Yabuta Journal of Inequalities and Applications (2015) 2015:107 DOI 10.1186/s13660-015-0630-7

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Triebel-Lizorkin space boundedness of rough singular integrals associated to surfaces

Kôzô Yabuta*

"Correspondence: kyabuta3@kwansei.ac.jp Research Center for Mathematical Sciences, Kwansei Gakuin University, Gakuen 2-1, Sanda, 669-1337, Japan

Abstract

In the present paper, we consider the boundedness of the rough singular integral operator T^<h0 along a surface V = {x = 0(|y|)y/|y|} on the Triebel-Lizorkin space

I for ^ e H](Sn 1)and ^ belonging to some class WFa(Sn 1), which relates to

the Grafakos-Stefanov class.

MSC: Primary 42B20; secondary 42B25; 47G10

Keywords: singular integrals; Triebel-Lizorkin spaces; rough kernel

ft Spri

ringer

1 Introduction

Let R" (n > 2) be the «-dimensional Euclidean space and S"-1 be the unit sphere in R" equipped with the induced Lebesgue measure da = da(■). Suppose that ^ e L1(Sn-1) satisfies the cancelation condition

f Q(y') da(y') =0. (1.1)

For a suitable function 0 and a measurable function h on [0, to), we denote by the singular integral operator along the surface

V = {x = 0(|y|)y':y e R"}

defined as follows:

TaMf (x) = p.v. i hman {y')f (x - 0(|y|)y') dy (1.2)

J Rn |y|

for f in the Schwartz class S(Rn). If 0 = 1, then is the classical singular integral operator T^,h, which is defined by

Ta,hf (x) = p.v. i hmQn {y')f (x -y) dy. (.3)

J R" |y|"

When h = 1, we denote simply Ta,ht0 and T^,h by and Ta, respectively.

The Lp boundedness of singular integrals along the surface has attracted the attention of many authors [1-3], etc. There are several papers concerning rough kernels associated to surfaces as above [4-6]. As one of them, we count the following one.

© 2015 Yabuta; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the originalworkis properly credited.

Theorem A ([5]) Let h e AY for some y > 2,1 <p < to, O e H1(Sn-1). Let $ be a nonnegative C1 function on (0, to) satisfying

(i) $(t) is strictly increasing and $(2t) > X$(t) for all t >0 and some X >1,

(ii) $(t) satisfies a doubling condition $(2t) < c$(t) for all t >0 and some c >1,

(iii) $'(t) > C1$(t)/tfor all t >0 and some C1. Then is bounded on Lp(Rn).

This is, in fact, stated in the more general setting, i.e., for a weighted case (Theorem 1 and Corollary 1 in [5]), but we state this as above for our purpose and for the sake of simplicity. We note here that condition (i) follows from (iii).

On the other hand, Triebel-Lizorkin space boundedness of rough singular integrals was also investigated by many authors, see [7, 8] and [9].

Before stating the following result, let us recall the definitions of some function spaces. First we give the definition of the Hardy space H 1(Sn-1):

— « e L1(Sn~1) If |lHi(sn-i) =

/ «(/)Pr(.)iy) da (/)

< œ h

L1(Sn-1)

where Pry (X) denotes the Poisson kernel on Sn defined by

Pry (X ^ —

1-r2 \ry' -x'l

-, 0 < r <1 and X,y1 e Sn 1.

For 1 < y <to, Ay (R+) is the collection of all measurable functions h :[0, to) ^ C satisfying

II Ay — SUp

jL |h(t)|>

Note that

Lto(R+) = Ato(R+) C Ap(R+) C Aa(R+) for a < ¡3,

and all these inclusions are proper.

As a result of boundedness on Triebel-Lizorkin spaces, we cite the following one, which is somewhat different from our setting, but closely related.

Theorem B ([9]) Let O e H1(Sn-1) satisfy the cancelation condition (1.1) and h e AY for some 1 < y <to. Let P = (P1, P2,...,Pd) be real polynomials in y. Then, for the singular integral

W (x) = p.v./ ^^f(x - P(y)) dy

(i) for a e R and |p - 21 < min(2, Y^) and | q - 21 < min(2, Yy), there exists a constant c > 0 such that \\Ta,P^hf llfp,q(Rd) < C\f Wpf^Rd);

(ii) for a e R and | - - 21 < min(±, -Yj) and 1 < q < to, there exists a constant C > 0 such

that \\Tnp,hf \^,q(K.d) < CWf Wb«^)•

Remark 1 We think that there is a gap in the proof of part (i) in the above theorem. Their proof works in the same region as in our Theorem 1.1 below.

Besides H1(Sn-1), there is another class of kernels which leads to Lp and Triebel-Lizorkin space boundedness of singular integral operators TQ,h. It is closely related to the class Fa introduced by Grafakos and Stefanov [10]. We say Q e WF3 = WF3(Sn-1) if

nilwr, := suJf i \a(y')a(z')\log\ 2e da(y')da(Z)

t'esn-l\ JSn-1JSn-1 1 \y - z) ■ § 1

§ 'eSn < TO.

We note that Ur>1 Lr(Sn-1) c WFp2(Sn-1) c WFPl(Sn-1) for 0 < ¡¡1 < ¡2 < to.

About the inclusion relation between Fe1 (Sn-1) and WFp2 (Sn-1), the following is known: when n = 2, Lemma 1 in [11] shows F (S1) c WFp (S1). It is also known that W^2a (S1) \ (Fa(S1) UH 1(S1))= 0, cf. [12].

Theorem C ([12]) Let h e Ay for some 1 < y <to. Suppose that Q e WFp = WFp(Sn-1) for some ¡3 > max(Y', 2), and it satisfies the cancelation condition (1.1). Then the singular integral operator TQih is bounded on Fpaq(Rn) if a e R, and (1/p, 1/q) belongs to the interior of the parallelogram P1P2P3P4, where P1 = , ^^), P2 = (^ + ^^ () -y_ ), ), P3 = (1 - max(p, 1 - max(p), andP4 = () - max2|^ (Y - Y7), 1 - maw21).

Let us recall the definitions of the homogeneous Triebel-Lizorkin spaces ip>q = Fpr(Rn)

). For 0< p, q <to (p = to) and a e R,

F0,„(Rn) is defined by

am n pr ~pr

and the homogeneous Besov spaces Bpr = B'

FpqK Rn)

= f e 5 '1

J22kaq№k *f lr

and Bap,q(Rn) is defined by

BFp ,q Rn

) = ff e S'(Rn): \f = (E 2kaq№k *f ill)^ < TO, (1.6)

keZ ' '

where 5'(Rn) denotes the tempered distribution class on Rn, ^k(f) = $(2-k§) for k e Z and $ e CTO(Rn) is a radial function satisfying the following conditions:

(i) 0 < $ < 1;

(ii) supp $ c {§ : 1/2 <|§|<2};

(iii) $ > c >0 if 3/5 < |f | < 5/3;

(iv) E $(2-§) = 1 (§ =0).

