Littlewood-Paley equivalence and homogeneous Fourier multipliers

0.3 MB·

by  Shuichi Sato

LITTLEWOOD-PALEY EQUIVALENCE AND HOMOGENEOUS
FOURIER MULTIPLIERS
6
1
SHUICHISATO
0
2
n Abstract. WeconsidercertainLittlewood-Paleyoperatorsandprovecharac-
a terizationofsomefunctionspacesintermsofthoseoperators. Whentreating
J weightedLebesguespaces,ageneralizationtoweightedspaceswillbemadefor
H¨ormander’stheoremontheinvertibilityofhomogeneous Fouriermultipliers.
3
Also,applications tothetheoryofSobolevspaceswillbegiven.
1

]
A
C 1. Introduction
h. Let ψ be a function in L1(Rn) such that
t
a
(1.1) ψ(x)dx=0.
m
ZRn
[ We consider the Littlewood-Paley function on Rn defined by

1 ∞ dt 1/2
v (1.2) gψ(f)(x)= |f ∗ψt(x)|2 ,
t
3 (cid:18)Z0 (cid:19)
7 whereψ (x)=t−nψ(t−1x). ThefollowingresultofBenedek,Caldero´nandPanzone
t
1 [2] on the Lp boundedness, 1<p<∞, of g is well-known.
ψ
3
0 Theorem A. We assume (1.1) for ψ and
.
1 (1.3) |ψ(x)|≤C(1+|x|)−n−ǫ,
0
6 (1.4) |ψ(x−y)−ψ(x)|dx≤C|y|ǫ
1 ZRn
v: for some positive constant ǫ. Then gψ is bounded on Lp(Rn) for all p∈(1,∞):
i
X (1.5) kgψ(f)kp ≤Cpkfkp,
r where
a 1/p
kfk =kfk = |f(x)|pdx .
p Lp
(cid:18)ZRn (cid:19)
By the Planchereltheorem, it follows that g is bounded on L2(Rn) if and only
ψ
if m ∈ L∞(Rn), where m(ξ) = ∞|ψˆ(tξ)|2dt/t, which is a homogeneous function
0
of degree 0. Here the Fourier transform is defined as
R
n
ψˆ(ξ)= ψ(x)e−2πihx,ξidx, hx,ξi= x ξ .
k k
ZRn k=1
X
2010Mathematics Subject Classification. Primary42B25;Secondary 46E35.
Key Words and Phrases. Littlewood-Paley functions, Marcinkiewicz integrals, Fouriermulti-
pliers,Sobolevspaces.
The author is partly supported by Grant-in-Aid for Scientific Research (C) No. 25400130,
JapanSocietyforthePromotionofScience.
1
2 SHUICHISATO
Let
t
P (x)=c
t n(|x|2+t2)(n+1)/2
bethePoissonkernelontheupperhalfspaceRn×(0,∞)andQ(x)=[(∂/∂t)P (x)] .
t t=1
Then, we cansee that the function Q satisfies the conditions (1.1), (1.3) and (1.4).
Thus by Theorem A g is bounded on Lp(Rn) for all p∈(1,∞).
Q
Let H(x) = sgn(x)χ (x) = χ (x)−χ (x) on R (the Haar function),
[−1,1] [0,1] [−1,0]
where χ denotes the characteristic function of a set E and sgn(x) the signum
E
function. Then g (f) is the Marcinkiewicz integral
H
∞ dt 1/2
µ(f)(x)= |F(x+t)+F(x−t)−2F(x)|2 ,
t3
(cid:18)Z0 (cid:19)
x
where F(x)= f(y)dy. Also, we can easily see that Theorem A implies that g
0 H
is bounded on Lp(R), 1<p<∞.
R
Further, we can consider the generalized Marcinkiewicz integral µ (f) (α > 0)
α
on R defined by
∞ dt 1/2
µ (f)(x)= |Sα(f)(x)|2 ,
α t t
(cid:18)Z0 (cid:19)
where
α t u α−1
Sα(f)(x)= 1− (f(x−u)−f(x+u)) du.
t t t
Z0 (cid:16) (cid:17)
We observe that µ (f)=g (f) with
α ϕ(α)
(1.6) ϕ(α)(x)=α|1−|x||α−1sgn(x)χ (x).
