The $n$ linear embedding theorem

0.17 MB·English

by  Hitoshi Tanaka

THE n LINEAR EMBEDDING THEOREM
HITOSHITANAKA
5
1
0 Abstract. Let σi, i = 1,...,n, denote positive Borel measures on Rd, let D denote the
usual collection of dyadic cubes in Rd and let K : D→[0,∞)be a map. In this paper we
2
giveacharacterizationofthenlinearembeddingtheorem. Thatis,wegiveacharacterization

n oftheinequality
a n n
J QX∈DK(Q)iY=1(cid:12)(cid:12)(cid:12)(cid:12)ˆQfidσi(cid:12)(cid:12)(cid:12)(cid:12)≤CiY=1kfikLpi(dσi)
0
interms of multilinearSawyer’s checking condition and discretemultinonlinear Wolff’s po-
1
tential,when1<pi<∞.

]
A
C
.
h
t
a 1. Introduction
m
[ The purpose of this paper is to investigate the n linear embedding theorem. We first fix

some notations. We will denote by D the family of all dyadic cubes Q = 2−k(m+[0,1)d),
1
v k ∈ Z, m ∈ Zd. Let K : D → [0,∞) be a map and let σi, i = 1,...,n, be positive Borel
4 measures on Rd. In this paper we give a necessary and sufficient condition for which the
0 inequality
3
2 n n
0 (1.1) K(Q) f dσ ≤C kf k ,
. ˆ i i i Lpi(dσi)
1 Q∈D i=1(cid:12) Q (cid:12) i=1
X Y(cid:12) (cid:12) Y
0 (cid:12) (cid:12)
5 to hold when 1<p <∞. (cid:12) (cid:12)
i
1
: For the bilinear embedding theorem, in the case 1 + 1 ≥ 1, Sergei Treil gives a simple
v p1 p2
proof of the following.
i
X
r Proposition 1.1 ([9, Theorem 2.1]). Let K : D → [0,∞) be a map and let σi, i = 1,2, be
a positive Borel measures on Rd. Let 1<p <∞ and 1 + 1 ≥1. The following statements are
i p1 p2
equivalent:
(a) The following bilinear embedding theorem holds:
2 2
K(Q) f dσ ≤c kf k <∞;
ˆ i i 1 i Lpi(dσi)
Q∈D i=1(cid:12) Q (cid:12) i=1
X Y(cid:12) (cid:12) Y
(cid:12) (cid:12)
(cid:12) (cid:12)
2010 Mathematics Subject Classification. 42B20,42B35(primary),31C45,46E35(secondary).
Key words and phrases. multinonlinear discrete Wolff’s potential; multilinear positive dyadic operator;
multilinearSawyer’schecking condition;nlinearembeddingtheorem.
TheauthorissupportedbytheFMSPprogramatGraduateSchoolofMathematicalSciences,theUniversity
ofTokyo,andGrant-in-AidforScientificResearch(C)(No.23540187), theJapanSocietyforthePromotionof
Science.
1
2 H.TANAKA
(b) For all Q∈D,
p′ 1/p′2
2
ˆQQX′⊂QK(Q′)σ1(Q′)1Q′p′1 dσ21/p′1 ≤c2σ1(Q)1/p1 <∞,
Moreover, theleaˆstQposQsX′i⊂blQeKc(Qan′)dσ2c(Qa′r)e1Qeq′uivaldeσn1t. ≤c2σ2(Q)1/p2 <∞.
1 2
Here, for each 1 < p < ∞, p′ denote the dual exponent of p, i.e., p′ = p , and 1 stands
p−1 E
for the characteristic function of the set E.
Proposition 1.1 was first proved for p = p = 2 in [4] by the Bellman function method.
1 2
Later in [3], this was proved in full generality. The checking condition in Proposition 1.1 is
called “the Sawyer type checking condition”, since this was first introduced by Eric T. Sawyer
in [5, 6].
To describe the case 1 + 1 <1, we need discrete Wolff’s potential.
p1 p2
Let µ and ν be positive Borelmeasures on Rd and let K : D→[0,∞) be a map. For p>1,
the discrete Wolff’s potential Wp [ν](x) of the measure ν is defined by
K,µ
p−1
1
Wp [ν](x):= K(Q)µ(Q) K(Q′)µ(Q′)ν(Q′) 1 (x), x∈Rd.
K,µ µ(Q)  Q
Q∈D Q′⊂Q
X X
 
