Hindawi Publishing Corporation Journal of Function Spaces and Applications Volume 2012, Article ID 431016,37 pages doi:10.1155/2012/431016

Research Article

Fourier Multipliers on Triebel-Lizorkin-Type Spaces

Dachun Yang, Wen Yuan, and Ciqiang Zhuo

School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, China

Correspondence should be addressed to Wen Yuan, wenyuan@bnu.edu.cn

Received 20 November 2011; Accepted 10 January 2012

Academic Editor: Hans Triebel

Copyright © 2012 Dachun Yang et al. 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.

The authors study the mapping properties of Fourier multipliers, with symbols satisfying some generalized Hormander's condition, on Triebel- Lizorkin-type spaces and Triebel-Lizorkin-Haus-dorff spaces. To this end, the authors first establish a new characterization of these spaces via some generalized (weighted) g*x functions, which essentially improves the known result for Triebel-Lizorkin spaces even when t = 0. Applying this new characterization, the authors then obtain the boundedness of Fourier multipliers on Triebel-Lizorkin-type spaces and Triebel-Lizorkin-Haus-dorff spaces, which also give a new proof of the Sobolev embedding theorems for these spaces.

1. Introduction

It is well known that many classical operators, including some convolution operators, fractional differential operators, and pseudodifferential operators with constant coefficients, fall into the framework of Fourier multipliers. The study of mapping properties of Fourier multipliers on Besov and Triebel-Lizorkin spaces has a long history; see, for example, [1-10]. Indeed, the best-known Fourier multiplier on Lp(Rn) for p e (1, to), which is nowadays called Hormander's multiplier theorem, was obtained by Hormander [3, Theorem 2.5], preceded by Mihlin [1, 2]. Triebel [4, Theorem 3.5] gave a very useful generalization of Hormander's multiplier theorem [3, Theorem 2.5] from the scalar-valued case to the vector-valued case, which further induced the introduction of the nowadays called Triebel-Lizorkin spaces; see also [5, pages 161-168] for more details including some history of the study on Fourier multipliers. Later, Triebel [9, Theorem 2] established a Fourier multiplier theorem for inhomoge-neous Triebel-Lizorkin spaces, which was even proved to be sharp in [9, Remark 12]; see also [10, pages 73-77] for a detailed discussion.

Recently, Cho and Kim [11] and Cho [12] introduced a new family of Fourier multipliers with symbols satisfying some generalized Hormander's condition and studied the mapping properties of these Fourier multipliers on the classical homogeneous Besov spaces Bpq(Rn) and Triebel-Lizorkin spaces Fpq(Rn) via first establishing some equivalent characterizations of these spaces. This family of Fourier multipliers contains the classical Riesz potential operator Ia and the differential operator da as special cases. As an application, Cho and Kim [11] and Cho [12] presented a new proof of the Sobolev embedding theorems for Besov and Triebel-Lizorkin spaces.

The main purpose of this paper is to clarify the behaviors of these Fourier multipliers in [11,12] on four new classes of function spaces: the Besov-type space Bpq (Rn), the Triebel-Lizorkin-type space Fpq(Rn), and their preduals, the Besov-Hausdorff space BHpq(Rn) and the Triebel-Lizorkin-Hausdorff space FHpq (Rn). These spaces were recently introduced and investigated in [13-18] and proved therein to cover many classical function spaces such as Besov spaces and Triebel-Lizorkin spaces (see, e.g., [10,19, 20]), Q spaces and Hardy-Haus-dorff spaces (see, e.g., [21-24]), Triebel-Lizorkin-Morrey spaces and Morrey spaces (see, e.g., [16, 25-28]). To study the boundedness of Fourier multipliers on Fpq(Rn) and FHpq(Rn), we first establish a new characterization of these spaces in terms of generalized (weighted) g*x functions, which essentially improve the known results in [12] for Triebel-Lizorkin spaces even when t = 0. Applying this new characterization, we then obtain the Fourier multiplier results on Fpq(Rn) and FHpq (Rn), which also essentially improve the known results for Triebel-Lizorkin spaces obtained by Cho in [12] and, moreover, give a new proof of the Sobolev embedding theorems, obtained in [14, 15], for these spaces. Besides, for the Besov-type space Bpq(Rn) and the Besov-Hausdorff space BHpq(Rn), some of the corresponding results are also presented.

We begin with some notions and notation. In what follows, let N := {1,2,...} and Z+ := N U{0}; letS(Rn) be the space of all the Schwartz functions on Rn with the classical topology and S'(Rn) its topological dual space, namely, the set of all continuous linear functionals on S(Rn) endowed with the weak-* topology.

Following Triebel [10], let

») := f y eS(Rn) :f y(x)xYdx = 0 V multi-indices y e (N U {0})"} (1.1)

I Jr» J

and consider S»(R") as a subspace of S(R"), including the topology. Use S^, (R") to denote the topological dual space of S»(R"), namely, the set of all continuous linear functionals on S„(R"). We also endow S^ (R") with the weak-* topology. Let P(R") be the set of all polynomials on R". It is well known that S^ (R") = S'(R")/P(R") as topological spaces. Similarly, for any N e Z+, the space SN (R") is defined to be the set of all Schwartz functions satisfying that JR„ y(x)xrdx = 0 for all multi-indices y e Z" with \y| < N and S'N(R") its topological dual space. We also let S-i(R") := S(R"). As usual, <j> denotes the Fourier transform of an integrable function < on R", which is defined as <f>(Q := JR„ e-i¿ x<(x)dx for all I e R".

The following notion of Fourier multipliers when a / 0 was originally introduced by Cho and Kim in [11] and Cho in [12]. For i e N and a e R, assume that m e C£(R" \ {0}) satisfies that for all \a\ < i,

Re(0,»)

R-n+2a+2|a|

R£|¿I<2R

\d¡m(£,)\ dl

< Aa < oo,

where for a := (a1,...,an) e Zn, da := (ô/ôx1)ai ■■■ (d/dxn)an .The Fourier multiplier Tm is defined by setting, for all f e 5„(Mn), (Tm?) := mf.

We remark that the condition (1.2) when a = 0 is just the classical Hormander condition (see [3, Theorem 2.5]) and, moreover, the condition (1.2) when a = 0 with maximum norms instead of L2 norms is called the Mihlin condition (see [1, 2]). One typical example satisfying (1.2) with a = 0 is the kernels of Riesz transforms Rj given by

fâf) (¿) := -i&mm (1.3)

for £ e Rn \{0} and j e {1,...,n}. When a / 0, a typical example satisfying (1.2) for any ê e N is given by

m(£) := |£|-a for £ e Rn \ {0}, (1.4)

another example is the symbol of a differential operator da of order a := a1 a := (ci,.. .,an) e Z+.

an with

To recall the notions of Bp'Jj(Rn) and FpJq(Rn) in [14] and, their predual spaces,

n) and FHs/q(Rn) in [13,14], we need the following notation. For j e Z and k e Zn, denote by Qjk the dyadic cube 2-j([0,1)n + k) and £(Qjk) its side length. Let Q := {Qjk : j e Z, k e Zn}, Qj := {Q eQ : £(Q) = 2-j} and jQ := -log2£(Q) for all Q eQ.

Let q e (0, to] and t e [0, to). The space Lp

, Z)) with p e (0, to) is defined to be

the set of all sequences G := {gj}of measurable functions on Rn such that

SUP PT

PeQ I1 I

r O p/q

H (x)\q dx

P j=JP

Similarly, the space £q(Lp(Rn, Z)) with p e (0, to] is defined to be the space of all sequences G := {gj }jeZ of measurable functions on Rn such that

llGIU(L?(Rn,Z)) := SUpTFT7

PeQ 11 1

I OO r A

\ E \gj(x)\pdx lj=jpUp

, a i/q

> < TO.

Throughout the whole paper, we denote by C a positive constant which is independent of the main parameters, but it may vary from line to line. Let A be the space of all functions y e S(Rn) such that

supp ip c\£ e Rn : 2 <I£I< №\> C > 0 if 5 <I£I< 5. (1.7)

Now we recall the notions of the Besov-type space Bpfq(Rn) and the Triebel-Lizorkin-type space F^ (Rn) from [14]. In what follows, for any j e Z and y eA, let yj (x) := 2jny(2jx) for all x e Rn.

Definition 1.1. Let s e M, t e [0, to), q e (0, to] and y eA.

(i) The Besov-type space Bp'Tq(Mn) with p e (0, to] is defined to be the space of all f e STO (Rn) such that ||f ||№) := ll{2js(yy * f ^ ^ < to.

Bp,q\:

(ii) The Triebel-Lizorkin-type space Fp'Tq(Rn) with p e (0, to) is defined to be the space of an f e S(Rn) such that \\f R^ := \\{2j%, * f »^U^^ z)) < to.

Obviously, B^(R») = Bp,q(Rn) and ipj(Rn) = ^(R»). We also remark that the spaces Rn) and Fp,q(R '

Bpq(Mn) and Fpq(Mn) are independent of the choice of y eA; see [14].

Remark 1.2. Let s e R.

(i) For p e (0, to), it was proved in [29, Theorem 1(i)] that F^(Rn) = FTO+,TO(T-1/p)(Rn)

when r e (0, to) and t e (1/p,to), and Fp^R») = FTO+TO(T-1/p)(Rn) when t e [1/p, to) with equivalent quasinorms. In [30, Corollary 5.7], it was proved that Fp^(Rn) = FTO,q(Rn) with equivalent quasinorms for p e (0, to) and q e (0, to].

(ii) For p e (0, to], it was proved in [29, Theorem 1(ii)] that Bs/r(Rn) = BTO»T-1/p)(Rn)

when r e (0, to) and t e (1/p, to), and Bp,TO(Rn) = BTO+»T-1/p) (Rn) when t e [1/p, to) with equivalent quasinorms.

