PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 135, Number 5, May 2007, Pages 1585–1595
S 0002-9939(07)08687-X
Article electronically published on January 9, 2007
NEW PROOF OF THE HÖRMANDER MULTIPLIER THEOREM
ON COMPACT MANIFOLDS WITHOUT BOUNDARY
XIANGJIN XU
(Communicated by Andreas Seeger)
Abstract. On compact manifolds (M, g) without boundary, the gradient estimates for unit band spectral projection operators χλ are proved for a second
order elliptic differential operator L. A new proof of the Hörmander Multiplier
Theorem (first proved by A. Seeger and C.D. Sogge in 1989) is given in this
setting by using the gradient estimates and the Calderón-Zygmund argument.
1. Introduction
Let (M, g) be a compact boundaryless manifold of dimension n ≥ 2, and let L be
a second order elliptic differential operator which is positive and self-adjoint with
respect to the C ∞ density dx. We shall consider the eigenvalue problem
(L + λ2 )u(x) = 0,
x ∈ M.
Recall that the spectrum of L: 0 ≤
≤ λ22 ≤ λ23 ≤ · · · → ∞, and let the
2
corresponding eigenfunctions {ej (x)} ⊂ L (M ) be an associated real orthogonal
basis. Define the unit band spectral projection operators,
χλ f =
ej (f ), where ej (f )(x) = ej (x)
f (y)ej (y)dy.
λ21
M
λj ∈[λ,λ+1)
In [7] and [8], Sogge proved the following Lp estimates on χλ for p ≥ 2, λ ≥ 1:
n − 1 n n − 1 1 1 − ,
( − ) .
(1) ||χλ f ||p ≤ Cλσ(p) ||f ||2 , with σ(p) = max
2
p
2
2 p
The special case of (1) where p = ∞ was proved in Hörmander [4]. The first result
of this paper is the following gradient estimate for χλ f .
Theorem 1.1. There is a uniform constant C such that
||∇χλ f ||∞ ≤ Cλ(n+1)/2 ||f ||2 ,
∀ λ ≥ 1.
∞
Given a function m(λ) ∈ L (R), define the multiplier operator, m(P ), by
∞
(2)
m(P )f =
m(λj )ej (f ),
j=1
Received by the editors September 15, 2005 and, in revised form, February 28, 2006.
2000 Mathematics Subject Classification. Primary 58J40, 35P20, 35J25.
Key words and phrases. Gradient estimate, eigenfunction, unit band spectral projection operator, Hörmander Multiplier Theorem.
c
2007
American Mathematical Society
Reverts to public domain 28 years from publication
1585
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√
where P = −L. Such an operator is always bounded on L2 (M ). However, for
any other Lp (M ), it is known that some smoothness assumptions on the function
m(λ) are needed to ensure that m(P ) is bounded on Lp (M ) (see [9]).
(R), let L2s (R) denote
Under the following assumption: Suppose that m ∈ L∞
∞
j
the usual Sobolev space and fix β ∈ C0 ((1/2, 2)) satisfying ∞
−∞ β(2 t) = 1, t > 0,
for s > n/2. Then there is is
sup ||β(·)m(λ·)||2L2s < ∞.
(3)
λ>0
By studying the parametrix of the wave kernel of m(P ), Seeger and Sogge [6] and
Sogge [8] proved the following Hömander Multiplier Theorem for L on (M, g).
Theorem 1.2. Let m ∈ L∞ (R) satisfy (3). Then there are constants Cp such that
(4)
||m(P )f ||Lp (M ) ≤ Cp ||f ||Lp (M ) ,
f or
1 < p < ∞.