The author prove the following.
Proposition 1.2 ([7, Theorem 1.3]). Let K : D → [0,∞) be a map and let σ , i = 1,2, be
i
positive Borel measures on Rd. Let 1<p <∞ and 1 + 1 <1. The following statements are
i p1 p2
equivalent:
(a) The following bilinear embedding theorem holds:
2 2
K(Q) f dσ ≤c kf k <∞;
ˆ i i 1 i Lpi(dσi)
Q∈D i=1(cid:12) Q (cid:12) i=1
X Y(cid:12) (cid:12) Y
(cid:12) (cid:12)
(b) For 1 + 1 + 1 =1, (cid:12) (cid:12)
r p1 p2
kWKp′2,σ2[σ1]1/p′2kLr(dσ1) ≤c2 <∞,
kWKp′1,σ1[σ2]1/p′1kLr(dσ2) ≤c2 <∞.
Moreover, the least possible c and c are equivalent.
1 2
In his excerent survey of the A theorem [2], Tuomas P. Hyt¨onen introduces another proof
2
of Proposition 1.1, which uses the “parallel corona” decomposition. In this paper, following
Hyt¨onen’s arguments in [2], we shall establish the following theorems (Theorems 1.3 and 1.4).
Theorem 1.3. Let K : D → [0,∞) be a map and let σ , i = 1,...,n, be positive Borel
i
measures on Rd. Let 1<p <∞ and n 1 ≥1. The following statements are equivalent:
i i=1 pi
P
THE n LINEAR EMBEDDING THEOREM 3
(a) The following n linear embedding theorem holds:
n n
K(Q) f dσ ≤c kf k <∞;
ˆ i i 1 i Lpi(dσi)
Q∈D i=1(cid:12) Q (cid:12) i=1
X Y(cid:12) (cid:12) Y
(cid:12) (cid:12)
(b) For all j =1,...,n and for(cid:12)all Q∈D(cid:12),
n n
K(Q′)σj(Q′) ˆ fidσi ≤c2σj(Q)1/pj kfikLpi(dσi) <∞.
Q′⊂Q i=1(cid:12) Q′ (cid:12) i=1
X Yi6=j(cid:12) (cid:12) Yi6=j
(cid:12) (cid:12)
(cid:12) (cid:12)
Moreover, the least possible c and c are equivalent.
1 2
Let the symmetric groupS be the set of all permutations of the set {1,...,n}, that is, the
n
set of all bijections from the set {1,...,n} to itself. Let K : D →[0,∞) be a map and let σ ,
i
i=1,...,n, be positive Borel measures on Rd. Let 1<p <∞ and n 1 <1.
i i=1 pi
Let φ∈S . Set P
n
1 1
+ =1,
r1φ pφ(1)
1 1 1
+ + =1,
r2φ pφ(1) pφ(2)
.
.
.
n−1
1 1
+ =1,
rnφ−1 i=1 pφ(i)
X
n
1 1
+ =1.
r p
φ(i)
i=1
X
Let, for Q∈D,
rφ−1
n 1
1
Kφ(Q):=K(Q)σ (Q) K(Q′) σ (Q′) ,
1 φ(1) σ (Q) φ(i) 
φ(1) Q′⊂Q i=1
X Y
 
let
rφ/rφ−1
n 2 1
1
Kφ(Q):=Kφ(Q)σ (Q) Kφ(Q′) σ (Q′)
2 1 φ(2) σ (Q) 1 φ(i) 
φ(2) Q′⊂Q i=2
X Y
and, inductively, for j =3,...,n−1,let 
n rjφ/rjφ−1−1
1
Kφ(Q):=Kφ (Q)σ (Q) Kφ (Q′) σ (Q′) .
j j−1 φ(j) σ (Q) j−1 φ(i) 
φ(j) Q′⊂Q i=j
X Y
 