Next we recall the Hausdorff-type counterparts of Lp(£q(Rn,Z)) and £q(Lp(Rn,Z)). To this end, for x e Rn and r e (0,to), let B(x,r) := {y e Rn : |x - y| < r}. For E c Rn and d e (0,n], the d-dimensional Hausdorff capacity of E is defined by

Hd(E) := if ^rf : E c (JB^r) (-, (1.8)

where the infimum is taken over all countable open ball coverings {B(xj,rj)}. of E; see, for example, [31, 32].

For any function f : Rn ^ [0, to], the Choquet integral of f with respect to Hd is then defined by

f (x)dHd(x) := f Hd({x e Rn : f (x) > X})dX. (1.9)

Jr» J 0

In what follows, we write M++1 := Mn x (0, to). For any measurable function w on M++1 and x e Mn, its nontangential maximal function Nw is defined by

Nw(x) := sup \w(y,t)|, x e Mn.

!y-x|<i

(1.10)

For p e (1, to) and t e [0, to), the space LPT(¿q(Rn, Z)) with q e (1, to) is defined to be the space of all sequences G := {gj }jeZ of measurable functions on Rn such that

f (Eg (x)|qUx,2-j)P) dx

JR" \ jeZ L J /

< TO, (1.11)

and the space (Lp(Rn, Z)) with q e [1, to) is defined to be the space of all sequences G := {gj }jez of measurable functions on Rn such that

"Gfez)) := Igj (X)\VHx,2- )}"dx)q/P\ < to, (1-12)

where the infimums are taken over all nonnegative Borel measurable functions w on R++1 satisfying

[Nw(x)](pVq)'dHnT(pVq)'(x) < 1, (1.13)

and with the restriction that for any j e Z, w(-, 2-j) is allowed to vanish only where gj vanishes. Here and, in what follows, for all a,b e R, the symbol a V b denotes max{a, b} and, for t e [1, to], the symbol t denotes its conjugate index, namely, 1/t + 1/t' = 1.

Remark 1.3. By [15, Remark 2.1], we know that if 0 < a < b < 1/t, then for all nonnegative measurable functions w on R»+!,

[N^(x)]fldHnTfl(x) < 1 implies that f [N^(x)]bdHnTb(x) < 1. (1.14)

J R" J R"

We now recall the notion of the spaces BH^(Rn) and FHsp'Jq(Rn) introduced in [17].

Definition 1.4. Let s e R, p e (1, to) and y eA.

(i) The Besov-Hausdorffspace BHp'Jq(Rn) with q e [1, to) and t e [0,1/(pVq)'] is defined to be the space of all f e STO (Rn) such that

BHS:rq (R")

{2jS(Vj * f)}

_ < to. (1.15)

eq(ifT( R",Z))

(ii) The Triebel-Lizorkin-Hausdorffspace FHp,'Jq(Mn) with q e (1, to) and t e [0,1/(p V q)'] is defined to be the space of all f e STO (Mn) such that

FHSp/q (Rn)

{2js(yj * f)}

_ < TO. (1.16)

LPp(£q(Rn,Z))

Recall that BHp^R») = Bspq(Rn) and FHfy(Rn) = Fp,q(Rn). Moreover, the dual spaces of BHpq (Rn) and FHpq(Rn) are, respectively, B^'^ (Rn) and F-^(Rn); see [13,14]. Now we present the main results of this paper as follows.

Theorem 1.5. Let a,j e R, t e [0, to), and r e (0, to]. Suppose that m satisfies (1.2) with £ e N.

(i) If £ > n[max(1/p, 1/r) + 1/2] and p e (0, to), then there exists a positive constant C such that for all f e Fp^R»), \\Tmf Wi;;^) < C\\f Wfp^R»).

(ii) If £ > n(1/p +1/2) and p e (0, to], then there exists a positive constant C such that for all f e (Rn), UmfW^ (Rn) < QfWsp^R").

We remark that the Fourier multiplier Tm is originally defined on STO (Rn). Although STO (Rn) may not be dense in Fpq(Rn) and Bp,q(Rn), Tm can still be defined on the whole spaces Fp'Jq(Rn) and Fpq(Rn) in a suitable way; see (3.10) and Lemma 3.4 below.

We also remark Theorem 1.5 when t = 0 completely covers the known results obtained in [12, Theorem 5.1]. The proof of Theorem 1.5 is given in Section 3.

From Theorem 1.5 and [14, Proposition 3.3], we immediately deduce the following conclusion. We omit the details.

Corollary 1.6. Let a,p e R,¡5 < a, p e (0,to), q,r e (0, to], and t e [0,to). Assume that m satisfies (1.2) with £ e N.

(i) If £ > n[max(1/p, 1/r)+1/2] and p* e (0, to) such that ¡5 - n/p* = a - n/p, then there exists a positive constant C such that for all f e Fp^ (Rn),

11 Tmf\|(Rn) < CllfllfpT(R»). (1.17)

(ii) If £ > n(1/p + 1/2) and p* e (0, to] such that 5 - n/p* = a - n/p, then there exists a positive constant C such that for all f e BpJr (Rn),

\TmfWBfl,(Rn) ^ ^llf IIbp,t(Rn). (1.18)

We point out that Corollary 1.6(ii) when t = 0 completely covers Cho and Kim [11, Theorem 1.1] and Cho [12, Theorem 7.1].

Moreover, the range of £ in Corollary 1.6(i) can be essentially improved as indicated by the following theorem.

Theorem 1.7. Let a,p e R, p e (0, to), t e [0,to) and r,q e (0, to] such that ¡5 < a. Let p, e (0, to) such that ¡¡-n/p, = a-n/p. Assume that m satisfies (1.2) with £ e N and £ > n/2. Then there exists a positive constant C such that for all f e FpT (Rn),

HTmfllft,(R») < (R»)• (1.19)

As an immediate consequence of Theorem 1.7 and the lifting property of the space Fpq(Rn) (see, [14, Proposition 3.5]), we have the following conclusion, which shows that Theorem 1.7 has variant for any s e R instead of s = 0.

Corollary 1.8. Given a,j e R, p e (0, to) and r,q e (0, to], let ¡5 be real number with ¡5 < a + y and p, e (0, to) such that p - n/p, = a + y - n/p. Assume that m satisfies (1.2) with £ e N and £ > n/2. Then there exists a positive constant C such that for all f e FpT(Rn), ||Tmf (R») < C\\f ||f^ (R»).

Remark 1.9. (i) We remark that, by taking fi = 0, a e (0,n), p e (1,n/a), q = r = 2, t = 0, and m(£) := |£|-a for all £ e Rn \ {0}, then Theorem 1.7 (and also Corollary 1.8 with y = 0) is just the well-known Hardy-Littlewood-Sobolev theorem for fractional integrals (see, e.g., [33, page 119, Theorem 1(b)]), namely, the Riesz potential Ia maps boundedly from LP(Rn) to Lp' (Rn), where 1/p, = 1/p - a/n. In this sense, Theorem 1.7 (and hence Corollary 1.8) is a generalization of the Hardy-Littlewood-Sobolev for fractional integrals.

(ii) Theorem 1.7 (resp., Corollary 1.8) is not true in the case that f = a and hence p, = p (resp., p = J + a and hence p, = p). Indeed, the assumption p < a (resp., p <j + a) and hence p, > p play a crucial role in the proof of Theorem 1.7 in Section 3, which is not valid for the case that f = a (resp., p = j + a) and hence p, = p.

For £ e (0, to), let W2£(Rn) be the well-known Sobolev-Slobodeckij space on Rn. Recall that Triebel [9, Theorem 2] proved that for all s e R, p,q e (0, to) and £ > n(1/min{p,q, 1} - 1/2), if m e LTO(Rn) and

IMIw|(r») + ^PlKK2'') ||w£(Rn) < TO, (1.20)

then the Fourier multiplier Tm is bounded from the inhomogeneous Triebel-Lizorkin space Fpq(Rn) to itself, where y and y are Schwartz functions satisfying that 0 < f, y < 1, supp y c B(0,2), y = 1 on B(0,1), supp y c B(0,4) \ B(0,1/2) and y = 1 onB(0,2) \ B(0,1). From this, together with the embedding theorem [10, Theorem 2.7.1], we further deduce that, under the above assumptions on m, Tm is also bounded from Fpq(Rn) to -Fp,+"/p,-n/p (Rn) with s e R, p,q e (0, to), p e (p, to) and r e (0, to].

Notice that, if m is as in Theorem 1.7 or Corollary 1.8, then m is not necessary to belong to Lto(Rn). For example, if af 0, then m as in (1.4) satisfies all the assumptions of Theorem 1.7 and Corollary 1.8, but m / LTO(Rn). Thus, the assumptions in both Theorem 1.7 (or Corollary 1.8) and Triebel [9, Theorem 2] are not comparable. This is quite natural, since we are considering the multiplier on homogeneous function spaces, while Triebel [9, Theorem 2] (see also [10, pages 73-77]) studied the multipliers on inhomogeneous function spaces. In some sense, Theorem 1.7 and Corollary 1.8 might be regarded as fractional variants of the homogeneous version of [9, Theorem 2] which corresponds to the case that a = 0of Theorem 1.7 and Corollary 1.8. This might also be the reason why the assumption on £ in [9, Theorem 2] is quite different from the requirement of £ in Theorem 1.7 and Corollary 1.8. Moreover,

the restriction i > n(1/ min{p, q, 1} - 1/2) in [9, Theorem 2] is sharp; see Triebel [9, Remark 12] or [10, pages 73-77].

