MIKHLIN-HÖRMANDER THEOREM FOR HIGHER ORDER
DIFFERENTIAL OPERATORS ON RIEMANNIAN MANIFOLDS
ATHANASIOS G. GEORGIADIS
Abstract. Let M be a Riemannian manifold which satisfies the doubling volume property. Let L be a 2m-order differential operator on M and m(λ), λ ∈
R, a multiplier satisfying the Mikhlin-Hörmander condition. We also assume
that the associated heat kernel satisfies a certain upper gaussian estimate and
we prove that the spectral multiplier m(L) is bounded on Lp , p ∈ (1, ∞) and
from L1 to L1 week.
1. Introduction and statement of the results
Let m(ξ) be a bounded measurable function in Rn and let Tm be the operator
b
b
defined by Td
m f (ξ) = m(ξ)f (ξ), where f denotes the Fourier transform of f .
The classical Mikhlin-Hörmander multiplier theorem [17, 20], asserts that if the
multiplier m(λ) satisfies the condition
sup |λ|α |∂ α m(λ)| < ∞,
λ∈Rn
£ ¤
for any multi-index α, with |α| ≤ n2 +1, then Tm is bounded on Lp , p ∈ (1, ∞) and
from L1 to L1 week. In the present paper we generalize this theorem for multipliers
of higher order differential operators on Riemannian manifolds.
1.1. The manifolds of our interest. Let M be a complete and noncompact
Riemannian manifold. We denote by d(., .) the Riemannian distance, by dx the
Riemannian measure, by B(x, r) the ball centered at x ∈ M with radius r > 0 and
by |B(x, r)| its volume. We assume that M satisfies the doubling volume property,
i.e. there is a constant c > 0, such that
(1.1)
|B(x, 2r)| ≤ c|B(x, r)|, for all x ∈ M, r > 0.
From (1.1) it follows that there exist constants c, D > 0, such that
³ r ´D
|B(x, r)|
(1.2)
≤c
, for all x ∈ M, r ≥ t > 0.
|B(x, t)|
t
Note that RD , Lie groups of polynomial volume growth and manifolds with RicM ≥
0 are examples of such a manifold.
1991 Mathematics Subject Classification. 42B15, 42B20, 42B25.
Key words and phrases. Spectral multipliers, Higher order differential operators, Heat kernels,
Riemannian manifolds, Calderon Zygmund decomposition, Hardy-Littlewood maximal function.
The author was partially supported by the Onaseio Foundation (Greece).
1
2
ATHANASIOS G. GEORGIADIS
1.2. The operators. Let us denote by L a formally symmetric and non-negative
higher order differential operator of order 2m, m ≥ 1, i.e.
X
L=
aα (x)∂yα .
|α|≤2m
Let pt (x, y), t > 0, x, y ∈ M , be the associated heat kernel , i.e. the kernel of the
heat semigroup e−tL . We assume that there is a constant c > 0 such that
³
d(x, y)2m/(2m−1) ´
(1.3) |pt (x, y)| ≤ c exp − c
|B(x, t1/2m )|−1 , x, y ∈ M, t > 0.
t1/(2m−1)
One interesting example of such operator is L = (−∆)m , i.e. the m−th power of
the Laplacian on RD , for D < 2m. A large class of Homogenous operators satisfying
our assumptions is in [3]. There is a lot of interest for higher order differential
operators as one can see also in [7, 8, 10, 11, 12, 23] and in their references.
1.3. Statement of the result. We denote by dEλ the spectral measure of L. By
the spectral theorem
Z ∞
L=
λdEλ ,
0
If m : R → R is a bounded Borel function, by the spectral theorem we can define
the spectral multiplier
Z ∞
m(L) =
m(λ)dEλ ,
0
2
which is a bounded operator on L (M ), with ||m(L)||2→2 ≤ ||m||∞ .
Let us denote by C A (R) the Lipschitz space of order A > 0, and let us fix a
function 0 ≤ φ ∈ C ∞ (R), with φ(t) = 1, for t ∈ [1, 2], φ(t) = 0, t ∈ (2−1 , 4)c . In
the present work we prove the following.