Theorem 1.4. With the notation above, the following statements are equivalent:
(a) The following n linear embedding theorem holds:
n n
K(Q) f dσ ≤c kf k <∞;
ˆ i i 1 i Lpi(dσi)
Q∈D i=1(cid:12) Q (cid:12) i=1
X Y(cid:12) (cid:12) Y
(cid:12) (cid:12)
(cid:12) (cid:12)
4 H.TANAKA
(b) For all φ∈S ,
n
1/rnφ−1
Kφ (Q)1 ≤c <∞.
(cid:13)(cid:13) n−1 Q (cid:13)(cid:13) 2
(cid:13) QX∈D (cid:13)
(cid:13)(cid:13)  (cid:13)(cid:13)Lr(dσφ(n))
(cid:13) (cid:13)
Moreover, the least pos(cid:13)sible c and c are equivalen(cid:13)t.
1 2
Even though Theorems 1.3 and 1.4 both characterize the same n linear embedding theo-
rem, it seems that the characterizations are very different. In very recent paper [1], Timo S.
Ha¨nninen,TuomasP.Hyt¨onenandKangweiLigiveaunifiedapproachsaying“sequentialtest-
ing” characterization, when n = 2,3. Especially, our Theorem 1.4 with n = 3 is obtained in
[1, Theorem 1.16]. (An alternative form of another unified characterization has been simulta-
neously obtained by Vuorinen [10].) In [8], the author gives a characterization of the trilinear
embedding theorem interms of Theorem 1.3 and Propositions 1.1 and 1.2.
The letter C will be used for constants that may change from one occurrence to another.
2. Proof of the necessity
Inwhatfollowsweshallprovethe necessityoftheorems. ThenecessityofTheorem1.3,that
is, (b) follows from (a) at once if we substitute the test function f = 1 . So, we shall verify
j Q
the necessity of Theorem 1.4. We need a lemma (cf. Lemma 2.1 in [7]).
Lemma 2.1. Let σ be a positive Borel measure on Rd. Let 1<s<∞ and {α } ⊂[0,∞).
Q Q∈D
Define, for Q ∈D,
0
s
α
Q
A := 1 dσ,
1 ˆ  σ(Q) Q
Q0 QX⊂Q0
  s−1
1
A2 := αQ αQ′ ,
σ(Q) 
QX⊂Q0 QX′⊂Q
  s
1 (x)
Q
A3 :=ˆ sup σ(Q) αQ′ dσ(x).
Q0Q⊂Q0 Q′⊂Q
X
 
