Dynamics of PDE, Vol.4, No.3, 227-245, 2007

Homothetic Variant of Fractional Sobolev Space with Application to Navier-Stokes System

Jie Xiao

Communicated by Y. Charles Li, received October 28, 2006.

n Abstract. It is proved that for α ∈ (0, 1), Qα(R ), not only as an interme- diate space of W 1,n(Rn) and BMO(Rn) but also as a homothetic variant of 2n ˙ 2 Rn n−2α Rn Sobolev space Lα( ) which is sharply imbedded in L ( ), is isomor- phic to a quadratic Morrey space under fractional differentiation. At the same n n time, the dot product ∇·(Qα(R )) is applied to derive the well-posedness of the scaling invariant mild solutions of the incompressible Navier-Stokes system R1+n ∞ × Rn in + = (0, ) .

Contents 1. Introduction and Summary 227 2. Proof of Theorem 1.2 233 3. Proof of Theorem 1.4 239 References 244

1. Introduction and Summary We begin by the square form of John-Nirenberg’s BMO space (cf. [13]) which plays an important role in harmonic analysis and applications to partial differen- tial equations. For a locally integrable complex-valued function f defined on the Euclidean space Rn, n 2, with respect to the Lebesgue measure dx, we say that f is of BMO class, denoted≥ f BMO = BMO(Rn), provided ∈ 1 n 2 2 f = sup `(I) − f(x) f dx < . k kBMO − I ∞  I ZI   1991 Mathematics Subject Classification. Primary 35Q30, 42B35; Secondary 46E30. 1,n n n n Key words and phrases. Homothety, W (R ), Qα(R ), BMO(R ), quadratic Morrey space, sharp Sobolev imbedding, incompressible Navier-Stokes system. The author was supported in part by both NSERC (Canada) and Dean of Science Startup Fund (MUN, Canada).

c 2007 International Press 227 228 JIE XIAO

Here and elsewhere sup means that the supremum ranges over all cubes I Rn I ⊂ with edges parallel to the coordinate axes; `(I) is the sidelength of I; and fI = n − `(I) I f(x)dx stands for the mean value of f over I. On the basis of the semi-norm , a large scale of function spaces has  R BMO been introduced in [11], as defined bkelo· kw.

Definition 1.1. For α ( , ), let Qα be the space of all measurable complex-valued functions f on∈ R−∞n obeying∞

1 2 2 2α n f(x) f(y) f = sup `(I) − | − | dxdy < . k kQα x y n+2α ∞ I  ZI ZI | − |   This Qα is a natural extension of BMO according to the following result (proved in [11] and [30]):

BMO, α ( , 0), 1,n ∈ −∞ Qα = New space between W and BMO, α [0, 1),  ∈ C, α [1, ). ∈ ∞ Here W 1,n = W1,n(Rn) is the homothetic energy space of all C1 complex-valued functions f on Rn with

1 n n f W 1,n = f(x) dx < . k k Rn |∇ | ∞ Z  More importantly, Qα, α (0, 1), can be regarded as the homothetically invariant ∈ ˙ 2 ˙ 2 Rn counterpart of the homogeneous Sobolev space Lα = Lα( ) which consists of all complex-valued functions f on Rn with the α-energy

1 f(x) f(y) 2 2 f L˙ 2 = | − n+2α| dxdy < . k k α Rn Rn x y ∞ Z Z | − | 

The reason for saying this is at least that Qα and ˙ 2 enjoy the following k · k k · kLα property: n α 2 f φ Qα = f Qα and f φ ˙ 2 = λ − f ˙ 2 k ◦ k k k k ◦ kLα k kLα n for any homothetic map x φ(x) = λx + x0; λ > 0, x0 R . In the six year period 7→since the paper [11] appeared,∈ it has been found that Qα is a useful and interesting concept; see also [1], [2], [24], [8], [9], [16], [23], [7] and [10]. This means that the study of this new space has not yet ended up – in fact, there are many unexplored problems related to Qα. In this paper, although not attacking one of those open problems in Section 8 in [11], we go well beyond the previous results by studying the relation between this space and the quadratic Morrey space, but also giving an application of the induced facts to the incompressible Navier-Stokes system. To deal with the former, it is necessary to consider the following variant of [8, Theorem 3.3] that expands Fefferman-Stein’s basic result for BMO in [12]: Given n a C∞ real-valued function ψ on R with

1 (n+1) n x ψ L , ψ(x) . (1 + x )− , ψ(x)dx = 0 and ψt(x) = t− ψ( ). ∈ | | | | Rn t Z HOMOTHETIC FRACTIONAL SOBOLEV SPACE WITH APPLICATION TO NS SYSTEM229