Theorem 1. Let M be a Riemannian manifold as above and let m(λ), λ ∈ R, be
a multiplier satisfying for some A > D/2 the condition
(1.4)
sup ||φ(.)m(t.)||C A < ∞
t>0
Then the operator m(L) is bounded on Lp , p ∈ (1, ∞) and from L1 to L1 week.
For generalizations of the Mikhlin-Hörmander multiplier theorem to abstract
contexts, we refer the reader for example
√ to [1, 2, 5, 9, 14, 16, 18, 19, 21].
In our case the wave operator cost L does not have the finite propagation speed
property (cf. [26]).So our proof is based on very good kernel estimates and a
Calderon-Zygmund decomposition. For our techniques see also [4, 14].
Throughout this article the different constants will always be denoted by the
same letter c. When their dependence or independence is significant, it will be
clearly stated.
2. Multipliers with compact support
HIGHER ORDER OPERATORS ON MANIFOLDS
3
2.1. An Approximation Lemma.
Lemma 1. (cf. [22]) For f ∈ C0n (R) and A = n + α with α ∈ (0, 1], n ∈ N, we set
||f ||C A =
n
X
½
||f (m) ||∞ + sup
m=0
¾
|f (n) (x + t) − f (n) (x)|
;
t
>
0,
x
∈
R
.
tα
Then for all λ > 0, there is a continuous and integrable function ψλ and a
constant c > 0, independent of λ and f , such that
||ψbλ ||∞ ≤ c, supp(ψbλ ) ⊂ [−λ, λ], and ||f − f ∗ ψλ ||∞ ≤ c||f ||C A λ−A .
2.2. Auxiliary estimates. We set Aq (y) = B(y, 2(q+1)/2m )\B(y, 2q/2m ), q ∈
Z, y ∈ M.
Lemma 2. There are constants c > 0 and δ ∈ (0, 1), such that for all q, j ∈ Z,
with q ≥ j, all x ∈ Aq (y) and |t| ≤ δ2(q−j)/2(2m−1)
¯ −2j L
¯ ce−c2(q−j)/2(2m−1)
¯ ite
¯
j
p2 (x, y)¯ ≤
,
¯e
|B(y, 2j/2m )|
(2.1)
−2j L
Proof. We have eite
¯ −2j L
¯ ite
p2j
¯e
=
P
n≥0
(it)n ¡ −2j L ¢n
e
n!
=
P
n≥0
(it)n −n2j L
.
n! e
So
¯ X |t|n
j
¯
(x, y) ¯ ≤
|e−n2 L p2j (x, y)|
n!
n≥0
X
=
n≤2(q−j)/2
(2.2)
:=
|t|n
j (x, y)| +
|p
n! (n+1)2
X
n>2(q−j)/2
|t|n
j (x, y)|
|p
n! (n+1)2
S1 + S2 .
Estimation of S1 . Using (1.3) we have for every x ∈ Aq (y) , n ≤ 2(q−j)/2
S1
≤
X
n≤2(q−j)/2
≤
∞
´
X
2q/(2m−1)
|t|n
j/2m −1
|B(y,
2
)|
(q−j)/2
j
1/(2m−1)
n!
(2 · 2
2 )
n=0
¡
¢
c exp − c2(q−j)/(2m−1) 2−(q−j)/2(2m−1) |B(y, 2j/2m )|−1 e|t|
=
c|B(y, 2j/2m )|−1 e−c2
(q−j)/2(2m−1)
≤
c|B(y, 2j/2m )|−1 e−c2
(q−j)/2(2m−1)
≤
(2.3)
³ −cd(x, y)2m/(2m−1) ´
|t|n
c exp
|B(y, ((n + 1)2j )1/2m )|−1
n!
((n + 1)2j )1/(2m−1)
³
c exp − c
Estimation of S2 . Using (1.3) and that
eδ2
(q−j)/2(2m−1)
.
1
n!