(iii) Recall that in [34], Marschall introduced a very general class SBkñ (r,p,v; N,1) of symbols a e S'(Rn x Rn) with k e R, 6 e [0,1], p,v e (0, to], r e [n/p, to) n (0, to), 1 e [1, to] and N e (n/1, to). For a symbol a e SBk(r,p,v; N,1), and any f e S'(Rn) and x e Rn, the nonregular pseudodifferential operator a(x, D) is defined as

a(x,D)f (x) :=—^f e**a(x,l)f №1- (1.21)

(2n) JRn

Then Marschall [34, Theorem 9(a)] proved that for all k e R, p,q e (0, to], 6 e [0,1], p,v e (0, to], r e (n/(1 - 6)p, to), 1 e [1, to], N e (n max{1/1,1/2,1/p, 1/q}, to) and

n^ ma^1, 1 + p^ - - (1 - 6)r = s<r - n ma^j p - p, 0j, (1.22)

if either p e (0,1] (p e (0,1) incase that p = to) or p e (p, TO]n[v, to], then the operator a(x, D) with a e SBk(r,p, v; N,1) is bounded from F^(Rn) to Fp,q(Rn), where Fp,q(Rn) denotes the inhomogeneous Triebel-Lizorkin space. This, together with the Sobolev embedding properties of Triebel-Lizorkin spaces, further implies that the operator a(x,D) is bounded from Fp+qk(Rn)

to Fp+n/p*-n/p(Rn) with p, e (p, to) and t e (0, to].

Notice that, if m satisfies the assumptions of Theorem 1.7 or Corollary 1.8, then m is not necessary to belong to S'(Rn); see, for example, m as in (1.4) with a e (0, to). Thus, by the same reason as in (ii), the assumptions in both Theorem 1.7 (or Corollary 1.8) and Marschall [34, Theorem 9(a)] are not comparable.

(iv) Recall that it was proved by Cho in [12, Theorem 5.2] that when i > n[max(1/p, 1/2)] if r e (0,2), or i > n[max(1/p, 1/r) + 1/2 - 1/r] if r e [2, to], the operator Tm maps Fp,r(Rn) boundedly into Fp„q(Rn). However, from Theorem 1.7, we deduce that this conclusion is also true when i > n/2 if r e (0, to]. Therefore, even when t = 0, Theorem 1.7 also essentially improves [12, Theorem 5.2]. Moreover, there exists a gap in the proof of [12, Theorem 5.2] in the endpoint case when p, = to, namely, the formula [12, (5.6)] seems not enough for the first inequality in [12, page 853]. The proof of Theorem 1.7 seals this gap and is given in Section 3.

Theorems 1.5 and 1.7 have the following counterparts for Hausdorff-type spaces.

Theorem 1.10. Let a, y e R, p e (1, to) and m satisfy (1.2) with i e N.

(i) If r e (1,to), t e [0,1/(p vr)'] and i > n[max(1 /p, 1/r)+t + 1/2], then there exists

a positive constant C such that for all f e FHpJr (Rn),

TmfWFñ;^ (R") ^ C\\f\(RT (i-23)

(ii) If r e [1, to), t e [0,1/(p v r)'] and i > n(1/p + t + 1/2), then there exists a positive

íp,r (

constant C such that for all f e BHpJ- (Rn),

Tmf\\Bn;;r-T(R") - Cllf IIbít^(R"). (1-24)

Differently from the spaces Fs/q(Rn) and Bp;q(Rn), it is known that Sto(R") is dense in the spaces FHpq(Rn) and BHp'Jq (Rn); see [13, Lemma 5.3] and [14, Lemma 6.3]. Thus, although Tmf is originally defined on STO(Rn), we can extend Tm into the whole spaces FHpq (Rn) and BHp?q(Rn) by a density argument.

We remark that Theorem 1.10 (i) when t = 0 coincides with [12, Theorem 5.1] in the case that p e (0, to). The proof of Theorem 1.10 is also given in Section 3.

From Theorem 1.10 and [15, Theorem 4.1], we immediately deduce the following conclusion and omit the details.

Corollary 1.11. Let a, ¡5 e R, ¡5 < a, p e (1, œ) and p, e (1, œ) such that ¡5 - n/p, = a - n/p. Assume that m satisfies the condition (1.2) with £ e N.

(i) Let r, q e (1, œ) and t e [0, min{1/(pVr)', 1/(p,Vq)'}] suchthat t (pv r)'< t (p,Vq)'. If £ > n[max(1/p, 1/r) + t + 1/2], then there exists a positive constant C such that for

all f e FHpT (R" ),

TmfWftitjK«) ^ C\\f\\FH°-Trm»y (l-25)

(ii) Let r e [1, to) and t e [0, min{1/(p v r)', 1/(p, v r)'}] such that t(p v r)' = t(p, v r)'. If £ > n(1/p + t + 1/2), then there exists a positive constant C such that for all f e BHpT (Rn),

llTmf IbA^ (R») < C\\f IIBA^ (R")^ (1.26)

Moreover, similar to Corollary 1.6(i), we can further improve the range of £ in Corollary 1.11 (i) as follows.

Theorem 1.12. Let a e R, p e (1, to), p e R with f < a and p, e (1, to) such that p - n/p, = a - n/p. Let r,q e (1, to) and t e [0, min{1/(p v r)', 1/(p,v q)'}] suchthat t (p v r)' < t (p,v q)'. Assume that m satisfies (1.2) with £ e N and

" ( t + ^, î/ r(p V ^ < 2t,

G + 2)' /P,r 6 (!'2), t/0.

(1.27)

Then there exists a positive constant C such that for all f e FApr (Rn),

llTmfl IfH^^R") < C\\f\\FH0p;;. (R") • (1.28)

The proof of Theorem 1.12 is given in Section 3.

Similar to Corollary 1.8, we have the following conclusion, which is an immediate consequence of Theorem 1.12 and the lifting property of the space FHpq(Rn) that can be deduced directly from [15, Theorem 4.1].

Corollary 1.13. Let a,j e R and p,r,q e (1, to). Let ß be a real number with ß < a + j and p« e (1, to) suchthat ß-n/p„ = a+j-n/p. Assume that t e [0, min{1/(p«vq)', 1/(pvr)'}] satisfies that t(p v r)' < t(p„v q)' and m satisfies (1.2) with £ as in (1.27). Then there exists a positive constant C such that for all f e FHpT (Rn) \\Tmf \\mß,T (Rn) < C\\f \\mr,

iprT(R").

Corollary 1.13 implies that Theorem 1.12 has variant for any s e R instead of s = 0.

Remark 1.14. Recall that when t = 0, the Triebel-Lizorkin-Hausdorff space FHpq (Rn) is just the classical Triebel-Lizorkin space Fpq(Rn). Thus, when t = 0, Theorem 1.12 coincides with Theorem 1.7. In this sense, Theorem 1.12 when t = 0 also essentially improves [12, Theorem 5.2]; see Remark 1.9(iv).

The proofs of Theorems 1.5,1.7, and 1.10 strongly depend on the Peetre-type maximal function characterizations of Bpq(Rn), FpJq(Rn), BHp'Jq(Rn), and FHpq(Rn) obtained in [18]. Additionally, to prove Theorems 1.7 and 1.12, we need first establish the generalized (weighted) g^-function equivalent characterizations of Fp'Jq(Rn) and FHpq(Rn), respectively, in Theorems 2.7 and 2.9 below. We point out that Theorems 2.7 and 2.9 consist of two parts: sufficiency part and necessary part. The proofs of the sufficiency part are essentially deduced from the corresponding generalized Lusin-area function characterizations, obtained in [18], of these function spaces. The approach used in the proofs of the necessary part of Theorems 2.7 and 2.9 is totally different from that used in the proof of [12, Lemma 3.2(3)] for Fpq (Rn), which induces an essential improvement of [12, Lemma 3.2(3)] such that we can replace the restriction X > n[max(1/p, 1/r)] in [12, Lemma 3.2(3)] by X> n/r. The proof of [12, Lemma 3.2(3)] strongly depends on the exact equivalent relations between the Lp(Rn) norms of the generalized Lusin-area functions with different apertures, which is not clear whether it is still true if Lp (Rn) norm is replaced by the Morrey norm. Instead of that, in the proofs of Theorems 2.7 and 2.9, we use the Lusin-area function characterization of these spaces and the homogeneity of the Euclidean space Rn. This improvement further induces an improvement of Theorems 1.7 and 1.12 even when t = 0, compared to [12, Theorem 5.2].

To prove Theorems 1.7 and 1.12, we need two technical lemmas from [12, Lemmas 4.1 and 4.2] (see also Lemmas 3.2 and 3.5 below). However, [12, Lemma 4.1(2)] therein is not accurate; see Remark 3.3 below. We give a corrected version in Lemma 3.2(ii) of this paper. We also remark that there exists a gap in the proof of [12, Theorem 5.2] for Triebel-Lizorkin spaces in the endpoint case when p, = to; see Remark 1.9 (iv). In this paper, we seal this gap via a subtle application of the equivalence between the Triebel-Lizorkin space F;TO/q(Rn) and

the Triebel-Lizorkin-type space Fp,q/p(Rn) obtained by Frazier and Jawerth [30, Corollary 5.7] (see also [14, Proposition 3.1]).

The paper is organized as follows. In Section 2, we present Theorems 2.7 and 2.9 and their proofs by first recalling some known characterizations, obtained in [18], of Bppjq(Rn), Fpq(Rn), BHp?q(Rn), and FHp?q (Rn) in terms of the Peetre-type maximal function and the Lusin-area function of local means. Section 3 is devoted to the proofs of Theorems 1.5, 1.7, 1.10, and 1.12. Finally in Section 4, as an application, we give a new proof of the Sobolev-type embedding theorems for Fsq (Rn) and FHpq(Rn).

We point out that so far, for the Besov-type space Bp'Tq(Rn) and the Besov-Hausdorff space BHpq (Rn), it is unclear whether the corresponding results of Theorems 1.7 and 1.12 are true or not. The proofs of Theorems 1.7 and 1.12 strongly depend on the generalized (weight-

ed) gX'function equivalent characterizations of Fpq(Rn) and FHpq (Rn), which are not available for Bpfq(Rn) and BHpq(Rn). Moreover, it is also interesting to establish the inhomoge-neous variants of these results.

