An Lp theory for outer measures.Application to singular integrals.II
Christoph ThieleSantander, September 2014
Recall Tents (or Carleson boxes)
X is the open upper half plane, generating sets are tents T(x,s) :
Define outer measure on X by
€
σ(T(x,s)) := s (= μ(T(x,s)))
Sizes
Sp size of function on a tent
Alos Sinfty
€
S p ( f )(T(x,s)) = (1
sf (y, t)p dy
dt
tT (x,s)
∫∫ )1/ p
€
S∞( f )(T(x,s)) = sup(y,t )∈T (x,s) f (y, t)
Outer essential supremum
Space of functions with finite out.ess.supremum
Outer essential supremum on a subset F:€
outess( f ) = sup{S( f )(E) : E ∈Σ}
€
L∞(X,μ,S)
€
outessF ( f ) = sup{S( f 1F )(E) : E ∈Σ}
Outer Lp spaces
Define super level measures
Define Lp norms
Also weak Lp (Lorentz space) €
μ( f > λ ) := inf{μ(F) : outessF c f ≤ λ}
€
fLp (X ,μ ,S )
:= ( pλ p
0
∞
∫ μ( f > λ )dλ /λ )1/ p
€
fLp ,∞ (X ,μ ,S )
:= supλ λ pμ( f > λ )
Embedding theorems
Thm: Define for fixed Schwartz function ϕ
Then we have:
If φ has integral zero:
€
Fφ ( f )(y, t) := f (z)t−1φ(t−1(z − y))dz∫
€
Fφ ( f )Lp (X ,μ ,S ∞ )
≤ Cφ fp
€
Fφ ( f )Lp (X ,μ ,S 2 )
≤ Cφ fp
Proof of embedding thm for Sinfty size
By Marcinkiewicz interpolation between
and
€
Fφ ( f )L∞ (X ,μ ,S ∞ )
≤ Cφ f∞
€
Fφ ( f )L1,∞ (X ,μ ,S ∞ )
≤ Cφ f1
Linfty-Sinfty estimate
To show for all tents T(x,s)
But this follows from
€
S∞(Fφ )(T(x,s)) ≤ C f∞
€
sup(y,t )∈T (x,s) Fφ (y, t) ≤ sup(y,t ) Fφ (y, t)
€
=sup(y,t ) f (z)t−1φ(t−1(z − y))dzR
∫ ≤ φ 1 f∞
Weak L1-Sinfty estimate
Ned for all lambda
Need to find a collection of tents T(xi,si) with union E such that for all x,s
€
S∞(Fφ1E c )(T(x,s)) ≤ Cλ
€
sii
∑ ≤ Cλ−1 f1
€
λ μ(Fφ > λ ) ≤ C f1
Weak L1-Sinfty estimate
Hardy Littlewood maximal function
There is an open set where Mf is larger than lambda. Let (xi-si,xi+si) be the collection of connected components of this set. These are the tents. By Hardy Littlewood maximal thm
€
Mf (x) = supε
1
2εf (x + y)dy
−ε
ε
∫
€
sii
∑ = x : Mf (x) > λ{ } ≤ Cλ−1 f1
More on weak L1-Sinfty estimate
We haveSince F(y,t) is testing f against bump function at y of width t, may estimate by Hardy Littlewood
€
S∞(Fφ1E c )(T(x,s)) ≤ sup
(y,t )∈E c Fφ (y, t)
€
≤C supy −t ≤z≤y +t Mf (z) ≤ Cλ
Proof of embedding thm for S2 size
By Marcinkiewicz interpolation between
and
€
Fφ ( f )L∞ (X ,μ ,S 2 )
≤ Cφ f∞
€
Fφ ( f )L1,∞ (X ,μ ,S 2 )
≤ Cφ f1
Use Calderon reproducing formula
€
Fφ (y, t)2
∫∫ dydt / t = f *ϕ t (y)2∫∫ dydt / t
€
= ˆ f ˆ ϕ t (η)2
∫∫ dη dt / t = ˆ f (η)2
ˆ ϕ (η / t)dt / t∫ dη ≤ C∫ ˆ f 2
dη∫
€
Fφ (y, t)2
T (x,s)
∫∫ dydt / t ≤ f 1[x −Cs,x +Cs]
2dη∫ ≤ Cs f
∞
2
Proof of Linfty-S2 estimate
Apply Calderon reproducing formula.If we integrate over arbitrary tent T(x,s), only
restriction of f to X-Cs,x+Cs matters if phi has compact support
Dividing by s yields the desired
€
Fφ (y, t)2
T (x,s)
∫∫ dydt / t ≤ f 1[x −Cs,x +Cs]
2dη∫ ≤ Cs f
∞
2
€
S(Fφ )(T(x,s)) ≤ C f∞
2
BMO estimate
Note that we have in fact proven for any m
Thus we have the stronger embedding theorem
For the space BMO, which is defined by
€
Fφ ( f )L∞ (X ,μ ,S 2 )
≤ Cφ fBMO
€
fBMO
2:= supI
1
I| f (x) − f I |2 dx
I
∫
€
S(Fφ )(T(x,s) ≤1
sf (y) − m
2dy
x −Cs
x +Cs
∫
Weak L1-S2 estimate
Let (xi-si,x-+si) be the connected components of the set where Mf(x) is larger than lambda/2.
Let E be the union of tents T(xi,3si).Do Calderon Zygmund decomposition of f
Where g is bounded by lambda, and bi is supported on interval xi-si,xi+si and has integral zero.
€
f = g + bii
∑
Weak L1-S2 estimate
For the good function use Linfty estimate.
For bi, do a careful estimate and accounting using1) Partial integration of bi to use mean zero when
paired with a phi-t of large support2) Support considerations of bi and phi when paired
with phi-t of small support. .