The second part of this paper gives a new proof for the above theorem based
on the gradient estimates on χλ , without using the parametrix of the wave kernel
of m(P ) as was done in [6] and [8]. As is known, the parametrix construction of
the wave kernel does not work well for compact manifolds with boundary unless
one assumes that the boundary is geodesic concave (see [3]). Here we use a new
approach to prove Theorem 1.2, which also works for the Laplacian operator on
compact manifolds with boundary (see [11] and [12]). The main idea is that we
make decompositions twice
to get {Kλ,l (x, y)} for the integral kernel K(x, y) of
m̃(P ), such that K(x, y) = j l K2j ,l (x, y). Also, for each operator Tλ,l (P ) with
integral kernel Kλ,l , there are L1 → L2 estimates by a scaling argument using
the L∞ estimates and gradient estimates on χλ . With the support properties and
the finite propagation speed properties, one has the relation (8) 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 [1], Duong-Ouhabaz-Sikora
gave another proof of the above Hörmander Multiplier Theorem using heat kernel
methods and the L∞ bounds of χλ .
In what follows we shall use the convention that C will denote a constant that
is not necessarily the same at each occurrence.
2. Gradient estimates: Theorem 1.1
We will apply the maximum principle to L in the cube centered at x0 ∈ M with
length d = (λ + 1)−1 , following the idea of interior gradient estimates for Poisson’s
equation in [2]. Define the geodesic coordinates x = (x1 , · · · , xn ) centered at point
basis {vi }ni=1 ⊂ Tx0 M , identify x = (x1 , · · · , xn ) ∈
x0 as follows: fix an orthogonal
n
n
R with the point exp( i=1 xi vi ) ∈ M . In this coordinate, the second order elliptic
operator L can be written as
L=
n
i,j=1
∂2
∂
+
bi (x)
+ c(x),
∂xi ∂xj
∂x
i
i=1
n
aij (x)
where aij (x), bi (x), c(x) ∈ C ∞ (M ), and c(x) ≤ 0, which comes from the fact that
L is an elliptic operator. Without loss of generality, we may assume aij (x0 ) =
aij (0) = δij , i.e., the principle term of L at point x0 is the Laplacian.
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HÖRMANDER MULTIPLIER THEOREM
1587
Denote u(x; f ) = χλ f (x); we have u ∈ C ∞ (M ), and
Lu(x; f ) = −
λ2j ej (f ) := h(x; f ).
λj ∈[λ,λ+1)
∞
From the L
estimate (1) on χλ and Cauchy-Schwarz inequality, we have
|h(x; f )|2 ≤ C(λ + 1)n+3 ||χλ f ||2L2 (M ) .
Proof of Theorem 1.1. 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. Define
Q = {x = (x1 , · · · , xn ) ∈ Rn | |xi | < d, i = 1, · · · , n} ⊂ M,
Q = {x = (x1 , · · · , xn ) ∈ Rn | |xi | < d, i = 1, · · · , n − 1, 0 < xn < d.} ⊂ M.
Consider the function
1
u(x , xn ; f ) − u(x , −xn ; f ) ,
2
where we write x = (x , xn ) = (x1 , · · · , xn−1 , xn ). We have that ϕ(x , 0; f ) = 0,
sup∂Q |ϕ| ≤ sup∂Q |u| := A ≤ C(λ + 1)(n−1)/2 , and
|Lϕ| ≤ sup |h| + sup (L(x ,xn ) − L(x ,−xn ) )u(x , −xn ; f ) := N, in Q .
ϕ(x , xn ; f ) =
Q
Q
Here we will estimate the bound of N . From Lemma 17.5.2 and Theorem 17.5.3 in
[5], we have the following crude estimates on the derivatives of u(x) = χλ f (x):
n
sup |Dα u| ≤ Cα λ 2 +α ,
∀α ∈ N ∪ {0}.
M
Applying the above estimates, we have
n
sup (L(x ,xn ) − L(x ,−xn ) )u ≤ Cd sup |D2 u| + sup |Du| + sup |u| ≤ Cdλ 2 +2 ,
Q
Q
Q
Q
n
and we have supQ |h| ≤ C(λ + 1)(n+3)/2 . Hence N ≤ C (λ + 1)(n+3)/2 + dλ 2 +2 .
