Scholarly article on topic 'Homogeneous Triebel-Lizorkin Spaces on Stratified Lie Groups'

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Academic research paper on topic "Homogeneous Triebel-Lizorkin Spaces on Stratified Lie Groups"

Hindawi Publishing Corporation Journal of Function Spaces and Applications Volume 2013, Article ID 475103,16 pages http://dx.doi.org/10.1155/2013/475103

Research Article

Homogeneous Triebel-Lizorkin Spaces on Stratified Lie Groups

Guorong Hu

Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Mekuro-ku, Tokyo 153-8914, Japan Correspondence should be addressed to Guorong Hu; hugr@ms.u-tokyo.ac.jp Received 7 January 2013; Accepted 3 March 2013 Academic Editor: Jozef Banas

Copyright © 2013 Guorong Hu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Homogeneous Triebel-Lizorkin spaces with full range of parameters are introduced on stratified Lie groups in terms of Littlewood-Paley-type decomposition. It is shown that the scale of these spaces is independent of the choice of Littlewood-Paley-type decomposition and the sub-Laplacian used for the construction of the decomposition. Some basic properties of these spaces are given. As the main result of this paper, boundedness of a class of singular integral operators on these function spaces is obtained.

1. Introduction

In recent years there were several efforts of extending Besov and Triebel-Lizorkin spaces from Euclidean spaces to other domains and non-isotropic settings. In particular, Han et al.

[1] developed a theory of these function spaces on spaces of homogenous type with the additional reverse doubling property. That setting is quite general and includes for example Lie groups of polynomial growth. However, the high level of generality imposes restrictions on the possible values of the parameters of the function spaces.

For the purpose of studying subelliptic regularity, Folland

[2] introduced fractional Sobolev spaces and Lipschitz spaces on stratified Lie groups. Later, Folland and Stein [3] established the theory of Hardy spaces on general homogeneous groups. Besov spaces on stratified Lie groups were first introduced by Saka [4], by means of the heat semigroup associated to the sub-Laplacian. Recently, Fiihr and Mayeli [5] introduced homogeneous Besov spaces on stratified Lie groups in terms of Littlewood-Paley-type decomposition and established wavelet characterization of them. However, the integrability parameter p and the summability parameter q of the function spaces studied in both [4, 5] are restricted to be no less than 1. Moreover, systematic treatment of Triebel-Lizorkin spaces on stratified Lie groups can not be found in the literature, to our best knowledge.

The purpose of this paper is to introduce and study homogeneous Triebel-Lizorkin spaces with full range of parameters on stratified Lie groups. Motivated by [5], we define these

function spaces via Littlewood-Paley-type decomposition. We find that a helpful way to treat the case that either the integrability parameter p or the summability parameter q is less than 1 is to take the Peetre type maximal function into consideration. With the help of the almost orthogonality estimate on stratified Lie groups (see Lemma 2), we show that our definition of homogeneous Triebel-Lizorkin spaces is independent of the choice of the Littlewood-Paley-type decomposition and the sub-Laplacian used for the construction of the decomposition. Thus, these function spaces reflect of properties of the group, not of the sub-Laplacian used for the construction of the decomposition.

Singular integral theory is a powerful tool for the study of partial differential equations. The If -boundedness of convolution operators with homogeneous distribution kernels on Lie groups endowed with suitable homogeneous structure was proved by Knapp and Stein [6] (for p = 2) and Koranyi and Vagi [7] (for 1 < p < to). In Section 4 of this paper, we prove the boundedness on homogeneous Triebel-Lizorkin spaces of a class of convolution type singular integral operators on stratified Lie groups, which includes convolution operators with homogeneous distribution kernels.

This paper is organized as follows. After reviewing some basic notions concerning stratified Lie groups and their associated sub-Laplaicans in Section 2, in Section 3 we introduce homogeneous Triebel-Lizorkin spaces F^(G) on stratified Lie groups, and give some basic properties of them. In Section 4 we show the F" (G)-boundedness of a class of convolution singular integral operators. Throughout this

paper the letter C will denote a positive constant which is independent of the main variables involved but whose value may differ from line to line. The notation a < b or b > a for some variable quantities a and b means that a < Cb for some constant C > 0; a ~ b stands for a < b < a. We agree that the set N of natural numbers contains 0.

2. Preliminaries

In this section we briefly review the basic notions concerning stratified Lie groups and their associated sub-Laplacians. For more details we refer the reader to the monograph by Folland and Stein [3]. A Lie group G is called a stratified Lie group if it is connected and simply connected, and its Lie algebra g may be decomposed as a direct sum g = V ® • • • ® Vm, with [V1; Vfc] = Vfc+1 for 1 < k < m - 1 and [V1; Vm] = 0. Such a group G is clearly nilpotent, and thus it may be identified with g (as a manifold) via the exponential map exp : g ^ G. Examples of stratified Lie groups include Euclidean spaces R" and the Heisenberg group H".

The algebra g is equipped with a family of dilations |<?t : t > 0} which are the algebra automorphisms defined by

**( ) = I(Xi 6 Vi) •

Under our identification of G with g, 8t may also be viewed as a map G ^ G. We generally write ix instead of <?t(x), for x e G. We shall denote by

A = Xi[dim (V,.)]

the homogeneous dimension of G.

A homogeneous norm on G is a continuous function x ^ |x| from G to [0, to) smooth away from 0 (the group identity), vanishing only at 0, and satisfying |x-1| = |x| and |ix| = i|x| for all x e G and t >0. Homogeneous norms on G always exist and any two of them are equivalent. We assume G is provided with a fixed homogeneous norm. It satisfies a triangle inequality: there exists a constant y > 1 such that |xy| < y(|x| + |y|) for allx, y e G.If x e G andr > 0 we define the ball of radius r about x by£(x, r) = |y e G : |y- x| < r}. The Lebesgue measure on g induces a bi-invariant Haar measure on G. As done in [3], we fix the normalization of Haar measure by requiring that the measure of B(0,1) be 1. We shall denote the measure of any measurable £cG by |E|. Clearly we have |5t(£)| = iA|£|. All integrals on G are with respect to (the normalization of) Haar measure. Convolution is defined by

We consider g as the Lie algebra of all left-invariant vector fields on G, and fix abasis X",..., X" of g, obtained as a union of bases of the Vj. Inparticular, X1;..., Xv,with v = dim(V1),

is a basis of Vj. We denote by Yj,..., Y„ the corresponding basis for right-invariant vector fields, that is,

If I = (i1;..., i„) e N" is a multi-index we set X7 = XÍ ••• and Y7 = Y"! • • • Y"n. Moreover, we set

= X^fc'fc > fc=1

where the integers d1 < • • • < d" are given according to that e Vdt. Then X7 (resp., Y7) is a left-invariant (resp., right-invariant) differential operator, homogeneous of degree d(7), with respect to the dilations <?t, t >0.

A complex-valued function P on G is called a polynomial on G if P o exp is a polynomial on g. Let ..., be the basis for the linear forms on g dual to the basis X1,..., X" for g, and set ^ = o exp-1. From our definition of polynomials onG,^1,...,^" are generators of the algebra of polynomials on G. Thus, every polynomial on G can be written uniquely as

P = I^j^7, flj 6 C,

where all but finitely many of the coefficients vanish, and = ^h •••rf". A polynomial of the type (6) is called of homogeneous degree L, where L e N, if d(7) < L holds for all multi-indices I with a7 = 0. We let P denote the space of all polynomials on G, and let PL denote the space of polynomials on G of homogeneous degree L. Note that PL is invariant under left and right translations (see [3, Proposition 1.25]). A function / : G ^ C is said to have vanishing moments of order L, if

VP 6 P

f /(*)P(*)

dx = 0,

with the absolute convergence of the integral. The Schwartz class on G is defined by

S (G) =

/ 6 Cm (G) : P_. . 6 L™ (G),

g^'i...