¡ ¢n
≤ c ne , we take
4
ATHANASIOS G. GEORGIADIS
S2
c|B(y, 2j/2m )|−1
≤
X
n>2(q−j)/2
c|B(y, 2j/2m )|−1
≤
|t|n
n!
³ eδ2(q−j)/2(2m−1) ´n
X
2(q−j)/2
n>2(q−j)/2
c|B(y, 2j/2m )|−1
≤
X
(eδ)n
n>2(q−j)/2
j/2m
≤
c|B(y, 2
)|
−1
X
e−cn
n>2(q−j)/2
(2.4)
c|B(y, 2j/2m )|−1 e−c2
≤
(q−j)/2(2m−1)
,
choosing δ small enough. Then (2.1) follows from (2.2),(2.3) and (2.4).
¤
Lemma 3. Let A > 0 and f ∈ C0A (0, ∞). Then there is a constant c > 0 such that
for all j ∈ Z and y ∈ M , we have
(i)
j
||f (e−2 L )p2j (., y)||2 ≤ c||f ||C A |B(y, 2j/2m )|−1/2
(ii)
j
||f (e−2 L )p2j (., y)||L1 (B(y,2j/2m )) ≤ c||f ||C A .
(2.5)
Proof. (i)Integrating (1.3) we get that ||p2j (., y)||2 ≤ c|B(y, 2j/2m )|−1/2 . By the
spectral theorem it is
j
||f (e−2 L )p2j (., y)||2 ≤ ||f ||∞ ||p2j ||2 ≤ c||f ||C A |B(y, 2j/2m )|−1/2
(ii)By Cauchy Scwartz and claim (i) we get (2.5).
¤
Lemma 4. Let A > 0 and f ∈ C0A (0, ∞). Then there is a constant c > 0 such that
for all q, j ∈ Z, q ≥ j and y ∈ M , we have
(i)
(2.6)
(ii)
(2.7)
j
||f (e−2 L )p2j (., y)||L2 (Aq (y)) ≤ c||f ||C A 2−A(q−j)/2(2m−1) |B(y, 2j/2m )|−1/2 .
¡
¢
D
j
||f (e−2 L )p2j (., y)||L1 (Aq (y)) ≤ c||f ||C A 2− A− 2 (q−j)/2(2m−1) .
Proof. (i)We consider the function ψj,q , given by Lemma 1 satisfying
(2.8)
||ψbj,q ||∞ < c, and ||f − f ∗ ψj,q ||∞ ≤ c||f ||C A 2−A(q−j)/2(2m−1) .
and supp(ψbj,q ) ⊆ [−δ2(q−j)/2(2m−1) , +δ2(q−j)/2(2m−1) ]. From (2.8) we take
j
||(f − f ∗ ψj,q )(e−2 L )p2j (., y)
(2.9)
||
≤ ||f − f ∗ ψj,q ||∞ ||p2j (., y)||2
L2 (Aq (y))
≤ c||f ||C A 2
−A(q−j)/2(2m−1)
|B(y, 2j/2m )|−1/2 .
HIGHER ORDER OPERATORS ON MANIFOLDS
5
For every x ∈ Aq (y) by (2.1) and (2.8) we get
Z
¯ −2j L
¯
j
¯ ite
¯
d
(f ∗ ψj,q )(e−2 L )p2j (x, y)| ≤ c
|fb(t)||ψ
p2j (x, y)¯dt
j,q (t)|¯e
|t|≤δ2(q−j)/2(2m−1)
≤ c||fb||∞ 2(q−j)/2(2m−1) e−c2
≤ c||f ||C A e−c2
(q−j)/2(2m−1)
(q−j)/2(2m−1)
|B(y, 2j/2m )|−1
|B(y, 2j/2m )|−1 .
Then by (1.2),
||
j
1
j
(f ∗ ψj,q )(e−2 L ) p2j (., y)||L2 (Aq (y)) ≤ |Aq (y)| 2 ||(f ∗ ψj,q )(e−2 L )p2j (., y)||∞
c||f ||C A e−c2
(q−j)/2(2m−1)
|B(y, 2j/2m )|−1 |B(y, 2(q+1)/2m )|1/2
≤
c||f ||C A e−c2
(q−j)/2(2m−1)
2D(q−j)/2m |B(y, 2j/2m )|−1/2
≤
c||f ||C A 2−A(q−j)/2(2m−1) |B(y, 2j/2m )|−1/2 .