(−1,1)
Thesquarefunctionµ coincideswiththeordinaryMarcinkiewiczintegralµ. When
1
ψ is compactly supported, relevant sharp results for the Lp boundedness of g can
ψ
be found in [6, 8, 20].
We can also consider Littlewood-Paley operators on the Hardy space Hp(Rn),
0 < p < ∞. We consider a dense subspace S (Rn) of Hp(Rn) consisting of those
0
functions f in S(Rn) which satisfy fˆ=0 near the origin, where S(Rn) denotes the
Schwartz class of rapidly decreasing smooth functions. Let f ∈ S (R). Then, if
0
2/(2α+1)<p<∞ and α>0, we have kµ (f)k ≃kfk , which means
α p Hp
(1.7) c kfk ≤kµ (f)k ≤C kfk
p Hp α p p Hp
with some positive constants c ,C independent of f (see [27], [19]).
p p
To stateresultsaboutthe reverseinequality of (1.5), wefirstrecallatheoremof
Ho¨rmander [12]. Let m∈L∞(Rn) and define
(1.8) T (f)(x)= m(ξ)fˆ(ξ)e2πihx,ξidξ.
m
ZRn
We say that m is a Fourier multiplier for Lp and write m ∈ Mp if there exists a
constant C >0 such that
kT (f)k ≤Ckfk
m p p
for all f ∈L2∩Lp. Then the result of Ho¨rmander [12] can be stated as follows.
Theorem B. Let m be a bounded function on Rn which is homogeneous of degree
0. Suppose that m ∈ Mp for all p ∈ (1,∞). Suppose further that m is continuous
and does not vanish on Sn−1 = {x ∈ Rn : |x| = 1}. Then, m−1 ∈ Mp for every
p∈(1,∞).
LITTLEWOOD-PALEY EQUIVALENCE 3
See [5, 2] for related results. Applying Theorem B, we can deduce the following
(see [12, Theorem 3.8]).
Theorem C. Suppose that g is bounded on Lp for every p∈(1,∞). Let m(ξ)=
ψ
∞|ψˆ(tξ)|2dt/t. If m is continuous and strictly positive on Sn−1, then we have
0
R kfkp ≤cpkgψ(f)kp,
and hence kfk ≃kg (f)k , f ∈Lp, for all p∈(1,∞).
p ψ p
In this note we shall generalize Theorems B and C to weighted Lp spaces with
A weights of Muckenhoupt (see Theorems 2.5, 2.9 and Corollaries2.6, 2.11). Our
p
proof of Theorem 2.5 has some features in common with the proof of Wiener-L´evy
theorem in [30, vol. I, Chap. VI]. We also consider a discrete parameter version of
g :
ψ
∞ 1/2
(1.9) ∆ (f)(x)= |f ∗ψ (x)|2 .
ψ 2k
!
k=−∞
X
We shall have ∆ analogues of results for g (see Theorem 3.5 and Corollary3.7).
ψ ψ
WeformulateTheorems2.9and3.5ingeneralformssothattheyincludeunweighted
cases as special cases, while Corollaries 2.11 and 3.7 may be more convenient for
some applications.
In the unweighted case, we shall prove some results on Hp analogous to Corol-
laries 2.11 and 3.7 for p close to 1, p ≤ 1, in Section 4 under a certain regularity
condition for ψ (Theorems 4.7 and 4.8). We shall consider functions ψ includ-
ing those which cannot be treated directly by the theory of [28]. As a result, in
particular, we shall be able to give a proof of the second inequality of (1.7) for
1/2<α<3/2 and 2/(2α+1)<p≤1 by methods of real analysis which does not
depend on the Poisson kernel.
