New Proof of Hörmander multiplier Theorem
on compact manifolds without boundary
Xiangjin Xu
Department of Mathematics
Johns Hopkins University
Baltimore, MD, 21218, USA
xxu@math.jhu.edu
Abstract
On compact manifolds (M, g) without boundary of dimension n ≥ 2, the
gradient estimates for unit band spectral projection operators χλ is proved for
any second order elliptic differential operators L by maximum principle. A
new proof of Hörmander Multiplier Theorem on the eigenfunction expansion
of the operator L is given in this setting by using the gradient estimates and
the Calderón-Zygmund argument.
Keywords: gradient estimate, spectral projection operator, Hörmander
Multiplier Theorem, Calderón-Zygmund decomposition
Mathematics Subject Classification 2000: 58J40, 35P20, 35J25,
1
1
Introduction
Let (M, g) be a compact boundaryless manifolds (M, g) of dimension n ≥ 2, and
suppose that L is a second order elliptic differential operator which is positive and
self-adjoint with respect to the C ∞ density dx associated to the Reimannian metric
g. The purpose of this paper is to give a simple proof of sharp gradient estimates
for the eigenfunctions of L and then to use these estimates to give a new proof
of Hörmander multiplier theorem for the eigenfunction expansions on the compact
boundaryless manifold (M, g). Thus, we shall consider the eigenvalue problem
(L + λ2 )u(x) = 0,
x ∈ M,
(1)
Recall that the spectrum of L is discrete and tends to infinity. Let 0 ≤ λ21 ≤ λ22 ≤
λ23 ≤ · · · denote the eigenvalues, so that {λj } is the spectrum of the first order
√
operator P = −L. Let {ej (x)} ⊂ L2 (M ) be an associated real orthogonormal
basis, define the unit band spectral projection operators,
χλ f =
X
ej (f ),
where ej (f )(x) = ej (x)
λj ∈[λ,λ+1)
Z
M
f (y)ej (y)dy.
In [5] and [6], Sogge proved the following L∞ estimates on χλ ,
||χλ f ||∞ ≤ Cλ(n−1)/2 ||f ||2 ,
λ ≥ 1,
for a uniform constant C, which is equivalent to
X
ej (x)2 ≤ Cλn−1 ,
∀x ∈ M,
λj ∈[λ,λ+1)
Here based on the above L∞ estimates on χλ f , by maximum principle, we show the
following gradient estimates for χλ f
2
Theorem 1.1 Fix a compact boundaryless Riemannian manifold (M, g), there is a
uniform constant C so that
||∇χλ f ||∞ ≤ Cλ(n+1)/2 ||f ||2 ,
λ ≥ 1,
which is equivalent to
X
|∇ej (x)|2 ≤ Cλn+1 ,
∀x ∈ M.
λj ∈[λ,λ+1)
We shall show the gradient estimates following the ideas of the interior gradient
estimates for Poisson’s equation in [1], where for Poisson’s equation ∆u = f , there
are gradient estimates for the interior point x0 as
|∇u(x0 )| ≤
C
sup |u| + Cd sup |f |,
d ∂B
B
by using maximum principle in a cube B centered at x0 with length d.
Our other main result is Hörmander Multiplier Theorem for the eigenfunction
expansions on compact manifolds without boundary. Given a bounded function
m(λ) ∈ L∞ (R), one can define the multiplier operator, m(P ), by
m(P )f =
∞
X
m(λj )ej (f )
(2)
j=1
such an operator is always bounded on L2 (M ). However, if one considers on any
other space Lp (M ), it is known that some smoothness assumptions on the function
m(λ) are needed to ensure the boundedness of
m(P ) : Lp (M ) → Lp (M ).
When m(λ) is C ∞ and, moreover, in the symbol class S 0 , i.e.,
|(
d α
) m(λ)| ≤ Cα (1 + |λ|)−α ,
dλ
3
α = 0, 1, 2, · · ·
(3)
It has been known for some time that (3) holds for all 1 < p < ∞ on compact
manifolds (see [10]). One assumes the following regularity assumption: Suppose
that m ∈ L∞ (R), let L2s (R) denote the usual Sobolev space and fix β ∈ C0∞ ((1/2, 2))
satisfying
P∞
−∞
β(2j t) = 1, t > 0, for s > n/2, there is
sup λ−1+s ||β(·/λ)m(·)||2L2s = sup ||β(·)m(λ·)||2L2s < ∞.
λ>0
(4)
λ>0
One can see this is the sharp assumption to ensure the boundedness of m(P ) on
Lp (M ) for all 1 < p < ∞, since for a special class of multiplier operators, Riesz
means Sλδ f =
P
λj ≤λ (1
− λ2j /λ2 )δ ej (f ), such an assumption is sharp (see in [6]).