Now consider the function
A
ψ(x , xn ) = 2 |x |2 + αxn (nd − (n − 1)xn ) + βN xn (d − xn ), ∀ (x , xn ) ∈ Q ,
d
where α ≥ 1 and β ≥ 1 will be determined below. One may check ψ(x , xn ) ≥ 0 on
xn = 0 and ψ(x , xn ) ≥ A in the remaining portion of ∂Q .
n
A
2tr(a
(x))
−
(2nα
−
2α
+
1)a
(x)
+
2
bi (x)xi + bn (x) nαd
Lψ(x) =
ij
nn
2
d
i=1
−(2nα − 2α + 1)xn + N β − 2ann (x) + bn (x)(d − 2xn ) + c(x)ψ(x).
Since M is compact, 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 an α, since d = (λ + 1)−1 , for large λ, we have
n
bi (x)xi + bn (x) nαd − (2nα − 2α + 1)xn < 1.
2
i=1
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Then the first term of Lψ(x) is negative. For the second term of Lψ(x), letting β
be large enough, we have
β − 2ann (x) + bn (x)(d − 2xn ) < −1.
For the third term, since c(x) ≤ 0 and ψ(x) ≥ 0 in Q , we have Lψ(x) ≤ −N in Q .
Now we have L(ψ ± ϕ) ≤ 0 in Q and ψ ± ϕ ≥ 0 on ∂Q , from which it follows
by the maximum principle that |ϕ(x , xn ; f )| ≤ |ψ(x , xn )| in Q . Letting x = 0 in
the expressions for ψ and ϕ, then dividing by xn and letting xn tend to zero, we
obtain
ϕ(0, x ; f ) αnA
n
+ βdN.
|Dn u(0; f )| = lim ≤
xn →0
xn
d
The same estimate holds for |Di u(0)|, i = 1, · · · , n − 1, hence we have
|∇u(0; f )| ≤
αn2 A
+ βndN.
d
By the bounds of d, A and N , we have the estimate
|∇u(0)| ≤ C(λ + 1)(n+1)/2 .
Since the estimate is for any x0 ∈ M , Theorem 1.1 is proved.
Remark 2.1. Our method can apply to gradient estimates of eigenfunctions on
compact manifolds with boundary as well; see details in [11] and [12]. After learning
of my gradient estimates on spectral functions of Laplacian on compact manifolds,
B. Xu in [10] studied the estimates of derivatives of the spectral function χλ on a
closed manifold by studying the Hadamard parametrix of the wave operator, which
gave another proof of the result of Theorem 1.1, but the method in [10] cannot
apply to compact manifolds with boundary.
3. Hörmander Multiplier Theorem: Theorem 1.2
Since the complex conjugate of m satisfies the same hypotheses as (3), we need
only to prove Theorem 1.2 for the exponents 1 < p ≤ 2. This will allow us to exploit
orthogonality, and also reduce Theorem 1.2 to show that m(P ) is weak-type (1, 1)
by the Marcinkiewicz Interpolation Theorem, i.e.,
µ{x : |m(P )f (x)| > α} ≤ α−1 ||f ||L1 ,
(5)
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
1
1
itP
m(P )f (x) =
m(t)e
f (x)dt =
m(t)
cos(tP )f (x)dt,
2π R
π R+
√
where P = −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 (8).
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HÖRMANDER MULTIPLIER THEOREM
1589
Proof of Theorem 1.2. The proof of the weak-type (1,1) estimate (5) will involve
a splitting of m(P ) into two pieces: a main piece which one need carefully study,
plus a remainder which has a strong (1,1) estimate by using the L∞ estimates for
χλ as was done in [8]. Specifically, define ρ ∈ C0∞ (R) as
(6)
ρ(t) = 1, for |t| ≤ ,
ρ(t) = 0, for |t| ≥ > 0,
2
where is a given
related to the manifold.