≤
(2.10)
The relation (2.6) follows by (2.9) and (2.10).
(ii)From (2.6), Cauchy- Swartz and (1.2) we have
j
j
||f (e−2 L )p2j (., y)||L1 (Aq (y)) ≤ ||f (e−2 L )p2j (., y)||L2 (Aq (y)) |Aq (y)|1/2
≤
c||f ||C A 2−A(q−j)/2(2m−1) |B(y, 2j/2m )|−1/2 |B(y, 2(q+1)/2m )|1/2
≤
c||f ||C A 2−A(q−j)/2(2m−1) 2 4 (q−j)/2m
¡
¢
D
c||f ||C A 2− A− 2 (q−j)/2(2m−1)
≤
D
and the proof is complete.
¤
2.3. Lp − boundedness. We are now ready to prove the following
Proposition 1. Let A >
all p ∈ [1, ∞].
D
2
and m ∈ C0A (0, ∞). Then m(L) is bounded on Lp , for
Proof. Let f (s) = m(− log s)s−1 . Then we take m(s) = f (e−s )e−s . Because m ∈
C0A (0, ∞) then so does f. In terms of L it is
m(L) = f (e−L )e−L , K(x, y) = m(L)δx (y) = f (e−L )p1 (x, y),
where δx (y) is the Diracc mass at x and K(x, y) is the kernel of m(L). It suffices
to show that The Kernel K(x, y) is integrable. Then we have the L∞ boundedness
and by interpolation and symmetry the Lp , boundedness for all p ∈ [1, ∞].
||K(., y)||1 = ||f (e−L )p1 (., y)||L1 (B(y,1)) +
∞
X
||f (e−L )p1 (., y)||L1 (Aq (y)) .
q=0
Using (2.5) and (2.7) we take because A > D/2,
||K(., y)||1 ≤ c||f ||C A + c||f ||C A
∞
X
q=0
and the proof is complete.
¡
2−
¢
A− D
2 q/2(2m−1)
<c
6
ATHANASIOS G. GEORGIADIS
¤
3. Preparation for the proof
3.1. Calderon-Zygmund decomposition. Following ([6] p73-75) we consider
bounded supported 0 ≤ f ∈ L1 ∩ L2 and a > 0. There are constants c = c(M ) >
0, k = k(M ) ∈ N∗ and a sequence of balls Bi = B(xi , ri ) such that
(1)
[
f (x) ≤ ca, a.e. in M −
Z
(2)
(3)
|Bi |
X
Bi .
i
−1
f (x)dx ≤ ca.
Bi
|Bi | ≤ (c/a)||f ||1 .
i
(4) each point belongs to at most k balls Bi .
Now following [13] we consider the functions
1B (x)
ηi (x) = P i
, x ∈ Bi , ηi (x) = 0, x ∈ M − Bi .
j 1Bj (x)
Also we set wi = ηi f and
2m
bi = e−ri
L
wi , θi = wi − bi , g = 1M −Si Bi .
We can write the function f in the form
X
X
f =g+
bi +
θi .
i
i
We continue with a lemma whose analogue in second order one can see in [4].
Lemma 5. Let b ∈ L1 supported in B(z, r), z ∈ M, r ≤ t1/2m .Then there is a
constant c > 0 independent of b such that for all u ∈ L2 ,
c||b||1
hM u, 1B(z,r) i.
|B(z, r)|
R
Where M u(y) = supr>0 |B(y, r)|−1 B(y,r) |u(x)|dx, is the Hardy-Littlewood maximal function.