The inhomogeneous versions of Triebel-Lizorkin space and Besov space, which are denoted by Fp^R") and Bapq(Rn) respectively, are obtained by adding the term ||$0 * f \\p to the right-hand side of (1.5) or (1.6) with ^keZ replaced by ^where $0 e S(R"), supp $o C{ : if | < 2}, and $o(f )> c > 0 if | < 5/3.

The following properties of the Triebel-Lizorkin space and the Besov space are well known. Let 1 < p, q < to, a e R, and 1/p + 1/p' = 1,1/q + 1/q' = 1:

(a) 10,2 = B 0,2= L2, Ip,2 = Lp and

Fapp = Bapp for 1 <p < to and Ico,2 = BMO;

(b) F;>q ~ FaM n Lp and

U + If IIlp (a > 0);

(c) El q ~ Ba q n Lp and

IB,q Bq + \\f \\lp (a>0);

(d) FpqY = Fv and F,/ = FpV ;

(e) (%<)* = B-q' and (^J = B-/q, ;

(f) (Fai F"2 ) = Ba

(i) \p,q\, Fp,q^ e ,q Bp,q

(ai = a2,0 <p < to, 0 < q,qi,q2 <TO,a = (1 - 0)a + Qa2,0 < Q < l).

See [13] and [14] for more properties of Ip and BapA. See Triebel [14], p.64 and p.244, for (f).

Now we can state our first result.

Theorem 1.1 Let $ be a positive increasing function on (0, to) satisfying $(2t) < ci$(t) (t > 0) for some c1 > 1

<p(t) = $(t)/(t$'(t)) e LTO(0, to).

(1.10)

Let h e AY for some 1 <y <to. Suppose ^ e H1(Sn 1) satisfying the cancelation condition (1.1). Then

(i) Ta,hi$ is bounded on F(Rn) for a e R andp, q with (p, 1) belonging to the interior of the octagon P1P2R2P3P4P5R4 P6 (hexagon P1P2P3P4P5P6 in the case 1 < y < 2), where P1 = (1--^tt, 1--^tt), P2 = (1,1--^tt), P3 = (1 + —^tt, 1),

1 v2 max{2,Y;^ 2 max^y'}7' 2 v2'2 max^y'}7' 3 v2 max^y'}' 2n

p4 = (1 +

1 __ _

2 1 max(2,y')' 2 1 max(2,y'

1 - 1 + ■ w n +

r)> p5 = (^,2 +

2 ' 2 max(2,y ')

2 max(2,y'), 2 )

^2 = (1 - ^ )> = (517,1- 27 );

(ii) 7^,$ is bounded on Bpq (R") for a e R and p, q satisfying 12 - p | < min( 2, 77) and 1 < q < to.

See Figures 1 and 2 for the conclusion (i) ofTheorem 1.1.

Example 1 As typical examples of $ satisfying conditions (1.9) and (1.10), we list the following three: ta logp(1 + t) (a > 0, p > 0), (2t2 - 2t + 1)t1+a (a > 0), and $(t) = 2t2 + t (0 < t < 2), $(t) = 2t2 + tsin t (t > f). Note that linear combinations with positive coefficients of functions $'s satisfying the above two conditions also satisfy them, cf. [15].

We shall state our following result, which relates to two function spaces L(logL)(Sn 1) and the block spaces BfV-1). Let L(logL)a(Sn-1) (for a > 0) denote the class of all measurable functions Q on Sn-1 which satisfy

\\Q\\L(log L)a(Sn-1) = \Q(y;)\ lOga (2+ \Q(y')\) da (y') < TO.

J Sn-1

Denote by L(logL)(Sn-1) for L(logL)1(Sn-1). A well-known fact is L(logL)(Sn-1) c H1 (Sn-1).

Next, we turn to the block space Bqo,v)(Sn-1). A q-block on Sn-1 is an Lq(Sn-1) (1 < q <to) function b which satisfies

(i) supp b c I;

(1.11)

(ii) llbllq <|I|-1/q,

where |I| = a (I), and I = B(x'0,00) n Sn-1 is a cap on Sn-1 for some x'0 e Sn-1 and 00 e (0,1]. For 1 < q < to and v > -1, the block space Bqo,v)(Sn-1) is defined by

BqM(Sn-1) = Jq e L1(Sn-1); q = E Xjbj, Mfv) (iXjl) < TO, (1.12)

where Xj e C and bj is a q-block supported on a cap Ij on Sn \ and

K'v\ {J = E ixj ^1+Wv+1)( ii-i-1)}. (i.i3)

For n e Bfv)(Sn-1), denote

limi^-i) = inf|M<0'v)({Xj}); n = ^Xjbj,bj is a q-block J.

Then || • ilB(o,v)(Sn-1) is a norm on the space Bqo,v)(Sn-1), and (Bqo,v)(Sn-1), || • ilB(o,v)(Sn-1)) is a Banach space.

Historically, the block spaces in Rn originated in the work of Taibleson and Weiss on the convergence of the Fourier series in connection with the developments of the real Hardy spaces. The block spaces on Sn-1 were introduced by Jiang and Lu [16] in studying the homogeneous singular integral operators. For further information about the theory of spaces generated by blocks and its applications to harmonic analysis, see the book [17] and survey article [18]. The following inclusion relations are known:

(a) BfV1)(Sn-1) c B(<W(Sn-1) if V1> v2 > -1;

(b) B<0v) (Sn-1) c B<0'v) (Sn-1) if 1 < q2 < q1 for any v > -1;

(c) y Lp (Sn-1) c B°v) (Sn-1) for any q > 1, v > -1;

(d) y Bqo'v)(Sn-1) c ULq(Sn-1) for any v >-1; (1.14)

q>1 q>1

(e) Bq0v)(Sn-1) c H1(Sn-1) + L(logL)1+v(Sn-1) for any q > 1, v > -1;

(f) y B(o0\Sn-1) c H 1(Sn-1).

The following theorem shows that if n belongs to L logL(Sn-1) or block spaces, then we can get better results than Theorem 1.1.

Theorem 1.2 Let $ be a positive increasing function on (0, to) satisfying the same condition as in Theorem 1.1. Let h e Ay for some 1 < y <to, and n e L1(Sn-1) satisfy the cancelation condition (1.1). Then if n e L(logL)(Sn-1) U (U1<q<TOBq0,0)(Sn-1)), then

(i) Tn,hi$ is bounded on Fp>q(Rn) for a e R andp, q with (p, 1) belonging to the interior of the octagon Q1Q2R2P3Q3Q4R4P6 (hexagon Q1 Q2P3Q3Q4P6 in the case 1 < y < 2), where Q1 = (0,0), Q2 = (^, 0), Q3 = (1,1), Q, = (±, 1), P3 = (2 + maxiT/), 2),

p6 = (21 maxfe/), r2 = (1 - 2Y, 2Y), andR4 = (2Y, 1 - 2Y);

(ii) Tn,hi$ is bounded on Bpq(Rn) for a e R and 1 <p, q < to.

See Figures 3 and 4 for the conclusion of Theorem 1.2 for the cases 1 < y <2 and 2 < y < to, respectively.

As a corresponding result to Theorem C, we have the following theorem.