) + r(P ),
constant
Write m(P ) = m(P
where m(P
) = m ∗ ρ̌ (P ) and r(P ) = m ∗ (1 − ρ)ˇ (P ). To estimate
the
main term
and remainder, for λ = 2j , j = 1, 2, · · · , define mλ (τ ) = β λτ m(τ ).
Part 1. Strong (1, 1) estimate on the remainder: ||r(P )f ||L1 ≤ C||f ||L1 .
Here we follow the first part in proof of Theorem 5.3.1 in [8]. Define
1
rλ (P ) = mλ ∗ (1 − ρ)ˇ (P ) =
λ (t)dt.
eitP (1 − ρ(t))m
2π
Note that r0 (P ) = r(P ) − j≥1 r2j (P ) is a bounded and rapidly decreasing function. Hence r0 (P ) is bounded from L1 to any Lp space. We need only show
||rλ (P )f ||L2 ≤ Cλn/2−s ||f ||L1 ,
λ = 2j , j = 1, 2, · · · .
Using the L∞ estimate (1), we have
||rλ (P )f ||2L2 ≤
∞
||rλ (P )χk f ||2L2 ≤ C
k=1
Hence we need only show
∞
sup
k=1 τ ∈[k,k+1]
∞
sup
k=1 τ ∈[k,k+1]
|rλ (τ )|2 (1 + k)n−1 ||f ||2L1 .
|rλ (τ )|2 (1 + k)n−1 ≤ Cλn−2s .
/ [λ/2, 2λ], both m
λ (τ ) and rλ (τ ) are
Note that since mλ (τ ) = 0 for τ ∈
/ [λ/4, 4λ]. Hence we need only show
O((1 + |τ | + |λ|)−N ) for any N when τ ∈
4λ
k=λ/4
sup
τ ∈[k,k+1]
|rλ (τ )|2 ≤ Cλ1−2s .
By the fundamental theorem of calculus and the Cauchy-Schwartz inequality,
4λ
2
2
sup |rλ (τ )| ≤ C( |rλ (τ )| dτ +
|rλ (τ )|2 dτ )
k=λ/4
τ ∈[k,k+1]
R
=
R
|m
λ (t)(1 − ρ(t))|2 dt +
C(
R
|tm
λ (t)(1 − ρ(t))|2 dt).
R
Since ρ(t) = 1, for |t| ≤ 2 , by a change variables, this is dominated by
|ts m
λ (t/λ)|2 dt = λ1−2s ||β(·)m(λ·)||2L2s ≤ Cλ1−2s ,
λ−1−2s
R
and the second inequality comes from our assumption (3). Hence we have the
estimate
||r(P )f ||L2 ≤ C||f ||L1 .
Also, since our manifold is compact, we have
||r(P )f ||L1 ≤ V ol(M )1/2 ||r(P )f ||L2 ≤ C||f ||L1 .
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Part 2. Weak-type (1, 1) estimate on the main term:
µ{x : |m(P
)f (x)| > α} ≤ α−1 ||f ||L1 .
The weak-type (1, 1) estimate on m(P
) would follow from the integral operator
1
itP
m(t)ρ(t)e
f (x)dt
m(P
)f (x) =
2π R
1
itλk
=
m(t)ρ(t)
e
eλk (x)
eλk (y)f (y)dydt
2π R
M
k≥1
1
=
m(t)ρ(t)
eitλk eλk (x)eλk (y)dt f (y)dy
2π M
R
k≥1
with the kernel
K(x, y) =
m(t)ρ(t)
eitλk eλk (x)eλk (y)dt =
(m ∗ ρ̌)(λk )eλk (x)eλk (y)
R
k≥1
k≥1
of weak-type (1,1). Now define the dyadic decomposition
Kλ (x, y) =
m
λ (t)ρ(t)
eitλk eλk (x)eλk (y)dt.