hu, e−tL bi ≤
(3.1)
Proof. Using (1.3) we take
Z
e−tL b(y) =
pt (x, y)b(y)dy
M
Z
≤
c
³
d(x, y)2m/(2m−1) ´
|B(y, t1/2m )|−1 |b(y)|dy
exp − c
1/(2m−1)
t
B(z,r)
Because d(y, z) < r ≤ t1/2m , we have that
³
³
2m/(2m−1) ´
0 d(x, z)
d(x, y)2m/(2m−1) ´
exp − c
≤ c exp − c
1/(2m−1)
1/(2m−1)
t
t
and by (1.1) we have also |B(y, t1/2m )|−1 ≤ c|B(z, t1/2m )|−1 . So
HIGHER ORDER OPERATORS ON MANIFOLDS
7
Z
³
d(x, z)2m/(2m−1) ´
1/2m −1
e−tL b(y) ≤ c exp − c
|B(z,
t
)|
|b(y)|dy
t1/(2m−1)
B(z,r)
Z
³ −c0 d(x, z)2m/(2m−1) ´
1/2m −1
−1
= c exp
|B(z,
t
)|
||b||
|B(z,
r)|
1B(z,r) (y)dy
1
t1/(2m−1)
M
Z
³
2m/(2m−1) ´
00 d(x, y)
c||b||1
exp − c
|B(y, t1/2m )|−1 1B(z,r) (y)dy
≤
|B(z, r)| B(z,r)
t1/(2m−1)
(q+1) 1/2m
t
) − B(y, 2q t1/2m ), then M = B(y, t1/2m ) ∪
S We set Dq (y, t) = B(y, 2
D
(y,
t).
We
take
that
q≥0 q
Z
³
d(x, y)2m/(2m−1) ´
exp − c
|u(x)||B(y, t1/2m )|−1 dx ≤ M u(y).
t1/(2m−1)
B(y,t1/2m )
³
´
2m/(2m−1)
|u(x)|
exp − c d(x,y)
t1/(2m−1)
Z
|B(y, t1/2m )|
Dq (y,t)
≤
≤
≤
ce−c2
q2m/(2m−1)
ce−c2
dx ≤
|B(y, t1/2m )|
Z
|u(x)|dx
Dq (y,t)
q2m/(2m−1)
|B(y, 2(q+1) t1/2m )|
M u(y)
|B(y, t1/2m )
ce−c2
q2m/(2m−1)
ce−c2
q2m/(2m−1)
2qD M u(y)
M u(y)
So we take
Z
³
d(x, y)2m/(2m−1) ´
exp − c
|u(x)||B(y, t1/2m )|−1 dx
t1/(2m−1)
M
X
q2m/(2m−1)
≤ M u(y) + c
e−c2
M u(y) ≤ cM u(y)
q≥0
and finally
hu, e−tL bi
≤
≤
c||b||1
|B(z, r)|
Z
B(z,r)
M u(y)1B(z,r) (y)dy
c||b||1
hM u(y), 1B(z,r) i
|B(z, r)|
¤
Corollary 1. There is a constant c > 0 independet of f such that
¯¯ X ¯¯2
¯¯
¯¯
(3.2)
bi ¯¯ ≤ ca||f ||1 .
¯¯
i
2
2m
Proof. Let u ∈ L2 . We recall that bi = e−ri L wi and supp(wi ) ⊆ B(xi , ri ). Then
by (3.1) and part (2) of Calderon-Zygmund decomposition we have
8
ATHANASIOS G. GEORGIADIS
hu,
X
bi i
≤
X
i
2m
hu, e−ri
L
wi i
i
≤
≤
X
||wi ||1
hM u, 1B(xi ,ri ) i
|B(xi , ri )|
i
X
cahM u,
1B(xi ,ri ) i
c
i
(3.3)
≤
ca||u||2 ||
X
1B(xi ,ri ) ||2
i
Now by parts (3) and (4) of Calderon-Zygmund decomposition,
¯¯ X
¯¯2
X
c
¯¯
¯¯
(3.4)
1B(xi ,ri ) ¯¯ ≤ c
|Bi | ≤ ||f ||1 .
¯¯
a
2
i
i
P
The relation (3.2) follows from (3.3) and (3.4) by setting u = i bi .