Figure 3 (1< y <2). l/« n

(0,1) Qi «3

Theorem 1.3 Let 0 be a positive increasing function on (0, to) satisfying the same condition as in Theorem 1.1. Let h e AY for some 1 < y <to. Suppose ^ e WFj = WFj(Sn-1) for some j > max(y ',2), and it satisfies the cancelation condition (1.1). Then

(i) the singular integral operator is bounded on Fp;q(Rn), if a e R and (p, 1) belongs to the interior of the octagon QiQ2R2P3Q3Q4R4P6 (hexagon Q1Q2P3 Q3Q4P6 in the case 1<y < 2), where Q1 = (mai(^, s^),

p. _ (1 . max(y;,2) / 1 max(Y;,2K -r, _ /1 1 _ 1 1 \

Q2 = V + j (2 y' " 2 j ), ' 3 = (2 + max(y',2) j , 2)

S/i max(Y1 max(Y;,2) ^ ^ /1 max(Y;,2) 1 1\ 1 max(Y;,2) \

3 = (1--2ji --2ji ), Q4 = (Y--j—(Y - 2),1--2ji ),

t> - ¡1 1 , 1 1\ -J? _ n 1 max(Y;,2) 1 max(Y;,2) \ » P6 = (2 - max(Y ',2) + j, 2), R2 = (1 - 57 - 2jY' , 2Y + 2jY' >' and

-n _ max(Y;,2) ^__1__max(Y;,2)\.

R4 = (2y + 2jY' ,1 2y 2jY' );

(ii) TQM is bounded on Bapq (Rn), if a e R, max2(jf,2) < p < 1 - max2(jf,2) and 1 < q < to.

This improves Theorem C sufficiently. See Figures 3 and 4 for the conclusion (i) of Theorem 1.3.

The proofs of Theorems 1.1 and 1.3 will be given in Sections 2 and 3, respectively, and the proof of Theorem 1.2 will be given in Section 4. The letter C will denote a positive constant that may vary at each occurrence but is independent of the essential variables.

2 Proof of Theorem 1.1 2.1 Some lemmas

In [19], the following atom-decomposition of H1(Sn-1) was given. If n e H1(Sn-1) satisfying (1.1), then

n = E kiai, (.1)

where \Xj | < C||n||H1(SH-1) and each aj is a regular H1(Sn-1) atom. Afunction a on Sn-1 is called regular TO-atom in H1(Sn-1) if there exist Z e Sn-1 and p e (0,2] such that

(i) supp(a) c Sn-1 n B(Z, p), where B(Z, p) = {y e Rn : \y - Z \ < p};

(ii) ||aNi~ < p-n+1;

(iii) fSn-1 a(y) da(y) = 0.

Let a be a regular TO-atom. When n > 3, set

Ea(s,f') = (1-s2)^X(-1,1)(s) f a(s,V^y)da(y), (2.2)

and when n = 2, set 1

ea(s, f 0 = ^^ X(-1,1)(s) [a(s, VT-2) + (2.3)

V1 -s2

Next we prepare two lemmas, whose proofs can be found in Fan and Pan [4].

Lemma 2.1 Let n be a regular TO-atom in H1(Sn-1) (n > 3). Then there exists a constant c >0, independent of n, such that cEn(s, f') is an TO-atom in H1(R). That is, cEn(s, f') satisfies

ycEn^LTO < , suppEn c (f1 - 2r(f;), f1 + 2r(f;)) and

) (2.4)

[ En(s, f') ds =0, J R

where r(f') = \f \-1\ATf \ andAx(f) = (t2f1,rf2,...,rfn).

Lemma 2.2 Let n be a regular TO-atom in H1(S1). Then, for 1<q <2, there exists a constant c >0, independent of n, such that cen(s, f') is a q-atom in H1(R), the center of whose support is f1 and the radius r(f') = \f \-1(t4f12 + t2f|)1/2.

For n e L1(Sn-1), h e Ay for some 1 < y <to, and a suitable function $ on R+, we define the maximal functions Mnhh,$ by

Mn,hJ (x) = sup i \n(y')h(\y\)f (x - $(\y\)y') \ dy. (2.5)

keZ 2 J2k-1<\y\<2k

Let $ be a positive increasing function on (0, to) satisfying $(2t) < c1$(t) (t > 0) for some c1 > 1, and ^(t) = $(t)/(t$'(t)) e LTO(0, to). Then, as in the proof of Lemma 2.3 in [20],

p.246, we have

MQMf(x) < \\h\\Ay (||^||l1(S«-1;

J^ J^(y')|My'(f|Y')(x)da(y')j y', (2.6)

where My'g is the directional Hardy-Littlewood maximal function of g defined by

My'g(x) = sup ^ f \g(x - ty1)1 dt.

r>0 2r J |t|<r

For this directional maximal function My, we know that for 1 <p, q < to,

X (BM(/')(x))^q

a q ip \p

CpJf (£fj(x)|

VR« L \jez

This is just (2.7) in the proof of Lemma 2.3 of [8], p.496. From (2.6) and (2.7), we get the following lemma.

Lemma 2.3 Let 0 be a positive increasing/unction on (0, to) satisfying 0(2t) < ci0(t) (t > 0) for some c1 > 1, and q>(t) = 0(t)/(t0'(t)) e Lto(0, to). Leth e AY for some 1 < y <to. For Y' < p, q < to, we have

£|MQ,f

LP(R«)

LP(R«)

Proof Let {gj}jeZ be a sequence of functions satisfying \\(XjeZ lgjlq')1/q'\\Lp' (R«) < 1. Then, notingp, q > y' and using (2.6), the duality, and Minkowski's inequality, we see that

/ V" MQ,h,0fj (x)gj (x) dx

jr« jez

< cf £(/ l^(y' )M f )(x) da (y')) 7'|g;(x)|dx

J R« jeZ\ JS«-1 /

< c(fRn (£(£l^O K'(f'lY0(x)da(y')) ^ q dx

c{/ J^lf f \fe(M(fjiY)(x))

lis«-1 \./R«.\icT

lp (R«)

Y- i P \ i \ q ' x)) ^^

y' \ P | y

dx I da (y'H .

Hence by (2.7) we have

^E MQMfj(x)gj(x) dx < cj^J^/)^ \fj(x)\qy do (y')j

j^z j—Z

LP(R«)

which implies our (2.8).

Now, for U e L1(5" we define the measures au^.k on Rn and the maximal operator

<,hj(x) by

i f (x) doQMik(x) = / f (0(|x|)x') )h(|x|) X2k-l<|x|<2k(x) dx,

J R« JR« |x|

OQ,hJ(x) = sup\|ow,k | *f (x)\,

(2.9) (.10)

where |au,h,$,k| is defined in the same way as 0Qth,$tk, but with U replaced by and h by |h|.

Then we have the following lemma.