Then for a measurable complex-valued function f on Rn, r 2α n 2 (1+2α) (1.1) f Qα sup r − f ψt(y) t− dydt < . ∈ ⇐⇒ x Rn,r (0, ) 0 y x

On the other hand, if 2 n xj x ψ (x) = (4π)− 2 exp | | for j = 1, ..., n, j − 2 − 4 then     t2∆ (ψj )t(x) = t∂j e (x, 0) and so, for α (0, 1) and a measurable complex-valued function f on Rn, ∈ r 2α n t2∆ 2 1 2α (1.3) f Qα sup r − e f(y) t − dydt < . ∈ ⇐⇒ x Rn,r (0, ) 0 y x

α (i) Qα = ( ∆)− 2 2,n 2α , BMO is proper with − L − → n+2α f n 2 sup k kBMO , f Q >0 f Qα ≤ s 2 k k α k k n where a measurable complex-valued function f on R belongs to 2,n 2α if and only if L − 1 2α n 2 − 2 f 2,n−2α = sup `(I) f(x) fI dx < . k kL | − | ∞ I  ZI  α 2n 2 2 −  (ii) L˙ = ( ∆)− 2 L , L n 2α is sharp with α − → 1 α 1 2 n 2n n 2α 2πy (1,0,...,0) 2 2 f − Γ − k kL n 2α −2 Γ(n) e− · 1 sup = n+2α n | n+2α − | dy . f 2 Rn y f L˙ 2 >0 L˙ Γ 2 ! Γ 2 ! k k α k k α  Z | |  1 n   Rn 1 (iii) Qα−; = (Qα) , where a tempered distribution f on belongs to Qα−; if and only∞if ∇ · ∞

1 r2 2 2α n t∆ 2 α f −1 = sup r − e f(y) t− dydt < . Qα;∞ k k x Rn,r (0, ) 0 y x

Note that 2,n 2α is the so-called Morrey space of square form (cf. [3] and L − [22]) and 2,n = BMO. So Theorem 1.2 (i) keeps true for α = 0 in the sense of ( ∆)0BLMO = BMO. Quite surprisingly, this part corresponds nicely to − α Strichartz’s ( ∆)− 2 BMO-equivalence [26, Theorem 3.3]: − 1 2 2 α n f(x) f(y) f ( ∆)− 2 BMO sup `(I) − | − | dxdy < . ∈ − ⇐⇒ x y n+2α ∞ I  ZI ZI | − |   The imbedding without best constant in Theorem 1.2 (ii) is well-known (see for example [21, Theorem 1] and the related references therein) and very useful in the study of the semi-linear wave equations (cf. [20]). A close look at both (i) and (ii) reveals that Qα behaves like a homothetic Sobolev space. In addition, Theorem 1.2 1 n 1 (iii) extends [15, Theorem 1]: BMO− = (BMO) that just says: f BMO− if and only if there are f BMO such that∇ · f = n ∂ f . ∈ j ∈ j=1 j j As with the latter, we recall that the Cauchy problem for the incompressible Navier-Stokes system on the half-space R1+n = (0P, ) Rn: + ∞ × ∂ u ∆u + (u )u p = 0, in R1+n; t − · ∇ − ∇ + R1+n (1.4)  u = 0, in + ; ∇ · n u t=0 = a, in R | is to establish theexistence of a solution (velocity)

u = u(t, x) = u1(t, x), ..., un(t, x) with a pressure p = p(t, x) of the fluid at time t [0,  ) and position x Rn ∈ ∞ ∈ that assumes the given data (initial velocity) a = a(x) = (a1(x), ..., an(x)). If the solution exists, is unique, and depends continuously on the initial data (with respect to a given topology), then we say that the Cauchy problem is well-posed in that topology. HOMOTHETIC FRACTIONAL SOBOLEV SPACE WITH APPLICATION TO NS SYSTEM231

Of particularly significant is the invariance of (1.4) under the scaling changes: u(t, x) u (t, x) = λu(λ2t, λx); 7→ λ p(t, x) p (t, x) = λ2p(λ2t, λx);  7→ λ a(x) aλ(x) = λa(λx). 7→ So if (u(t, x), p(t, x), a(x)) satisfies (1.4) then (uλ(t, x), pλ(t, x), aλ(x)) is a solution of (1.4) for any λ > 0. This leads to a consideration of the well-posedness for (1.4) with a Cauchy data being of the scaling invariance. Through the scale invariance n n n a n n = (a ) n = (a ) n = a n = a n n , k λk(L ) k λ j kL k j λkL k jkL k k(L ) j=1 j=1 j=1 X X X Kato proved in [14] that (1.4) has mild solutions locally in time if a (Ln)n and ∈ globally if a (Ln)n is small enough (for some generalizations of Kato’s result, see e.g. [28] andk k[31]). Furthermore, in [15], Koch-Tataru found, among other results, 1 n that (1.4) still has mild solutions locally in time if a (V MO− ) and globally once ∈ 1 n n r2 2 n t∆ 2 aj BMO−1 = sup r− e aj(y) dydt k k Rn | | j=1 j=1 x ,r (0, ) 0 y x