Many authors have studied this problem under different setting. Hörmander
[2] first proved the boundedness of m(P ) for Rn under the assumption (4), using
the Calderón-Zygmund decomposition and the estimates on the integral kernel of
the multiplier operator. Stein and Weiss [9] studied the boundedness of m(P ) for
multiple Fourier series, which can be regarded as the case on the flat torus Tn .
By studying the paramatrix of the wave kernel of m(P ), Seeger and Sogge [4] and
Sogge [6] proved the following Hömander multiplier Theorem for the eigenfunction
expansions on compact manifolds without boundary:
Theorem 1.2 Let m ∈ L∞ (R) satisfy (4), then there are constants Cp such that
||m(P )f ||Lp (M ) ≤ Cp ||f ||Lp (M ) ,
1 < p < ∞.
(5)
Here we shall give a new proof for above theorem by using the L∞ estimates and
the gradient estimates on χλ f , without using the paramatrix of the wave kernel of
m(P ) as done in [4] and [6]. Since the complex conjugate of m satisfies the same
hypotheses (4), we need only to prove Theorem 1.2 for exponents 1 < p ≤ 2. This
4
will allow us to exploit orthogonality, and since m(P ) is bounded on L2 (M ), also
reduce Theorem 1.2 to show that m(P ) is weak-type (1, 1) by the Marcinkiewicz
Interpolation Theorem. The weak-type (1,1) estimates of m(P ) will involve a splitting of m(P ) into two pieces m(P ) = m̃(P ) + r(P ). For the remainder r(P ), one
can obtain the strong (1,1) estimates by the L∞ estimates on χλ f as done in [6].
For the main term m̃(P ), in [4] and [6], the authors used the paramatrix of the
wave kernel of m̃(P ) to get the required estimates on the integral kernel of m̃(P ),
and applying these estimates, proved the weak-type (1,1) estimates on m̃(P ). As
people know, the the paramatrix construction of the wave equation does not work
well for general compact manifolds with boundary unless one assume the boundary
is geodesically concave. Here we use another approach to prove the weak-type (1,1)
estimates on m̃(P ), which works for any compact manifolds with boundary (One
may see it from [11] and [12] for the proof of Hörmander Multiplier Theorem of
Dirichlet Laplacian and Neumann Laplacian on compact manifolds with boundary).
We make a second decomposition {Kλ,l (x, y)}∞
l=−∞ for each term Kλ (x, y), which
comes from the dyadic decomposition of the integral kernel K(x, y) of m̃(P )
K(x, y) =
∞
X
K2k (x, y) + K0 (x, y),
(∗)
k=1
such that Kλ (x, y) =
m̃(P )f (x) =
P∞
∞ X
∞
X
l=−∞
Kλ,l (x, y) and
T2k ,l (P )f (x), where Tλ,l (P )f (x) =
k=0 l=−∞
Z
M
Kλ,l (x, y)f (y)dy.
We shall give the definitions of Kλ,l (x, y) in the proof of Theorem 1.2. And for each
operator Tλ,l (P ), we shall prove the L1 → L2 estimates by a rescaling argument of
the proof for the remainder r(P ) using the L∞ estimates and gradient estimates on
5
χλ f which we obtained in Theorem 1.1. With the support properties and the finite
propagation speed properties, one has the relation (10) between λ and l, which is
one key observation when we apply the Calderón-Zygmund decomposition to show
the weak-type (1,1) estimates on m̃(P ).
In what follows we shall use the convention that C will denote a constant that
is not necessarily the same at each occurrence.
Acknowledgement: The author would like to thank his advisor, Professor
C.D. Sogge, brings the problems to him and a number helpful conversations on
these problems and his research. The results in this paper come from part of the
author’s Ph.D. thesis [12] in Johns Hopkins University.
2
Gradient estimates
In this section, we shall prove Theorem 1.1. by using maximum principle.
Proof of Theorem 1.1.
Now we fix x0 ∈ M . We shall use the maximum
principle in the cube centered at x0 with length d = (λ + 1)−1 to prove the same
gradient estimates for χλ f as for Poisson’s equation.
Define the geodesic coordinates x = (x1 , · · · , xn ) centered at point x0 as following,
fixed an orthogonormal basis {vi }ni=1 ⊂ Tx0 M , identity x = (x1 , · · · , xn ) ∈ Rn with
Pn
the point exp(
i=1
xi vi ) ∈ M . In this coordinate, the elliptic operator L can be
written as
L=
n
X
i,j=1
aij (x)
n
X
∂2
∂
+
bi (x)
+ c(x),
∂xi ∂xj i=1
∂xi
where aij (x), bi (x), c(x) ∈ C ∞ (M ), and c(x) ≤ 0 which comes from L is an elliptic
6
operator.