Lemma 2.4 Lei $ be a positive increasing function on (0, to) satisfying $(2t) < c1^(i) (t > 0) for some c1 > 1, an<i ^(t) = $(t)/(t$'(t)) e Lto(0, to). Let h e Ay for some 1 < y <to, U e L1(5n-1). Then:

(i) If (p, 1) belongs to the interior of the octagon P1P2R2P3P4 P5R4P6, there exists C >0 such that

E * gk,j| '

j—Z k—Z

< C\h\Ay ||^y£i(SK-i)

q. i 2\ q

LP(Rn)

E(E|gk,j|:

(2.ll)

LP(R«)

where Pl = (^ max(2,/) ,2 max(2,/) ^ = ( 1, l mf,x(9 ,/) )' P3 = (,+

P4 = (2 +

^2'2 max(2,Y

1 1 , 1 \ p _ cii, 1 \ p _ ci _

+----n ../) )> 1 5 2 + max(2,/) ^ 1 6 = (2

. ^ 1)

2 max(2,Y') 2>' 1 1\

2 max(2,Y') ' 2 max(2,y

2 max(2,Y')' 2 '

r2 = (1 - 2Y,2Y)' and r4 = (2Y,1"21Y)•

(Note that if 1 < y < 2, the octagon P1P2R2P3P4P5R4P6 reduces to the hexagon P1P2P3P4P5P6.)

(ii) If (i, i) belongs to the interior of Q1Q2Q3Q4, there exists C >0 such that p q

EE|0^,h,^,k *gj

j — Z k— Z

< C\\hUy \\n\Ll(S»-l)

LP(R«)

EE |gk,j|q

j—Z k—Z

(2.12)

LP(R«)

where Q1 = (0,0), Q2 = (Jr, 0), Q3 = (1,1), and Q4 = (±,1).

Proof (a) Let 1 < y <to. Since

sup lanA0,k * gk,j l < sup lan,h,0,k | * sup Lgiy | < MQ,h,J sup lgi,jl),

keZ keZ ieZ v'eZ '

we get using Lemma 2.3

V sup lan,h,0,k * gk,jl)

LP(R«)

£(Mn,h,^sup lgk,jl))

Z ke Z

£(sup lgk,jl)

LP(R«)

LP(R«)

(2.13)

On the other hand, there exists {hj} e Lp(iq) with \\{hj}\LP'(iq) = 1 such that

E( E la^,h,0,k * gk,jl

LP(R«)

E / y2\aa,h,0J< *gkj(x) \hj(x) dx

¡^m JR"

< E / E \gk,j (x) \ l a^,h,0,k l * hj (x) dx

jeZ R« keZ

< £ / £ ^gk,j(x) |Mi,h,0hj(x) dx

jeZ R« keZ

E(Elgk,jl

je Z ke Z

LP(R«)

E(M^,h,0 hj(x))

LP (R«)

where i(y') = i(-y'). So by Lemma 2.3 we obtain for y' <p', q' < to, i.e., 1 <p, q < y,

E( E lai,h,0,k * gk,jl

LP(R«)

lgk,jl

jeZ keZ

E(E lgk,jl

je Z ke Z

LP(R«)

LP(R«)

E(|hj(x)|)

LP (R«)

Now let R1 = (±, ) r2 = a - 2Y, 2Y) R3 = a - 2Y, 1 - 27), and R4 = (2;, 1 - 2Y). Then

if (p, q) belongs to the interior of the square R1R2R3R4, there are two points (p^, and

(—,—) suchthat

>2 q2'

11111 11111

p 2 p1 2 p2 q 2 q1 2

1<p1, q1 < Y and y' <P2, q2 < to.

Hence, interpolating (2.13) with (2.14), we obtain (2.11) if (¿, ¿) belongs to the interior of

the square R1R2R3R4. (b) Let 1 < y <2. Using the Cauchy-Schwarz inequality, we get

\oa,h,<t>,k * gkj(x) \ < i

I V(y') I1 h(|y |) | y IyIn

\ i '\\2 I^(y/)I\h(IyI)I2-Y , \gk,Ax - n Wjy) \-yn-dy

-1<IyI<2k IyI

C\\h\\lY ll^y]21(Sn-1^a^I,IhI2-y 4,,k *Igk,j PK*)2.

So, we have

EIa^,h,^,k *gkjY

_ 1 2 \ q

LP(Rn)

< C\\h\\i Y \\n\L21(Sn-1

EaIsI,IhI2-y4kk * Lgkj'r

jeZ vkeZ

Lp(Rn)

Hence, noting Ih12 Y e lY/(2-Y) and using (2.14) for y/(2 - y), p/2 and q/2 in place of y,P, q, respectively, we see that (2.11) holds provided 1 <p/2, q/2 < y/(2 - Y), 1/2 - 1/y' < 1/p, 1/q < 1/2. By duality, it holds also provided 1/2 < 1/p, 1/q < 1/2 + 1/y'. Interpolating these two cases, we see that (2.11) holds if (p, q) belongs to the interior of the hexagon P1P2P3P4P5P6.

(c) Noting lY c l2 for y >2, and interpolating cases (a) and (b) above, we see that (2.11) holds if (p, 1) belongs to the interior of the octagon PiP2R2P3P4P5R4P6. This completes the proof of Lemma 2.4(i).

(d) We shall prove Lemma 2.4(ii). If (p, 1) belongs to the interior of the parallelogram Q1Q2Q3Q4, there are two points (pp q^) and (p2, q2) such that

i-(1-i) 1 + ii.

p q p 1 q p 2

1.(1-i) ^ + i -1,

q q q1 q q2

1<pi, qi < Y and y' <p2, q2 < to.

Hence, interpolating (2.13) with (2.14), we obtain (2.12). Thus, we finished the proof of Lemma 2.4. □

About the Fourier transform estimates of with ^ e H1(Sn-1), we have the follow-

Lemma 2.5 Let 1 < q < +to and ^ be a regular TO-atom in H1(Sn-1) supported in Sn-1 n £(e1, t ), where e1 = (1,0,..., 0). Let $ be a positive increasing function on (0, to) satisfying q>(t) = 0(t)/(t0'(t)) e Lto(0, to), and h e lY for some 1 < y <to. Then there exist positive constants C's such that

(2.15)

(2.16)

^Qh^k(f )| <

C\\h\\Ay (0(2k-1)lAr (f )l)1/ max{ Y',2} .

These are shown by using Lemmas 2 1 and 2 2 as in the proofs of Lemmas 3 3 and 3 4 in [20], pp.247-248. There these are stated for the case where a parameter p of positive number arises, but one sees easily that these hold in our case (p = 0), too.

To show Theorem 1.1, we need a characterization of the Triebel-Lizorkin space in terms of lacunary sequences. Let {aj}jeZ be a lacunary sequence with lacunarity a > 1,

> a for j e Z.

Let n be a radial function in CTO(Rn) satisfying x\f l<1(f) < n(f) < Xf l<a(f) and ldan(f )l < ca(a - 1)-al for f e Rn and a e Z+. We define functions f j on Rn by

jf ) = K7^) - (f e Rn).

Then observe that

f j(f) =

0, 0 <lf l < ay, lf l > aaj+1,

1, aa j <lf l < aj+1,

(2.20)

and that

supp f j c {a j <lf l < aaj+1}, supp fj n supp fi = 0 for lj- i l >2, |fadafj(f)|< Ca for a e Z+,

J2fj(f ) = 1 (f e Rn \{0}).