Summary of proof of embedding thm
Encodes much of singular integral theory:Hardy Littlewood maximal theorem, Vitali covering argumentsCalderon reproducing formula, BMO estimates, square function techniques, Calderon Zygmund decomposition
Use to prove boundedness of operators
Suppose we have operator T mapping functions on real line to functions on real line.
Want to prove
(If T is commutes with dilations, it is forced thatBoth exponents are the same)
€
Tfp
≤ C fp
Use to prove boundedness of operators
Duality implies
Hence it suffices to prove
€
Tfp
= supg p ' =1
Tf ,g
€
Tf ,g ≤ C fp
gp'
Use to prove boundedness
Express <Tf,g> by Fphi and Gphi and prove
Where either S is S2 or Sinfty, as the case may beBy outer Hoelder suffices to prove
Which itself my be result of outer triangle ineq.
€
Tf ,g ≤ C Fφ Lp (X ,μ ,S )Gϕ Lp ' (X ,μ ,S )
€
Tf ,g ≤ C FφGϕ L1 (X ,μ ,S )
Example identity operator
By polarization of Calderon reproducing f.:
Provided phi has mean zero and
Triangle ineq. and outer triangle ineq. imply€
f ,g = Fφ ,Gφ dydt / t
€
ˆ ϕ (η / t)2dt / t∫ =1
€
f ,g ≤ FφGφ L1 (X ,μ ,S1 )
Cauchy projection operator
By polarization of Calderon reproducing f.:
Provided phi has mean zero and
Boundedness of identity operator and of Cauchy projection imply that of Hilbert transform
€
Πf ,g = Fφ ,Gφ
€
ˆ ϕ (η / t)2dt / t∫ =1(0,∞)(η)
€
Hf (x) = p.v. f (x − t)dt / t∫
Paraproduct estimates
For three Schwartz functions
If two Schwartz functions have integral zero
€
Λ( f1, f2, f3) := ( Fφ i( f i)(y, t))dy
dt
ti=1
3
∏∫∫
€
Λ( f1, f2, f3) ≤ Fφ i( f i)
i=1
3
∏L1 (X ,μ ,S1 )
≤ C Fφ i( f i)
i=1
3
∏Lpi (X ,μ ,Sqi
)
≤ C f ii=1
3
∏Lpi
Special paraproducts
Paraproducts ard trilinear forms which are dual to bilinear operators
If phi-2 has mean 1 and f2 is 1, then F2 is 1 And we are redued to previous case. In
particular there is a paraproduct withFixing h and considering as operator in f we haveAn operator with
€
Λ( f1, f2,.)
€
Λ(h,1,.) = h
€
T( f ) = Λ(h, f ,.)
€
T(1) = h
Basic T(1) Theorem
Let T be a bounded operator in L2 with
whereFor some nonzero test function phi with mean
zero. Then for 1<p<infty we have for all f,
with universal constant Cp independent of T
€
Tφx,s,φy,t ≤min(t,s)
max(t,s, x − y )
€
Tfp
≤ Cp fp
€
φx,s(z) = s−1φ(s−1(z − x))
Why T(1) theorem?
Usually have a different set of assumptions.If s<t<|x-y| then we write
and demand suitable pointwise estimates on the partial derivative of K (Calderon Zygmund kernel).
Symmetrically if t<s<|x-y|€
Tφx,s,φy,t = K(u,v)φx,s(u)φy,t (v)dudv∫∫
€
=− ∂1K(u,v)Φx,s(u)φy,t (v)dudv∫∫
Why T(1) theorem?
We further demand T(1)=0If |x-y|<s and t<s, then we write
And again demand suitable bounds on K. Similarly we ask T*(1)=0 to address the case|x-y|<t, s<t
€
= K(u,v)(u − v)(φx,s(u) − φx,s(v))
u − vφy,t (v)dudv∫∫
€
K(u,v)(φx,s(u) − φx,s(v))φy,t (v)dudv∫∫
More general T(1) theorem
It suffices to demand T(1)=h, T*(1)=0, for some h in BMO. One reduces to the previous case by subtracting a suitable paraproduct:
Similarly one can relax the condition on T* by subtracting a dual paraproduct.
€
′ T f ,g = Tf ,g − P(h, f ,g)
Proof of T(1) theorem
By Calderons reproducing formula
At which time we use the assumption. To apply outer Hoelder, we break up the region, e.g
|x-y|<s, t<s where we obtain€
F(x,s) Tφx,s,φy,t∫∫ G(y, t)dxdyds /sdt / t∫∫€
Tf ,g =
€
≤ F(x,s)x −s
x +s
∫0
s
∫ G(y, t)dydt / tR
∫ dx0
∞
∫ ds /s3
Proof of T(1) theorem
Applying Fubini
Setting Gab(x,s)=G(x+as,bs) and using outer triangle inequality
€
≤ F(x,s)R
∫0
∞
∫ G(x + as,bs)dxds /s−1
1
∫ da0
1
∫ db
€
≤ FGa,b L1 (X ,S1 )−1
1
∫ da0
1
∫ db
€
FGa,b L1 (X ,S 2 )
Proof of T(1) theorem
Applying outer Hoelder
Integrating trivially:
€
≤ F Lp (X ,S 2 ) Ga,b Lp ' (X ,S 2 )−1
1
∫ da0
1
∫ db
€
≤C F Lp (X ,S 2 ) Ga,b Lp ' (X ,S 2 )
Proof of T(1) theorem
Using embedding theorem (a modified one with tilted triangles for Gab)
Similarly one proves this estimate for the other regions other than|x-y|<s, t<s. €
≤C fp
gp'
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