Now define the cube
Q = {x = (x1 , · · · , xn ) ∈ Rn | |xi | < d, i = 1, · · · , n} ⊂ M.
Denote u(x; f ) = χλ f (x), we have u ∈ C 2 (Q), and
λ2j ej (f ) := h(x; f ).
X
Lu(x; f ) = −
λj ∈[λ,λ+1)
From the L∞ estimate in [5], and Cauchy-Schwarz inequality, we have
|h(x; f )|2 = (
Z
(λ2j ej (x))(
X
λj ∈[λ,λ+1)
≤
λ4j e2j (x)
X
ej (y)f (y)dy))2
X
≤ (λ + 1) (
X
Z
(
λj ∈[λ,λ+1)
λj ∈[λ,λ+1)
4
M
M
ej (y)f (y)dy)2
e2j (x))||χλ f ||2L2 (M )
λj ∈[λ,λ+1)
≤ C(λ + 1)
n+3
||χλ f ||2L2 (M )
Here we estimate |Dn u(0; f )| = | ∂x∂n u(0; f )| only, and the same estimate holds
for |Di u(0; f )| with i = 1, · · · , n − 1 also. Now in the half-cube
Q0 = {x = (x1 , · · · , xn ) ∈ Rn | |xi | < d, i = 1, · · · , n − 1, 0 < xn < d.} ⊂ M,
Consider the function
1
ϕ(x0 , xn ; f ) = [u(x0 , xn ; f ) − u(x0 , −xn ; f )],
2
where we write x = (x0 , xn ) = (x1 , · · · , xn−1 , xn ). One sees that ϕ(x0 , 0; f ) = 0,
sup∂Q0 |ϕ| ≤ sup∂Q |u| := A, and |Lϕ| ≤ supQ |h| := N in Q0 . Now consider the
function
ψ(x0 , xn ) =
A 02
[|x | + αxn (nd − (n − 1)xn )] + βN xn (d − xn ) ≥ 0
d2
7
defined on the half-cube Q0 , where α ≥ 1 and β ≥ 1 will be determined below.
Obviously ψ(x0 , xn ) ≥ 0 on xn = 0 and ψ(x0 , xn ) ≥ A in the remaining portion of
∂Q0 .
Lψ(x) =
n
X
A
[2tr(a
(x))
−
(2nα
−
2α
+
1)a
(x)
+
2
bi (x)xi + bn (x)(nαd
ij
nn
d2
i=1
−(2nα − 2α + 1)xn )] + N β[−2ann (x) + bn (x)(d − 2xn )] + c(x)ψ(x).
Since in M , tr(aij (x)) and bi (x) are bounded uniformly, c(x) ≤ 0 and ann (x) is
positive, then for a large α, we can make
2tr(aij (x)) − (2nα − 2α + 1)ann (x) ≤ −1,
Fix such a α, since d = (λ + 1)−1 , for large λ, we have
2
n
X
bi (x)xi + bn (x)(nαd − (2nα − 2α + 1)xn ) < 1.
i=1
Then the first term is negative. For second term, let β large enough, we have
β[−2ann (x) + bn (x)(d − 2xn )] < −1.
For the third term, we have c(x) ≤ 0 and ψ(x) ≥ 0 in Q0 . Hence we have Lψ(x) ≤
−N in Q0 .
Now we have L(ψ ± ϕ) ≤ 0 in Q0 and ψ ± ϕ ≥ 0 on ∂Q0 , from which it follows by
the maximum principle that |ϕ(x0 , xn ; f )| ≤ |ψ(x0 , xn )| in Q0 . Letting x0 = 0 in the
expressions for ψ and ϕ, then dividing by xn and letting xn tend to zero, we obtain
|Dn u(0; f )| = lim |
xn →0
αnA
ϕ(0, xn ; f )
|≤
+ βdN.
xn
d
Note that d = (λ + 1)−1 , A ≤ C(λ + 1)(n−1)/2 , and N ≤ (λ + 1)(n+3)/2 , then we
have the estimate
|Dn u(0; f )| ≤ C(λ + 1)(n+1)/2 .
8
The same estimate holds for |Di u(0)|, i = 1, · · · , n − 1. Hence we have
|∇u(0)| ≤ C(λ + 1)(n+1)/2 .
Since the estimate is for any x0 ∈ M , Theorem 1.1 is proved.
Q.E.D.