Here we recall some more background materials on µ . When p<1 and 1/2<
α
α < 1, we know proofs for the first and the second inequality of (1.7) which use
pointwiserelationsµ (f)≥cg (f)andµ (f)∼g∗(f)withλ=1+2α,respectively,
α 0 α λ
and apply appropriate properties of g and g∗. Also, we note that a proof of the
0 λ
inequalitykµ(f)k ≤Ckfk usingatheoryofvectorvaluedsingularintegralscan
1 H1
befoundin[10,Chap. V](seealso[17]). Wehaveassumedthatsupp(fˆ)⊂[0,∞)in
stating µ (f)∼g∗(f) and g (f), g∗(f) are the Littlewood-Paley functions defined
α λ 0 λ
by
∞ 1/2
g (f)(x)= |(∂/∂x)u(x,t)|2tdt ,
0
(cid:18)Z0 (cid:19)
λ 1/2
t
g∗(f)(x)= |∇u(y,t)|2dydt
λ ZZR×(0,∞)(cid:18)t+|x−y|(cid:19) !
withu(y,t)denotingthePoissonintegraloff: u(y,t)=P ∗f(y)(see[27],[19]and
t
references therein, and also [13], [15] for related results).
In [28], a proof of kfk ≤ Ckg (f)k on Rn is given without the use of har-
Hp Q p
monicity (see [9] for the original proof using properties of harmonic functions).
Also,whenn=1,asimilarresultisshownforg . Itistobenotedthat,combining
0
this with the pointwise relation between g and µ mentioned above, we can give
0 α
a proof of the first inequality of (1.7) for the whole range of p, α in such a manner
4 SHUICHISATO
that a special property of the Poisson kernel is used only to prove the pointwise
relation.
In Section 5, we shall apply Corollaries 2.11 and 3.7 to the theory of Sobolev
spaces. In [1], the operator
1/2
2
∞ dt
(1.10) U (f)(x)= f(x)−− f(y)dy , α>0,
α Z0 (cid:12)(cid:12) ZB(x,t) (cid:12)(cid:12) t1+2α
(cid:12) (cid:12)
wasstudied,where− f(y)d(cid:12)(cid:12)yisdefinedas|B(x,t)|(cid:12)(cid:12)−1 f(y)dywith|B(x,t)|
B(x,t) B(x,t)
denoting the Lebesgue measure of a ball B(x,t) in Rn of radius t centered at x.
R R
The operator U was used to characterize the Sobolev space W1,p(Rn).
1
Theorem D. Let 1<p<∞. Then, the following two statements are equivalent:
(1) f belongs to W1,p(Rn),
(2) f ∈Lp(Rn) and U (f)∈Lp(Rn).
1
Furthermore, from either of the two conditions (1),(2) it follows that
kU (f)k ≃k∇fk .
1 p p
This may be used to define a Sobolev space analogous to W1,p(Rn) in metric
measure spaces. We shall also consider a discrete parameter version of U :
α
1/2
∞ 2
(1.11) E (f)(x)= f(x)−− f(y)dy 2−2kα , α>0,
α
k=X−∞(cid:12)(cid:12) ZB(x,2k) (cid:12)(cid:12) 
(cid:12) (cid:12)
and prove an analogueof Theor(cid:12)em D for E . Further,(cid:12)we shall consider operators
(cid:12) α (cid:12)
generalizing U , E and show that they can be used to characterize the weighted
α α
Sobolev spaces, focusing on the case 0<α<n.
2. Invertibility of homogeneous Fourier multipliers and
Littlewood-Paley operators
We say that a weight function w belongs to the weight class A , 1<p< ∞, of
p
Muckenhoupt on Rn if
p−1
[w] =sup |B|−1 w(x)dx |B|−1 w(x)−1/(p−1)dx <∞,
Ap
B (cid:18) ZB (cid:19)(cid:18) ZB (cid:19)
where the supremum is taken over all balls B in Rn. Also, we say that w ∈ A if
1
M(w)≤Cw almost everywhere, with M denoting the Hardy-Littlewood maximal
operator
M(f)(x)= sup|B|−1 |f(y)|dy,
x∈B ZB
where the supremum is taken over all balls B in Rn containing x; we denote by
[w] the infimum of all such C.
A1
Let m ∈ L∞(Rn) and w ∈ A , 1 < p < ∞. Let T be as in (1.8). We say that
p m
m is a Fourier multiplier for Lp and write m ∈ Mp(w) if there exists a constant
w
C >0 such that
(2.1) kT (f)k ≤Ckfk
m p,w p,w
LITTLEWOOD-PALEY EQUIVALENCE 5
for all f ∈L2∩Lp, where
w
1/p
kfkp,w =kfkLpw =(cid:18)ZRn|f(x)|pw(x)dx(cid:19) .