(2.21) (2.22) (..3) (2.24)

Let ^j be defined on Rn by j) = fj(f) for f e Rn, i.e., ^,(x) = anj+1n(aj+1x) - ajn(ajx). Lemma 2.6 Define the multiplier Sj by Sf = ^j *f. Then, for 1 < p, q < to, we have

UElSkfji-

jeZ keZ

q/2\ 1/q

LP(R«)

LP(Rn)

where C is independent of {f }jeZ.

This is a consequence of Proposition 4.6.4 in Grafakos [21]. For the sake of completeness, we will give a proof in the Appendix. From this lemma we have the following lemma with minor change of the proof of Lemma 2.2 in [8].

Lemma 2.7 Let fyj be as in Lemma 2.6. Denote AT(f) = (r2f^ rf2,...,rfn)/or r >0 and f e Rn. Define the multiplier Sjr by Sj,/(f) = f (akAr (f ))/(f). Then,/or 1 <p, q < to, we have

£(£m:

'S'gZ keZ

q/2s 1/q

ZP(Rn)

ZP(Rn)

w^ere C ¿5 independent of {f j;eZ.

We need one more lemma. If {ak}keZ satisfies furthermore ak+1/ak < d for some d > a, we can characterize Triebel-Lizorkin spaces in terms of this lacunary sequence. Denote by P the set of all polynomials in Rn. Let 1 < p, q < to, and a e R. For/ e

S'(Rn)/V, we define the norm

<>f*k IkeZ/

"¿¡T* lkeZ (Rn)

(2.25)

IP(Rn)

Lemma 2.8 Let a e R and 1 < p, q < to. Let {ak}keZ be a lacunary sequence o/positive numbers with d > ak+1/ak > a > 1 (k e Z). Then |[/||.a,{*k}keZm„> is equivalent to the usual

Fpq (R )

homogeneous Triebel-Lizorkin space norm |/||pa (Rn).

This is stated in Proposition 1 in [22] for a = 0, but the proof of this part works also for a = 0.

2.2 Proof of Theorem 1.1

We have only to show Theorem 1.1 in the case ^ is a regular atom with supp ^ c Sn-1 n B(f, r), where B(f, r) = {y e Rn; |y - f | < r}. Using the definition of aa,h,4>,k, we see that

Tqm/(x) = p.v. f h(ly|)^(x - 0(|y|)y') dy = V*/(x). (2.26)

•/Rn |y| keZ

Let ak = 1/0(2-k), k e Z. Thenasisknown, {ak}keZ is a lacunary sequence with lacunarity a = 21/|M|lto(e+) . This follows from (1.10) (see, for example, [22]). Also, we have ak+1/ak < C1, which follows from (1.9).

Let fk e CCTO(Rn) be radial functions defined by (2.19). Set fkr(f) = fk(At(f)) and Skj(f) = fkAf/f), f e Rn. Then, noting j jf) = 1 (f =0) and £l=-1 jf) = 1 on supp fyj, we have

Tqm/(x) = £ £ £ Sj-k+ir (°qm,k * Sj-k,r/)(x) = £ Q/(x), (2.27)

keZ jeZ t=-1 jeZ

Q/(x) = £ £ Sj-k+i,r (anm,k * Sj-kr/)(x). (2.28)

keZ i=-1

We follow the proof of Theorem 1 in [8], using our Lemma 2.7 and Lemma 2.4 in place of Lemma 2.2 and Lemma 2.4 in [8], respectively, and we see that if a e R and (p, 1) belongs to the interior of the octagon P1P2R2P3P4P5R4P6, then we have

\Q/IIf^R") — C\f Ilfp^R«). About L2 estimate, we have

(2.29)

HQjf llf

f2,2(r«) —

Ca-\j\l max(y',2)\

In fact, by Lemma 2.5, we get

(H )| — C\h\Ai \\^\\l1(S»-1), (H, — C\h\\Ai ${2k )A (H )|

(1) (2.32)

&QM4kk(H )| —

(^(2k-1)\Ar (H )\)17 max(Y ',2)'

Also, we have

\\Q/\\i72',2(R«)

y^ y^ Sj-k+i,t (ffQ,h,^,k * Sj-k,rf)(x)

keZ £=-1 1

2 x 1/2

E E /+£ A (h )) ¿nM,k (h )fi-Mr (H )/(H )

keZ £=-1

2 1/2 dA .

So, for / > 0, we have, using (2.33) and <£(2£) = 1/a-t and a^/at > a = 21/\H\L<x,<R+>,

\\Q/\\jF0,2(R«)

— c( W |<ra,h^,k (H )f(H )|2 dH

^keZ aj-k — A(H aj-k+2

— C\h\a I v i

Y keZJaj-k — \Ar(H )\—aj-k+A a-k+1

A (H )\\-2/max(Y'2)^£.,|2 \

— C\\h\\Ay a

— C\h\Ay a-j/ max(Y

— Ca-// max(Y ',2)\f W,

-(/'-1)/max(Y',2)/f ^keZ a/

2)(f f(H)|

keZ'a/-k—\ar (h) \—a/-k+A ai-k

lf(H )|2 dHj

A (H )\x-2/max(Y ',2)

lf(H )|2 dH

In the fourth inequality we used a/+1 — c1a/.

F',2 (R«)

For j < -1, using (2.32) we get as before

HQ/IIf02(RH) < c(W |a^,A,k(f/(f)|2

< cHaHa, (E/ (^)>(f)|2df)1

VeZ1' aj-k <|Ar (f )l<aj-k+A a-k ' '

2 \ 1/2

Thus we have

< <*( £/ Mf« )i2 ds

aj-k (S )i<«j-k+2 \ aj-k /

< Cdi W f(S)|2 dA

^keZ aj-k <|ar(s )i<aj-k+2 /

< Ca>([ f (S)|2 dS

< Ca'\f \\l2(R,F°2(R» )). -iji/max(y',2) i

IIQ/if02(R„) < Ca-1 j/max(Y ,2)|/!f02(R«),

which shows the required estimate (2.30).

Interpolating these two cases (2.29) and (2.30), we see that if a e R and (1,1) belongs to

the interior of the octagon P1P2Q2P3P4PsQ4P6, then Ta,h,^ is bounded on Faq(Rn). This

completes the proof of Theorem 1.1(i).

Next, we prove (ii). Let 12 - p | < min{ 1, ^7}, 1 < q < to, and a e R. Then, by Theorem 1.1(ii), TQM is bounded on F^R") and Fap1(Rn). Since (F;,-1(Rn), Fap1(Rn)) 1,q = Baq(Rn), we see by interpolation that is bounded on Ba,q(Rn). This shows (ii) and completes the proof of Theorem 1.1.