Then
A1 ≤c(s)A2, A2 ≤c(s)s−11A3 and A3 ≤(s′)sA1.
Here,
s, 1<s≤2,
c(s):= s−1
((s(s−1)···(s−k))s−k−1 , 2<s<∞,
where k =⌈s−2⌉ is the smallest integer greater than s−2.
We will use fdσ to denote the integral average σ(Q)−1 fdσ. The dyadic maximal
Q Q
operator Mσ isffldefined by ´
D
1 (x)
Mσf(x):= sup Q |f(y)|dσ(y).
D σ(Q) ˆ
Q∈D Q
THE n LINEAR EMBEDDING THEOREM 5
Suppose that (a) of Theorem 1.4. Then, for φ∈S ,
n
n n
(2.1) K(Q) ˆ fφ(i)dσφ(i) ≤c1 kfφ(i)kLpφ(i)(dσφ(i)).
Q∈D i=1(cid:12) Q (cid:12) i=1
X Y(cid:12) (cid:12) Y
(cid:12) (cid:12)
Recall that 1 + 1 =1. By du(cid:12)ality, we see t(cid:12)hat
r1φ pφ(1)
rφ
n 1 n
rφ rφ
K(Q) f dσ 1 dσ ≤c 1 kf k 1 ,
ˆRdQ∈D i=2(cid:12)ˆQ φ(i) φ(i)(cid:12) Q φ(1) 1 i=2 φ(i) Lpφ(i)(dσφ(i))
X Y(cid:12) (cid:12) Y
 (cid:12) (cid:12) 
which implies by Lemma 2.1(cid:12) (cid:12)
n
K(Q)σ (Q) f dσ
φ(1) ˆ φ(i) φ(i)
Q∈D i=2(cid:12) Q (cid:12)
X Y(cid:12) (cid:12)
(cid:12) (cid:12) rφ−1
(cid:12) (cid:12)n 1
1
× K(Q′)σ (Q′) f dσ
σφ(1)(Q)Q′⊂Q φ(1) i=2(cid:12)ˆQ′ φ(i) φ(i)(cid:12)
X Y(cid:12) (cid:12)
 n (cid:12) (cid:12)
rφ rφ (cid:12) (cid:12)
≤Cc 1 kf k 1 .
1 φ(i) Lpφ(i)(dσφ(i))
i=2
Y
It follows from this inequality that
n
Kφ(Q) g dσ
1 ˆ φ(i) φ(i)
Q∈D i=2(cid:12) Q (cid:12)
X Y(cid:12) (cid:12)
(cid:12) (cid:12) rφ−1
(cid:12) n (cid:12) n 1
1
= K(Q)σ (Q) g dσ K(Q′) σ (Q′)
φ(1) ˆ φ(i) φ(i) σ (Q) φ(i) 
Q∈D i=2(cid:12) Q (cid:12) φ(1) Q′⊂Q i=1
X Y(cid:12) (cid:12) X Y
n (cid:12)(cid:12) (cid:12)(cid:12) 1/r1φ 
= K(Q)σ (Q) σ (Q) g dσ
φ(1) φ(i) φ(i) φ(i)
Q∈D i=2 (cid:12) Q (cid:12)
X Y (cid:12) (cid:12)
(cid:12) (cid:12) rφ−1
(cid:12) n (cid:12) 1/rφ 1
1 1
× K(Q′)σ (Q′) σ (Q′) g dσ
σ (Q) φ(1) φ(i) φ(i) φ(i) 
φ(1) Q′⊂Q i=2 (cid:12) Q (cid:12)
X Y (cid:12) (cid:12)
 n (cid:12) (cid:12) 
≤ K(Q)σ (Q) Mσφ(i)g 1/r1φ dσ (cid:12) (cid:12)
φ(1) ˆ D φ(i) φ(i)
Q∈D i=2 Q
X Y (cid:0) (cid:1)
rφ−1
n 1
× 1 K(Q′)σ (Q′) Mσφ(i)g 1/r1φ dσ
σφ(1)(Q)Q′⊂Q φ(1) i=2ˆQ′ D φ(i) φ(i)
X Y (cid:0) (cid:1)
 n 
≤Ccr1φ kMσφ(i)g k
1 i=2 D φ(i) Lpφ(i)/r1φ(dσφ(i))
Y
n
rφ
≤Cc 1 kg k ,
1 i=2 φ(i) Lpφ(i)/r1φ(dσφ(i))
Y
where we have used the boundedness of dyadic maximal operators. Thus, we obtain
n n
(2.2) Kφ(Q) f dσ ≤Ccr1φ kf k .
Q∈D 1 i=2(cid:12)ˆQ φ(i) φ(i)(cid:12) 1 i=2 φ(i) Lpφ(i)/r1φ(dσφ(i))
X Y(cid:12) (cid:12) Y
(cid:12) (cid:12)
(cid:12) (cid:12)
6 H.TANAKA
Notice that
rφ rφ
i−1 + i−1 =1, i=2,...,n−1,
(2.3)  riφ pφ(i)
rnφ−1 + rnφ−1 =1.
r p
φ(n)
Bythesamemannerastheabovebutstartingfrom(2.2),insteadof (2.1),andusing(2.3)with
i=2, we obtain
n n
Kφ(Q) f dσ ≤Ccr2φ kf k .
Q∈D 2 i=3(cid:12)ˆQ φ(i) φ(i)(cid:12) 1 i=3 φ(i) Lpφ(i)/r2φ(dσφ(i))
X Y(cid:12) (cid:12) Y
(cid:12) (cid:12)
By being continued inductively(cid:12) until the n−(cid:12)1 step, we obtain
Kφ (Q) f dσ ≤Ccrnφ−1kf k .
Q∈D n−1 (cid:12)ˆQ φ(n) φ(n)(cid:12) 1 φ(n) Lpφ(n)/rnφ−1(dσφ(n))
X (cid:12) (cid:12)
(cid:12) (cid:12)
Notice that the last equation(cid:12)of (2.3). Then(cid:12)by duality
Kφ (Q)1 ≤Ccrnφ−1
(cid:13) n−1 Q(cid:13) 1
(cid:13)(cid:13)QX∈D (cid:13)(cid:13)Lr/rnφ−1(dσφ(n))
(cid:13) (cid:13)
(cid:13) (cid:13)
and, hence, (cid:13) (cid:13)
1/rnφ−1
Kφ (Q)1 ≤Cc ,
(cid:13)(cid:13) n−1 Q (cid:13)(cid:13) 1
(cid:13) QX∈D (cid:13)
(cid:13)(cid:13)  (cid:13)(cid:13)Lr(dσφ(n))
which completes the nece(cid:13)ssity of Theorem 1.4. (cid:13)
(cid:13) (cid:13)
3. Proof of the sufficiency
In what follows we shall prove the sufficiency of theorems.
Let Q ∈D be taken large enough and be fixed. We shall estimate the quantity
0
n
(3.1) K(Q) f dσ ,
ˆ i i
QX⊂Q0 iY=1(cid:18) Q (cid:19)
wherefi ∈Lpi(dσi)isnonnegativeandissupportedinQ0. Wedefinethecollectionofprincipal
cubes F for the pair (f ,σ ), i=1,...,n. Namely,
i i i
∞
F := Fk,
i i
k=0
[
where F0 :={Q },
i 0
Fk+1 := ch (F)
i Fi
F[∈Fik
and ch (F) is defined by the set of all “maximal” dyadic cubes Q⊂F such that
Fi
f dσ >2 f dσ .
i i i i
Q F
THE n LINEAR EMBEDDING THEOREM 7
Observe that
σ (F′)
i
F′∈cXhFi(F)
−1
≤ 2 f dσ f dσ
i i ˆ i i
(cid:18) F (cid:19) F′∈cXhFi(F) F′
−1
σ (F)
i
≤ 2 f dσ f dσ = ,
i i ˆ i i 2
(cid:18) F (cid:19) F
which implies
σ (F)
(3.2) σ (E (F)):=σ F \ F′ ≥ i ,
i Fi i  2
F′∈c[hFi(F)
 