3.2. Kernel estimates. Following [5] since by (1.3) lim ||e−tL f ||2 = 0, we cont→∞
clude that lim ||
t→∞
1/t
R
dEλ (f )||2 = 0. So the point λ = 0 may be neglected in the
0
spectral resolution of L and we can consider m(λ) defined in (0, ∞).
Following [17, 14] we consider a C ∞ function φ, satisfying
X
supp(φ) ⊂ (2−1 , 2) and
φ(2j t) = 1, t > 0.
j∈Z
We divide the multiplier m into pieces mj , j ∈ Z, by setting
(3.5)
mj (s) = m(s)φ(2j s), j ∈ Z.
We have
(3.6)
supp(mj ) ⊂ (2−j−1 , 2−j+1 ), and m(s) =
X
mj (s).
j∈Z
We denote by
m and mj respectively. Then
P K(x, y) and Kj (x, y) the kernels of the
−j
K(x, y) =
K
(x,
y).
We
set
h
(s)
=
m
(−2
log s)s−1 , j ∈ Z. Then the
j
j
j
j∈Z
functions hj are compactly supported and ||hj ||C A ≤ c. Note that
(3.7)
j
mj (L) = hj (e−2 L )e−2
j
L
j
, Kj (x, y) = hj (e−2 L )p2j (x, y) , j ∈ Z.
Lemma 6. There is a c > 0 such that for every q, j ∈ Z, q ≥ j, y ∈ M
¡
¢
D
(3.8)
||Kj (., y)||L1 (Aq (y)) ≤ c2− A− 2 (q−j)/2(2m−1) .
Proof. By (3.7) and (2.7) follows (3.8).
We consider the functions gj,τ (s) = hj (s)s2
τ
kernel of mj (L)e−2 L . Then we obtain that
(3.9)
||gj,τ ||C A ≤ ce−c2
τ −j
¤
τ −j
, τ ≥ j. Let Sj,τ (x, y) be the
j
, Sj,τ (x, y) = gj,τ (e−2 L )p2j (x, y).
Lemma 7. There is a c > 0 such that for every τ, j ∈ Z, τ ≥ j, y ∈ M
(3.10)
||Sj,τ (., y)||1 ≤ ce−c2
τ −j
.
HIGHER ORDER OPERATORS ON MANIFOLDS
9
τ −j
Proof. By (2.5) and (3.9) it is ||Sj,τ (., y)||L1 (B(y,2j/2m )) ≤ ce−c2 . Also by (2.7)
we take because A > D
2,
X −¡A− D ¢(q−j)/2(2m−1)
τ −j
−c2τ −j
2
||Sj,τ (., y)||L1 (B(y,2j/2m )c ) ≤ ce
2
≤ ce−c2 .
q≥j
¤
Let now Λj,τ (x, y) be the kernels of (I − e−2
τ
L
)mj (L).
Lemma 8. There is a c > 0 such that for every τ, j ∈ Z, τ ≥ j, y ∈ M
¡
¢
D
(3.11)
||Λj,τ (., y)||L1 (B(y,2τ /2m )c ) ≤ c2− A− 2 (τ −j)/2(2m−1) .
Proof. We have that
(3.12)
Λj,τ (x, y) = (I − e−2
By (3.8) we have because A >
τ
L
)mj (L)δx (y) = Kj (x, y) − Sj,τ (x, y).
D
2,
||Kj (., y)||L1 (B(y,2τ /2m )c
X
=
||Kj (., y)||L1 (Aq (y))
q≥τ
≤
c
X
¡
¢
D
2− A− 2 (q−j)/2(2m−1)
q≥τ
(3.13)
¡
¢
D
c2− A− 2 (τ −j)/2(2m−1)
≤
Now (3.11) follows from (3.13), (3.12) and (3.10).
¤
We consider the functions fj,τ (s) = (1 − s2
(3.14)
τ −j
)hj (s), τ ≤ j. We observe that
j
||fj,τ ||C A ≤ c2τ −j , Λj,τ (x, y) = fj,τ (e−2 L )p2j (x, y), τ ≤ j.