R
k≥1
∞
We have K(x, y) = j=1 K2j (x, y) + K0 (x, y), 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:
m
λ (t)β 2−l λ|t| ρ(t)
eitλk eλk (x)eλk (y)dt.
Kλ,l (x, y) =
R
k≥1
∞
We have Kλ (x, y) = l=−∞ Kλ,l (x, y). Define Tλ,l (P )f (x) =
From above two dyadic decompositions, we have
∞
∞ T2k ,l (P )f (x).
(7)
m(P
)f (x) =
M
Kλ,l (x, y)f (y)dy.
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 log λ. 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 sums of (7),
there are uniform constants c, C > 0 such that
cλdist(x, y) ≤ 2l ≤ Cλ
(8)
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:
||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) ,
y,y0 ∈Ω
where Ω = supp(g), Ω g(y)dy = 0 and n/2 < s0 < min{s, n/2 + 1}.
(a)
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HÖRMANDER MULTIPLIER THEOREM
1591
We first show estimate (a). Note 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 did to prove the estimate on
the remainder r(P ) in Part 1, where 1 − ρ(t) = 0, for |t| ≤ 2 . Using orthogonality
of χk for k ∈ N, and the estimates (1) on χk for p = ∞, we have
∞
||Tλ,l (P )f ||22 ≤
||Tλ,l (P )χk f ||22 ≤ C
k=1
∞
sup
k=1 τ ∈[k,k+1]
|Tλ,l (τ )|2 (1 + k)n−1 ||f ||21 .
Hence we need only show
∞
sup
k=1 τ ∈[k,k+1]
|Tλ,l (τ )|2 (1 + k)n−1 ≤ C(2l )−2s λn .
Note 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
sup |Tλ,l (τ )|2 (1 + k)n−1 ≤ C
(1 + k + λ)−2N (1 + k)n−1
k∈[λ/4,4λ]
/
τ ∈[k,k+1]
k∈[λ/4,4λ]
/
≤ C(1 + λ)n−2N .
Since 2l ≤ Cλ from our observation (8) above, we need only show
4λ
sup
k=λ/4
τ ∈[k,k+1]
4λ
i.e.
sup
k=λ/4
τ ∈[k,k+1]
|Tλ,l (τ )|2 (1 + k)n−1 ≤ C(2l )−2s λn ,
|Tλ,l (τ )|2 ≤ C(2l )−2s λ.
Using the same argument as in Part 1, note that β(2−l λ|t|)ρ(t) = 0 when |t| ≤
2 λ ; we have the estimate (a).
1
Next
we prove the estimate (b). Given function g ∈ L (M ) with Ω = supp(g)
and M g(y)dy = 0, fix a point y0 ∈ Ω. Using the cancellation of g, we have
l−1 −1
||Tλ,l (P )g||2L2
2
=
Kλ,l (x, y)g(y)dy dx
M Ω
2
=
[Kλ,l (x, y) − Kλ,l (x, y0 )]g(y)dy dx
M
Ω
2
=
Tλ,l (λk )eλk (x) eλk (y) − eλk (y0 ) g(y)dy dx
M
=
Ω k≥1
k≥1
M
2
Tλ,l (λj )eλj (x) eλj (y) − eλj (y0 ) g(y)dy dx,
Ω λ ∈[k,k+1)
j
where we use the orthogonality of {eλj (x)}, which is dominated by
2
2 Tλ,l (λj )eλj (x) eλj (y) − eλj (y0 ) dx
max |g(y)|dy k≥1
=
M y∈Ω
||g||2L1
k≥1
M
λj ∈[k,k+1)
Ω
2
Tλ,l (λj )eλj (x) eλj (y1 ) − eλj (y0 ) dx,
λj ∈[k,k+1)
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1592
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where the maximum achieves at some point y1 ∈ Ω,
2
= ||g||2L1
Tλ,l (λj )eλj (x)eλj (ȳ), y1 − y0 ) dx
(∇y
k≥1
M
= ||g||2L1
k≥1
= ||g||2L1
λj ∈[k,k+1)
2
Tλ,l (λj )eλj (x)(∇eλj (ȳ), y1 − y0 ) dx
M
λj ∈[k,k+1)
2
Tλ,l (λj )(∇eλj (ȳ), y1 − y0 )
k≥1 λj ∈[k,k+1)
≤ ||g||2L1
k≥1
≤ ||g||2L1
≤
max
τ ∈[k,k+1)
|Tλ,l (τ )|2
max dist(y, y0 )
2 y,y0 ∈Ω
C||g||2L1
k≥1
max dist(y, y0 )
λj ∈[k,k+1)
max
τ ∈[k,k+1)
2 y,y0 ∈Ω
|∇eλj (ȳ)|2 dist(y1 , y0 )2
k≥1
|Tλ,l (τ )|2
max
τ ∈[k,k+1)
|∇eλj (ȳ)|2
λj ∈[k,k+1)
|Tλ,l (τ )| (1 + k)n+1 .