where the sets E (F), F ∈ F , are pairwise disjoint. We further define the stopping parents,
Fi i
for Q∈D,
π (Q):=min{F ⊃Q: F ∈F },
Fi i
(π(Q):=(πF1(Q),...,πFn(Q)).
Then we can rewrite the series in (3.1) as follows:
= .
QX⊂Q0 (F1,...,FnX)∈(F1,...,Fn) XQ:
π(Q)=(F1,...,Fn)
We notice the elementary fact that, if P,R ∈ D, then P ∩R ∈ {P,R,∅}. This fact implies, if
π(Q)=(F ,...,F ), then
1 n
Q⊂F ⊂···⊂F for some φ∈S .
φ(1) φ(n) n
Thus, for fixed φ∈S , we shall estimate
n
n
(3.3) K(Q) f dσ .
ˆ φ(i) φ(i)
(Fφ(i))X∈(Fφ(i)): XQ: Yi=1(cid:18) Q (cid:19)
Fφ(1)⊂···⊂Fφ(n)π(Q)=(Fφ(i))
Proof of (a) of Theorem 1.3. It follows that, for fixed F ∈F ,
φ(n) φ(n)
n
K(Q) f dσ
ˆ φ(i) φ(i)
Fφ(1)⊂X···⊂Fφ(n) XQ: Yi=1(cid:18) Q (cid:19)
π(Q)=(Fφ(i))
n−1
≤ 2 f dσ K(Q)σ (Q) f dσ .
φ(n) φ(n) φ(n) ˆ φ(i) φ(i)
Fφ(n) !Fφ(1)⊂X···⊂Fφ(n) XQ: Yi=1(cid:18) Q (cid:19)
π(Q)=(Fφ(i))
We need two observations. Suppose that F ⊂ ··· ⊂ F and π(Q) = (F ). Let
φ(1) φ(n) φ(i)
i=1,...,n−1. If F′ ∈ch (F ) satisfies F′ ⊂Q. Then
Fφ(n) φ(n)
F , when f′ ∈/ F ,
(3.4) π π (F′) = φ(n) φ(i)
Fφ(n) Fφ(i) (F′, when f′ ∈Fφ(i).
(cid:0) (cid:1)
By this observation, we define
chφ(i) (F ):={F′ ∈ch (F ): F′ satisfies (3.4)}.
Fφ(n) φ(n) Fφ(n) φ(n)
8 H.TANAKA
We further observe that, when F′ ∈chφ(i) (F ), we can regard f as a constant on F′ in
F φ(n) φ(i)
φ(n)
the above integrals, that is, we can replace f by fFφ(n) in the above integrals, where
φ(i) φ(i)
fφF(φi()n) :=fφ(i)1EFφ(n)(Fφ(n))+F′∈chFφ(Xφi()n)(Fφ(n))(cid:18) F′fφ(i)dσφ(i)(cid:19)1F′.
It follows from (b) of Theorem 1.3 that
n−1
K(Q)σ )(Q) fFφ(n)dσ
φ(n) ˆ φ(i) φ(i)
Fφ(1)⊂X···⊂Fφ(n) XQ: iY=1(cid:18) Q (cid:19)
π(Q)=(Fφ(i))
n−1
≤c2σφ(n)(Fφ(n))1/pφ(n) kfφF(φi()n)kLpφ(i)(dσφ(i)).
i=1
Y
Thus, we obtain
n−1
(3.3)≤Cc2Fφ(nX)∈Fφ(n) iY=1kfφF(φi()n)kLpφ(i)(dσφ(i)) Fφ(n)fφ(n)dσφ(n)!σφ(n)(Fφ(n))1/pφ(n).
Since n 1 ≥1, we can select the auxiliary parameterss , i=1,...,n−1, that satisfy
i=1 pφ(i) φ(i)
P
n−1
1 1
+ =1 and 1<p ≤s <∞.
s p φ(i) φ(i)
φ(i) φ(n)
i=1
X
It follows from Ho¨lder’s inequality with exponents s ,...,s ,p that
φ(1) φ(n−1) φ(n)
n−1 1/sφ(i)
(3.3)≤Cc kfFφ(n)ksφ(i)
2  φ(i) Lpφ(i)(dσφ(i))
iY=1 Fφ(nX)∈Fφ(n)
 pφ(n) 1/pφ(n)
× f dσ σ (F )
Fφ(nX)∈Fφ(n) Fφ(n) φ(n) φ(n)! φ(n) φ(n) 
n−1 1/pφ(i) 
≤Cc kfFφ(n)kpφ(i)
2  φ(i) Lpφ(i)(dσφ(i))
iY=1 Fφ(nX)∈Fφ(n)
 pφ(n) 1/pφ(n)
× f dσ σ (F )
Fφ(nX)∈Fφ(n) Fφ(n) φ(n) φ(n)! φ(n) φ(n) 
 