Finally, we make more conventions on the notation. Throughout the whole paper, the symbol A < B means that A < CB, where C is a positive constant independent of the main parameter. If A < B and B < A, then we write A ~ B.If E is a subset of R, we denote by xe the characteristic function of E.

2. Some Equivalent Characterizations of Fpq(Rn), Bp,q (Rn), FHSq (Rn), and BHSq (Rn)

In this section, we first recall some equivalent characterizations, established in [18], of Fpq(Rn), Bpq(Rn), FHp,q(Rn), and BHpq(Rn), in terms of the Peetre-type maximal function and the Lusin-area function of local means. Using these characterizations, we further establish some new characterizations of these spaces in terms of the generalized (weighted) g*k-functions, which play a key role in the proofs of Theorems 1.7 and 1.12 in Section 3.

Let e e (0, to), R e Z+ u{-1} and O e S(Rn) satisfy that

> 0 on {£ e Rn : 2 < j£j < 2^,

(ô) (0) = 0 Vjaj< R. (2.1)

In what follows, for any function y,t e (0, to) and x e Rn, yt(x) := t ny(x/t). For all y e Sn(Rn), f e SN(Rn), t e (0, to), 1 e (0, to), and x e Rn, let

(Vtf)x(x) := suP

\(<pt * f (x + y) I (1 + \y\/t)A

which is called the Peetre-type maximal function of local means; see, for example, [18]. The following characterization of Fpq(Rn) was obtained in [18].

Theorem 2.1. Let s e R, t e [0, to), p e (0, to), q e (0, to], 1 e (n(1/p v 1/q), to) and R e Z+ u {-1} such that s + nr <R + 1 and O be as in (2.1). Then the space Fpq(Rn) is characterized by

) = f eSR(Rn) : f j Fpq(Rn)||. < to}, i e {1,2,3},

\\f j t;q(Rn)||!:= sup —

PeQ jP j' ^P

f£(p) dtY/q 1

Jo t-sq\(®t *f (x)\qj dx\

1 if Tf^(p)

llf j tpq (Rn)|| 2 := sup-J t-sqK®*fi(x)\

PeQ jPj \JP J0

p/q Ï 1/p dx

p/q Ï 1/p dx,

|f j Fspq(Rn)||3:= sup-^

PeQ jPf

ft(P ) c

t-sq\ \(G>t * f (x + z)\

Jo J\z\<t

,dz dt tn+1

p/q 1/p

with the usual modification made when q = to.

Remark 2.2. Recall that when t e [0,1 /p), the Triebel-Lizorkin-type spaces are just the Triebel-Lizorkin-Morrey spaces, that is, in the definition of Triebel-Lizorkin-type space, the sum ^ °=jp can be replaced by see [16, Theorem 1.1]. By an argument similar to that used in [18, Theorem 3.1], we can prove that Theorem 2.1 is also true with £(P) replaced by to in \\/ | Fpq(R»)\\1, \\f | psq(R»)\\2 and \\f | Fpq(R»)\3 when t e [0,1/p). We omit the details. The following Theorems 2.3 through 2.5 were established in [18].

Theorem 2.3. Let s e R, t e [0, to), p,q e (0, to], 1 e (n/p, to), and R e Z+ u {-1} such that s + nT < R + 1 and ® be as in (2.1). Then the space Bp,q(Rn) is characterized by

BpTq(R") = {/ eS'R(Rn) : f | Bs/q(R")|| ^ to}, i g{1,2}, (2.5)

1 f r£(p) u

f | Bspq(Rn)\\i := sup — t-sq \(Ot *f (x)\pdx

pgq |P | I jo up

q/pdt t

1 r(p) c \\f | Btww 2:= sup — t-sq \(®*/)A(x)\<

pgq |P | I Jo Up

q/p^1/q t

with the usual modifications made when q = to or p = to.

Theorem 2.4. Let s e R, p,g e (1, to), t e [0,1/(p V q)'], 1 e (n[max{1/p, 1/q} + t], to) and R e Z+ u {-1} such that s + nT < R + 1 and ® be as in (2.1). Then the space FHpq(Rn) is characterized by

FHpq(Rn) = {/ e SR(Rn) : 11/ | FHpq(Rn)||t < to}, i e {1,2,3}, (2.7)

\\f | FH^q(Rn) || 1:= inf

{J"Vq\®t * /\X■ ,t)]-q

11/ | FH

pq (Rn)i 2:= inf

{[rsqm/)JX

qr^( ■ t)]-q

(Rn)|| 3:= inf

LP(R") LP(R")

dzdt\ /q

t-sq\ \Ot * f (• + z)\qM• + z,t)]-qdz#

0 J\z\<t t J

LP (R")

where the infimums are taken over all nonnegative Borel measurable functions w on R++1 satisfying (1.13).

■ /.

Theorem 2.5. Let s e R, p e (1, to), a e [1, to), t e [0,1/(p v a)'], 1 e (n(1/p + t), to) and R e Z+ u{-1} such that s+ut<R+1 and O be as in (2.1). Then the space BHp'rTa (Rn) is characterized by

BHpq(Rn) = / eSR(Rn) : 11/ | BHpq(R")||t< to}, i e {1,2}, (2.9)

( Cto m ua dM

11/ I BHP« (Rn) II1 := f J0 ^W0 * / [-(',t)]-1| Ir.) Tj ,

(2.10)

C CTO a At- ^ 1/a

ii/impq(Rn)i2:= iS4J0 t-sa\\m)iM-^Wap(R.)fj ,

where the infimums are taken over all nonnegative Borel measurable functions w on R++1 satisfying (1.13).

Remark 2.6. (i) The space FHpq(Rn) is a quasi-Banach space; see [13,14,17]. Indeed, by [17, Remarks 7.1 and 7.3], we know that for any // e FHs/a(Rn),

f1 + h\\FHpaa (Rn) ^ 2l/(pva)[\IfrW Ffjs,r (R„) + 11/2 || Ffjs^ (R„ )] . (2.11)

(ii) By the Aoki-Rolewicz theorem ([35, 36]), there exists v e (0,1] such that

(2.12)

for all {fj }jeZ c FHpq (R"). Indeed, v := (p V q)'/(1 + (p V q)') does the job.

(iii) The conclusions in (i) and (ii) are also true for the space BHspJq (R").

Next we establish a new characterization of the spaces Fpq (R") and FHpq (R"). Let q e (0, to], X e (0, to), and w be a nonnegative Borel measurable function. In what follows, for R e Z+ u {-1}, f e SR(R") and y in (2.1), set

u(x,t) := (f * yt)(x), uX(x,t) := sup j |u(y,t)| f 1 + I^-Mj L, (2.13)

for all x g Rn and t g (0, to). For all b g (0, to), s g R, and x g Rn, recall that the generalized weighted Lusin-area function Ssbq(w,u)(x) and the generalized weighted g*x-function GsXq(w,u)(x)

are defined, respectively, by

f fTO r dp1/q

Sl (w,u)(x) := t-sq\ \u(y,t)\q[v(y,t)}~q(btyndy—

^0 Jy-x^bt t

Gsx,q (w,u)(x) := {|TO t-s^^\u(y,t)\^ 1 + J^-M^ q [w(y,t)]-qdyt—t} .

(2.14)

If w(x,t) = 1, then Ssbq(w,u) and Gs1q(w,u) are called, respectively, the generalized Lusin-area function, denoted by Ssbq(u), and the generalized g*xfunction, denoted by GsXq (u).

In what follows, for t e [0, to) and p e (0, to), let Lp (Rn) be the set of all functions / e Lp, (Rn) such that

Lp(Rn) := S|P|T

I" \f(x)\pdx Jp

< to. (2.15)

Theorem 2.7. Let s g R, p g (0, to), t g r0,1/p), q g (0, to], 1 g (n/q, to) and R g Z+ U{-1} such that s+nt <R +1. Then f g FpJq (Rn) if and only if f g SR (Rn) and GsKq (u) g L?(Rn), where u is as in (2.13). Moreover, there exists a positive constant C such that for all f g Fp'rTq (Rn),

C-'W fh- (Rn) < | |G1q (u)|| LP (R„) < C\\f\kq (Rn). (2.16)

Proof. Assume / e SR(Rn) and GsX/ (u) e Lp(Rn). Notice that for any X e (0, to) and x e Rn,

xq ^ 1/q

Sshq(u)(x) < xJt-s\u(y,t)\]q( 1 + t-ndy

0 Jy-x^t" ..... \ t / ' t J (2.17)

= 2AG!q (u)(x).

Then, from Remark 2.2, we deduce that f g Fpq (Rn) and

S? q(u) p < G«(u) p , (2.18)

I WLp(R») II 1q lp(R") v '

which completes the proof of the sufficiency of the theorem.