We also write Lp(w) for Lp. Define
w
kmk =infC,
Mp(w)
wherethe infimumis takenoverallthe constantsC satisfying (2.1). Since L2∩Lp
w
is dense in Lp, T uniquely extends to a bounded linear operator on Lp if m ∈
w m w
Mp(w). In this note we shall confine our attention to the case of Lp boundedness
w
of T with w ∈ A . We note that Mp(w) = Mp′(w−p′/p) by duality, where
m p
1/p+1/p′ =1 and w(x)=w(−x).
If w ∈ A , 1 < p < ∞, then ws ∈ A for some s > 1 and r < p (see [10]).
p r e
In applying interpoleation arguments it is useful if sets of those (r,s) are specially
notated.
Definition 2.1. Let w ∈A , 1<p<∞. For 0<σ <p−1, τ >0, set
p
U(w,p)=U(w,p,σ,τ)=(p−σ,p+σ)×[1,1+τ).
We say that U(w,p) is a (w,p) set if ws ∈ A for all (r,s) ∈ U(w,p). We write
r
m∈M(U(w,p)) if m∈Mr(ws) for all (r,s)∈U(w,p).
We need a relation of kmk and kmk in the following.
Mp(w) ∞
Proposition 2.2. Let w ∈ A , 1 < p < ∞. Suppose that m ∈ M(U(w,p)) for a
p
(w,p) set U(w,p). Then
kmk ≤kmk1−θkmkθ , (p+δ,1+ǫ)∈U(w,p), if p>2;
Mp(w) ∞ Mp+δ(w1+ǫ)
kmk ≤kmk1−θkmkθ , (p,1+ǫ)∈U(w,p), if p=2;
Mp(w) ∞ Mp(w1+ǫ)
kmk ≤kmk1−θkmkθ , (p−δ,1+ǫ)∈U(w,p), if 1<p<2,
Mp(w) ∞ Mp−δ(w1+ǫ)
for some θ ∈(0,1) and some small numbers δ,ǫ>0.
Proof. Let 1<p<2 and w ∈A . Then, there existǫ ,δ >0 such that (p−δ,1+
p 0 0
ǫ) ∈ U(w,p) for all ǫ ∈ (0,ǫ ] and δ ∈ (0,δ ]. Let 1/p = (1−θ)/2+θ/(p−δ),
0 0
δ ∈(0,δ ]. Then,sincem∈Mp−δ(w1+ǫ),byinterpolationwithchangeofmeasures
0
of Stein-Weiss (see [3]) between L2 and Lp−δ(w1+ǫ) boundedness, we have
kmkMp(wpθ(1+ǫ)/(p−δ)) ≤kmk1∞−θkmkθMp−δ(w1+ǫ).
Note that pθ/(p−δ) = (2−p)/(2−p+δ). Thus we can choose ǫ,δ > 0 so that
pθ(1+ǫ)/(p−δ)=1. This completes the proof for p∈(1,2).
Suppose that 2<p<∞ and w ∈A . Then there exist ǫ >0 δ >0 such that
p 0 0
(p+δ,1+ǫ)∈U(w,p)forallǫ∈(0,ǫ ]andδ ∈(0,δ ]. So,m∈Mp+δ(w1+ǫ)forall
0 0
ǫ ∈ (0,ǫ ] and δ ∈ (0,δ ]. Similarly to the case 1 < p < 2, applying interpolation,
0 0
we have
kmk ≤kmk1−θkmkθ
Mp(wpθ(1+ǫ)/(p+δ)) ∞ Mp+δ(w1+ǫ)
with pθ/(p+δ)=(p−2)/(p+δ−2). Taking ǫ,δ so that pθ(1+ǫ)/(p+δ)=1, we
conclude the proof for p∈(2,∞). The case p=2 can be handled similarly. (cid:3)
To treat Fourier multipliers arising from Littlewood-Paley functions in (1.2)
and(1.9) simultaneously, we slightly generalize the usual notion of homogeneity.
6 SHUICHISATO
Definition 2.3. Let f be a function on Rn. We say that f is dyadically homoge-
neous of degree τ, τ ∈ R, if f(2kx) = 2kτf(x) for all x ∈ Rn\{0} and all k ∈ Z
(the set of integers).