=:Cc (I )×···×(I ),
2 1 n
where we have used k·klpφ(i) ≥k·klsφ(i).
For (I ), using σ (F )≤2σ (E (F )) (see (3.2)), the fact that
n φ(n) φ(n) φ(n) Fφ(n) φ(n)
f dσ ≤ inf Mσφ(n)f (y)
Fφ(n) φ(n) φ(n) y∈Fφ(n) D φ(n)
THE n LINEAR EMBEDDING THEOREM 9
and the disjointness of the sets E (F ), we have
Fφ(n) φ(n)
1/pφ(n)
(I )≤C Mσφ(n)f pφ(n) dσ
n  ˆ D φ(n) φ(n)
Fφ(nX)∈Fφ(n) EFφ(n)(Fφ(n))(cid:0) (cid:1)
≤Cˆ MDσφ(n)fφ(n) pφ(n) dσφ(n) 1/pφ(n) ≤Ckfφ(n)kLpφ(n)(dσφ(n)).
(cid:20) Q0 (cid:21)
(cid:0) (cid:1)
It remains to estimate (I ), i=1,...,n−1. It follows that
i
(Ii)pφ(i) = ˆ fφp(φi()i)dσφ(i)
Fφ(nX)∈Fφ(n) EFφ(n)(Fφ(n))
pφ(i)
+ f dσ σ (F′).
φ(i) φ(i) φ(i)
Fφ(nX)∈Fφ(n)F′∈chFφ(Xi)) (Fφ(n))(cid:18) F′ (cid:19)
φ(n)
By the pairwise disjointness of the sets E (F ), it is immediate that
Fφ(n) φ(n)
fpφ(i)dσ ≤kf kpφ(i) .
Fφ(nX)∈Fφ(n)ˆEFφ(n)(Fφ(n)) φ(i) φ(i) φ(i) Lpφ(i)(dσφ(i))
For the remaining double sum, there holds by the uniqueness of the parent
pφ(i)
f dσ σ (F′)
φ(i) φ(i) φ(i)
Fφ(nX)∈Fφ(n)F′∈chFφX(n)(Fφ(n)):(cid:18) F′ (cid:19)
F′ satisfies(3.4)
pφ(i)
≤2 f dσ σ (F′)
φ(i) φ(i) φ(i)
Fφ(nX)∈Fφ(n) F∈XFφ(i): F′∈chFφX(n)(Fφ(n)):(cid:18) F′ (cid:19)
πFφ(n)(F)=Fφ(n) πF (F′)=F
φ(i)
pφ(i)
≤2 2 f dσ σ (F)
φ(i) φ(i) φ(i)
F∈XFφ(i)(cid:18) F (cid:19)
≤CkMσφ(i)f kpφ(i) ≤Ckf kpφ(i) .
D φ(i) Lpφ(i)(dσφ(i)) φ(i) Lpφ(i)(dσφ(i))
Altogether, we obtain
n
(3.3)≤Cc2 kfφ(i)kLpφ(i)(dσφ(i)).
i=1
Y
This yields (a) of Theorem 1.3.
Proof of (a) of Theorem 1.4. We shall estimate (3.3) by use of multinonlinear Wolff’s
potential. We first observe that if F ∈ F , i = 1,...,n, satisfy F ⊂ ··· ⊂ F and,
φ(i) φ(i) φ(1) φ(n)
for some Q∈D, π(Q)=(F ), then
φ(i)
(3.5) π (F )=F for all 1≤i<j ≤n.
Fφ(j) φ(i) φ(j)
Fix F ∈F , i=1,...,n, that satisfy (3.5). Then
φ(i) φ(i)
n
K(Q) f dσ
ˆ φ(i) φ(i)
Q: i=1(cid:18) Q (cid:19)
X Y
π(Q)=(Fφ(i))
n n
≤ 2 f dσ K(Q) σ (Q).
φ(i) φ(i) φ(i)
i=1 Fφ(i) ! Q: i=1
Y X Y
π(Q)=(Fφ(i))
10 H.TANAKA
Recall that
j
1 1
+ =1, j =1,...,n−1,
(3.6) 1rjφ+ nXi=1 p1φ(i)=1.
r p
φ(i)
i=1
Inthe followingestimates, Fφ(1X) runs overall Fφ(1) ∈Fφ(1) that satisfy (3.5)for fixedFφ(i) ∈
F , i=2,...,n.
φ(i) P
n
f dσ K(Q) σ (Q)
φ(1) φ(1) φ(i)
FXφ(1) Fφ(1) ! XQ: Yi=1
π(Q)=(Fφ(i))
n
≤ f dσ K(Q) σ (Q)
φ(1) φ(1) φ(i)
FXφ(1) Fφ(1) !Q⊂XFφ(1) Yi=1
= f dσ σ (F )1/pφ(1)
φ(1) φ(1) φ(1) φ(1)
FXφ(1) Fφ(1) !
n
×  K(Q) σφ(i)(Q)1Q dσφ(1)σφ(1)(Fφ(1))1/r1φ,
Fφ(1) Q⊂XFφ(1) Yi=2
   