Conversely, suppose that / e Fp,'Ja(Rn). Then by Theorem 2.1, / e SR(Rn). Moreover, similar to the proof of [18, Theorem 3.1], for any k e N, we see that

suPrpF , ,

PeQ \P\ | JP

, dt p/q } 1/p

{t-s\n-H * f (y)\)qrndyd dx\

JO J\y—x\<t t _

PeQ \P \

Z2isqWk+j * f (x)\q

j=—O

(2.19)

where C is a positive constant independent of k and /. Then by changing variables, we conclude that

SUP P . ,

PeQ \P \ ^P

< 2—kssup

PeQ \P \

J0 J\y—x\<t 1

Z2isqWj * f (x)\q

j=—O

O -tp/q } 1/P

X f (t—>2—4 * f {y)\)qt-ndyddt dx\

Jo J\y—x\<t t

FpURn)'

(2.2o)

where the last inequality follows from the equivalence between Triebel-Lizorkin spaces and Triebel-Lizorkin-Morrey spaces when t e [0,1/p); see [16] and also Remark 2.2. By changing variables, we know that for all x e Rn,

Glq (u)(x) =

f rWt * f (y)|)Yl + l\-ndy

|y—x|<t t

E ■■■dy

k=1 J 2k—1t<\y—x\<2kt

OO /"O r

¿V'kXq (t-sWt * f (y)\)qt-ndy

k=0 J0 j\y—x\<2kt

<1 Z2—k(Xq—sq—n)n (t-sW2—H * f (y)\)qt-ndyddt

k=o o y—x <t t

(2.21)

Thus, when p < g, from the well-known inequality that for all d e (0,1] and {aj }j c C,

a ')d <?1

(2.22)

it follows that

Rq^llL^R-)

PgQ |P| i JP

-k(lq-sq-n)

0 Jy-x<t

(i-s\y2-k.t * fy \)qt-ndy

S ¡pHJpIO2

-k(X-s-n/q)p

0 |y-x|<t

(t-s\V2-kt * f(y) \) qt-ndyT

< J ^p-k(1-s-n/q)p

* sup IPr.P

c —if\p/q 1

(t-s\^2-kt * fy\)qrndy— dxl

0 |y-x|<t t

p/q ^ 1/p dx

< ^ ^2-k(1-n/q)^|f F fc=0

F?2 (rt

(2.23)

where the last inequality follows from (2.20) and X> n/q. Similarly, when p > q, by Minkowski's inequality and (2.20), we see that

sup 1 I J]2-

PgQ |P| \ k=0

k(lq-sq-n)

0 ./ |y-x|<t

(t-s\^2-kt * f (y)\)qt-ndy—

q/p 1/q

<1 y^2-k(iq-sq-n) k=0 PgQ

0 ^ |y-x|<t

< ^ ^2-k(i?-n)ii ziq-,k=0

(t-s\^2-kt *f (y)\)qt-n dy—^

Fp,q (r-).

(2.24)

These estimates, together with Remark 2.2, imply the necessity of the theorem and hence complete the proof of Theorem 2.7. □

Remark 2.8. We point that, by an argument similar to the proof of Theorem 2.7, one can characterize Fp^ (Rn) via a discrete version of the generalized weighted ^-function. More precisely, for all s e R, p e (0, to), t e [0,1/p), q e (0, to], 1 e (n/q, to), and R e Z+ u {—1}

such that s + nr < R + 1, then f e

[£ teZ JRn 2i(sq+n) | f * fi(z) | q(1 + 2' | z - ■ | )-1q dzfq

r b'Tnon

) if and only if f e SR(R") and

e LP(Rn). Moreover,

W 2i(sq+n) |f * ^t(z)|V 1 + 2'|z --i)-Aqdz

fe^R" X 7

(2.25)

We omit the details.

We also obtain the following analogy of Theorem 2.7 for the space FHp'Jq (Rn).

Theorem 2.9. Let s e R, p e (1, to), q e (1, to], t e [0,1/(p V q)'], 1 e (n/q, to) and R e Z+ u{-1} such that s + nr <R +1. Then f e FHp,q (Rn) if and only if f e SR (Rn) and Gs1q(w, u) e LP(Rn), where u is as in (2.13). Moreover, there exists a positive constant C such that for all f e FHpq (Rn),

< inf | GSXq(w,U)

< cifb

| LP(R") _ 1 '■' (Rn)/

(2.26)

where the infimum is taken over all nonnegative Borel measurable functions w on R++1 satisfying (1.13).

Proof. Assume f e SR(Rn) and Gs1q(w,u) e Lp(Rn). For any 1 e (0, to) and x e Rn, similar to the proof of Theorem 2.7, we know that

Ss1q(w,u)(x) < 2XqGslM(w,u)(x).

(2.27)

Then by Theorem 2.4, we see that

FHZ(R") ~ inf SS,q(^'U)Tp(„") Z inf Glq(W'U)\

(2.28)

Fp,:'q (R"

FH^ (R")

which completes the proof of the sufficiency of the theorem.

Conversely, suppose that f e FHsp'Jq(R"). Then by Theorem 1.12, f e SR(R"). By an argument similar to the proof of [18, Theorem 3.3], we see that for any k e N,

f t-sq\«n)t */)(z)\qL(z,2-kt)|~qdz

0 J\z-\<t L V /J

q , dt tn+1

2-ksinf

'K )f 1

^^ * /\q [,(•, 2-j)]-q 1

L;eZ j

LP (Rn)

LP (Rn)

(2.29)

FHS,Tq (Rn) '

Let R+ := (0, œ). For all measurable functions F on Rn x R+ x Rn, let

||F||f := inf

\\ t-sq\ \F(y,t, •)\[M(y,t)]-qdy1dL

LP(Rn)

(2.30)

where the infimum is taken over the same set as in (1.13). We claim that || ■ | is a quasinorm with respect to F, precisely, for any measurable functions Fi, F2 on Rn x R+ x Rn,

l|Fi + F2|f< 21/(pVq)'(|Fi|F + IIF2ll?)• (2.31)

To see this, without loss of generality, we may assume that ||Fi\\? + \\F2\\f < to. Then, for any e e (0, to), choose nonnegative Borel measurable functions w1, w2 on R++1 satisfying (1.13) such that

{ f rsq \

dt ^ 1/q

\ Fi(y, t, •) \ My,t)} -qdydl)

< (1 + eWillf, (2.32)

LP(Rn)

for i e {1,2}. Notice that w := 2-1/(pVq)' max{w1,w2} still satisfies (1.13). Then by (2.22) and Minkowski's inequality, we see that

IF + F2^f<

i (W |F1 (y, t, ■) + F2(y,f, -)|Ky,t)]-qdy

I J 0 J R"

dt x/q

< 21/(Pvq)'

t"+1 dt 1/q

LP (R")

{[ rsq\jF1(y,t, •)|[^1(y,t)]-qdyt^ 1

T T dt 11/q

{Jo rsqJR |F2(y,t,.)|[^2(y,t)]-qdytd+T

LP (R") (2.33)

< 21/(pVq)'(1 + C)(HF1|f + IIF2II?).

Letting e ^ 0 then concludes the above claim.

Thus, by the Aoki-Rolewica theorem [35, 36], we know that

LP(R")

(2.34)

for all measurable functions {Fj}jeZ on Rn x R+ x Rn, where v = (p V q)'/(1 + (p V q)').

Choosing 1> n/q, by (2.34), (2.29), and an estimate similar to (2.21), we conclude that

inf I Gx,q(w,u)\

LP(R")

f t-sq\ |w * f (y)|^1

| \ -Aq 11/q

[-(y,t)]-qdytd|

LP (R")

if t-sqt( 2~Xkq | Wt * f (y)rKy,t)rqdydr

[Jo k=0J B(v2kt) tn+s

LP(R")

< y 2-xkv inf

< ¿0 -

{{>! |W * f (y)|qXB(v2kt)(y)Ny,t)]-qdyA

dt 1/q

< 2-k(^-s-"/q)v k=0

[ f t-sq\(<Pk)t * f (y)\qHy,2-kt)\^dy^i

'0 J\y—\<t

< ^ 2_k(A-n/q)v k=0

FHpT (R»)

Ftf/q (R»)'

(2.35)

which implies the necessity of the theorem and hence completes the proof of Theorem 2.9. □

3. Proofs of Theorems 1.5,1.7,1.10, and 1.12

In this section, we give the proofs for Theorems 1.5,1.7,1.10, and 1.12.

In what follows, K always denotes the distribution whose Fourier transform is the function m in (1.2). Then we have the following observation.

Lemma 3.1. Let m be as in (1.2) and K its inverse Fourier transform. Then K e SO (Rn). Proof. Let y e Soo(Rn). Then (0) = 0 for all y e Z" and hence for any L e N,

(K'V) =\ m(l)(p(l)dl

m(l)(p(l)dl + f m(l) •'\l\>1 •'K

«{) - 2 ^i

dl =: Ii + I2.

For I1, by Holder's inequality and (1.2), we see that

CO /•

l Ill <X l m(l)|m)\dl

k=0J2k < \ l \ <2k+1

¡£0(1 + 2k)M J 2k< \ l\ <2k+i 2»k/2

! m(l) d

ke0(1 + 2k)

2k< l <2k+1

\ m(l) fdl

^ ok(n-a)

< y —-< 1,

~ Zj ^ „iaM ~ '

fc=0 (1 + 2k)

where M e [0, oo) is chosen large enough such that M > n - a.

For I2, by the mean value theorem, there exists d e [0,1] such that

|I2| < £ f M£)| sup \drmMtM

k=-xJ 2k< | i| <2k+1 | y |=L+1

< 2fc(L+1) f

2k < | i | <2k+1

\m(i)\di

< ^^ 2k(L+1+n-a) < i

k=—<x>

where L is chosen large enough such that L > a - n - 1. This finishes the proof of Lemma 3.1. □

The following estimates play an important role in the proofs of Theorems 1.5,1.7,1.10, and 1.12.

Lemma 3.2. Let f, Z be Schwartz functions on Rn such that f, £ are supported in the annulus {£ e Rn : 1/2 < ||| < 2}. Assume that m satisfies (1.2).

(i) If 1 e (0, to) and £ > 1 + n/2, then there exists a positive constant C such that for all

t e (0, to),

i + y ) Kk * ft) (z)\dz < cta.

(ii) Let k, N be any two positive integers. If £ > 1 > 0, then there exists a positive constant C such that for all s, t e (0, to),

1 + -j) \(K * * fd (z)\2dz < C(min(t,s})

if S < t, if t < S.