For Riemannian manifolds without boundary, in [8], the authors proved that for
(n−1)/2
generic metrics on any manifold one has the bounds ||ej ||L∞ (M ) = o(λj
) for L2
normalized eigenfunctions. For Laplace-Beltrami operator ∆g , there are eigenvalues
{−λ2j }, where 0 ≤ λ20 ≤ λ21 ≤ · · · → ∞ are counted with multiplicity. Let {ej (x)}
be an associated orthogonal basis of L2 normalized eigenfunctions. If λ2 is in the
spectrum of −∆g , let Vλ = {u | ∆g u = −λ2 u} denote the corresponding eigenspace.
We define the eigenfunction growth rate in term of
L∞ (λ, g) =
||u||L∞ ,
sup
u∈Vλ ;||u||L2 =1
and the gradient growth rate in term of
L∞ (∇, λ, g) =
sup
||∇u||L∞ .
u∈Vλ ;||u||L2 =1
In [8], Sogge and Zelditch proved the following results
(n−1)/2
L∞ (λ, g) = o(λj
)
for a generic metric on any manifold without boundary. Here we have the following
estimates on the gradient growth rate
(n+1)/2
Theorem 2.1 L∞ (∇, λ, g) = o(λj
) for a generic metric on any manifold with-
out boundary. And the bounds are uniform if there is a uniformly bound on the norm
of tr(g ij (x)) for (M, g).
9
Proof. For a compact Riemannian manifold (M, g) without boundary, we can
apply Theorem 1.1 to any point in M . From Theorem 1.4 in [8], we have
(n−1)/2
L∞ (λ, g) = o(λj
)
for a generic metric on any compact manifold without boundary. Fix any such a
metric on the manifold M and a L2 normalized eigenfunction u(x), apply Theorem
1.1 to u(x) at each point x0 ∈ M , we have
|∇u(x0 )| ≤
αnA
+ βdN.
d
where α and β are constants depending on the norm of tr(g ij (x)) at M only, which
(n−1)/2
can been seen in the above proof of Theorem 1.1, and A = sup∂Q |u| = o(λj
(n+3)/2
and N = supQ |λ2 u| = (λj
),
), where the cube
Q = {x = (x1 , · · · , xn ) ∈ Rn | |xi | < d, i = 1, · · · , n} ⊂ M,
√
we choose d = (λ + 1)−1 / n. Hence we have
(n+1)/2
|∇u(x0 )| = o(λj
)
holds for all L2 normalized eigenfunction u(x) ∈ Vλ , furthermore, the bounds are uniform when those metrics of (M, g) have a uniformly bound on the norm of tr(g ij (x))
from the proof. Hence we have our Theorem.
3
Q.E.D.
Hörmander Multiplier Theorem
In this section, we shall see how the L∞ estimates and gradients estimates for χλ
imply Hörmander multiplier Theorem. As discussed in Introduction, we reduce
10
Theorem 1.2 to show that m(P ) is weak-type (1, 1), i.e.,
µ{x : |m(P )f (x)| > α} ≤ α−1 ||f ||L1 ,
(6)
where µ(E) denotes the dx measure of E ⊂ M . Since the all eigenvalues of L are
non-negative, we may assume m(t) is an even function on R. Then we have
m(P )f (x) =
where P =
√
1Z
1 Z
m̂(t)eitP f (x)dt =
m̂(t) cos(tP )f (x)dt,
2π R
π R+
−L, and the cosine transform u(t, x) = cos(tP )f (x) is the solution of
the following Cauchy problem of the wave equation:
(
∂2
− L)u(t, x) = 0,
∂t2
u(0, x) = f (x), ut (0, x) = 0.
We shall use the finite propagation speed of solutions of the wave equation in Part
2 of the proof to get the key observation (10).
Proof of Theorem 1.2. The proof of the weak-type (1,1) estimate will involve
a splitting of m(P ) into two pieces: a main piece which one need carefully study,
plus a remainder which has strong (1,1) estimate by using the L∞ estimates for the
unit spectral projection operators as done in [6]. Specifically, define ρ ∈ C0∞ (R) as
ρ(t) = 1,
f or |t| ≤ ,
2
ρ(t) = 0,
f or |t| ≥ .
(7)
where > 0 is a given small constant related to the manifold, which will be specified
later. Write m(P ) = m̃(P ) + r(P ), where
1 Z itP
e ρ(t)m̂(t)dt
m̃(P ) = (m ∗ ρ̌)(P ) =
2π
1 Z itP
e (1 − ρ(t))m̂(t)dt
r(P ) = (m ∗ (1 − ρ)ˇ)(P ) =
2π
11
To estimate the main term and remainder, for λ = 2j , j = 1, 2, · · ·, define
τ
mλ (τ ) = β( )m(τ ).
