=1 |2k +1 B | 2k+1 B\2k B
Justin Feuto
∞
2k −2 r
dt
B(x,t) du t 3n+1
1
2
dt
dz
2n+1
t
R
1
2
dz
|f (z)| dz.
Intrinsic square functions on functions spaces including weig
∗
Introduction Norms inequalities for intrinsic square function Norm inequalities for intrinsic Littlewood-Paley gλ
-function
Outline
1
Introduction
2
Norms inequalities for intrinsic square function
3
Norm inequalities for intrinsic Littlewood-Paley
gλ∗ -function
4
Commutator of those operators
Justin Feuto
Intrinsic square functions on functions spaces including weig
∗
Introduction Norms inequalities for intrinsic square function Norm inequalities for intrinsic Littlewood-Paley gλ
-function
Intrinsic Littlewood-Paley gλ∗ -function
∗ (f ) is defined by
gλ∗ -function gλ,γ
∗
gλ,γ
(f )(x) =
"Z
Rn+1
+
t
t + |x − y|
Justin Feuto
λn
dydt
fγ∗ (y , t)2 n+1
t
#1
2
,
Intrinsic square functions on functions spaces including weig
∗
Introduction Norms inequalities for intrinsic square function Norm inequalities for intrinsic Littlewood-Paley gλ
-function
A norm control of Litlewood-Paley intrinsic
operator
Theorem [Theorem 1.3, Wang]
Let 0 < γ ≤ 1, 1 < q < ∞, 0 < κ < 1 and w ∈ Aq . If
λ > max {q, 3}, then
∗ g f q,κ 4 kf k q,κ .
λ,γ L
Lw
w
Theorem [F.]
Let 0 < γ ≤ 1, 1 < q ≤ α < p ≤ ∞ and w ∈ Aq . If
λ > max {q, 3} then
∗
g (f )
4 kf kqw ,p,α , f ∈ (Lqw , Lp )α (Rn )
λ,γ
q ,p,α
w
Justin Feuto
Intrinsic square functions on functions spaces including weig
∗
Introduction Norms inequalities for intrinsic square function Norm inequalities for intrinsic Littlewood-Paley gλ
-function
A norm control of Litlewood-Paley intrinsic
operator
Theorem [Theorem 1.3, Wang]
Let 0 < γ ≤ 1, 1 < q < ∞, 0 < κ < 1 and w ∈ Aq . If
λ > max {q, 3}, then
∗ g f q,κ 4 kf k q,κ .
λ,γ L
Lw
w
Theorem [F.]
Let 0 < γ ≤ 1, 1 < q ≤ α < p ≤ ∞ and w ∈ Aq . If
λ > max {q, 3} then
∗
g (f )
4 kf kqw ,p,α , f ∈ (Lqw , Lp )α (Rn )
λ,γ
q ,p,α
w
Justin Feuto
Intrinsic square functions on functions spaces including weig
∗
Introduction Norms inequalities for intrinsic square function Norm inequalities for intrinsic Littlewood-Paley gλ
-function
Relation between the two operators
Let 0 < γ ≤ 1 and β > 0.
Define Sγ,β (f ) by
"Z
Sγ,β (f )(x) =
dydt
fγ∗ (y , t)2 n+1
t
Γβ (x)
#1
2
,
n
o
n+1
where Γβ (x) = (x, t) ∈ R+
/ |x − y | < βt .
We have
∗
gλ,γ
(f )(x)2
2
4 Sγ (f )(x) +
∞
X
2−jλn Sγ,2j (f )(x)2 .
j=1
Justin Feuto
Intrinsic square functions on functions spaces including weig
∗
Introduction Norms inequalities for intrinsic square function Norm inequalities for intrinsic Littlewood-Paley gλ
-function
Relation between the two operators
Let 0 < γ ≤ 1 and β > 0.
Define Sγ,β (f ) by
"Z
Sγ,β (f )(x) =
dydt
fγ∗ (y , t)2 n+1
t
Γβ (x)
#1
2
,
n
o
n+1
where Γβ (x) = (x, t) ∈ R+
/ |x − y | < βt .
We have
∗
gλ,γ
(f )(x)2
2
4 Sγ (f )(x) +
∞
X
2−jλn Sγ,2j (f )(x)2 .
j=1
Justin Feuto
Intrinsic square functions on functions spaces including weig
∗
Introduction Norms inequalities for intrinsic square function Norm inequalities for intrinsic Littlewood-Paley gλ
-function
Boundedness on weighted Lebesgue space
Lemma [Wang]
Let 0 < γ ≤ 1, 1 < q < ∞ and w ∈ Aq . Then for all non
negative integers j, Sγ,2j is bounded on Lqw (Rn ). Moreover
njq
S j (f ) 4 (2nj + 2 2 ) kf k .
γ,2
qw
q
w
Justin Feuto
Intrinsic square functions on functions spaces including weig
∗
Introduction Norms inequalities for intrinsic square function Norm inequalities for intrinsic Littlewood-Paley gλ
-function
Sketch of the proof
The same arguments we use to estimate Sγ (f2 )(x) for x ∈ B,
i.e., Minkowsky’s integral inequality and the fact that for k ∈ N∗ ,
z ∈ 2k+1 B \ 2k B
Z
B(x,2j t)
χB̃(z,t) (u)du 6= 0 ⇒ t ≥
2k−1
r,
2j + 1
allow us to get the following
∞
X
1
S j (f2 )(x) 4 23jn/2
γ,2
k+1
2 B k =1
Z
|f (z)| dz
2k +1 B\2k B
for all x ∈ B(y , r ).