Remark 3.3. We remark that Lemma 3.2(i) is just [12, Lemma 4.1(1)]. It was also claimed in [12, Lemma 4.1(2)] that the inequality in Lemma 3.2(ii) is valid with (min{t, s})—n+2a replaced by s—n+2a. However, the proof of [12, Lemma 4.1(2)] is problematic. Indeed, the last inequality in [12, page 849] seems to be true only when s < t. We give a correct version in Lemma 3.2(ii).

Proof of Lemma 3.2(ii). Since X < i, by the Plancherel theorem, we see that

lzl\2X 2

^ * Zs * №)(z)'2

(1 + \(K * * yt)(z)\2 dz

JR» \ S /

< 2 s~2H f k (m(l)£(l)<t(l))\2dl

< 2 s-2H 2 s2\"2^f \(a^1 m)(l)(öf l)(sl)(ö?v)(tl)\2dl.

\ a \ <£ 01+02+03=0 •'R»

When s < t, by the support of Z and (1.2), we see that

(1 + \ä) \(k * Zs * <t)(z)\2dz

< Y1 s-2 \ a \ V s2 \ a2 \t2 \ a3 \

2 s-2 \ a \ 2 s2 \ a2 \t2 \ a3 \[ Udl1 m) (l)(df <) (tl)\2dl

a\ <£ a1+a2+a3=a J 1/(2s) < \l \ <2/s 1 V 7 7 1

<2.....Lis) ' ~1"+t/^j2N+27s-2'a1' J^/^sw l|<2/s Kd^ ^ (l)|^dl

a \<£ a1+a2+a3=aVs/ (1 + t/s) + J 1/(2s)<\l\<2/s 1

,-n+2a

(1 + t/s)2N'

When t < s, by the support of < and (1.2), we conclude that

f (1 + — y'KK * Zs * ft)(z)\2dz

< 2 2 s-2\ a \s2 \ a2\t2\a3 \[ \(dfm)(l)(dfz)(sl)\2dl

\ a \ <£ 01+02+a3=a J1/(2t)<\l \ <2/t 1 X 7 7 1

t 2 1 +2 3 1

2 1 +2 3 \ \ 2

< 2 2 (M -1-^t-2 \a1\| Md?m)(l)\2dl

\ a \ <i a1+ a2+a3=a \s/ (1 + (s/t)) J 1/(2t)< \ l \ <2/t 7 1

< ^ s r

which completes the proof of Lemma 3.5(ii). □

Recall that So(Rn) is dense in the spaces FH^ (Rn) and BHsp'Jq (Rn) (see [13, Lemma 5.3] and [14, Lemma 6.3]). Then the definition of Fourier multiplier Tm can be extended to the whole spaces FHpq(Rn) and BHpq(Rn) via a dense argument. Next we show that,

via a suitable way, Tm can also be defined on the whole spaces Fpq(Mn) and Bp,q(Mn). To this end, let y eA. Then by [37, Lemma (6.9)], there exists $ eA such that

Xy(2-t£)$ (2-t0 = V£ e M" \ {0}. (3.9)

For any f e Fl'l(Rn) or Bs'Z(Rn), we define Tmf by setting, for all < e 5o(Rn),

(Tmf,$) := £f * y * fi * < * K(0).

(3.10)

In this sense, we say Tm/ e STO (Mn). The following result shows that Tmf in (3.10) is well defined.

Lemma 3.4. Let £ e (n/2, to), s e M, t e [0, to) and q e (0, to], f e Fp'Jq(Mn) with p e (0, to) or f e Bpq(Mn) with p e (0, to]. Then Tmf in (3.10) is independent of the choices of the pair (y,$) of functions in A satisfying (3.9). Moreover, Tmf e STO(Mn).

Proof. Assume first that f e FpTO(Mn). Let y and $ be another pair of functions in A satisfying (3.9). Since $ e STO(Mn), by the Calderon reproducing formula (see [13, Lemma 2.1]), we know that

$ = £yj * $ * $ (3.11)

in So(Rn). Thus,

* yi * fi * < * K(0) = * yi * fi * (^Jfj * fj * * ^(0)

ieZ ieZ \ jeZ /

N 7 (3.12)

= X X f * y * j * (fi * * < * ^(0)'

ieZ j=i-1

where the last equality follows from the fact that yi * y = 0 if |i - j | > 2.

Let Z := y * ffi and ^ := $ * $. Then Z, ^ eA. If t e [0,1/p) and q = to, we see that for all x e Mn,

X If * y * f 0 * (fi * f) * < * *(0) I

= X f * Ci(~z) ni * < * K(z)dz

sup sup 2is|/* &(-z)|( 1 + 2i| Z + x|Y

ieZ zeR" ^ '

W 2-is( 1 + 2i| z + x | Yk * m * $(z^dz =:Ii(x)l2(x),

iaZJR"

(3.13)

where X is an arbitrary positive number.

For Ii, by Remark 2.8, we know that ||I1 NLp(Rn) ^ 11/Nf^o^) < o, which implies that there exists x e B(0,1) such that | I1(x)| < o.

For I2, choosing X e (0, o) and p e (n/2, o) such that X+p< £,by Holder's inequality and Lemma 3.2(ii), we see that for all x e B(0,1),

(1 + 2i| z + x |)A|K * m * $(z^dz

< (1 + 2)X [ (1 + 2i| z | yk * m * $(z^dz

< (1 + 2i)X2-in/2U (1 + 2i | z | f'V * ni * $(z)|2dz

< 2-in/2(min{1,2-i})~n/2+"(min{1,?})*( 1 + 2^^.

(3.14)

Thus, by choosing k and N large enough such that k > n/2 +1 s| and N > X +1 a| +1 s|, we know that

I2(x) < ^ 2i(A-a-s-N) + £ 2i(k-n/2-s) < 1.

(3.15)

Therefore,

2|/ * (yi * fa) * (fi * * $ * K(0)| < œ.

(3.16)

By an argument similar to the above, we see that

X X 1/ * i^i * <Pi) * (fi * $1) * § * K(0) | < o,

ieZ 1=i-1

which, together with the Calderon reproducing formula, further induces that yj * yi * * y * K(0)

(3.17)

= X 2 / * (fai * j * * j * $ * K(0)

jeZ i=j-1

= y1 f * yj * fj * ( * fi * <1 * K(0)

jeZ ieZ

= y1 f * yij * (fj * y * K(0).

(3.18)

Thus, Tmf in (3.10) is independent of the choices of the pair (y,$). Moreover, the previous argument also implies that Tmf e STO (Mn).

If t e [0,1/p) and q e (0, to), from the embedding Fpq(Mn) c Fp;TO(Mn), we deduce that Tmf is also well defined in Fpq(Mn) and Tmf e STO(Mn).

If t e (1/p, to) and q e (0, to), by Remark 1.2 and [29, Corollary 1], we know that

ts-Tnon\ _ rs+n<T-1/p^_ Rs+n(T-1/p)mn\ Fp,q(R ) = (R ) = Bo,o (R ).

(3.19)

Then, by Theorem 2.3, we know that

E f * Zi(-z) ni * < * K(z)dz fe z •'r"

sup sup 2i[s+n(T-1/p)]|f * Zi(z)1

ieZ zeR"

~ 11 ri| ^^(R")

^2-i[s+n(T-1/p)] f n * < * K(z)|dz

ieZ Rn

^2-i[s+n(T-1/p)] f ^i * < * K(z)|dz.

ie Z J R"

(3.20)

Choosing X > n/2 such that £ > X, by Holder's inequality and Lemma 3.2(ii), we conclude that

^2-i[s+n(T-1/p)] f n * < * K(z)|dz

ieZ Rn

< ^2-i[s+n(T-!/P)]

— 2A " 1/2 " 2A.

(1 + ?|z|) dz (1 + ?|z|) |ni * < * K(z)|2dz

Jr" ^ ' LJr" ^ '

< £2-i[s+n(T-1/P+1/2)](min{1'2-^)^(min^1 + 2i)

^2-i[s+"T-n/p+a+N] ^ 2-i(s+"T-n/p+n/2-k) < 1

(3.21)

where k and N are chosen large enough such that k > |s| + n(r - n/p + n/2) and N > |s| + |a| + n/p. By an argument similar to the above, we see that

E H |/ * (yi * 1 * * 1 * § *K(0) 1 £ II/llFrof-1/p>(R") ~ IjmR«y (3.22)

ieZ j=i-1 ' '

which, together with the Calderon reproducing formula, further induces that

yj * y; * yi * § * K(0) = E/ * y * ijfj * § * K(0). (3 23)

ieZ jeZ

Thus, in the case that t e (1 /p, œ), Tmf in (3.10) is also independent of the choices of the pair (y,y). Moreover, Tmf e Sœ (R").

Finally, if t = 1/p and q e (0, œ), since (R") = (R") c Fœ^R") (see [30, Corollary 5.7]), from the previous argument, we deduce that Tm is also well defined in F^R"). Therefore, we obtain the desired conclusion for the space FspJq(R") for all admissible indices.

Assume now that f e Bpq(R"). If p e (0, œ), by the obtained conclusion for Fs/q(R"), the embedding Bp^(R") c Fspfq(R") when q < p (see [14, Proposition 3.1(vii)]) and

Bp;q(R") c Bp;pe;T+e/"(R") c Fp;qe;T+e/"(R") (3.24)

for some e when q > p (see (iii) and (vii) of [14, Proposition 3.1]), we know that Tm is well defined in Bp^(R"). This, together with the embedding BsJT,q(R") c Bspl+1/po (R") for some p0 e (0, œ) (see [14, Proposition 3.1(ii)]), further induces the corresponding result for BsJfrq(R"), and hence completes the proof of Lemma 3.4. □

Now we have the following technical lemma.

Lemma 3.5. Let a e R, 1 e (0, œ), r e [2, œ], £ e N, y , y eA, and u, u*k be as in (2.13). Assume that m satisfies (1.2) and f e Sœ (R") such that Tmf e Sœ(R").