V7 6 N", VP 6 P

that is,/ e S(G) if and onlyif/o exp e S(g).Asisindicated in [3, p. 35], S(G) is a Frechet space and several different choices of families of norms induce the same topology on S(G). In this paper, for our purpose we use the family Ill • ii(l,n) : L, N e N} of norms given by

M(LN) = sup (1 + |x|)N (|X7<Mx)| + |X7<Mx)|). (9)

d(7)<L,xeG ' '' iv/

Here and in what follows, we use the notation convention 0(x) = 0(x-1) for any function 0 : G ^ C. The dual space

S'(G) of S(G) is the space of tempered distributions on G. If f e S (G) and 0 e S(G) we shall denote the evaluation of f on 0 by (f,$).

Using the above conventions for the choice of the basis X1,..Xn, and ] = dim^ ), the sub-Laplacian is defined by L = -T] , X2,. When restricted to smooth functions with compact support, L is essentially self-adjoint. Its closure has domain D = [u e L2(G) : Lu e L2(G)}, where Lu is taken in the sense of distributions. We denote this extension still by the symbol L. By the spectral theorem, L admits a spectral resolution

XdE (X),

where dE(X) is the projection measure. If m is a bounded Borel measurable function on [0, to), the operator

(L)=\ m(X)dE(X) jo

is bounded on L2(G), and commutes with left translations. Thus, by the Schwartz kernel theorem, there exists a tempered distribution M on G such that

i(L)f = f*M, Vf e S (G).

Note that the point X = 0 may be neglected in the spectral resolution, since the projection measure of {0} is zero (see [8, p. 76]). Consequentlywe should regard m as functions on R+ = (0, to) rather than on [0, to).

Let S(R+) denote the space of restrictions to R+ of functions in S(R). An important fact proved by Hulanicki [9] is as in the following lemma.

Lemma 1. If m e S(R+) then the distribution kernel M of m(L) is in S(G).

Moreover, from the proof of [10, Corollary 1] we see that if m is a function in S(R+) which vanishes identically near the origin, then M is a Schwartz function with all moments vanishing.

In the sequel, if not other specified, we will generally use Greak alphabets with hats to denote functions in S(R+), and use Greek alphabets without hats to denote the associated distribution kernels; for example, for (f> e S(R+) we shall denote by 0 the distribution kernel of the operator <p(L), where L is a sub-Laplacian fixed in the context.

3. Homogeneous Triebel-Lizorkin Spaces on Stratified Lie Groups

For any function h on G and t > 0, we define the L1-normalized dilation of h by

Dth (x) = tAh (tx).

Before we introduce the homogeneous Triebel-Lizorkin spaces on stratified Lie groups, we prove the following basic estimate, which is a generalization of [11, Lemma B.1] and will be frequently used throughout this paper.

Lemma 2. Let L,N e N with N> L + A + 2. Suppose both <p,f e S(G) have vanishing moments of order L. Then there exists a constant C > 0 such that for all j,£ e Z and all x e G,

\(D2J<i>)*{D2ty) (x)| where j Al := minjj, l}.

(1 + VAe

Proof. Using dilations and the facts (D2j$>) * (D2ef)(x) = (D2ef) * (D21 $)(x-1 ) and \\$\\ 9

may assume l > j = 0. To proceed we follow the idea in the proof of [11, Lemma B.1]. Let D1 = {y : \y\ < 1}, D2 = {y : \y\ > 1 and \xy-1 \ < \x\/2y} and D3 = {y : \y\ > 1 and \xy-1\ > \x\/2y}. Let y ^ Px$(y) be the left Taylor polynomial of 0 at x of homogeneous degree L (see [3, pp. 26-27]). Then using vanishing moments of y

\$*(D2iy)

< | \$(xy-1) - Px4 (y-1)\\(D*y) (y)\dy (15)

= i +i +i •

jdj jd2 jd3

For y e D1, the stratified Taylor formula (cf. [3, Corollary 1.44]) yields that, with b a suitable positive constant,

\$(Xy-1)-Px4 (y-1)|

sup |(x70)(xz)|

ki^1 H

d(I)=L+1

\\(L+1,N) SUP

Mib^1 (1 + M)

\\(L+m (i + Mf'

since 2bL+1 (1 + Ixzl) > 2bL+1 + M > 2bL+1 + IxHy - lzl > bL+1 + Ixlly > (1 + lxl)ly if Izl < bL+1. Hencewehave

\\(L+1,N)\I

\\(L+1,N)\I

\\(L+1,N)\I

"(0N) (1 + lxl)N Jdj (1+211

-e(L+1)

Iii 2 7

\\(0N) (1 + lxl)N Jg (1 + 2i I

2-i(L+1)

\\(0N) (1 + lxl)N'

where for the last inequality we used [3, Corollary 1.17] and that N - L- 1> A+ 1.

For y e D2,wehave |y| > |x|/y-|xy 11 > |x|/y-|x|/2y = |x|/2y. On the other hand, |y| < y(|x| + |xy-1|) < y(|x| + |x|/2y) = (y + (1/2))|x|. Thus, we have

1+2i|y|>2i|y|>21 (l + |yD>21 (1 + |x|)

f <(y+2)LwL-

Also, we note that by [12, Proposition 20.3.14] the left Taylor polynomial is of the form

¿w (y)

2-e(N-A) ll(L,N)IMI(Q,N) (1 + |X|)N

L (7+bnNdy

(1 + |x|) J|y|<y(M + M/2y) 2~e(N-A)

ll(L,N)l|r ||(Q,N)

(1 + |x|)

(1 + W)

"(L'N)^"(Q'N) (1 + |x|)N'

where we used that |{y : |y| < y(|x| + |x|/2y)}| ~ |x| and N - A > L.

For y e D3 we have |xy-11 > |x|/2y, and, hence

L h % (y)---^ (y)

= <M*) + II I

h=1 fc=1 1< ij,..., it<M

where the integers d1 < • • • < d" are given according to that

X; e V . From these remarks, it follows that

II(l,n)I|t ||(Q,N)

(1 + |xy 11) Q<k<L (1 + M)

(1 + |2ly|)

ll(L,N)llr ll(Q,N)

L(1 + |xy-1 |)N (1 + |x|)

[2l (1 + |x|)]

-£(N-A)

(L'N)^"(Q'N) (1 + |x|)N 1

Jo (1 + |xy-1|)]

(1 + |x|)N J|xy-j|<|x|/2y

J|xv 1 |<|x|/2v

ll(L,N)Hr ||(Q,N)

L(1 + |xy-1|)N Q<t<L (1 + |^|)N

(1 + |2ly|)

ll(L,N)llr ll(Q,N)

(1 + |x|)N J|y|>1 (1+211 2-lL f |2ly|L

"(L'N)^"(Q'N) (1 + |x|)N Jo (1 + 211

fd(2ly)

ll(L,N)imi(Q,N) (1 + |x|)N'

where for the last inequality we used [3, Corollary 1.17] and N - L > A + 1.

Combining the above estimates, we arrive at

* (x)| < |0|(L+1,N)l^l(L+1>N)2-lL (1+|x|)N .