3 Proof of Theorem 1.3

Let , ak, , and Sk be the same as in the proof of Theorem 1.1. Then, noting

E/eZ fj(S) = 1 (S = °) and EL-i fj+t (S) = 1 on supp fj, we have

TuMf (x) = E E ^Shk+i(oaM,k * Sj-kf )(x) = E ¿2¡f (x), (3.1)

keZ jeZ £=-1 jeZ

¿2jf(x) = E E Sj-k+t (°aM,k * Sj-kf)(x). (3.2)

keZ t=-1

Using our Lemma 2.6 and Lemma 2.4(i) in place of Lemma 2.2 and Lemma 2.4 in [8], respectively, we see, as in the proof of Theorem 1.1, that if a e R and (1,1) belongs to the interior of the octagon P1P2R2P3P4P5R4P6, then we have

HQ/HFja q(W) < CH/biUR«). (3.3)

Next, we approach the above estimate (3.3) by another method. We calculate the Fp,q norm of Qj more directly. Considering the support property of tyk, we have

WQif W^R") =

Sm^^^2Sj-k+t(aQM,k * Sj-kf)

keZ l=-1 1

Sm / , Sm+l (oq ,h,$,j-m * Smf)

Sm Sm+l (oq ,h,$,j-m-1 * Sm+ 1f)

Sm ^ ; Sm+l (oQ,h,$,j-m-1 * Sm-1f)

Lp(Rk)

LP(RK) 1

Lp(Rn)

LP(RK)

By Fefferman-Stein's vector-valued inequality for maximal functions, Lemma 2.4(ii), and

am+1/c1 < am < am+1/a, we get

WO/ Wf^R") < ^E ( E amIaaM,j-m * Sm+lf ^

l=-1 meZ 1

Lp(Rn)

< ^ (EamqISm+^I

E amqISmf I

lp (Rk)

lp(Rk )

< C^Wf \U

if a e R and (-1, -1) belongs to the interior of the parallelogram Q1Q2Q3Q4.

Interpolating (3.3) and (3.4), we obtain

HQjfllFUR") < C^Wf llF

Fp,q(Rn)

if a e R and (p, q) belongs to the interior of the octagon Q1Q2R2P3Q3Q4R4P6 (hexagon Q1Q2P3Q3Q4P6 in the case 1 < y < 2). About L2 estimate, we have

\\q/Wj^R") < C(^

P/max(Y ',2)

In fact, let ok = OQ,h,4,,k. Then we have

f 2k f , dr

Ok(f )=/ Q(y<)h(r)e-i$(r)yf do(y') -.

2k-1 Sn-1 r

First we have

|Ok(f)\ < 2\\h\\iy wQwl1(sk-1).

Next, using Holder's inequality and assuming \\Q\\L1(Sk-1) < 1 without loss of generality, we have

* « )i <( L ihwr di)\ L LL ^y) e-^?(y)

< nhWAy^j^ i jsn i «(/K^^ ? da (/)

/ dr^ VY'

i f ) f

= 2\\h\\A I / nty)e-iry/• ? da(y')

\j"(2k-1) A«-

/ r "(2k ) r

< 2Wh\AY \Mi~(R+) / / «(/)e-iry? da(y') \,/0(2k-1) Js«-1

dA max(2,Y')

2 "("-1(r)) dA max(2,Y')

"-1(r)"'("-1(r))

' dr\ max(2,Y')

= 2\\h\\Ay

If mW)l dida (/) da (^ --

jS«~1 j S«-1 j "(2k-1) r

We see that

ç"(2 )

j"(2k-1)

-ir(J-Z) • ?

/"(2k-1) We see also

f"(2k )

/•"(2")

j"(2k-1)

-ir(y'-Z) • ?

"(2k )

< log —;—j—r— < log Cl.

"(2k-1)|?||?' •(*'-y')!'

So, as in [10], p.458 (using Lemma 3.1 in [12]), we have for P >1,

p"(2k ) j"(2k-1)

e-ir(y'-z!) • ?

log^„ 2_1 ,„ for"(2k)|?|> C1

logp(logC1)e"(2k-1)?| 6 |(y' -z') •?'| Hence we have

2 \ \h \ \ ay {wfp (^))2/max{Y ',2} \ \ « \\^2S/ma1f ',2}

log c1

a (? )l <

(log(e(log C1)"(2k ) |? |/ d))^/max{Y ',2}

for $(2k)If I> ^ > e. On the other hand, using the cancelation property of Q, we get easily

\Ok(f)\< 2\hWi1 HQWL1(SK-1)$(2k)If I. Now we can estimate the L2 norm of Qf:

\\Qif \\f° (R«)

E E Sj-k+i (a«,h,k * Sj-kf )(x)

keZ £=-1

2 1/2 dx I

E E )aa,h,",k(? (?)f (?)

keZ £=-1

2 1/2 d?l .

Note that 0(2k)|f | = |/a-k > a;|f |/a;-k > a' for aj-k < 11 < aj-k+2 and j > 0, where al+1lat > a = 21/|HliTO<R+). So, for j > 2 loga(d/log C1) and aj-k <|f |< aj-k+2, we have

^0(2k)|f | > a2 > 1, C1

and hence we have

WQjf ||jffl2(R,) < c(W |âaAM(?)/(?)|2d?)

' HeZ aj-k <? l<aj-k+2 /

/ f /1 \2^/ max(y ', 2) n 1/2

< ^UE |/(?)|2d?

/ 1 max(Y ', 2) / . \ 1/2

< cw^waJ ^ E/ W2 d?

X1^/ VeZ"^ aj-k <? l<aj-k+2 /

/ 1 \ 0/max(Y',2W » x 1/2

(1 max(y ',2)

1+j) Wfl

For j < -2, we have 0(2k)|f | = |/a-k < a2+j for aj-k <|f |< aj-k+2. So, using (3.9) we get as before

WQf W^2(RK) < c( w |<Wm(?)/(?)|2

' keZ ai-k<? l<aj-k+2 /

< cW^WaJW a4+2j|/(?)|2d?)

\ keZ ai-k <? l<aj-k+2 '

< caY W |/(? )|2 d^1/2

\b^ir'ai-k <? l<ai-k+2 /

vkeZ ai-k <?l <aj-k+2

if \ 1/2

< Caj[ f (?)|2 d?

VR« /

< Cai|l/WJF°2(R«)-

For -2 < j < 2loga(d/log C1), using (3.7), we get

WQf Wf02(R«) < c(E / |/(?)|2d^ ' < C||f Wffo

' keZ ai-k <? l<aj-k+2 / '

Thus we have (3.6) for j e Z. Now, let Q1 = (^^, ^^), Q2 = (7 + ^^ (2 "

1 \ max(y', 2) \ -j-, /1 1 1 1 \ ^ max(y' ,2) 1 max(y', 2) \ ^ /1 max(y' ,2) / 1

7), 2J ), P = (2 + max(y',2) - J, 2), Q3 = (1--2^,1--2T~), Q4 = (Y--J-( Y ~

1\ 1 max(y;,2) x jy /1 1 1 1? - il__1__max(y;,2) ^ max(y;,2) x ,

2), 1 2J ), ' 6 = (2 max(y',2) + J ^^ /V2 = (1 2y 2Jy' , 2y + 2J/ ), and

R4 = (27 + 1 - 2Y - marY2)). Then, for (p, q) belonging to the interior of the oc-

tagon Q1Q2R2P3Q3Q4R4P6 (hexagon Q1Q2P3Q3Q4P6 in the case 1 < y < 2), we can find (pp q1) in the interior of the octagon Q1Q2R2P3Q3Q4R4P6 (hexagon Q1Q2P3Q3Q4P6

in the case 1 < y < 2) such that 1 = % + i-%, 1 = % + i-% ,and 1 > Q > max(J',2). Hence, for

' — ' p 2 pi ' q 2 qi ' /3

a e R, taking a with a = (1 - Q)a1 and interpolating between (3.6) and (3.5), we obtain the desired estimate

\\Qjf 11J«q(RK) < C(1 + |j|)-Q//max{Y'2) If hf«^). (310)

Summing up this with respect to j , we finish the proof of Theorem 1.3(i). The proof of (ii) is the same as in Theorem 1.1(i).