λ
(8)
Part 1: Strong (1, 1) estimate on the remainder
||r(P )f ||L1 ≤ C||f ||L1 .
We first show
||r(P )f ||L2 ≤ C||f ||L1 .
Here we follow the first part in proof of Theorem 5.3.1 in [6]. Define
1 Z itP
rλ (P ) = (mλ ∗ (1 − ρ)ˇ)(P ) =
e (1 − ρ(t))m̂λ (t)dt
2π
Notice that r0 (P ) = r(P )−
P
j≥1 r2j (P )
is a bounded and rapidly decreasing function
of P . Hence r0 (P ) is bounded from L1 to any Lp space. We need only to show
||rλ (P )f ||L2 ≤ Cλn/2−s ||f ||L1 ,
λ = 2j , j = 1, 2, · · · .
Using the L∞ estimate on χk , see [5] or [6], we have
||rλ (P )f ||2L2
≤
∞
X
||rλ (P )χk f ||2L2
k=1
≤C
∞
X
sup |rλ (τ )|2 (1 + k)n−1 ||f ||2L1
k=1 τ ∈[k,k+1]
Hence we need only to show
∞
X
sup |rλ (τ )|2 (1 + k)n−1 ≤ Cλn−2s
k=1 τ ∈[k,k+1]
Notice since mλ (τ ) = 0, for τ ∈
/ [λ/2, 2λ], we have
m̃λ (τ ) = O((1 + |τ | + |λ|)−N )
rλ (τ ) = O((1 + |τ | + |λ|)−N )
12
for any N when τ ∈
/ [λ/4, 4λ]. Hence we need only to show
4λ
X
sup |rλ (τ )|2 (1 + k)n−1 ≤ Cλn−2s
k=λ/4 τ ∈[k,k+1]
that is
4λ
X
sup |rλ (τ )|2 ≤ Cλ1−2s
k=λ/4 τ ∈[k,k+1]
Using the fundamental theorem of calculus and the Cauchy-Schwartz inequality,
we have
4λ
X
Z
2
sup |rλ (τ )|
≤ C(
R
k=λ/4 τ ∈[k,k+1]
Z
= C(
R
2
|rλ (τ )| dτ +
Z
R
|rλ0 (τ )|2 dτ )
|m̂λ (t)(1 − ρ(t))|2 dt +
Recall that ρ(t) = 1, for |t| ≤
,
2
Z
R
|tm̂λ (t)(1 − ρ(t))|2 dt)
by a change variables shows that this is
dominated by
λ
−1−2s
Z
R
|ts m̂λ (t/λ)|2 dt = λ−1−2s ||λβ(·)m(λ·)||2L2s
= λ1−2s ||β(·)m(λ·)||2L2s
≤ Cλ1−2s
Here the first equality comes from a change variables, the second equality comes
from the definition of Sobolev norm of L2s (M ) and the third inequality comes from
our assumption (4). Hence we have the estimate for the remainder
||r(P )f ||L2 ≤ C||f ||L1 .
And since our manifold is compact, we have
||r(P )f ||L1 ≤ V ol(M )1/2 ||r(P )f ||L2 ≤ C||f ||L1 .
13
Part 2: weak-type (1, 1) estimate on the main term
µ{x : |m̃(P )f (x)| > α} ≤ α−1 ||f ||L1 .
The weak-type (1, 1) estimate on m̃(P ) would follow from the integral operator
1 Z
m̃(P )f (x) =
m̂(t)ρ(t)eitP f (x)dt
2π R
Z
X
1 Z
itλk
=
m̂(t)ρ(t)
e eλk (x)
eλk (y)f (y)dydt
2π R
M
k≥1
X
1 Z Z
{ m̂(t)ρ(t)
eitλk eλk (x)eλk (y)dt}f (y)dy
2π M R
k≥1
=
with the kernel
K(x, y) =
Z
m̂(t)ρ(t)
R
=
X
X
eitλk eλk (x)eλk (y)dt
k≥1
(m ∗ ρ̌)(λk )eλk (x)eλk (y)
k≥1
is weak-type (1,1). Now define the dyadic decomposition
Kλ (x, y) =
Z
R
m̂λ (t)ρ(t)
X
eitλk eλk (x)eλk (y)dt
k≥1
We have
K(x, y) =
∞
X
K2j (x, y) + K0 (x, y)
j=1
where K0 is bounded and vanishes when dist(x, y) is larger than a fixed constant.