This is exactly what we need. □

Let Z(G) denote the space of Schwartz functions with all moments vanishing. We then consider Z(G) as a subspace of S(G), including the topology. It is shown in [5, Lemma 3.3] that Z(G) is a closed subspace of S(G), and the topologydual Z'(G) of Z(G) can be canonically identified with the factor space S'(G)/P.

We now have the following Calderon type reproducing formula.

Lemma 3. Suppose L is a sub-Laplacian on G, and ф e S(R+) is a function with compact support, vanishing identically near the origin, and satisfying

Z4t(2-2jX) = 1, УХе R+

Then for all д e Z(G), it holds that

9= ,lim Z 9*(D2> Ф)'

к — œ. ,

with convergence in Z(G). Duality entails that, for all u e S'(G)/P,

= lim Zu* (D2j ф),

к — œ

and the convergence is in S'(G)/P.

Proof. First note that the 2-homogeneity of L implies that the distribution kernel of ¡p(2-2^L) coincides with D2j ifi. Let I e N" and N e N be arbitrarily chosen. Then take L e N such that L > N + d(I) + 1. Since both g and <p are Schwartz functions with all moments vanishing, it follows by Lemma 2 that

(1 + \x\)N \X1 [g*(D2i$)](x)\

= 2dW(1 + \x\f \g*[D2J (X'$)](x)\

< C2d(i)j2-\J\L2(J^o)A(1 + 2(Ja0) \x\)

< C2d(I»2-l»L2W(1 + 2{^0) MpU + \x\)

(L+A+2) N

(1 + \x\)

-\j\(L-N-d(I))

where the constant C is a suitable multiple of \\9\\(L+1,L+A+2)\\Xl$W(L+1MA+2). This implies that Xjez(1 + \x\)N\XI[g * (D2><p)](x)\ converges uniformly in x, for every I e Nn and every Ne N. Consequently there exists h e S(G) such that ^j^k 0 * (D2j fi) converges in the topology of S(G) to h, as k ^ to. On the other hand, by (23) and the spectral theorem (cf. [13, Theorem VII.2]),

g=Z^P(2-2jL)g=Z9*(Dv ф)

holds in L2-norm. Therefore, h = g, which completes the

proof.

Let A denote the class of all functions (f> in S(R+) satisfying

supp ф с [2 2,22],

\ф(Х)\>С>0 for Ae

Definition 4. Let a e R, 0 < p < >x> and 0 < q < >x>. Let L be a sub-Laplacian on G and ф e A. We define F^ (L, ф) as the space of all f e S'(G)/P such that

^\f*(D2, ф)\У ) (29)

with the usual modification for q = то.

We then introduce the Peetre type maximal functions: Given f e S'(G), ф e S(R+), L a sub-Laplacian, and a,t > 0, we define

V (f w л I f*(Dt-i Ф) (x) I Ma,t ( f, L, Ф) (X) = sup .-u -1 ha yeG (1 +t 1 ¡у

aWw ^ I Хк [f * (Dt-iФ)] (x)I

Maa {/' L,(t>) (x) = sup f, 1 I 1 .v. •

yeG (1 + t 1 jy 1x|)

1<к<]

Lemma 5. Suppose L is a sub-Laplacian and ф e A. Then for every a > 0 there is a constant С > 0 such that for all f e S'(G)/P, all j e Z, and all x e G,

Kx> (f> L, ф) (x) < C2jM:a-, (f, L, ф) (x). (31)

Proof. Because of (28) it is possible to find a function f e S(R+) supported in [2-2,22] such that £j6Z 4>(2-2jX)f(2-2jX) = 1 for X e R+. Set Ш) = j}j=-l ф(2-2Х)у(2-2Я), X e R+. Then ф(Х) = ф(Х)((Х) for all X e R+. Consequently, for je Z and 1 < к < v,

\Xk [f * (Dvф)] = \Xk [f * (D21■ ф) * (D21■ С)] (>0| <\\f*(D2> Ф)(*)\\[Хк (D2S С)](z-ly)\dz = 2№ j I f * (D2iф) (z)I j (Xk0 (2 (z-1y))\ dz (32) <2>m;x, (f, L,$)(x) x j 2*Д(1 + 2* \z-1^\)a(l + 2* \z-'yiy^dz

< 2!M;xi (f, L,$)(x)(l+2j j y-1 x\)a,

where for the last inequality we used [3, Corollary 1.17] and that (l + 2j\z-1 x\)a(l+2j\z-1y\)-a < (1+2j|y-1x|)a. Dividing both sides of the above estimate by (1 + 2j\y-1x\)a, and then taking the supremum over у e G and 1 < к < v, we obtain the desired estimate. □

Lemma 6. Suppose L is a sub-Laplacian, ф e A, and r > 0. Then there exists a constant С such that for all f e S' (G)/P, all j e Z, and all x e G,

<2-* (f,L,$)(x)<c[M(\f *(D24)\r)(x)]Ur, (33)

where a = A/r, and M is the Hardy-Littlewood maximal operator on G.

Proof. Let g e Cm(G) and y0 e G. The stratified mean value theorem (cf. [3, Theorem 1.41]) gives that for every S > 0 and every y e G with |y- y0| < S,

|g(y) -g(yo)l

<C|Vv| suP |(x/c#)(Voz)|

izisfcly/yl 1<fc<]

<C5 sup |(Xfc^)(7oZ)|,

|z|<M 1<fc<]

where b isa suitable positive constant. Hence we have

|g(yo)|<CS s up |Xfcg(y)|

1<fc<]

Wf 1 1/r.

VJir 'yoiss /

Puttingg = / *(d2j0), dividingboth sides by (1 + 2J|y01x|)a, and using Lemma 5, we have

|/*(£>2^)(yo)|

(1 + 2i|yo-1x|)a

<CS X [/*(P2J0)] (y)| (1 + 2' |y-1x|)a " (1 + 2i |y-1x|)a (1 + 2i |yo1x|)a

i<fc<]

LMD^OOf^

J|y-'Voisä

(l +2-> IV^D^ V'yoisä <C5 sup (l + 2j |y-Vo|)X2- (/,L,£)(*)

|y Vo|sM

(1 + 2^*1)"

xif ] |/*GD2^)MI>

% 1xi<y(5+iy01xi)

< C<$( 1 + 2jbS)a2j(/, L, <£) (x)

<Taya(S+Iv-1*!)" . . r. .

^^ [M(I/-№»«I')<*>]

< Ce(1 + be)aM*2-i (/, L,£)(x)

+ ya(l+e-1)a[M(|/,(D2J0)D(x)]1/r;

wherewehavesetS = 2 Jeandusedthat(1+2Je |y0 x|)/(1+ 27|y-1x|) < 1 + e-1. Finally, taking e sufficiently small (such

that Ce(1 + be)a < 1/2), and taking the supremum over y0 e G, we get the desired estimate. □

Theorem 7. Suppose L(1), L(2) are any two sub-Laplacians on G, and are any two functions in A. Then, for a e

R, 0 < _p < to and 0 < q < to, we have the (quasi-)norm

equivalence

Ik(L<1),0<1)) ~ ll/IU (L^r / 6 S (G) /p. (37)

Proof. First we note that if a > A/ min{_p, then we have

Z(2j" |m;,2-j (/, l('),^))|)?

for « = 1,2, where 0(,) denotes the distribution kernel of

cp(L(i)). Indeed, the direction' 38

other direction follows by Lemma 6 and the Fefferman-Stein

vector-valued maximal inequality on spaces of homogeneous

type (see, e.g., [14]). Thus, to prove (37), it suffices to show

X(2-|M;,rJ(/, L(1),0<1))|)?