1 r2 2 − 2α n t∆ 2 α f Q 1 = sup r − e f(y) t− dydt < ; k k α;T x Rn,r (0,T ) 0 y x

∞ g Xα;T = sup √t g(t, ) L k k t (0,T ) k · k ∈ 1 r2 2 2α n 2 α + sup r − g(t, y) t− dydt < . x Rn,r2 (0,T ) 0 y x

In particular, we write

1 1 1 1 Q0;−T = BMOT− , V Q0− = V MO− and X0;T = XT . Two immediate comments are given below: If f (x) = λf(λx) and g (t, x) = λg(λ2t, λx) for λ, t > 0 and x Rn, λ λ ∈ then

− − fλ 1 = f 1 and gλ X ∞ = g X ∞ ; k kQα;∞ k kQα;∞ k k α; k k α; − that is, 1 and X ∞ are scaling invariant. Second, we have k · kQα;∞ k · k α; n 1 1 L Q− BMO− and X X . ⊆ α;1 ⊆ 1 α;1 ⊆ 1 To see this, note that f BMO−1 = f −1 f −1 . Additionally, recall that k k k kQ0;∞ ≤ k kQα;∞ n n−p 1+ p t∆ f B˙ p,− , p > n if and only if e f Lp . t 2p for t (0, 1]; see also [4]. Using ∞ n ∈ k k 1+ p ∈ H¨older’s inequality, we obtain that if f B˙ p,− , p > n and r (0, 1), then ∈ ∞ ∈ r2 r2 t∆ 2 α n(p 2) t∆ 2 α n 2α e f(y) t− dydt . r − e f Lp t− dt . r − 0 y x n, yields the desired inclusion. ∞ ⊆Below is our result on the well-posedness for the incompressible Navier-Stokes system. Theorem 1.4. Let α (0, 1). Then (i) The Navier-Stokes system∈ (1.4) has a unique small global mild solution in n (Xα; ) for all initial data a with a = 0 and a −1 n being small. ∞ ∇ · k k(Qα;∞) (ii) For any T (0, ) there is an  > 0 such that the Navier-Stokes system (1.4) ∈ ∞ n n has a unique small mild solution in (Xα;T ) on (0, T ) R when the initial data × 1 n − a satisfies a = 0 and a (Q 1 )n . In particular for all a V Qα− with ∇ · k k α;T ≤ ∈ a = 0 there exists a unique small local mild solution in (X )n on (0, T ) Rn. ∇ · α;T  × In the case of α = 0, Theorem 1.4 goes back to Theorems 2-3 of Koch-Tataru in [15]. However, it is perhaps worth pointing out that their Theorems 2-3 do 1 n 1 n not deduce our Theorem 1.4 even though (Qα−;T ) and V Qα− are subspaces of 1 n 1 n n (BMOT− ) and V MO− respectively, since the (Xα;T ) is contained properly n  in (XT ) when 0 < α < 1. In the forthcoming two sections, we provide the proofs of the above-stated theorems. The argument of Theorem 1.2 (i) follows from a chain of integral es- timates for operators (see e.g. [5] and [6]) with being partially inspired by Wu-Xie’s work [29], while in the demonstration of Theorem 1.2 (ii) we formulate the integral involved in the Sobolev space as weighted integral of the Fourier transform of the given function and take Lieb’s sharp estimate for the Hardy-Littlewood- into account. The justification of Theorem 1.2 (iii) is an extension of Koch-Tataru’s argument for the BMO-setting in [15]. In showing Theorem 1.4 (i)-(ii), we improve Lemari´e-Rieusset’s treatment (cf. [17, Chapter 16]) of Koch-Tataru’s proof of settling the case α = 0 (see again [15, The- orems 2 and 3]) in order to handle any value α (0, 1). More precisely, our proof ∈ HOMOTHETIC FRACTIONAL SOBOLEV SPACE WITH APPLICATION TO NS SYSTEM233 rests on two technical lemmas of which Lemma 3.1 brings Schur’s lemma into play and so makes a difference.