2
Now using the same computation as in the estimate (a), for some constant s0
satisfying n/2 < s0 < min{s, n/2 + 1}, we have
max |Tλ,l (τ )|2 (1 + k)n+1 ≤ C(2l )−2s0 λn+2 .
k≥1
τ ∈[k,k+1)
Combining the above two estimates, we prove 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
Next we use the estimates (a) and (b) to show that
m(P
)f (x) =
K(x, y)f (y)dy
M
is weak-type (1,1). We let f (x) = g(x)+ ∞
k=1 bk (x) := g(x)+b(x) be the CalderónZygmund decomposition of f ∈ L1 (M ) at the level α. Let Qk (⊃ supp(bk )) be the
cube associated to bk on M, and we have
||g||L1 +
∞
||bk ||L1 ≤ 3||f ||L1 ;
|g(x)| ≤ 2n α
almost everywhere,
k=1
and for certain non-overlapping cubes Qk ,
bk (x) = 0 for x ∈
/ Qk and
bk (x)dx = 0;
M
µ|Qk | ≤ α−1 ||f ||L1 .
k=1
|g|dx and
α α
)b(x)| > } ,
{x : |m(P
)f (x)| > α} ⊂ {x : |m(P
)g(x)| > } ∪ {x : |m(P
2
2
Since
|g| dx ≤ 2 α
∞
2
M
n
M
by L2 boundedness of m(P
) and Tchebyshev’s inequality, we get
µ{x : |m(P
)g(x)| > α/2} ≤ Cα−2 ||g||2L2 ≤ C α−1 ||f ||L1 .
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HÖRMANDER MULTIPLIER THEOREM
1593
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∗ : |m(P
)b(x)| > α/2} ≤ 2α−1
∞ k=1
Hence we need only show
|m(P
)bk (x)|dx =
x∈Q
/ ∗
k
x∈Q
/ ∗
k
x∈Q
/ ∗
k
|m(P
)bk (x)|dx.
K(x, y)bk (y)dy dx ≤ C
Qk
|bk |dx.
M
From the double dyadic decomposition (7), we show two estimates of Tλ,l (P )bk (x)
on the set {x ∈ M : x ∈
/ O∗ }:
(I) ||Tλ,l (P )bk ||L1 (x∈O
/ ∗)
(II) ||Tλ,l (P )bk ||L1 (x∈O
/ ∗)
≤ C(2l )n/2−s ||bk ||L1 (Qk ) ,
≤ C(2l )n/2−s0 λ max dist(y, y0 ) ||bk ||L1 (Qk ) ,
y,y0 ∈Qk
From our observation (8), it suffices to show that for all geodesic balls BRλ,l of
radius Rλ,l = 2l λ−1 , one has the bounds
(I) ||Tλ,l (P )bk || 1 L
(II) ||Tλ,l (P )bk || 1 L
{x∈O
/ ∗ }∩BRλ,l
{x∈O
/ ∗ }∩BRλ,l
≤ C(2l ) 2 −s ||bk ||L1 (Qk ) ,
n
≤ C(2l ) 2 −s0 λ max dist(y, y0 ) ||bk ||L1 (Qk ) ,
n
y,y0 ∈Qk
To show (I) , using the estimate (a) and Hölder inequality, we get
||Tλ,l (P )bk || 1 L
{x∈O
/ ∗ }∩BRλ,l
≤ V ol(BRλ,l )1/2 ||Tλ,l (P )bk ||L2
≤ C(2l λ−1 )n/2 (2l )−s λn/2 ||bk ||L1
= C(2l )n/2−s ||bk ||L1 .