For m∈Mp, 1<p<∞, w ∈A , we consider the spectral radius operator
w p
ρ (m)= lim kmkk1/k .
p,w Mp(w)
k→∞
To prove a weighted version of Theorem B, we need an approximation result for
Fourier multipliers in Mp(w).
Proposition 2.4. Let 1<p< ∞, w ∈A and m ∈L∞(Rn). We assume that m
p
is dyadically homogeneous of degree 0 and continuous on the closed annulus B =
0
{ξ ∈ Rn : 1 ≤ |ξ| ≤ 2}. We further assume that there exists a (w,p) set U(w,p)
such that m ∈ M(U(w,p)). Then, for any ǫ > 0, there exists n ∈Mp(w) which is
dyadically homogeneous of degree 0 and in C∞(Rn\{0}) such that km−nk <ǫ
∞
and ρ (m−n)<ǫ.
p,w
Proof. Let {ϕ }∞ be a sequence of functions on O(n) such that
j j=1
• each ϕ is infinitely differentiable and non-negative,
j
• for any neighborhood U of the identity in O(n), there exists a positive
integer N such that supp(ϕ )⊂U if j ≥N,
j
• ϕ (A)dA=1, where dA is the Haar measure on O(n).
O(n) j
Also, letR {ψj}∞j=1 be a sequence of non-negative functions in C∞(R) such that
supp(ψ )⊂[1−2−j,1+2−j] and ∞ψ (t)dt/t=1. Define
j 0 j
∞ R dt
m (ξ)= m(tAξ)ϕ (A)ψ (t)dA .
j j j
t
Z0 ZO(n)
Then m is dyadically homogeneousof degree 0, infinitely differentiable andm →
j j
muniformlyinRn\{0}bythecontinuityofmonB . This canbe shownsimilarly
0
to [12, pp. 123-124],where we can find the case when m is homogeneous of degree
0. Also, for a positive integer k, the derivatives of mk satisfy
j
(2.2) |∂γm (ξ)k|≤C kmkk |ξ|−|γ|, ∂γ =(∂/∂ξ )γ1...(∂/∂ξ )γn
ξ j j,k,M ∞ ξ 1 n
forallmulti-indicesγwith|γ|≤M,whereM isanypositiveinteger,γ =(γ ,...,γ ),
1 n
|γ|=γ +···+γ , γ ∈Z, γ ≥0 and we have C ≤C kM with a constant
1 n j j j,k,M j,M
C independent of k. By (2.2), if M is sufficiently large, it follows that
j,M
kmkk ≤CC kmkk , w ∈A ,1<p<∞
j Mp(w) j,k,M ∞ p
witha constantC independent ofk (see[4,14]). Thus,bythe evaluationofC
j,k,M
we have
(2.3) ρ (m )≤kmk .
p,w j ∞
Since m,m ∈ M(U(w,p)), by Proposition 2.2, we can find r close to p, s > 1
j
with (r,s)∈U(w,p) and θ ∈(0,1) such that
k(m−m )kk ≤k(m−m )kk1−θk(m−m )kkθ .
j Mp(w) j ∞ j Mr(ws)
It follows that
ρ (m−m )≤km−m k1−θρ (m−m )θ.
p,w j j ∞ r,ws j
LITTLEWOOD-PALEY EQUIVALENCE 7
Thus, by (2.3) we have
ρ (m−m )≤km−m k1−θ(ρ (m)θ+ρ (m )θ)
p,w j j ∞ r,ws r,ws j
≤km−m k1−θ(ρ (m)θ+kmkθ ).
j ∞ r,ws ∞
This completes the proof since km−m k →0 as j →∞. (cid:3)
j ∞
Applying Proposition 2.4, we can generalize Theorem B as follows.
Theorem 2.5. Suppose that 1<p<∞,w ∈A and that m∈L∞(Rn) fulfills the
p
hypotheses of Proposition 2.4. Also, suppose that m(ξ) 6= 0 for every ξ 6= 0. Let
ϕ(z) be holomorphic in C\{0}. Then we have ϕ(m(ξ))∈Mp(w).