Justin Feuto
Intrinsic square functions on functions spaces including weig
∗
Introduction Norms inequalities for intrinsic square function Norm inequalities for intrinsic Littlewood-Paley gλ
-function
Outline
1
Introduction
2
Norms inequalities for intrinsic square function
3
Norm inequalities for intrinsic Littlewood-Paley
gλ∗ -function
4
Commutator of those operators
Justin Feuto
Intrinsic square functions on functions spaces including weig
∗
Introduction Norms inequalities for intrinsic square function Norm inequalities for intrinsic Littlewood-Paley gλ
-function
∗
Commutator of Sγ,λ and gλ,γ
Let b ∈ L1loc . The commutator [b, Sγ ] is defined by
Z
dydt
[(b(x) − b)f ]∗γ (y , t)2 n+1
t
Γ(x)
[b, Sγ ] (f )(x) =
!1
2
,
h
i
∗
and b, gλ,γ
by
∗
b, gλ,γ
(f )(x) =
(Z
Rn+1
+
t
t + |x − y|
Justin Feuto
λn
dydt
[(b(x) − b)f ]∗γ (y, t)2 n+1
t
)1
2
Intrinsic square functions on functions spaces including weig
∗
Introduction Norms inequalities for intrinsic square function Norm inequalities for intrinsic Littlewood-Paley gλ
-function
Norm inequalities
Theorem [Wang]
Let
h 0 <iγ ≤ 1, 1 < q < ∞ and w ∈ Aq . Then [b, Sγ ] and
∗
b, gλ,γ
are bounded on Lqw (Rn ) whenever b ∈ BMO(Rn ).
Theorem [F.]
Let 0 < γ ≤ 1, 1 < q ≤ α < p ≤ ∞ and w ∈ Aq . Suppose that
b ∈ BMO(Rn ), then
k[b, Sγ ] (f )kqw ,p,α 4 kf kqw ,p,α .
Theorem [F.]
Let 0 < γ ≤ 1, 1 < q ≤ α < p ≤ ∞ and w ∈ Aq . If
b ∈ BMO(Rn ) and λ > max {q, 3} then
b, g ∗ (f )
4 kf kqw ,p,α .
λ,γ
q ,p,α
w
Justin Feuto
Intrinsic square functions on functions spaces including weig
∗
Introduction Norms inequalities for intrinsic square function Norm inequalities for intrinsic Littlewood-Paley gλ
-function
Norm inequalities
Theorem [Wang]
Let
h 0 <iγ ≤ 1, 1 < q < ∞ and w ∈ Aq . Then [b, Sγ ] and
∗
b, gλ,γ
are bounded on Lqw (Rn ) whenever b ∈ BMO(Rn ).
Theorem [F.]
Let 0 < γ ≤ 1, 1 < q ≤ α < p ≤ ∞ and w ∈ Aq . Suppose that
b ∈ BMO(Rn ), then
k[b, Sγ ] (f )kqw ,p,α 4 kf kqw ,p,α .
Theorem [F.]
Let 0 < γ ≤ 1, 1 < q ≤ α < p ≤ ∞ and w ∈ Aq . If
b ∈ BMO(Rn ) and λ > max {q, 3} then
b, g ∗ (f )
4 kf kqw ,p,α .
λ,γ
q ,p,α
w
Justin Feuto
Intrinsic square functions on functions spaces including weig
∗
Introduction Norms inequalities for intrinsic square function Norm inequalities for intrinsic Littlewood-Paley gλ
-function
Norm inequalities
Theorem [Wang]
Let
h 0 <iγ ≤ 1, 1 < q < ∞ and w ∈ Aq . Then [b, Sγ ] and
∗
b, gλ,γ
are bounded on Lqw (Rn ) whenever b ∈ BMO(Rn ).
Theorem [F.]
Let 0 < γ ≤ 1, 1 < q ≤ α < p ≤ ∞ and w ∈ Aq . Suppose that
b ∈ BMO(Rn ), then
k[b, Sγ ] (f )kqw ,p,α 4 kf kqw ,p,α .
Theorem [F.]
Let 0 < γ ≤ 1, 1 < q ≤ α < p ≤ ∞ and w ∈ Aq . If
b ∈ BMO(Rn ) and λ > max {q, 3} then
b, g ∗ (f )
4 kf kqw ,p,α .
λ,γ
q ,p,α
w
Justin Feuto
Intrinsic square functions on functions spaces including weig
∗
Introduction Norms inequalities for intrinsic square function Norm inequalities for intrinsic Littlewood-Paley gλ
-function
Some references on these spaces
References
J. Feuto, Intrinsic square functions on functions spaces
including weighted Morrey spaces, Bull. Korean Math. Soc.
50 (2013), No. 6, pp. 1923-1936
J. Feuto, I. Fofana and K. Koua,Integrable fractional mean
functions on spaces of homogeneous type, Afr. Diaspora J.
Math. 9 1 (2010), 8-30.
Y. Komori and S. Shirai, Weighted Morrey spaces and a
singular integral operator, Math. Nachr. 2822 (2009),
219-231
H. Wang, Intrinsic square functions on the weighted Morrey
spaces, J. Math. Anal. Appl, 396 (2012), 302-314.
M. Wilson,The intrinsic square function, Rev. Mat.
Iberoamericana, 23 (2007), 771-791.
Justin Feuto
Intrinsic square functions on functions spaces including weig
∗
Introduction Norms inequalities for intrinsic square function Norm inequalities for intrinsic Littlewood-Paley gλ
-function
THANKS
Justin Feuto
Intrinsic square functions on functions spaces including weig