Z(2j" |Ma*2-j (/,L(2),0<2))|)?

To this end, let i/^(1) be a function in S(R+) with support in [2-2,22] such that £j6Z c^(1)(2-2jA)i/^(1)(2-2jA) = 1 for A e R+. For / e S'(G)/P, by Lemma 3 we have

with convergence in S'(G)/P. Here ^(1) is the distribution kernel of v>(1)(L(1)). Hence, since c0(2) e Z(G), we have the pointwise representation

(36) / * (D^®) (V) = I/ * (Ö2.0(1)) * (Ö2^T(1))

(D^(2))(y), VyeG.

It follows that

|/*(D2i0(2))(y)|

< X i|/*(^0(1))(z)|

x|(d^(1) )*(D2l0(2))(Z-1y)|dZ <!m;,2-j(/) l(1),0(1) )(y)

X (1+ 2j |z|)B |(D^(1)) * (^210(2) ) (z)| dz

= ZM«.2-J (/>L(1),0(1)

Since both f(1) and 0(2) are Schwartz functions with all moments vanishing, we can use Lemma 2 to estimate that, with L sufficiently large,

h* < c

(1+2'" |z|)V|j-l|L2(jAl)A

x (1 + 2ja1 |Z|)-(L+A+2)dZ

(1+2' |z|)V|j-l|L2(jAl)A

x (1 + 2'a1 |Z|)-(s+A+1)dZ

C2_|''"l|(i_a) i 2('a1)a (1 + 2jAl |Z|)-(A+1)dZ

-|j-£|(L-«)

Here, the constant C is a suitable multiple of

|^(1)|(L+1>L+A+2)IC(2) ||(L+1,L+A+2). On the other hand we

observe that

m;,2-(/, L(1),c(1))(y)

< m;>2-j (/ L(1),r)(x)(1 + 27 |y-1x|)

< m;>2-j (/, L(1),r)(x)

x (1 + 2l |y-1x|)a max (1,2<^fl).

Putting these estimates into (42), multiplying both sides by 2l", dividing both sides by (1 + 2l|y-1x|)a and then taking the supremum over y e G,we obtain

2feMa*ri (/,L(2),0<2))(x)

< l2-|i-l|(L-2a-|a|)2jaMa*rJ (/, L(1),<^t1))(x)-

In view of [15, Lemma 2], taking L > 2a + |a| in the above inequality yields the direction "<" of (39). By symmetricity, (39) holds, and the proof is complete. □

Remark 8. From Theorem 7 we see that the space F"^(L, 0) is actually independent of the choice of L and 0. Thus, in what follows we don't specify the choice of L and 0 and write F" (G) instead ofF" (L, 0). Henceforth we shall fix any sub-Laplacian L. Moreover, for the sake of briefness, we will write Ma*t(/ 0)(x) instead of Ma*t(/, L, 0)(x).

Proposition 9. For a e R, 0<_p<œ> and 0 < ^ < œ>, one foas tfoe continuous inclusion maps Z(G)

S'(G)/P.

¿m(g)

Proof. Let g e Z(G) and 0 e A. Choose N > (A + 1)/p and L > N + |a| + 1. Since both g and 0 are Schwartz functions with all moments vanishing, it follows by Lemma 2 that

(L+1.L+A+2)

(L+1.L+A+2)

2-|'|L2('AQ)A(1+2'AQ |x|)-(L+A+2) 2-|'|L2(jAQ)A(1+2jAQ |x|)-N

(L+1.L+A+2)

2-|'|(L-N)(1 + |x|)-

for all j e Z and all x e G, where the constant C is a suitable multiple of ll0ll(L+1>L+A+2). This together with [3, Corollary 1.17] give that

IIf?„(G) ~ IMI(L+1,L+A+2)|

-|j|(L-N-|«|)?

(L+1.L+A+2)'

which implies that Z(G) ^ F^^(G) continuously.

Now we show the other embedding. Let / e F"^(G) and take any f e Z(G). Let 0> e A, and then let £ e S(R+) be a function with support in [2-2,22] such that £j6Z 0(2-2jA)C(2-2jA) = 1 for A e R+. Then by Lemma 3 we have

|</^>| =

< I |(/* (°2i0),^* (D2iC))|

To proceed we claim that

¡/,(02,0)1,. < C2jA/-p ILMD^Hy (49)

for all 0 < p < to. Assuming the claim for a moment, it follows from (48) that

I (f,r)\< (sup2ja\\f*(D2,fi)\L

xZ2^-«]\\f*(D2J t)\\Ll.

It is easy to see that

SUp2'a\\f * (Ü2, fi)^ <MPUG).

To estimate the sum in (50), we note that if we choose N > A + 1 and L> N + (A/p) + \a\ + 1 then similarly to (46) we have

I f*(D2i ï)(x) \ <

(L+1,L+&+2)2

-\j\(L-N)

(1 + lxl)-N,

where the constant C is a suitable multiple of \\C\\(L+1>L+A+2). From this it follows that

z^'^wy *(d21 ^

< Z2-ljl[L-N-(A'^)-l^l^lYll(L+1L+A+2) (53)

Since all the necessary tools are developed in the above arguments, the following proposition can be proved in the same manner as its Euclidean counterpart; see, for example, the proof of [16, Theorem 2.3.3].

(50) Proposition 10. For a e R, 0 < p < to and 0 < q < to, Fpq(G) is a quasi-Banach space.

Let us introduce a class of functions. We say that f e R(G), if there exists fi e S(R+) whose support is compact and which vanishes identically near the origin, and g e S(G), such that f = fi(L)g. Clearly R(G) c Z(G).

Lemma 11. Let a e R and 0 < p,q < to. Then M(G) is dense in Fpq(G). Inparticular, Z(G) is dense in F"a(G).

Proof. Take any u e F^ (G) and any e > 0. In the appendix we show that F^^(G) admits smooth atomic decomposition. By the smooth atomic decomposition, we see that C™(G) n F" (G) is dense in F" (G), for a e R and 0 < p,q < to. Thus,

y,q y,q ^ -i

we can find g e C0°(G)nF^^(G) such that \\^-w\\p* (G) < e/2. On the other hand, the argument in Step 5 of the proof of [16, Theorem 2.3.3] shows that there exists a sufficiently large Ne N such that \\g-jNL-N 9* put

\\(L+1,L+&+2).

Therefore,

I (f>f) l<\\/\\ F"(G)\W\\(L+1,L+&+2).

This implies that Fp (G) — S'(G)/P continuously.

We are left with showing the claim. Indeed, if r > 0 is fixed then by Lemma 6 we have, for all x e G,

\ f*(Dv fi)(x) I- inf

I f*(Dy ï)(x) I

yeB(x,2->)(1 + 2j \x-1y\)

< inf .,Ml/r,2-> (M)(y)

< inf [M( \ f*(Dv fi) \ r)(y)]l'r.

yeB(x,2

Taking r < p in the above estimate and using Hardy-Littlewood maximal inequality, we have

| f*(Dv fi)(x)

I B(x:

m^» [M{lf*(D2> wW*

JA \\f*(D2,*)(j)\Pdy. Since x is arbitrary, the claim follows.