Proof. Define ǫ >0 by
0
4ǫ = min |m(ξ)|= min |m(ξ)|.
0
ξ∈Rn\{0} 1≤|ξ|≤2
By Proposition2.4,there is n∈Mp(w) which is dyadicallyhomogeneousof degree
0andinfinitelydifferentiableinRn\{0}suchthatkm−nk <ǫ andρ (m−n)<
∞ 0 p,w
ǫ . If we consider a curve C : n(ξ)+2ǫ eiθ,0 ≤ θ ≤ 2π, Cauchy’s formula can be
0 0
applied to represent ϕ(m(ξ)) by a contour integral as follows:
1 ϕ(ζ) ǫ 2π ϕ(n(ξ)+2ǫ eiθ)
ϕ(m(ξ))= dζ = 0 0 eiθdθ, ξ 6=0.
2πi ζ−m(ξ) π 2ǫ eiθ +n(ξ)−m(ξ)
ZC Z0 0
Note that
eiθ 1 ∞ m(ξ)−n(ξ) k
= ;
2ǫ eiθ+n(ξ)−m(ξ) 2ǫ 2ǫ eiθ
0 0 k=0(cid:18) 0 (cid:19)
X
the series converges uniformly in θ ∈[0,2π] since |m(ξ)−n(ξ)|<ǫ . Thus
0
∞ k
1 m(ξ)−n(ξ)
ϕ(m(ξ))= M (ξ)
k
2π 2ǫ
k=0(cid:18) 0 (cid:19)
X
uniformly in Rn\{0}, where
2π
M (ξ)= ϕ(n(ξ)+2ǫ eiθ)e−ikθdθ.
k 0
Z0
Since |n(ξ)+2ǫ eiθ| ≥ ǫ , we can see that M (ξ) is dyadically homogeneous of
0 0 k
degree 0 and infinitely differentiable in Rn\{0}; also the derivative satisfies
|∂γM (ξ)|≤C |ξ|−|γ|
ξ k γ
for every multi-index γ with a constant C independent of k. This implies that
γ
kM k ≤ C with a constant C independent of k (see [4, 14]). Thus we have
k Mp(w)
ϕ(m)∈Mp(w) and
∞
1
kϕ(m)k ≤ (2ǫ )−kk(m−n)kk kM k ,
Mp(w) 2π 0 Mp(w) k Mp(w)
k=0
X
since the series converges, for k(m−n)kk ≤ǫk if k is sufficiently large. This
Mp(w) 0
completes the proof.
(cid:3)
Theorem 2.5 in particular implies the following.
8 SHUICHISATO
Corollary 2.6. Let 1 < p < ∞ and w ∈ A . Let m be a dyadically homogeneous
p
function of degree 0 such that m ∈ Mr(v) for all r ∈ (1,∞) and all v ∈ A . We
r
assumethatmiscontinuousonB anddoesnotvanishthere. Thenm−1 ∈Mp(w).
0
WehaveapplicationsofTheorem2.5andCorollary2.6tothetheoryofLittlewood-
Paley operators. Let w ∈ A , 1 < p < ∞. We say that g of (1.2) is bounded on
p ψ
Lp if there exists a constant C such that kg (f)k ≤ Ckfk for f ∈ L2 ∩L2.
w ψ p,w p,w w
The unique sublinear extensiononLp is alsodenoted by g . The Lp boundedness
w ψ w
for ∆ of (1.9) is considered similarly.
ψ
Let H be the Hilbert space of functions u(t) on (0,∞) such that kukH =
∞|u(t)|2dt/t 1/2 < ∞. We consider weighted spaces Lp of functions h(y,t)
0 w,H
with the norm
(cid:0)R (cid:1) 1/p
khkp,w,H = khykpHw(y)dy ,
(cid:18)ZRn (cid:19)
where hy(t)=h(y,t). If w =1 identically, the spaces Lp will be written simply
w,H
as Lp .
H
Define
∞ dt
(2.4) Eǫ(h)(x)= ψ (x−y)h (y,t)dy ,
ψ Z0 ZRn t (ǫ) t
where h ∈ L2H and h(ǫ)(y,t) = h(y,t)χ(ǫ,ǫ−1)(t), 0 < ǫ < 1, and we assume that
ψ ∈L1(Rn) with (1.1).