(i) If £ >1 + "/2 and ® = y * y, then for all x,y e R" and t e (0, œ),

| (Tmf * ®t)(y)|< Ct^1 + iflM^ ul(x,t). (3.25)

(ii) If £ >1 + n(1/2 - 1/r ), then for all x,y e R" and t e (0, œ) satisfying that x - y| < t, KTm/ * ^t)(y)| < CtaG0Xr(u)(x). (3.26)

Proof. (i) is just [12, Lemma 4.2(1)]. The proof of (ii) is similar to the proofs of (2) and (3) of [12, Lemma 4.2], but with [12, Lemma 4.1(2)] replaced by Lemma 3.2(ii). This finishes the proof of Lemma 3.5. □

We remark that by Lemma 3.4, Tmf e STO(Rn) when f e Fp,'fq(Rn) or Bp;q(Rn) with all indices as in Lemma 3.4. Thus, Lemma 3.5 is also true for all f e Fpq(Rn) or f e Bp,q(Rn) with all indices as in Lemma 3.4.

Now we are ready to prove Theorems 1.5,1.7,1.10, and 1.12.

Proof of Theorem 1.5. Let d be as in Lemma 3.5.

(i) By the assumption that £ > n[max(1/p, 1/r)+1/2], there exists X > n[max(1/p, 1/r)] such that £ > X+n/2. Then by Lemma 3.5(i),we see that for all x e Rn and t e (0, to),

t-(a+Y) (dt (Tmf))x(x) < t-ruX(x,t), (3.27)

which yields the desired result in view of Theorem 2.1.

(ii) By the assumption that £ > n(1/p +1/2), there exists X > n/p such that £ > X + n/2. Then by Lemma 3.5(i), we also see that for all x e Rn and t e (0, to), (3.27) holds, which yields the desired result in view of Theorem 2.1 and hence completes the proof of Theorem 1.5.

Now we give the proof of Theorem 1.7.

Proof of Theorem 1.7. To prove the theorem, by the monotone embedding property on the parameter q of the spaces i>,'Tq(Rn) (see [14, Proposition 3.1(i)]), namely, F^ (Rn) c F^(Rn) if qi < q2, it suffices to consider the case q e (0, to). We show the desired result in two cases for

Case 1 (t e [0,1/p)). Assume first that f e Fp^(Rn) with r e [2, to]. By assumption that £ > n/2, we know that there exists X> n/r such that £ > X + n/2 - n/r. Then from Lemma 3.5(ii), we deduce that for all x,y e Rn and t e (0, to) satisfying that |x - y| < t,

\U(y,t)\< taG°Xr(u)(x), (3.28)

where and in what follows, U(x, t) := (Tmf * ft)(x) for all x e Rn and t e (0, to).

If llf Hi^T(r») = 0, by Theorem 2.7, we know that ||GXr(u)||Lp(R») = 0, and hence GX r(u)(x) = 0 for almost every x e Rn, which, together with (3.28), implies that U(y,t) = 0 for all y e Rn. We then conclude that ||Tmf||tp,T = 0.

Fp* ,q(R )

If ||f Ht^T(R») > 0, we know that ||GXr(u)|Lp(R») > 0. Let P be a dyadic cube and t e (0,£(P)). Then, there exist 3" dyadic cubes {Pi>3"1, with £(P{) = £(P), such that

{y :dis^y,p)<t} ^ Pi.

(3.29)

Then, raising (3.28) to the power p and integrating over the ball B(y , t), we see that

B(y , t)

U(y,t) fdx\ < t

u3=»1 p

Gl (u)(x)\ dx

3» c f 11/p

< ta2{jpjGX,r(u)(x)\PdA ,

(3.30)

which further implies that

\U(y,t) \ < ta-»/p g j Gl^ (u)(x)fdx

(3.31)

For any fixed x e P and A := A(x) e (0, o) which is determined later, by (3.28), (3.31), a > p and a - p - n/p = -n/p*, we see that

r(p) r dt

t-ßq \U(y,t)\qt-»dyd-t

J0 J\ y-x \ <t t

= f f X(0,e(P))(t)t-ßq\U(y,t)\qt-»dydt + C f

J0 J\y-x\ <t t JAJ\

A y-x <t

< [Gl,r(u)(x)]qA(a-ß)q + \Gl,r(u)(x)\pdxj A(a-ß-»/p)q

< [Gl r(u)(x)]qA(a-ß)q + 11Glr(u)\\qp \P^A*-?-"^.

(3.32)

Take A such that

A~u/p =

Gl (u)(x)

\PGlr(u)

X,r\ -1 rp(

(3.33)

Then we see that

■£(P)

rßq\ \U(y,t)\qt-»dydtt

\y-x\<t t

< P\T(1-

p.) g0

GXr(um p

X,r "Lp (R»)

* I" _ 0

G°ir (u)(x)]p

(3.34)

Then, by Theorem 2.7 and X> n/r, we conclude that

i fr /reP) r dt\p"/q 11/p*

llT-/ll*>« = sup ¡FT {Ja/. LJr'U(y-t} 'I ,r"dllT)

iii-(p/p.) (3.35)

<|sup ]1f(jplGir («)(x)] Pdx) \ \\G{r («)

Lp (R")

FpT (R")

When f e FpT(R") with r e (0,2), the desired conclusion is a direct consequence of the case r e [2, œ], together with the the embedding F(R") c FpT(R") (see [14, Proposition 3.1 (i)]).

Case 2 (t e [1/p, œ)). In this case, since p, > p, we see that t > 1/p > 1/p,.

If t e (1/p, œ), by the assumption that i > n/2, we know that there exists X > 0 such that i > X + n/2. Then from Remark 1.2, Theorem 2.3, Lemma 3.5(i) and the fact that Fœ,œ(R") = éœ,œ(R"), it follows that

Mf*™ ~ I |Tmf|fiP+»(T-1/p.)(R„) ~ supsup f-P-"(T-(1/p')) | Tmf * <&i(x)|

œ,œ t>0 xeR"

< sup sup ttt-^-"(T-(1/p*))|MX(x,t)1 ~ sup sup t-"(T-(1/p)) 1 u*X(x, t)1 (3.36)

t>0 xeR" t>0 xeR"

B"jœ1/p)(Rn) '

If t = 1/p, we only consider the case r = œ in view of the embedding FpT (R") c Fp^(R") (see [14, Proposition 3.1(i)]). Then, similar to the above argument, we see that

IIWIU ~ sup sup r^-^lTmf * cDt(x)| < | IflIfoœcR") ~ | |f|fpœtRn)' (3.37)

p*,q t>0 xeR" > pœ ) \ I

which completes the proof of Theorem 1.7. q

Now we give the proof of Corollary 1.8.

Proof of Corollary 1.8. The result follows from either a minor modification of the proof of Theorem 1.7 or considering the symbols m(£) := m(£)|£|-r for all £ e R" \ {0} and the lifting property. We omit the details. □

Next, we give the proof of Theorem 1.10.

Proof of Theorem 1.10. Let cC be as in Lemma 3.5.

(i) Since i > "[max(1/p, 1/r) + t + 1/2], there exists X > "[max(1/p, 1/r) + t] such that i > X + "/2. Then by Lemma 3.5(i), we see that for all x e R" and t e (0, œ), (3.27) holds, which yields the desired result in view of Theorem 2.4.

(ii) Since V > n(1/p + t + 1/2), there exists 1 > n(l/p + t) such that V > X + n/2. Then by Lemma 3.5(i), we also see that for all x e Rn and t e (0, to), (3.27) holds, which yields the desired result in view of Theorem 2.5 and hence completes the proof of Theorem 1.10. □

Now we give the proof of Theorem 1.12. We begin with a technical lemma proved in [15, Lemma 3.2], which reflects the geometrical properties of Hausdorff capacities.

Lemma 3.6. Let ¡5 e [1, to), 1 e (0, to), and w be a nonnegative Borel measurable function on R++1. Then there exists a positive constant C, independent of ¡¡, w, and X, such that

Hd({x e Rn : Npw(x) > X}) < CpdHd({x e Rn : Nw(x) > X}), (3.38)

where Npw(x) := sup| |<^tw(y, t) for all x e Rn.

Proof of Theorem 1.12. Since when t = 0, the Triebel-Lizorkin-Hausdorff space is just the Triebel-Lizorkin space, we only give the proof for the case t e (0, min{1/(p Vr)', 1/(p* Vq)'}]. Assume first that f e STO(Rn) and \\f \\ptfp-T(R») > 0. Choose X > n/r and w be a

nonnegative function on Rn++1 with

[N¿3(x)](pVr)'dHnT(pVr)' (x) < 1 (3.39)

such that

< \\G{r (w,u)\\Lp (Rn) < 21 f\\FH°l (R")' (3.40)

Then \\GX,r(¿3,u)\\lp(r») > 0.