4 Proof of Theorem 1.2

In this section we shall give the proof of Theorem 1.2.

(A) L logL case. Let & e L logL(Sn-1) satisfying the cancelation property. Then letting Am = \\^x2M-1<|S(y)|<2M HL1(S«-1) and A = {m e N: Am > 2-m}, we can construct &m e L2(Sn-1) (m e A) and &0 e H1<r<2 Lr(Sn-1) such that

H^mHL2(S«-1) < C2m, H^mHL1(Sn-1) < C, (.1)

J^mAm < Ch^hllogL(Sn-1), (.2)

/ ^(y1) da(/) =0 (m = 0, m e A), & = &0 + ^ Am&m. (4.3)

JSn-1 meA

From the above, we see that

TQ,f,hf = T&0,h^f + E AmT&m . (4.4)

So, we consider T&m. We use the notations in Section 3 with minor change such as Qmj for &m instead of Qj for Since \\&m \\L1(S n-1) < C, we have as in Section 3 that

\\Qm,jf \\F«q(Rn) < C|f \\f«q(Rn) (4.5)

if a e R and (p, q) belongs to the interior of the octagon Q1Q2R2P3Q3Q4R4P6 (hexagon Q1Q2P3Q3Q4P6 in the case 1 < y < 2). About L2 estimate we have

~ 3 1 -i

II Qm ,jf HF202(Rn) < Ca m jl \f H^Rn) (4.6)

for some / with 0 < / < 1/2. In fact, let am,k = a&m,h>^k. Since \\&m\\L1(Sn-1) < C and H&mHL2(Sn-1) < C2m, we get by Lemma 3.1 in [23], p.1567,

\am,k(H)|< CHhHM, (.7)

HhHAy (1+ IMU

\amk(H)\ < C-Y-3—, (.8)

10 (2k-1)H lm

\am,k(H)\< CHhHA1\0(2k)H\m, (.9)

where J is a fixed constant with 0 < J < 1/2. Using Plancherel's theorem, (4.8), the support property of 0j, and ak+1/c1 < ak < ak+1/a, we get for j > 0,

HQ»/IIf°2(r«) < c( w |am,k(?)f(?)|2 d?) 2

, ^keZ aj-k|<aj-k+2 /

< cfe/ -^)|2d^

VeZJaj-k<|? |<aj-k+2 |0(2k-1)f | » /

< Ca-j W |/(? )|2 d?) 7

\ kJ ' aj-k <№ |<a/-k+2 /

keZ aj-k <|f |<aj-k+2 - J •

< Ca-^l/1IIf20,2(r«).

For j < 0, using (4.9) in place of (4.8), we get

HQM,jf HF202(R") < Camj[f Hjf0,2(R«).

This shows (4.6). Interpolating (4.6) and (4.5), we obtain for some 0 < 0 <1,

HQmjf HFpq(R«) < Ca-»j| Hf H^q(Rn) (4.10)

provided a e R and (1,1) belongs to the interior of the octagon Q1Q2R2P3Q3Q4R4P6

(hexagon Q1Q2P3Q3Q4P6 in the case 1 < y < 2). From (4.10) and the definition of Qmj it follows

HT^„ ,hj HFa,q (R») < CJ2a~ HFap,q(Rn) < -C"J£ IHf ^(R^ < C JO Hf F^).

jeZ 1 -a m jo

We can see that the same estimate holds for n0. Thus, by (4.4) we have

||Tn,h,/ HFa,q(Rn) < C( 1 + E Ammj |f ^(R^ < C|f ||JFp,q(R«)

^ mcA '

provided a e R and (1,1) belongs to the interior of the octagon Q1Q2R2P3Q3Q4R4P6

(hexagon Q1Q2P3Q3Q4P6 in the case 1 < y < 2). This completes the proof of Theorem 1.2 in the case n e L log L(Sn-1).

(B) Block space case. Let r >1. Then if n e B[0,0)(Sn-1) and satisfies the cancelation condition, it can be written as n = ^TO1 i, where Xi e C and ¿2i is an r-block supported on a cap Bi = B(xi, t1) n Sn-1 on Sn-1 and

EM 1 + log( |Bi |-1)} < 21| n IIb(0,0)(s«-1) < TO. (411)

To each block ¿2i, we define

ni(yf) = ¿2i(y) - jgn j ni(x')da(x').

Let A = {l e N; lBtl< 1/2} and set

&0 = & -Ekt&t. (.2)

Then there exists a positive constant C such that the following hold for all l e A:

/ &i(x') da(x') = 0, (4.13)

i -1/r

II&i HLr(S"-1) < C|Bi|-1/r, (4)

\\&i\\L1(sn-1) < 2, (4.15)

& = &0 + E xl&l- (.6)

Moreover, from (4.11) and the definition of &l it follows that

II&0HLr(s"-1) < C E 2-Wlkil < CH&HBro,o)(Sn_1), (4.17)

/ &0(x') da (x') = 0. (4.18)

By (4.16), we have

T&mJ(x)= E xiT&iA4>f(x). (4.19)

So, we have only to show the boundedness of T&[ih^f. We use the notations in Section 3 with minor change such as Qj for &l instead of Qj for &. Since \\&l \\L1(Sn-1) < C, we have as in Section 3 that

HQl/l^q (R") < C\f \\P«qRn) (4.20)

if a e R and (p, q) belongs to the interior of the octagon Q1Q2R2P3Q3Q4R4P6 (hexagon Q1Q2P3Q3Q4P6 in the case 1 < y < 2). About L2 estimate we have

HQijf \\^2(R") < Ca 3 jl \f \\^2(Rn) (4.21)

for some / with 0 < / < r. In fact, let al,k = a&,tih$,k. For l e A U {0}, we set ml = [log2 lBl l-1/^] + 1, where [ • ] denotes the greatest integer function. Since \\&l HL1(Sn-1) < 2 and \\&l HL2(Sn-1) < C2mi , we get by Lemma 3.1 in [23], p.1567

\ai,k(H)\< CHhHA1, (2)

(1+ MU

\ai,k(H) \ < C-Y-—, (.23)

l0(2k-1)H l mi

\ai,k(H)\< C\\h\\A1 \^(2k)H\&, (4)

where ff is a fixed constant with 0 < ff < r. Using Plancherel's theorem, (4.23), the support property of and ak+ilci < ak < ak+ila, we get for j > 0,

iiQj ii^r») < c( W (?f (? )|2 d?\

' Vtri "i-k<1? |<ai-k+2 /

F 2(R") < ^¿W ?