Then we have the following.
Lemma 2.7. Let 1<r <∞ and v ∈A . We assume that
r
kgψ(f)kr′,v−r′/r ≤C0(r,v)kfkr′,v−r′/r.
Then, if h∈Lr ∩L2 , we have
v,H H
sup kEψeǫ¯(h)kr,v ≤C0(r,v)khkr,v,H,
ǫ∈(0,1)
where ψ¯ denotes the complex conjugate.
Proof. For f ∈S(Rn), we see that
ǫ−1 dt
Eǫ(h)(x)f(x)dx = ψ¯ (x−y)h(y,t)dy f(x)dx
(cid:12)(cid:12)(cid:12)(cid:12)ZRn ψe¯ (cid:12)(cid:12)(cid:12)(cid:12) =(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)ZRǫn− 1 Zǫ ψ¯Z∗Rnf(eyt)h(y,t)dy dt t ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
t
(cid:12)(cid:12)Zǫ ZRn t (cid:12)(cid:12)
(cid:12) (cid:12)
≤(cid:12)(cid:12) gψ(f¯)(y)khykHdy. (cid:12)(cid:12)
ZRn
Thus, by Ho¨lder’s inequality, we have
1/r
(cid:12)ZRnEψeǫ¯(h)(x)f(x)dx(cid:12) ≤kgψ(f¯)kr′,v−r′/r(cid:18)Z khykrHv(y)dy(cid:19)
(cid:12) (cid:12) 1/r
(cid:12) (cid:12)
(cid:12) (cid:12) ≤C0(r,v)kfkr′,v−r′/r khykrHv(y)dy .
(cid:18)Z (cid:19)
Taking the supremum over f with kfkr′,v−r′/r ≤1, we get the desired result. (cid:3)
LITTLEWOOD-PALEY EQUIVALENCE 9
By applying Lemma 2.7, we have the following.
Proposition 2.8. Suppose that g satisfies the hypothesis of Lemma 2.7 with r ∈
ψ
(1,∞) and v ∈A . Also, we assume that
r
kg (f)k ≤C (r,v)kfk .
ψ r,v 1 r,v
Put
∞ dt
m(ξ)= |ψˆ(tξ)|2 .
t
Z0
Then kmk ≤C (r,v)C (r,v).
Mr(v) 0 1
Proof. We first note that an interpolation with change of measures between the
Lr(v) and Lr′(v−r′/r) boundedness of g implies the L2 boundedness of g . Thus
ψ ψ
we have m∈L∞(Rn).
Let F(y,t)=f ∗ψ (y), f ∈Lp ∩L2. Then
t w
ǫ−1 dt
Eǫ(F)(x)= ψ ∗f(y)ψ¯(y−x)dy = Ψ(ǫ)(x−z)f(z)dz,
ψe¯ Zǫ ZRn t t t ZRn
where
ǫ−1 dt
Ψ(ǫ)(x)= ψ (x+y)ψ¯(y)dy .
t t
Zǫ ZRn t
We see that
ǫ−1 dt ǫ−1 dt
Ψ(ǫ)(ξ)= ψˆ(tξ)ψ¯(−tξ) = |ψˆ(tξ)|2 .
t t
Zǫ Zǫ
Thus d b
ǫ−1 dt
Ψ(ǫ)(x−z)f(z)dz =T f(x), m(ǫ)(ξ)= |ψˆ(tξ)|2 .
ZRn m(ǫ) Zǫ t
From Lemma 2.7 and the Lr boundedness of g it follows that
v ψ
(2.5) kT fk =kEǫ(F)k ≤C (r,v)kg (f)k ≤C (r,v)C (r,v)kfk .
m(ǫ) r,v ψe¯ r,v 0 ψ r,v 0 1 r,v
Lettingǫ→0,weseethatm∈Mr(v)andkmk canbeevaluatedby(2.5). (cid:3)
Mr(v)
Now we can state a weighted version of Theorem C.