(56) □

h= Zfi(2-2jL)g= Z9*(D2> fi)- (57)

j=-N j=-N

Then h e R(G), and we have

\\h-u\\pUG) < Wh- gllp^G) + llg-This proves the claimed statement.

u\\p«(G) < £. (58)

We next consider lifting property of F^q(G). For a e R, the power L is naturally given by

L°f = \ XadE (X) f, f e Dom (La) c L2 (G). (59) Jo

Remark 12. By [17, Theorem 13.24], we have R(G) c Dom(La) for all o e R. As a consequence, Dom(LCT) n F" (G) is dense in F"„(G), for all a,o e R and 0 < p,q < to.

y,q y,q ^ -i

We now have the lifting property of F^^(G).

Theorem 13. Let a,ae R and 0 < p, q < to.

(i) The operator L°, initially defined on Dom(LCT) n Fpa(G), extends to a continuous operator from F^^(G)

(ii) Let T, : F" (G) — F" a(G) denote the continuous extension of La. Then T, is an isomorphism, and \\Taf\\p^-2„(G) is an equivalent quasi-norm of F^^(G).

Proof. (i) Let fie A. Set ((X) = Xa and y(X) = fi(X)i^(X) = fi(X)Xa. Clearly {p is also in A, and

22jfff(2-2jA) = 0(2-2jA)£(A). By [17, Theorem 13.24], we have 0(2-2jL) o C(L) c (0(2-2jOCO)(L), and moreover

Dom (0(2-2jL)oC(L))

= Dom (C(L)) n Dom ((0~(2-2^) £(•)) (L)) (60) = Dom (La) n L2 (G) = Dom (Lff).

Hence, for every / e Dom(Lff), we have 22jfff(2-2jL)/ = 0(2-2jL)LCT/.

Now let / e Dom(LCT) n F* (G). By the above remarks,

we have 22jff [/ * (D2if)] = (Lff/) * (D2i0). It follows that

llLff/lk"-2<7(G)

^^(L^M^l)?)

Our next goal is to show the Lusin and Littlewood-Paley function characterizations of F" (G). If a e R, 0 < a < to, A > 0, b e R, and u(x, k) is a function on G x Z, we define

fc=-œ

GL (и) (x) =

œ /•

¡t=-œ JG

x(1+2 ly xl) 2 ay

(l(2ja|/*(ö2JT)|)q

S",q (и) (x)

œ /•

I I (2fca Ky,fc)|)q2(")Ady fc= œ V1^2"

Since Dom(LCT)nF" (G) is dense in F" (G) (see Remark 12),

the mapping L : Dom(L) П F"(G) ^ F"(G) extends to

a continuous operator from F"(G) to F«-2ct (G). We denote

The following proposition shows that the spaces F«?(G) are characterized by Lusin and Littlewood-Paley functions.

Proposition 14. Lef a e R, 0 < p, q < œ>, fee R and Я > max{A/p, A/qj. Lef / e S'(G)/P and ф e A. Pmî m(x, fc) = / * (D2k<p)(x),for x e G and fc e Z. Then one foas

p>qv ' p>q

this extension by Ta : F" (G) ^ F"-2ff(G).

(ii) Let us first show that the mapping Ta : F"?(G)

К (")!lp ~IKq (")!lp ~Rq (")!lp•

F" (G) is injective. Indeed, assume f e F" (G) such that

TCTf is the zero element of F""2ct(G). By Remark 12 we can

find a sequence f in Dom(Lff) П F£ (G) which converges Proof. Step 1. Show that if Я > max{A/p, A/^j then in F£ (G) to f. Then applying (i) to L yields that

converges in F*?CT(G) to the zero element. Since L"/e e Dom(L-CT) n f"-2ct(G) and / = L-ff(Lff/£), applying (i) to the operator L-CT we see that / converges in F"?(G) to the zero element. Therefore, / is the zero element in F"?(G). This proves that TCT is injective.

а" (и) „ <

L-1« 11 GA,q

( и) P < ( )Hlp <

S",q (И)|

Next we show that T : F"q(G) — F"q2CT(G) is surjective.

Indeed, given / e F"? CT(G), we let / be a sequence in Dom(L-ff) n F£-2ff(G) which converges in F£-2ff(G) to /. Then from (i) we see that L-0/ converges in F"?(G). Denote this limit by g. We claim that TCTg = /. Indeed, since / = Lct(L-ct/£), it follows from (i) that / converges to TCTg in F" (G). Hence Tffg = / in F£-2<T(G). This proves that Tff is surjective.

The above arguments also show that both T-0. o TCT and TCT o T-0. are identity operators on F"?(G). Furthermore, by an easy density argument we see that (61) holds for all / e F"?(G), provided that L in (61) is replaced by TCT. Thus, TCT :

FL(G) ^ F'"-2ff(G) is an isomorphism, and ||Tff/||p*-2„(G) is

Indeed, the proof of the first inequality in (64) is essentially the same as that of [18, Theorem 2.3]. To see the second inequality in (64), one only needs to examine the proofs of Theorems 1, 2 and 4 in [19, Chapter 4] and observe that, although the function u considered in [19] is defined on the half space R++1 = R" x (0, to), the arguments there can also be adapted to functions u which are defined on G x Z.

Step 2. We prove that ||S£ (u)^ < |g"(u)|Li.Firstnote that, by an argument similar to the proofs of [19, Theorems 1 and 2], it follows that ||S"'?(u)||Lf. - ||S" (u)|| for everyfixedb e R. Hence, in view of (38), it is enough to show that

an equivalent quasi-norm of F"q(G).

(и) P < ( )Hlp <

But this is a consequence of the following elementary estimate:

f , (2ja \f*(D2Jfi) (y)\)q2jAdy

J\y-Ix\<2->

(2j«\f*(Dvfi) (y)\)q

< sup ---

yeB(x,2-') (1 + 2j \y-1x\r (66)

(2j»\f*(D2>fi) (y)\)q

< sup--5-—

yeG (1 + 2j \y-1x\)Xq

= [2jaMirJ (f,fi)(x)]q. The proof of Proposition 14 is thus complete. □

Corollary 15. Let G be a stratified Lie group. Then F° 2(G) = HP(G) with equivalent (quasi-)norms, for 0 < p < to. Here HP(G) are Hardy spaces on G.

Proof. In [3, Chapter 7], Folland and Stein proved the characterization of Hardy spaces Hp (G) by continuous version Lusin function, for 0 < p < to. Note that the arguments in [3, Chapter 7] are still valid if we replace the continuous version Lusin function by discrete version one defined above; see also [20] for a treatment of discrete version Lusin funcion. This fact together with Proposition 14 yield the identification of F0pa(G) with Hp(G) for 0 < p < to. □

4. Convolution Singular Integral Operators on

In this section we study boundedness of convolution singular integral operators on homogeneous Triebel-Lizorkin spaces on stratified Lie groups. Motivated by [21, Section 5.3 in Chapter XIII], we introduce a class of singular convolution kernels as follows.

Definition 16. Let r be a positive integer. A kernel of order r is a distribution K e S'(G) with the following properties:

(i) K coincides with a Cr function K(x) away from the group identity 0 and enjoys the regularity condition:

| XIK(x)| < CI\x\-Q-d(I), for \I\ <r, x = 0. (67)

(ii) K satisfies the cancellation condition: For all normalized bump function fi and all R > 0,we have

| (K,fiR)\<C, (68)

where fiR(x) = fi(Rx), and C is a constant independent of fi and R. Here, by a normalized bump function we mean a function fi supported in B(0,1) and satisfying

| XIfi(x)| <1, V\I\<N,VxeG, (69)

for some fixed positive integer N.