where we have used (3.6) with j =1. By Ho¨lder’s inequality, we have further that
pφ(1) 1/pφ(1)
≤ f dσ σ (F )
FXφ(1) Fφ(1) φ(1) φ(1)! φ(1) φ(1) 
  rφ 1/r1φ
n 1
× K(Q) σ (Q)1 dσ σ (F ) .
   φ(i) Q φ(1) φ(1) φ(1) 
FXφ(1) Fφ(1) Q⊂XFφ(1) iY=2
 
     
By the same way as the estimate of (I ), we see that the last term is majorized by
n
rφ 1/r1φ
n 1
C K(Q) σ (Q)1 dσ .
ˆ  φ(i) Q φ(1)
Fφ(2) Q⊂XFφ(2) Yi=2
 
   
By Lemma 2.1, we have further that
1/rφ
n 1
≤C Kφ(Q) σ (Q) .
 1 φ(i) 
Q⊂XFφ(2) Yi=2
 
By (2.3), we notice that
1 1 1
(3.7) + = , i=2,...,n−1.
riφ pφ(i) riφ−1
Inthefollowingestimates, runsoverallF ∈F thatsatisfy,forfixedF ∈F ,
Fφ(2) φ(2) φ(2) φ(i) φ(i)
i=3,...,n,
P
(3.8) π (F )=F for all 2≤i<j ≤n.
Fφ(j) φ(i) φ(j)