Let y be as in (2.1). Then there exists a Schwartz function £ such that £ has compact support away from the origin and

rTO _ ds

m)m)— = 1, i=0, (3.41)

see, for example, [30,37]. By the Calderon reproducing formula, we know that for all y e Rn,

rTO ds

(Tmf * ft) (y) = (f * ys * K * Zs * ft) (y)—- (3.42)

Then, applying Holder's inequality, we conclude that for all nonnegative functions w on R++1 and x e B(y,t),

I (Tmf * vOMIKy-O]"1

s -H C l( 1+"

x \ (K * & * ft) (y - z)\r' [¿3(z, s)^y, 0_1]r's^'-^dz—

< G°(¿3,u)(x)

ij" j (i+

^.>0 JR" \

x 1 (K * £s * ft)(y - z)|r' [¿3(z, s)w(y, t)-1]r sn(r'-1)dz

=: Glr (¿3,u)(x)^y/^'

(3.43)

Raising this inequality to the power p and integrating over the ball B(y, t) with respect to x, we see that

\(Tmf * ft)(y) | [w{y,t) ]-1 < ||g° (¿3,u)||„ rn/ph(y,t)'

(3.44)

Since \\f Wph^-t > 0, then \\GXr(w,u)\\ > 0. Thus, in this case, for any fixed x and D := D(x) e (0, to) which is determined later, applying (3.43), (3.44), and the Aoki-Rolewicz theorem (see [35, 36]), we know that

lTmfllIfh^r")

JT+L%Ktrß,KTmf * ^w^nf^}

D^ ^ [g^(¿5,u)h(y,t)]'dyd-i

fD t-(ß+("/P))^y |<j||Gl,r^^LP(R")h(y,t)

LP* (R")

y t"+1

fW fix/ f . 0+^

JO J\y— | <t . J s~2lt J | z-y | ~Vs \ /

LP• (R") lr'

x ^K * £s * ft)(y - z)|r [¿3(z,s)w(y/t) 1

x s"(r'-1)dz-

- q/r'

I LP (Rn)

LP« (R")

j=0 teZ (

0 J\y-\<t

J\y-\<t J s~2lt J I

s~2H J \z-y\~2js

|z - v\

x ^K « Zs « yt)(y - z)1 ¿3(z,s)w(y,t)

x sn(r'-1)dz

dytÊi [Gl,r ((,u)] '

t-ßq-(n/p)q f

JD •'\y-\<t

^'dy^l |G°r (¿3 ,u)Wq

LP(R")

LP« (Rn)

(3.45)

where v is as in Remark 2.6, |z - y|~2js means 2j 1s <|z - y| < 2js for j e N and 0 <|z - y| < s for j = 0, s ~ 2/ means 2i-1t < s< 2it for i e Z. For (y,t) e Rn x (0, to), let

(y,t) := 2-(i+j)nT sup-j ¿3(£,6) : ^ - y\ < 2j+16,2-i-1 < 6 < 2i+1

(3.46)

Then by Lemma 3.6 and Remark 1.3, Wj satisfies that

f [Nwlrj(x)](p*vq)'dH"T(p*vq) < 1

(3.47)

modulo a positive constant.

Observing that t(p v r)' < t(p, vq)' and p < p,, we know that r > p, > p. We now show the desired conclusion in two cases for r and p.

Case 1 (r e [2, œ) and p e (1, œ)). By (1.27), there exist X > u/r and y > "(t + 1/2 - 1/r) such that i > X + y. Since r e [2, œ), then by Holder's inequality and Lemma 3.2(ii), we have

f A + Jz-yl)

J \z-y\~2j^ \ S /

1 + Jl-li^ KK « Zs «y)(y- z) |rdz

\z-y\~2js ^ s

J\ z \ ~2Js f

J\ z \ ~2Js

1 +z V71 +z^ (l+^)r 1 (k « Zs « y)(z)|r'dz

|z|X -^r'(2/(2-r'))

1 + s) dz

(2-r )/2

2(l+^)

/ izi\ 2( w 2 (1 + \(K * Zs * ft)(z)|2dz

'-21 s \ s /

< 2.(-^r'+n-n/2r')s"(1-r'/2)(min{t,s})((-n/2)+a)r' mini 1, (s)

(3.48)

where k, N are arbitrary positive integers, which are determined later. Hence, choosing k > n(T + 1/2) and N > X + |a|, we see that

J s~2H J \z

' s~2H J \z-y\~2is ^ s

z - y1

(1 + t) KK * Zs * ft)(y - z)\rsn(r-1)dz

< 2;(-,r'+n-(n/2)r') J s(n/2)r'(min{t,s})(-n/2+a)r' (1 +

< t^'2j(-pr'+n-(n/2)r') min^2-i(k-n/2)rJ, 2-i(l-N-a)r'J.

t\(l-N)r' ds

(3.49)

Thus, by choosing w := w^, we conclude that

IWIIfhP^cr")

< 2j(nT-^+(n/r')-n/2)v

y'2i(nT-k+(n/2))v + ^^ 2i(nT+a-l+N)v

D(«-B)q[Glr (¿3 ,u)J q + D(a-B-(n/p))^|Glr (¿3 ,u)

I LP(R")

LP* (Rn)

Take D such that

D-n/P =

Glr (¿3 ,u)(x)

Glr ,u)

LP(R")

(3.50)

(3.51)

We then see that

^^fh* (Rn)'

< V"1 jj(nT-p+n/r'-n/2)v

^^2_i(nT-k+n/2)v + ^^ 2i(nT+a-l+N)v i=0 i=-<x>

Gl r (¿^ U)

lp(R")

< II Gl r (¿3, u) I

lp(R")

(3.52)

which, together Theorem 2.9, implies that \\Tmf Wp^* (R„) < \\f \\fh0'T(R») for all f e STO(Rn).

Case 2 (1 <p <r < 2). By the assumption that £ > n(1/r +1/2), there exists X> n/r such that £ > X + n/2. Then by Lemma 3.2(i), we see that

(l + S) \(K * Zs * ft)(z)\

<f (l + Z) V * ft)(z - y)\\Zs(y)\dy

Jk" \ s / { (

<| (1 + \(K * ft)(z - y)\(l + M) \Zs(y)\dy

< sn\sup(1 + \y\)A\Z(y)\} (l + \(K * y)(z - y)\dy

< S-(maxjl, S})X\Kn (l + \ (K * ft)(z - y) \dy

< s-nt4maxil,-^

(3.53)

From p <r and t e (0, min{1/(p V r)', 1/(p, V q)'}], it follows that rr' < 1. Thus, by £ > X+n/2, there exists p > nrr'/2 such that £ > X + p, which, together with Lemma 3.2(ii), implies that

J\z-y| ~2Js \

l + \(K* Zs * ft)(y - z)\rdz

\z-y\~2Js \ s

< 2-2w

s-nt"(maxi l,-

x\ (l + \(K * Zs * ft)(y - Z)\2dz

J\z-y\ ~2js \ s J

l(r'-2)

< 2-2vs-n(r'-2)ta(r'-2) ma^l, |}) (min{t,s})-n+2a

x miMi.(s) 2i 6+s;-

(3.54)

Then, choosing k > nrr'/2 + n/2 and N > (Xr'/2) + | a|, we see that

Xr' ■ - Xr'

f i V + s) i1 + SV^*^*^)(y-z)|r'sn(r'-1)dzT

Js~2HJ\z-y|~2j s \ s/

< 2-2vta(r'-2) j ( '(min{ t,s})-

's~2t \ I !>!/ (3.55)

« m-uj)'} (1 + S) ^

< 2-2m tar' min |2-i(2k-n), 22i(a+X+N)}.

This, together with the fact that y > nrr'/2 and an argument similar to Case 1, further implies that for all f eSoo(Rn),

It fir „ < V2j(nT-2y/r>

^^Wfh^(Rn) < 2u2

< \\f\\FHP't(Rn),

^2'(nT-2k/r'+n/r')v + ^ 2'(nT+2(a+X+N)/

In II v

GXr (¿3 ,u)\\ I X'r IIlp(R")

(3.56)

namely, ||Tmf ^hP^R») < llf hH^(R»).

Next we assume that f e So(Rn) and ||f Hfh°,j(Rn) = 0. Then, for any e e (0, o), there exists a nonnegative function ¿3 on R++1 such that 0 < ||GXr(¿3,u)||Lp(Rn) < e. If ||GXr(¿3,u)||Lp(Rn) = 0, then G°Xr(¿¿,u)(x) = 0 for almost every x e Rn, which, together with an argument similar to (3.43), further implies that ||Tmf !FH{e,T (R») = 0. If ||GXr(¿3,u)||Lp(R») is positive, repeating the previous argument, we see that

II^U^R») < e (3.57)

for any e e (0, o), and hence ||Tmf !FlFe,T (R„) = 0. Thus, in this case, we also have

||Tmf ||Ffi^,(Rn) < ||f ^H^(r»).

Finally, by the fact that So(Rn) is dense in FHpT(Rn) (see [13, Lemma 5.3]), together with a density argument, we know that the inequality ||Tmf Hfi1{ii,t (R„) < ||f H?*(R„) is true for

all f e FHpT (Rn), which completes the proof of Theorem 1.12. □

4. Applications to Sobolev Embeddings

As an application of Theorems 1.7 and 1.12, we give new direct proofs for the following Sobolev embedding theorems (see also [14, Proposition 3.3] and [15, Proposition 2.2]).

Theorem 4.1. Let a,p e R, a > ¡¡, q,r e (0, to], p e (0, to), and t e [0, to). If p* e (0, to) such that ¡5 - n/p* = a - n/p, then (Rn) ^ Fp^(Rn).

Proof. If we take m(£) := |£|-a for all £ e Rn \{0} in Theorem 1.7 and then apply the lifting property (see [14, Proposition 3.5]), we immediately obtain the desired conclusion of Theorem 4.1, which completes the proof of Theorem 4.1. □

Theorem 4.2. Let a,¡5 e R, a > ¡¡, and p e (1, to). Assume that p* e (1, to) satisfies ¡5 - n/p* = a - n/p. Let r,q e (1, to), and t e [0, min{1/(p* V q)', 1/(p V r)'}] such that t(p V r)' < t(p* V q)'. Then FHpar (Rn) ^ FH^(Rn).

Proof. If we take m(£) := |£|-a for all £ e Rn \ {0} in Theorem 1.12 and then apply the lifting property which can be deduced directly from [15, Theorem 4.1], we immediately obtain the desired conclusion of Theorem 4.2, which completes the proof of Theorem 4.2. □

Acknowledgments

D. Yang would like to thank Professor Hans Triebel and Professor Winfried Sickel for some suggestive and helpful discussions on this paper. D. Yang is supported by the National Natural Science Foundation (Grant no. 11171027) of China and Program for Changjiang Scholars and Innovative Research Team in University of China. W. Yuan is supported by the National Natural Science Foundation (Grant no. 11101038) of China.

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