The convolution operator T with kernel of order r is called a singular integral operator of order r.

Remark 17. Using [3, Proposition 1.29], it is easy to verify that (67) is equivalent to the following condition:

| YIK(x) \<CI\x\-Q-d(I), for \I\<r,x = 0. (70)

Examples of such kernels include the class of distributions which are homogeneous of degree -A (see Folland and Stein [3, p. 11] for definition) and agree with Cm functions away from 0. Indeed, assume is such a distribution, then

it is easy to verify that K satisfies the regularity condition (i) in Definition 16; moreover, from [3,Proposition 6.13] wesee that K is a principle value distribution such that Je<\x\<L K(x)dx = 0 for all 0 < £ < L < to. Hence, for every normalized bump function fi, by the homogeneity of K we have

| (K,fiR)\ = \ {K, fi)\

= lim f K(x)[fi(x)-fi(0)]dx (71) I £^oJe<\x\<2 (71)

< f \K(x)\| fi(x)-fi(0)\dx.

\ x\<2

Using stratified mean value theorem (cf. [3, Theorem 1.41]) and (67)-(69), it is easy to verify that the last integral converges absolutely and is bounded by a constant independent of fi and R. Hence K satisfies the condition (ii) in Definition 16.

Now we state the main result of this section.

Theorem 18. Let a e R, 0 < p,q < to, and let r be a positive integer such that r > A/ min{p,q} + \a\ + 2. Suppose T is a singular integral operator of order r. Then T extends to a bounded operator on F'^^(G).

If K e S (G) and t > 0, we define DtK as the tempered distribution given by (DtK,fi) = (K,fi(t-1-)) (Vfi e S(G)). For the proof of Theorem 18, we will need the following lemma, in which b is the positive constant as in [3, Corollary 1.44].

Lemma 19. Let r be a positive integer. Suppose K is a kernel of order r, and fi is a smooth function supported in and having vanishing moments of order r-1. Then, thereexists a constant C > 0 such that for all j e Z, all I e N" with \/\ < r, and all x e G

max { I X1 [fi * (D2i K)] (x) \, \ Y1 [fi * (Dv K)] (x) \ }

<C(1 + M)-A-r.

Moreover, fi * (D2j K) have vanishing moments of the same order as fi.

Proof. Recall that the convolution of fie S(G) with K e S'(G) is defined by fi * K(x) := (K,(xfi)~), where xfi is the function given by xfi(z) = fi(xz), and as before f(x) := f(x-1) for any function f : G — C. From [3, p. 38] we see that fi * (D2j K) are Cm functions, j e Z. We claim that for every x with \x\ < 1/2y, the function z — (xfi)~(z)

is a normalized bump function multiplied with a constant independent of x. Indeed, using the quasi-triangle inequality satisfied by the homogeneous norm it is easy to verify that the function z ^ (Xfi)~(z) is supported in B(0,1); moreover, since \x\ < 1/2y and since

Y1 = I Pu*J

d(J)>d(I)

where PIJ are polynomials of homogeneous degree d(J)-d(I) (see [3, Proposition 1.29]), we have

I X1 [(Xfi)l(z) \

= \ Y1 (xï)(z-1 ) \

Pu (z-1 ) || XJ (xfi)(z-1) \

\J\i\I\ d(J)> d(I)

= I I PiJ (z-1)\\(xJfi)(xz-1)\<Ci.

\!\<\I\

d(J)>d(I)

Here CI is a constant depending on I but not on x. Hence the claim is true. The above argument also shows that, for every x with \x\ < 1/2y and for every I e N", [x(YIfi)]~ is a normalized bump function multiplied with a constant C[ independent of x. Thus, by the condition (ii) in Definition 16, there exits a constant C > 0 such that for all j e Z, all I e N" with \/\ < r, and all x with \x\ < 1/2y

I Y1 [fi * (D2,K)] (x)\

= \(YIfi)*(D2> K)](x) j

= 1 ^ (YIfi)]~)\

=\ (K'ix (yw (2-m<c.

From this and [3, Proposition 1.29], we also get that, for all j e Z, all I e N" with \I\ < r, and all % with M < 1/2y

| X1 [fi * (D2iK)] (x)\

<C Z \x\d(f)-d(I) \YI' [fi * (D2jK)] (x)| < C. (76) \i'\<\i\

d(I')>d(I)

Let now \x\ > 1/2y. Let y e supp fi. Let I e N" with !il < r, and denote by PxtY'(D jk) the right Taylor polynomial

of Yi(D2i K) at x of homogeneous degree r-\I\-1 (see [3, pp. 26-27]). Then by the right-invariant version of [3, Corollary 1.44], we have

| X1 (D21■ K)(y-1x)-Px,xl(D2jk) (y-1)|

<C I y\

\z\ibr-mlyl d(J)=r-\I\

I YJXI (D2iK)(ZX)\. (77)

Observe that D2j K satisfies (67) with the bound C independent of j e Z. Also note that for y e supp fi and \z\ < br-^\y\ we have zx e G \ {0}. Thus, for all J with d(J) = r-\I\ and all z with \z\ < br-^\y\, by using (73) and (67) (with K replaced by D2j K) we have

Y'X1 (D2j K) (zx) j

< I I Pjj, (zx)\\xJ'+I (D2, K)(zx)\

lJ'li\J\ d(J )>d(J)

lzxld(J')-d(J)lzxl-&-d(J'+I)

d(j' )>d(J)

&-r+\I\-d(I)

Here, for the second inequality we also used the observation that when d(J) = r - \I\ and \j'\ < \J\, we have \j' + I\ < \J + I\< d(J) + \I\=r- \I\ + \I\ = r. Inserting (78) into (77) we obtain

I X1 (D2iK)(y-1x)-Px^(D2jK) (y-1)\ <C\y\r-m sup lzxr&-r+m-d(I).

\z\ibr-'"lyl

Notice that for \x\ > 1/2y, y e supp fi and \z\ < br-^\y\, we have \zx\ ~ \x\. Thus, by using the vanishing moments of fi and (79), we have

\ X1 [fi*(D21 K)](x) \

= \ \fi(y)XI (Dv K)(y-1x)dy

< \ I fi (y)11 X1 (DvK) (y-1x) - Px,x'(D2jK) (y-1)| dy \ lzxl-&-r+m-d{I)\y\r-m\fi(y)\dy

jl y\r-m\fi(y)\dy

< C sup \ lzx

\z\ibr-mly

-&- r+\ I\ -d( I) r-\ I\

-&- r+\ I\ -d( I)

Combining (76) and (80), we see that, for all j e Z, all I e N" with \/\ < r, and all x e G

\XI [fi * (D2jK)] (x) < C(1 + lxl)-&-r+m-d(I). (81)

From this and (73), we also get that, for all j 6 Z, all 7 6 N" with |7| < r, and all x 6 G

|YJ [0 * (D^K)] (x)|

< I (x)|XJ' [0 * (Dj^K)](x)|

|j'|<Ui d(j' )>d(J)

<C I |x|d(i')-d(i)(1 + |x|)-A-r+ii'i-d(i')

|j'|<Ui d(j')>d(j)

< C(1 + |x|)

-A-r+iJi-d(J)

Since |I| < d(I), (81) alongwith (82) yield (72).

It is straightforward to verify that 0 * (D2JK) have vanishing moments of the same order as 0. The proof of Lemma 19 is therefore complete. □

The proof of Theorem 18 also relies on the existence of smooth functions with compact support and having arbitrarily high order vanishing moments.