In order to estimate Kλ (x, y), we make a second dyadic decomposition as follows
Kλ,l (x, y) =
Z
R
m̂λ (t)β(2−l λ|t|)ρ(t)
X
eitλk eλk (x)eλk (y)dt
k≥1
We have
Kλ (x, y) =
∞
X
l=−∞
14
Kλ,l (x, y)
Define
Tλ,l (P )f (x) =
Z
M
Kλ,l (x, y)f (y)dy.
From above two dyadic decompositions, we have
m̃(P )f (x) =
∞ X
∞
X
T2k ,l (P )f (x).
(9)
k=0 l=−∞
Note that, because of the support properties of ρ(t), Kλ,l (x, y) vanishes if l
is larger than a fixed multiple of logλ. Now we exploit the fact that the finite
propagation speed of the wave equation mentioned before implies that the kernels
of the operators Tλ,l , Kλ,l must satisfy
Kλ,l (x, y) = 0,
if dist(x, y) ≥ C(2l λ−1 ),
since cos(tP ) will have a kernel that vanishes on this set when t belongs to the
support of the integral defining Kλ,l (x, y). Hence in each of the second sum of (9),
there are uniform constants c, C > 0 such that
cλdist(x, y) ≤ 2l ≤ Cλ
(10)
must be satisfied for each λ = 2k , we will use this key observation later.
Now for Tλ,l (P )s, we have the following estimates:
(a).
||Tλ,l (P )f ||L2 (M ) ≤ C(2l )−s λn/2 ||f ||L1 (M )
(b).
||Tλ,l (P )g||L2 (M ) ≤ C(2l )−s0 λn/2 [λ max dist(y, y0 )]||g||L1 (Ω)
where Ω = support(g),
y,y0 ∈Ω
R
Ω
g(y)dy = 0 and n/2 < s0 < min{s, n/2 + 1}.
We first show estimate (a). Notice that β(2−l λ|t|)ρ(t) = 0 when |t| ≤ 2l−1 λ−1 ,
we can use the same idea to prove estimate (a) as we prove the estimate on the
15
remainder r(P ) in Part 1, where 1 − ρ(t) = 0, for |t| ≤ 2 . Using orthogonality of
χk for k ∈ N, and the L∞ estimates on χk in [5], we have
||Tλ,l (P )f ||2L2 ≤
∞
X
||Tλ,l (P )χk f ||2L2
k=1
∞
X
≤ C
sup |Tλ,l (τ )|2 (1 + k)n−1 ||f ||2L1
k=1 τ ∈[k,k+1]
Hence we need only to show
∞
X
sup |Tλ,l (τ )|2 (1 + k)n−1 ≤ C(2l )−2s λn
k=1 τ ∈[k,k+1]
Notice since mλ (τ ) = 0, for τ ∈
/ [λ/2, 2λ], we have
Tλ,l (τ ) = O((1 + |τ | + |λ|)−N )
for any N when τ ∈
/ [λ/4, 4λ]. Then we have
X
sup |Tλ,l (τ )|2 (1 + k)n−1 ≤ C
τ ∈[k,k+1]
k∈[λ/4,4λ]
/
X
(1 + k + λ)−2N (1 + k)n−1
k∈[λ/4,4λ]
/
≤ C
Z
x>1,x∈[λ/4,4λ]
/
xn−1
dx
(x + λ)2N
≤ C(1 + λ)n−2N
Since 2l ≤ Cλ from our observation (10) above, we need only to show
4λ
X
sup |Tλ,l (τ )|2 (1 + k)n−1 ≤ C(2l )−2s λn
k=λ/4 τ ∈[k,k+1]
that is
4λ
X
sup |Tλ,l (τ )|2 ≤ C(2l )−2s λ
k=λ/4 τ ∈[k,k+1]
16
Using the same argument as in Part 1, Notice that β(2−l λ|t|)ρ(t) = 0 when
|t| ≤ 2l−1 λ−1 , we have the estimate (a),
||Tλ,l (P )f ||L2 (M ) ≤ C(2l )−s λn/2 ||f ||L1 (M )
Next we prove the estimate (b). We will use the orthogonality of {ej }j∈N ,