Lemma 9. There is a c > 0 such that for every τ, j ∈ Z, τ ≤ j, y ∈ M
||Λj,τ (., y)||1 ≤ c2τ −j .
(3.15)
Proof. By (2.5), (2.7) and (3.14) we have because A >
||Λj,τ (., y)||1
≤
≤
||Λj,τ (., y)||B(y,2j/2m ) +
τ −j
c2
τ −j
+ c2
X
¡
X
D
2,
||Λj,τ (., y)||L1 (Aq (y))
q≥j
¢
− A− D
2 (q−j)/2(2m−1)
2
≤ c2τ −j
q≥j
¤
Let Λτ (x, y) be the kernel of (I − e−2
τ
L
)m(L).
Lemma 10. There is a c > 0 such that for every τ ∈ Z, y ∈ M
(3.16)
||Λτ (., y)||L1 (B(y,2τ /2m )c ) ≤ c.
10
ATHANASIOS G. GEORGIADIS
P
P
Proof. It is Λτ (x, y) = j≤τ Λj,τ (x, y) + j>τ Λj,τ (x, y). By (3.11) and (3.15) we
take using that A > D
2,
X −¡A− D ¢(τ −j)/2(2m−1) X
2
c2
||Λτ (., y)||L1 (B(y,2τ /2m )c ) ≤
+
c2τ −j ≤ c.
j≤τ
j>τ
¤
4. proof of theorem 1
We will proove that there is a constant c > 0 such that for every f ∈ L1 ,
¯
¯
¯{|m(L)f | > a}¯ ≤ c||f ||1 .
a
P
P
We recall that f = g + i bi + i θi , so
(4.1)
(4.2)
X
X
¯
¯ ¯
¯ ¯
a ¯ ¯
a ¯
¯{|m(L)f | > a}¯ ≤ ¯{|m(L)g| > a }¯ + ¯{|m(L)
bi | > }¯ + ¯{|m(L)
θi | > }¯
3
3
3
i
i
By the property (1) of Calderon-Zygmund decomposition we take
¯
¯
¯{|m(L)g| > a }¯ ≤ 9 ||m(L)g||22 ≤ c ||g||22 ≤ c || g g||1 ≤ c ||f ||1
3
a2
a2
a a
a
By (3.2) we take
(4.3)
(4.4)
X
¯
a ¯
c X 2
c
¯{|m(L)
bi | > }¯ ≤ 2 ||
bi ||2 ≤ ||f ||1
3
a
a
i
i
By (3.16) it is ||m(L)θi ||L1 (B(xi ,cri )c ) ≤ c||wi ||1 and then by the property (3) of
Calderon-Zygmund decomposition we take
(4.5)
X
[
X
¯
a ¯ ¯¯
a ¯¯ ¯¯ [ ¯¯
c
¯{|m(L)
θi | > }¯ ≤ ¯{x ∈ M − Bi : |m(L)
θi | > }¯ + ¯ Bi ¯ ≤ ||f ||1
3
3
a
i
i
i
i
The theorem follows from (4.1),(4.2),(4.3),(4.4) and (4.5)
¤
5. Final remarks
It is also interest the problem of H p − boundedness for the multipliers m(L).
One has to suppose that M has bounded geometry. Also that the estimate
(5.1) |pt (x, y) − pt (x, z)| ≤
³
c
d(x, y)2m/(2m−1) ´³ d(y, z) ´γ
exp − c
1/2m
|B(x, t
)|
t1/(2m−1)
t1/2m
holds for every x, y, z ∈ M, t > 0 with d(y, z) ≤ t1/2m , for some γ ∈ (0, 1].
Then using
the¢ strategy of [14] follows that if m(λ) satisfies¡ the condition
(1.4)
¡
¤
D
for A > D p1 − 12 then m(L) is bonded on H p , for every p ∈ D+γ
,1 .
HIGHER ORDER OPERATORS ON MANIFOLDS
11
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Department of Mathematics, Aristotle University of Thessaloniki, Thessaloniki 54.124,
Greece
E-mail address: [email protected]