Lemma 20. Given any nonnegative integer L and any positive numberS, there exists a function £ e S(R+) with the following properties:

(i) |£(A)| > C > 0 for A e [2-2(*°+1),2-2(*°-1)], with k0 some (large) positive integer;

(ii) C is a Schwartz function on G having vanishing moments of order L;

(iii) suppC c £(0,S).

Proof. From the appendix of [22] we see that there exists Q e S(R+) such that 0(0) = 1 and Q has compact support. Now let us define £(A) = (r2A)fc%-2A), A e R+. Here t > 0 and k is a nonnegative integer. Then C(x) = Dt(LfcQ)(x) = iA(LfcQ)(ix). Hence, if we take i, k sufficiently large, then (ii) and (iii) follow immediately. Moreover, since 0(0) = 1, it is easy to see that (i) is also satisfied, provided that k0 is sufficiently large. □

We are now ready to prove Theorem 18.

Proof of Theorem 1. Choose a function C e S(R+) which satisfies conditions (i)—(iii) in Lemma 20 with L = r - 1 and S = 1/4y2br. The condition (i) guarantees the existence of a function f e S(R+) with the following properties:

suppf c [2-2(fco+1),2-2(fco-1)], |f(A)|>0 for A e (2-2(fco+1),2-2(fco-1)), (83) (2-2jA) C(2-2jA) = 1, VA e R+.

Note that f(2-2fc° •) e A. For / e Z(G), by Lemma 3 we have

/ = I / * (D^) * (D^)

with convergence in Z(G). Let 0 e A, and let K be the convolution kernel of the operaotor T. Then we have the representation

/ * K * (D2i0)

= I / * (D^) * (D^) * K * (Dji0)

= I/ * (D^y) * D^ [C * (D2-JK)] * (DJI0) ,

which holds pointwise and also in the sense of S'(G). Since (by Lemma 19) C * (D2-jK) satisfies the decay condition (72) (with the bound C independent of j e Z) and has vanishing moments of the same order as C, from the proof of Lemma 2 we see that

D [C*(D2-JK)]*(D2i0)(v)|

2(jA£)A

-ij-li(r-2) _

(1 + 2jAl |y|rfl

This together with (85) gives that |/*K*(D2i0) (x)|

<1 [l/*(D2Jf)(^)|

x |d2j [£ * (D2-jK)] * (D210) (z-1x)| dz

< I2-ij-fi(r-2-a)

j-ii(r-2) f2(jAl)A |/*(D2jy)(Z)| (1 + 2jAl |z-1x|)r+A

|/*(D^)(z)|

dz (87)

(1 + 2j |z-1x|)a

2(jAl)A

J (1 + 2jAl |z-1x|)r+A-a

where for the last inequalitywe used that (1 + 2jAl|z-1x|)-a < 2|j-l'a(1 + 2j|z-1x|)-a. By the hypothesis we can choose a such that a > A/ min|_p, a} and r - 2 - a - |a| > 0. From [3, Corollary 1.17] we see that the last integral converges absolutely. Consequently, we obtain

2& |/*K*(D*0) (x)|

< I2-ij-li(r-2-«-i«i)2jaM*>2-i (/, y) (x).

Hence, it follows by [15, Lemma 2] that

I(2fe |/*K*(D2i0)(x)|)i

I(2j>ayJ(/,y)|)'i

This together with (38) imply that ||/ * K||p« <G) < ||/||p* <G) for all / e Z(G). Since Z(G) is dense in F" (G), / ^ / * K extends to an bounded operator on F"?(G). This completes the proof of Theorem 18.

Corollary 21. Let a e R, 0 < _p, q < to, and let k be a nonnegative integer. Then

d(J)=fc

Proof. Note that by the Poincare-Birkhoff-Witt theorem (cf. [23, I.2.7]), the operators X7 form a basis of the algebra of the left-invariant differential operators on G. By this fact and the stratification of G, it suffices to show that

Hfm(g)

■IllvlU-i( j=1 (

To this end, we first note that when restricted to Schwartz functions, XjL-1/2 are convolution operators with distribution kernels homogeneous of degree -A and coincide with smooth functions in G\{0}. This follows from the fact that the operator L-1/2 is a convolution operator whose distribution kernel is homogeneous of degree -A + 1 and coincides with a smooth function in G \ {0} (see [2, Proposition 3.17]). Hence, by Theorem 18, X;L-1/2 extend to bounded operators on F"?(G). From this fact and the lifting property (Theorem 7), we deduce that

vlt-i(G) = ll(*,L-1/2) L1/2/IU-i(

J Hfji,-1(g)

Hence £]=1 l|Xj/|p«-i(G) < ||/||p« <G). To see the converse, we need to use [2, Lemma 4.12], which asserts that there exists tempered distributions K1,...,K] homogeneous of degree -A + 1 and coinciding with smooth functions in G \ {0} such that / = Z]=1(Xj/) * K for all / e S(G). By this result and

Theorem 7, we have, at least for / e R(G) (c Dom(L-1/2)), H/k(G) = ||L (L-1/2/)l^.-i(G)

= IIL (/^OIIl^G)

L (I(V)*Kj *«1

I(Xj/)* L (Kj

¿«(G)

where is the convolution kernel of the operator L-1/2. As is indicated in [2, p. 190], L(Kj * are distributions homogeneous of degree -A and coincide with smooth functions away from 0. Thus it follows by Theorem 18 that

ll(V)* L (Kj ^OIUg) <IIVL-1

,J "FP,-1(G)*

Inserting this into (93), we obtain ||/||p* <G) < £]=1 |Xj/|pr-i(G) for all / e R(G). Since R(G) is dense in F^G), the latter inequality also holds for all / completes the proof.

6 f;>9(g). This □

Appendix

Smooth Atomic Decomposition of F" (G)

In this appendixwe show that homogeneous Triebel-Lizorkin spaces on stratified Lie groups admit smooth atomic decomposition. We follow the proof of [24, Theorem 6.6.3] with necessary modifications.

Equipped with Haar measure and the quasi-distance defined by the homogeneous norm, the group G is a space of homogeneous type in the sense of Coifman and Weiss [25]. On such type of spaces, Christ [26] constructed a dyadic grid analogous to that of the Euclidean space as follows.

Lemma A.1. Let G be a stratified Lie group. There exists a collection {Q" : k e Z, a e Afc} of open subsets ofG, where Afc is some (possibly finite) index set, and constants S e (0,1) and A 1,A 2 > 0 such that

(i) |G \ U«eAt Q"| = 0 for each fixed k and Q nQ* =0 if a = p;

(ii) for any a, p, k, I with I > k, either Q* c Q" or Q* n

Q" = 0;

(iii) for each (k, a) and I < k, there exists a unique p such that Q" c Q*;

(iv) diam(Q") < A 1Sk, where diam(Q") := sup{|x-1y| : x, y e Q"};

(v) each Q" contains some ball , A2Sfc), where z^ e G.

The set Qa can be thought of as a dyadic cube with diameter roughly Sfc and centered at z". We denote by D the family of all dyadic cubes on G. For k e Z,we set Dfc = {Q" : a e Afc}, so that D = UfceZ D^. For any dyadic cube Q e D, we denote by Zg the "center" of Q and by kg theuniqueinteger k such that Q e Dfc.

Without loss of generality, in what follows we assume S = 1/2. Otherwise we need to replace 2j in Definition 4 by S-j, and also make some other necessary changes; see [27, pp. 9698] for more details.