Z
M
eλk (x)eλj (x)dx = δkj ,
and the gradient estimates on χk for all k ∈ N as in Theorem 1.1. Given function
g ∈ L1 (M ) with Ω = support(g) and
R
M
g(y)dy = 0, fix a point y0 ∈ Ω, we have
||Tλ,l (P )g||2L2
=
=
Z
ZM
|
|
Z
ZΩ
M
Ω
Kλ,l (x, y)g(y)dy|2 dx
[Kλ,l (x, y) − Kλ,l (x, y0 )]g(y)dy|2 dx
(here use the cancellation of g)
=
Z
|
Z X
M
=
Ω k≥1
XZ
k≥1 M
|
Tλ,l (λk )eλk (x)[eλk (y) − eλk (y0 )]g(y)dy|2 dx
Z
Ω
{Tλ,l (λj )eλj (x)[eλj (y) − eλj (y0 )]}g(y)dy|2 dx
X
λj ∈[k,k+1)
(here use the orthogonality)
≤
XZ
y∈Ω
k≥1 M
= ||g||2L1
X
max |
XZ
k≥1 M
{Tλ,l (λj )eλj (x)[eλj (y) − eλj (y0 )]}|2 dx|
Z
λj ∈[k,k+1)
X
|
Ω
{Tλ,l (λj )eλj (x)[eλj (y1 ) − eλj (y0 )]}|2 dx
λj ∈[k,k+1)
(where the maximum achieves at y1 )
=
||g||2L1
= ||g||2L1
XZ
k≥1 M
XZ
k≥1 M
Tλ,l (λj )eλj (x)eλj (ȳ), y1 − y0 )|2 dx
X
|(∇y
λj ∈[k,k+1)
|{
X
Tλ,l (λj )eλj (x)(∇eλj (ȳ), y1 − y0 )}|2 dx
λj ∈[k,k+1)
17
|g(y)|dy|2
= ||g||2L1
X
|Tλ,l (λj )(∇eλj (ȳ), y1 − y0 )|2
X
k≥1 λj ∈[k,k+1)
≤
||g||2L1
X
k≥1
max |Tλ,l (τ )|2 {
τ ∈[k,k+1)
|∇eλj (ȳ)|2 dist(y1 , y0 )2 }
X
λj ∈[k,k+1)
(here use the orthogonality)
≤ ||g||2L1 [ max dist(y, y0 )]2
y,y0 ∈Ω
≤
max |Tλ,l (τ )|2 {
X
k≥1
C||g||2L1 [ max dist(y, y0 )]2
y,y0 ∈Ω
τ ∈[k,k+1)
|∇eλj (ȳ)|2 }
λj ∈[k,k+1)
2
max |Tλ,l (τ )| (1 + k)n+1
X
k≥1
X
τ ∈[k,k+1)
Now using the same computation as to the estimate (a), for some constant s0
satisfying n/2 < s0 < min{s, n/2 + 1}, we have
X
k≥1
max |Tλ,l (τ )|2 (1 + k)n+1 ≤ C(2l )−2s0 λn+2 .
τ ∈[k,k+1)
Combine above two estimates, we proved the estimate (b),
||Tλ,l (P )g||L2 (M ) ≤ C(2l )−s0 λn/2 [λ max dist(y, y0 )]||g||L1 (Ω)
y,y0 ∈Ω
Now we use the estimates (a) and (b) to show
Z
m̃(P )f (x) =
K(x, y)f (y)dy
M
is weak-type (1,1). We let f (x) = g(x) +
P∞
k=1 bk (x)
:= g(x) + b(x) be the Calderón-
Zygmund decomposition of f ∈ L1 (M ) at the level α using the same idea as Lemma
0.2.7 in [6]. Let Qk ⊃ supp(bk ) be the cube associated to bk on M, and we have
||g||L1 +
∞
X
k=1
n
||bk ||L1 ≤ 3||f ||L1
|g(x)| ≤ 2 α
almost everywhere,
and for certain non-overlapping cubes Qk ,
bk (x) = 0 f or x ∈
/ Qk
∞
X
µ|Qk | ≤ α−1 ||f ||L1 .
k=1
18
and
Z
M
bk (x)dx = 0
Now we show the weak-type (1, 1) estimate for m̃(P ). Since
{x : |m̃(P )f (x)| > α} ⊂ {x : |m̃(P )g(x)| > α/2} ∪ {x : |m̃(P )b(x)| > α/2}
Notice
Z
|g|2 dx ≤ 2n α
Z
M
|g|dx.
M
Hence we use the L2 boundedness of m̃(P ) and Tchebyshev’s inequality to get
µ{x : |m̃(P )g(x)| > α/2} ≤ Cα−2 ||g||2L2 ≤ C 0 α−1 ||f ||L1 .