To show (II) , using the cancellation property Qk bk (y)dy = 0, the estimate (b),
and Hölder inequality, we have
||Tλ,l (P )bk || 1 L
{x∈O
/ ∗ }∩BRλ,l
≤ V ol(BRλ,l )1/2 ||Tλ,l (P )bk ||L2
2l n2 l −s0 n ≤ C
(2 ) λ 2 λ max dist(y, y0 ) ||bk ||L1 (Qk )
y,y0 ∈Qk
λ
l n
−s0
2
λ max dist(y, y0 ) ||bk ||L1 (Qk ) .
= C(2 )
y,y0 ∈Qk
From our observation (8), and estimates (I), we have
∞
l=−∞
||Tλ,l (P )bk ||L1 (x∈O
/ ∗)
≤ C
(2l )n/2−s ||bk ||L1 (Qk )
2l ≥cλdist(x,y)
n/2−s
≤ Cs λdist(x, y)
||bk ||L1 (Qk )
≤ Cs (λR)n/2−s ||bk ||L1 (Qk ) ,
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1594
X. XU
and from maxy,y0 ∈Qk dist(y, y0 ) ≤ CR, estimate (II), and
∞
l=−∞
≤ C
n
2
< s0 < min{s, n2 + 1},
||Tλ,l (P )bk ||L1 (x∈O
/ ∗)
n
(2l ) 2 −s0 λ max dist(y, y0 ) ||bk ||L1 (Qk )
y,y0 ∈Qk
2l ≥cλdist(x,y)
n −s ≤ Cs0 λdist(x, y) 2 0 λ max dist(y, y0 ) ||bk ||L1 (Qk )
y,y0 ∈Qk
≤ Cs0 (λR)
n
2 +1−s0
||bk ||L1 (Qk ) .
Therefore, we combine the above two estimates to conclude that
x∈Q
/ ∗
k
|m(P
)bk (x)|dx ≤
∞ ∞
||T2j ,l (P )bk ||L1 (x∈O
/ ∗)
j=0 l=−∞
≤
Cs
(2j R) 2 −s ||bk ||1 + Cs0
n
2j R>1
≤
n
(2j R) 2 +1−s0 ||bk ||1
2j R≤1
Cs ||bk ||1 .
Hence we have the weak-type (1, 1) estimate on the main term
µ{x : |m(P
)f (x)| > α} ≤ α−1 ||f ||L1 .
Combining Case 1 and Case 2, we have the weak-type estimate of m(P ) and we
finish the proof of Theorem 1.2.
Acknowledgments
The results in this paper are part of the author’s Ph.D. thesis [12] during his
stay at Johns Hopkins University. The author would like to thank his advisor,
Professor C.D. Sogge, for bringing the problems to him and for a number of helpful
conversations on his research. Finally, but not least, the author would like to thank
the referee for his/her careful reading and valued suggestions on this paper.
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HÖRMANDER MULTIPLIER THEOREM
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Mathematical Sciences Research Institute, 17 GaussWay, Berkeley, California 94720
E-mail address: xiangjxu@msri.org
Current address: Department of Mathematics, University of Virginia, P.O. Box 400137, Charlottesville, Virginia 22904
E-mail address: xx8n@virginia.edu
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