Definition A.2. Let Q be a dyadic cube and let L be a nonnegative integer. A smooth function ag on G is called a smooth L-atom for Q if it satisfies

(i) ag is supported in B(zg, (y(1 + A 1)/A2) 2-fcQ);

(ii) J ag(x)P(x)dx = 0 for all P e PL;

(iii) |X7ag(x)| < |Q|-(d(7)/A)-(1/2) for all multi-indices I with | I| < L + 1.

Definition A.3. Let a e R and 0 < p,q < to. The sequence space f^ consists of all sequences {sq}q6D such that the function

¡r (-mq) = (i(iqi

-(a/A) -(1/2)

is in LP(G). For such a sequence s = {sq}q we set

- \\na'1 I

Wip ■

The smooth atomic decomposition of homogeneous Triebel-Lizorkin spaces on stratified Lie groups can be stated as follows.

Thus, it follows from Lemma 2 that

I aQ * (D2i fi) (X) \ = \ (D2e a'Q ) * (D2i fi) (X) \

2-(jAt)& (A.9)

< 2-eA/22-\j-e\L_

1 + (2^1 |zQ1x|)i

where fie A, and N can be taken to be arbitrarily large. Consequently,

Theorem A.4. Let a e R, 0 < p,q < to, and let L be a nonnegative integer satisfying L > [A max(1,1/p, 1/q) - A -a] + 1. Then there is a constant CA p q a such that for every sequence of smooth L-atoms {uq}q£D and every sequence of complex scalars {sq}q6D one has

^ CA>i>9>aWiSQ}QesWf.' ■ (A.3)

Conversely, there is a constant C'Apqa such that given any distribution f e Fpq(G) and any L> 0, there exists a sequence of smooth L-atoms {aQ}QeD such that

f= Z SQaQ'

where the sum converges in S'(G)/P and moreover

||{Sq}q||^ < CA,p,qJlfllp^(G).

Proof. Let Q e De for some l e Z, and let aQ be a smooth L-atom for Q. We set aQ(x) = oq(zqx). Then aQ is supported in B(0, (y(1 + Aj)/A2)2-e); moreover (since 1 + 2e\y\ : 1 for y e supp aQ),

\ XIaQ (y) \<2-£A/2 2

(1 + 2l\

VIII<L+1, (A.6)

where N can be chosen to be arbitrarily large. Set a'Q(x) = 2~tAaQ(2-ex), that is, aQ = D2eaQ. Then the above inequality can be rewritten as

| X^'Q (y) \<2"

(1 + \

VIII<L+L (A.7)

Using this estimate, (73), and that supp a'Q c B(0, (y(1 + Ai))/A2), we also deduce that

| ¿(a'Q) (j)\ = \(YIa'^(y-1)\

_1,d(J)-d(I)

)(y-1) \

\j\<\i\

d(J)>d(I)

(l + \

V|/| <L+ 1■

\ Vq * (D2j fi) (x) \

= \ aQ * (D2'(zQ1x) \

2-(jM)A

(A.10)

< 2-eA/22-\j-e\L_

1 + (2j*e | zQ x |)

Note that if r e (0,1] and N > A/r then we have

(1+2j*e | zQ^xl)

(1+2j*e | zQ%|) ^

JaQi3f (1

SQI iQI-1^ (z)

(1+2jA£ | z^x^

l ( I I SQ \ r*Q (Z) ]dz

hx-^ncyA^-^yQ^ (1 + 2jM |zq1x|)n"

k=0 J(2yA1)2t2-(>Ae'><\x-1z\<(2yA1 )2k+12-(>Ae)

(1 + 2j*e \z-1x\) J

\x-1z\<(2yA1)2-(>Ae'1

+ I(1 + A 12ky

1 I SQ \ VXq (z) ) dz

KQt3e J

FP,„(G)

J (2yA 1 )2* 2-(->Af) <|x-1 zi<(2yA 1 )2t+12-(J'Af) 1/r

x ( I I^QI'xQ (z)) dz

\QeSf /

2-(jAl)A r

(x, 2yA 12-(jAl))| JB(x>2yA12-<iAi))

x ( I I^qI'xq (z)) dz

\QeSf /

+ I(1+A 12fc)

_№ 2(fc+1)A2-(jA«)A |ß(x,2yA 12fc+12-(jAl) )|

Jß(x,2yA12fc+12-(->Af))

x ( I I^qI'xq (z) )dz

\QeSf /

TO ikt

2-(jAl)A + I(1 + A 12fc)-Nr2(fc+1)A2-(jAl)A

Mi Z W^x)

\Qe3f )

2lA/r2-(jA£)A/r _

M( I |5Q|r^Q )(x)

\Qe3f /

_ 2max(l-j,o)A/r

M( IlSQ|rXQ)(x)

\Qe3f /

(A.11)

Here we used the fact that for z e Q and |x 1z| > (2yA 1)2fc2-(jAl) one has (1 + 2jAl|zg1x|)N > (1 + A12-fc)Nr, which can be easily verified by using Lemma 19 (iv) and the quasi-triangle inequality satisfied by the homogeneous norm. The above estimate and (A.10), along with the argument in [24, pp. 80-81], yield (A.3).

Now we show the converse statement of the theorem. By Lemma20, there exists C e S(R+) such that |£(A)| > C > 0 for A e [2-2(fco+1),2-2(fco-1)] for some (large) positive integer k0, and that C is supported in B(0,1) and has vanishing moments of order L. Then it is possible to find a function f e S(R+) with the properties that suppf c [2-2(fco+1),2-2(fco-1)], |f(A)| > 0 for A e (2-2(fco+1),2-2(fco-1)), and £ieZf(2-2jA)C(2-2jA) = 1 for A e R+. Note that

f(2 2fc°•) e A. Given / e F" (G), it follows from Lemma 3

/ _ I / * (D^y) * (Dj^C)

(A.12)

with the convergence in us decompose / as

/_ I I f /*(D2JT)(v)(D2,C)(v-1x)dy

jeZ QeS^ jQ

_ I I sQaQ,

jeZ QeSj-

where we have set, for Q 6 Dj,

Sq ^ |Q|1/2sup 1/ * (D^y) (y)| sup |xJC|L1,

(A.13)

|J|<L+1

(x) ^ - f / * (D^y) (y) (D2.C) (v-1x) dy. Sq Jq

(A.14)

It is straightforward to verify that ag is supported in B(zg, (y(1 + A 1)/A2) 2-j), and that ag has vanishing moments of the same order as C. Moreover, for Q e Dj we have

2Aj 2d(J)j |X AqI < -

sq Jq 2d(J)j

f |/*(D^)(y)(XJi)(2j (y-1x))|dy

(XJC)|L1 sup |/*(D2^)(y)|

-(d(J)/A)-(1/2)

(A.15)

for all |I| < L+ 1. Hence the function ag a smooth L-atom for Q. Choose any A > A/ min{_p, q}. We note that

I (|Q|-WAH1/%Xg (x))?

- I ( 2jasup |/*(D^)(y)|xQ (x) < sup [2j" |/*(D^)(y)|f

yeB(x,A12-J)

(A.16)

[2jaM*rJ (/,^)(x)f. We thus obtain, using (38),

q}qIU <

This proves (A.5).

(A.17) □

Acknowledgments

The author would like to thank Professor Hitoshi Arai for his

patient guidance and constant support. He is also grateful to

Professor Yoshihiro Sawano for his valuable comments.

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