XkeZ ai-k <? l<aj-k+2

< ClE/ -^xf(?)|2d^ '

vkeZ Jai-kl<aj-k+2 |0(2k-1)? l

< Ca-mj(W |/(£)|2d?

^keZ aj-k |<aj-k+2

< Ca"^ 11/\\p0 2(RH).

For j < 0, using (4.9) in place of (4.8), we get

\\~Qt,jf Hf02(R«) < Ca^'\f \\p0 2(Rk). This shows (4.21). Interpolating (4.21) and (4.20), we obtain for some 0 < 0 <1,

\\Qijf\\Pp,f (R") < Ca" ^ jl \fWpa^Rn) (4.25)

provided a e

R and (p, i) belongs to the interior of the octagon Q1Q2R2P3Q3Q4R4P6 (hexagon QiQ2P3Q3Q4P6 in the case 1 < y < 2). From (4.25) and the definition of Qi,j it follows

_ ^ fo |j| c m.£

W^W \\Fpqm < C2^a mi' \\\PaM(R") <-\\ppq(R") < C£0\\\pM(R").

jeZ 1-a mi f0

We can see that the same estimate holds for Thus, by (4.14) we have

\\TQMf Wpp^R") < C(1 + E ktmt) \\f \\ppq (R") < C\f Wpf^R")

^ leA '

provided a e R and (1,1) belongs to the interior of the octagon QiQ2R2P3Q3Q4R4P6 (hexagon QiQ2P3Q3Q4P6 in the case 1 < y < 2). This completes the proof of Theorem 1.2.

Appendix

In this section we shall prove Lemma 2.6. Let |a;};ez, Vj, ^j, and Sj be the same as in Lemma 2.6. Set nj(?) = n(?/ai+1) and ?) = j?). Then we have

(-ix)a jx) = c„f danj(?)eix ? d?,

so we have

|xa||$j(x)| < C i |9%<?)|d? < cf ldan j(?)|d?.

JR« J supp danj

From this and the definition of nj we get

\*j(x)\< Canan+V (A.1)

and for N e N,

lxN * ^ < C W-jNiC r"-1 dr) < C j • <">

Thus, for N e N we have

\*j(x)\< c(-"—Y ((a - y'. (A.3)

1 ;W|" \a-1/ (1 + l(a-1)aj+1xl)N v '

Next we have

(-ix)adxk *j(x) = 0nt da (iHknj)(H)eixH dH,

so we have

\xa\\3Xk*j(x)\< C i \da(Hknj)(H)\dH < cf \da(Hknj)(H)\dH.

JRn Jsupp da(Hknj)

From this and the definition of nj we get

\v*j(x)\< C(aaj+1)n+1, (A.4)

and for N e N,

\ 1 f aaj+1 1 f aai+1

lxlN\V*,(x)\< C----— r"- dr + C----— rndr

111 j()|- ((a -1)aj+l)N_1 Jaj+1 ((a -1)aj+1)N Jaj+l

C (aaj+1)n + C (aaj+1)"+1 (A 5) " ((a -1)aj+l)N_1 ((a -1)aj+1)N' ( . )

Thus, for N e N we have

^(x)\< ci^Y\ «- . (A.6)

1 ;W|" \a-1/ (1+ l(a-1)aj+1xl)N v '

Let b = a -1 and B = (¿-O^1. Taking N = n + 1 and using (A.3) and (A.6), we obtain

f ( \ 1/2 1 = (E\*k(x-y)-*k(x)\ ) dx

J lxl>2lyl V keZ /

< I E\*k(x - y)-*k(x)\ dx

'lxl>2lyl keZ

^hv 1-1

\ *k(x - y)- *k(x) \ dx

, , 1 - lx l >2 ly l

ak+1 <l by l-1

ak+1<|by|

V / |$k(x -y)| + |$k(x)| dx

1> | by | V |x | >2 y |

V i |y||V(x - 0y) | dx

,<lby,|x|>2|y|

+ E I |^k (x - y) + |^k (x)| dx

(bak+1)

, , 1 « |x|>2 |y| ak+1>|by|-1

< CB V |bak+1y| f

1 u i-1 "'R

au. 1 < | by | 1

+1 >|by |x

< CB V |by|ak+^

, 1 Jr." (1 + |x|)

ak+1<|by| 1

+ CB j2 - dr

ak1>\by\-1}2bak+1y r

< CB J2 |by|ak+i + CB J2

„ 1 _ (1+ |bak+1x|)"+1

ak+1<|by| 1

+c^ v 1 , ebak+1)", dx

ak+1^y|-;|x|>2|y| (|bak+1x|)"+1

ak+1>|by|

1 1 - 1 |by| ak+1.

ak+1<|by| 1 ak+1>|by|

Let k0 be the integer satisfying ak0 < | (a - 1)y|-1 < ak0+1. Then we have

I <— > ak+1 + CBak0+1 } -.

ak0 k<k0-1 ktk0 ak+1

From ak+1lak > a it follows that ak+1 < a-1ak+2 <• • •< ak-k0+1ak0 for k < k0. Hence we get

E ak+1 < E ak0 ak-k0+1= a^^— =<

ak k0 a 1

k<k0 -1 k<k0- 1 k=0

From ak+1lak > a it follows that ak+1 > aak > ••• > ak k0 ak0+1 for k > k0. Hence we get

1 ^ 11 1 ^ 1 1 a

ak+1 ^ ak0+1 ak-k0 ak0+1 k=0 ak ak0+1 a - 1.

Thus we have

i < ci A

If we define by ) = laj), we get = Q,^. So, for we have the same estimate as for Qj. Therefore, we obtain

/| 2| ^E^k (x-y)-*k (x)|2)112 dx < "+2. (A.?)

Now let Bl = C, B2 = I2, define an £2-valued function II(x) by K(x) = (x)}*eZ, and the linear operator T by T(f) = II * f for f g Lœ (Rn) with compact support. Then we have \\\\T(f)\\B2 IL (M" ) = \\(I] ^eZ * f I2)22 IlLr (Rn ), and so by Littlewood-Paley theory this is equivalent to \ f \\Lr(M") for any I < r < œ.By (A.7), the kernel K(x) satisfies the Hormander condition. Thus, we can apply Proposition 4.6.4 in Grafakos [21], and get the conclusion of Lemma 2.6.

Competing interests

The author declares that he has no competing interests.

Author's contributions

The author contributed to the writing of this paper. He read and approved the final manuscript.

Acknowledgements

The work is partially supported by Grant-in-Aid for Scientific Research (C) (No. 23540228), Japan Society for the Promotion

of Science.

Received: 12 December 2014 Accepted: 12 March 2015 Published online: 21 March 2015

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