Theorem 2.9. Let g be as in (1.2). Let w ∈A , 1<p<∞. Suppose that there
ψ p
exists a (w,p) set U(w,p) such that g fulfills the hypotheses of Proposition 2.8 on
ψ
the weighted boundedness for all r, v =ws, (r,s)∈U(w,p). Further, suppose that
m(ξ)= ∞|ψˆ(tξ)|2dt/t is continuous and does not vanish on Sn−1. Then we have
0
R kfkp,w ≤Cp,wkgψ(f)kp,w
for f ∈Lp.
w
Obviously, this implies Theorem C when w =1.
Proof of Theorem 2.9. WefirstnotethatbyProposition2.8m∈M(U(w,p)). Thus
fromTheorem2.5withϕ(z)=1/zandourassumptions,weseethatm−1 ∈Mp(w).
Since f =Tm−1Tmf, f ∈Lpw∩L2, we have
kfk ≤CkT fk .
p,w m p,w
Also, by (2.5) it follows that
kT fk ≤Ckg (f)k .
m p,w ψ p,w
10 SHUICHISATO
Combining results we have the desired inequality. (cid:3)
From Theorem 2.9 the next result follows.
Theorem 2.10. Suppose the following.
(1) kg (f)k ≤C kfk for all r∈(1,∞) and all v ∈A ;
ψ r,v r,v r,v r
(2) m(ξ)= ∞|ψˆ(tξ)|2dt/t is continuous and strictly positive on Sn−1.
0
Then, if f ∈Lp, we have
wR
kfk ≤C kg (f)k
p,w p,w ψ p,w
for all p∈(1,∞) and w ∈A .
p
The following result is known (see [18]).
Theorem E. Suppose that
(1) B (ψ)<∞ for some ǫ>0, where B (ψ)= |ψ(x)||x|ǫdx;
ǫ ǫ |x|>1
(2) C (ψ)<∞ for some u>1 with C (ψ)= |ψ(x)|udx;
u u |Rx|<1
(3) H ∈L1(Rn), where H (x)=sup |ψ(y)|.
ψ ψ |y|≥|x| R
Then
kg (f)k ≤C kfk
ψ p,w p,w p,w
for all p∈(1,∞) and w ∈A .
p
By Theorem 2.10 and Theorem E we have the following result, which is useful
in some applications.
Corollary 2.11. Suppose that ψ satisfies the conditions (1),(2),(3) of Theorem E
and the non-degeneracy condition: sup |ψˆ(tξ)|>0 for all ξ 6=0. Then kfk ≃
t>0 p,w
kg (f)k , f ∈Lp, for all p∈(1,∞) and w ∈A .
ψ p,w w p
Proof. Let m(ξ) = ∞|ψˆ(tξ)|2dt/t. Then by our assumption m(ξ) 6= 0 for ξ 6= 0.
0
Thus by Theorem E and Theorem2.10, it suffices to show that m is continuous on
Sn−1. From [18] itRcan be seen that ǫ−1|ψˆ(tξ)|2dt/t → m(ξ) uniformly on Sn−1
ǫ
as ǫ → 0. Since ǫ−1|ψˆ(tξ)|2dt/t is cRontinuous on Sn−1 for each fixed ǫ > 0, the
ǫ
continuity of m follows. (cid:3)
R
Remark 2.12. Let 1 < p < ∞, w ∈ A . Suppose that g is bounded on Lp and
p ψ w
ψ is a radial function with ∞|ψˆ(tξ)|2dt/t = 1 for every ξ 6= 0. Then we have
0
kfk ≤ Ckg (f)k if g is also bounded on Lp′ . This is well-known and
p,w ψ p,w ψR w−p′/p
followsfromtheproofsofLemma2.7andProposition2.8. Also,thiscanbeproved
by applying arguments of [10, Chap. V, 5.6 (b)].
3. Discrete parameter Littlewood-Paley functions
Let ψ ∈ L1(Rn) with (1.1) and let ∆ be as in (1.9). We first give a criterion
ψ
for the boundedness of ∆ on Lp analogous to Theorem E.
ψ w
Theorem 3.1. Let B (ψ), H be as in Theorem E. Suppose that
ǫ ψ
(1) B (ψ)<∞ for some ǫ>0;
ǫ
(2) |ψˆ(ξ)|≤C|ξ|−δ for all ξ ∈Rn\{0} with some δ >0;
(3) H ∈L1(Rn).
ψ