Let Q∗k be the cube with the same center as Qk but twice the side-length. After
possibly making a translation, we may assume that Qk = {x : max |xj | ≤ R}. Let
O∗ = ∪Q∗k , we have µ|O∗ | ≤ 2n α−1 ||f ||L1 , and
∗
µ{x ∈
/ O : |m̃(P )b(x)| > α/2} ≤ 2α
−1
≤ 2α−1
Z
x∈O
/ ∗
∞
XZ
|m̃(P )b(x)|dx
/ ∗k
k=1 x∈Q
|m̃(P )bk (x)|dx
Hence we need only to show
Z
x∈Q
/ ∗k
|m̃(P )bk (x)|dx =
Z
x∈Q
/ ∗k
≤ C
Z
M
|
Z
Qk
K(x, y)bk (y)dy|dx
|bk |dx.
From the double dyadic decomposition (9), we show two estimates of Tλ,l (P )bk (x)
on set {x ∈ M : x ∈
/ O∗ },
l n/2−s
(I) ||Tλ,l (P )bk ||L1 (x∈O
||bk ||L1 (Qk )
/ ∗ ) ≤ C(2 )
l n/2−s0
[λ max dist(y, y0 )]||bk ||L1 (Qk )
(II) ||Tλ,l (P )bk ||L1 (x∈O
/ ∗ ) ≤ C(2 )
y,y0 ∈Qk
19
From our observation (10), as did in [7], it suffices to show that for all geodesic balls
BRλ,l of radius Rλ,l = 2l λ−1 , one has the bounds
(I)0
(II)0
l n/2−s
||Tλ,l (P )bk ||L1 ({x∈O
||bk ||L1 (Qk )
/ ∗ }∩BRλ,l ) ≤ C(2 )
l n/2−s0
||Tλ,l (P )bk ||L1 ({x∈O
[λ max dist(y, y0 )]||bk ||L1 (Qk )
/ ∗ }∩BRλ,l ) ≤ C(2 )
y,y0 ∈Qk
To show (I)0 , using the estimate (a), and Hölder inequality, we get
1/2
||Tλ,l (P )bk ||L1 ({x∈O
||Tλ,l (P )bk ||L2
/ ∗ }∩BRλ,l ) ≤ V ol(BRλ,l )
≤ C(2l λ−1 )n/2 (2l )−s λn/2 ||bk ||L1
= C(2l )n/2−s ||bk ||L1
To show (II)0 , using the cancellation property
R
Qk
bk (y)dy = 0, the estimate (b),
and Hölder inequality, we have
1/2
||Tλ,l (P )bk ||L1 ({x∈O
||Tλ,l (P )bk ||L2
/ ∗ }∩BRλ,l ) ≤ V ol(BRλ,l )
2l
≤ C( )n/2 (2l )−s0 λn/2 [λ max dist(y, y0 )]||bk ||L1 (Qk )
y,y0 ∈Qk
λ
l n/2−s0
= C(2 )
[λ max dist(y, y0 )]||bk ||L1 (Qk )
y,y0 ∈Qk
From our observation (10), and estimates (I), we have
∞
X
X
||Tλ,l (P )bk ||L1 (x∈O
/ ∗) ≤ C
(2l )n/2−s ||bk ||L1 (Qk )
2l ≥cλdist(x,y)
l=−∞
≤ Cs (λdist(x, y))n/2−s ||bk ||L1 (Qk )
≤ Cs (λR)n/2−s ||bk ||L1 (Qk ) ,
and from maxy,y0 ∈Qk dist(y, y0 ) ≤ CR, estimate (II), and n/2 < s0 < min{s, n/2 +
1}, we have
∞
X
l=−∞
||Tλ,l (P )bk ||L1 (x∈O
/ ∗) ≤ C
(2l )n/2−s0 [λ max dist(y, y0 )]||bk ||L1 (Qk )
X
2l ≥cλdist(x,y)
20
y,y0 ∈Qk
≤ Cs0 (λdist(x, y))n/2−s0 [λ max dist(y, y0 )]||bk ||L1 (Qk )
y,y0 ∈Qk
n/2+1−s0
≤ Cs0 (λR)
||bk ||L1 (Qk ) .
Therefore, we combine the above two estimate we conclude that
Z
x∈Q
/ ∗k
|m̃(P )bk (x)|dx ≤
∞ X
∞
X
||T2j ,l (P )bk ||L1 (x∈O
/ ∗)
j=0 l=−∞
≤ Cs
X
(λR)n/2−s ||bk ||L1 + Cs0
2j R>1
X
(λR)n/2+1−s0 ||bk ||L1
2j R≤1
≤ Cs ||bk ||L1
Hence we have the weak-type (1, 1) estimate on the main term
µ{x : |m̃(P )f (x)| > α} ≤ α−1 ||f ||L1 .
Combine Case 1 and Case 2, we have the weak-type estimate of m(P ) and we
finish the proof of Theorem 1.2.
Q.E.D.
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