2. Proof of Theorem 1.2 To verify Theorem 1.2 (i), we must understand the quadratic Morrey space in spirit of (1.2).

Lemma 2.1. Given α (0, 1). Let f be a measurable complex-valued function on Rn. Then ∈ (2.1) r 2 2α n t√ ∆ f 2,n 2α sup r − ∂te− − f(y) tdydt < . ∈ L − ⇐⇒ x Rn,r (0, ) 0 y x

Proof. Assume f 2,n 2α. Recall ∈ L − n+1 2 n+1 2 1 + x (n + 1)Γ( 2 )π− t√ ∆ ψ0(x) = | | − n+3 and (ψ0)t(x) = t∂te− − (x, 0). (1 + x 2) 2 | | For any ball B = y Rn : y x < r in Rn, let 2B be the double ball of B and 1 { ∈ | − | } f2B = 2B − 2B f be the mean value of f on 2B. Note that E stands for the Lebesgue| measure| of a set E. Let also | | R

f1 = (f f2B)12B , f2 = (f f2B)1Rn 2B and f3 = f2B, − − \ where 1E stands for the characteristic function of a set E. Since Rn ψ0(x)dx = 0, we conclude R t√ ∆ t∂ e− − f(y) = (ψ ) f(y) = (ψ ) f (y) + (ψ ) f (y). t 0 t ∗ 0 t ∗ 1 0 t ∗ 2 Concerning the first term (ψ ) f (y), we have the following estimates 0 t ∗ 1

r 2 1 ∞ 2 1 (ψ ) f (y) t− dydt (ψ ) f (y) t− dtdy | 0 t ∗ 1 | ≤ | 0 t ∗ 1 | Z0 ZB ZB Z0 ∞ 2 1 (ψ0)t f1(y) t− dt dy ≤ Rn | ∗ | Z Z0  2 . f(y) f2B dy 2B | − | Zn 2α 2 . r − f 2,n−2α , k kL 2 where we have used the L -boundedness: (f1) L2 . f1 L2 for the Littlewood- Paley -function of f kG k k k G 1 1 2 2 ∞ t√ ∆ 1 (f1)(y) = t∂te− − f1(y) t− dt . G 0 Z 

At the same time, if (t, y) (0, r) B and Bk is the ball with center x and radius 2kr, then we take the Cauc∈ hy-Sc×hwarz inequality into account, and obtain the following inequalities for (ψ ) f (y): 0 t ∗ 2 234 JIE XIAO

t f(z) f2B (ψ0)t f2(y) . | − n+1| dz | ∗ | Rn 2B (t + x z ) Z \ | − | t f(z) f2B . | − n+1| dy Rn 2B (r + x z ) Z \ | − | ∞ t f(z) f2B . t | − n+1| dz Bk (r + x z ) Xk=1 Z | − | ∞ k (n+1) . t (2 r)− f(z) f dz | − 2B| k=1 ZBk Xα n 1 . tr − − f 2,n−2α . k kL Consequently,

r 2 1 n 2α 2 (ψ0)t f2(y) t− dydt . r − f 2,n−2α . | ∗ | k kL Z0 ZB The above-established estimates yield that the supremum in (2.1) is finite. To handle the converse part, denote by

S(I) = (t, x) R1+n : t (0, `(I)] and x I . ∈ + ∈ ∈ n o the Carleson box based on a given cube I Rn. Suppose the supremum condition in (2.1)⊂ is satisfied. Then

2α n t√ ∆ 2 (2.2) sup `(I) − ∂ e− − f(y) tdydt < . t ∞ I ZS(I) 

In order to verify f 2,n 2α, we consider the projection operator ∈ L −

1 Πψ0 F (x) = F (t, y)(ψ0)t(x y)t− dydt, R1+n − Z + and prove that if

1 2 2α n − 2 1 F α = sup `(I) F (t, y) t− dydt < k kC | | ∞ I ZS(I) !  then for any cube J Rn, ⊂

n 2α 2 − 2 (2.3) Πψ0 F (x) Πψ0 F dx . `(J) F α . | − J | k kC ZJ   Given a cube J Rn and λ > 0, define λJ as the cube concentric with J and with sidelength `(λJ)⊂= λ`(J). Let F = F and F = F F . Then by a result of 1 |S(2J) 2 − 1 Coifman-Mayer-Stein [6, p. 328-329], HOMOTHETIC FRACTIONAL SOBOLEV SPACE WITH APPLICATION TO NS SYSTEM235

2 2 Πψ0 F1(x) dx Πψ0 F1(x) dx | | ≤ Rn | | ZJ Z 2 1 . F1(t, y) t− dydt R1+n | | Z + 2 1 . F (t, y) t− dydt | | ZS(2J) n 2α 2 − . F α `(J) . k kC

On the other hand, from the definition of Πψ0, the boundedness of ψ0 and the Cauchy-Schwarz inequality it follows that

2 Πψ0 F2(x) dx J | | Z 2 1 = (ψ0)t(x y)F2(t, y)t− dydt dx J R1+n − Z Z + 2 1 (ψ0)t(x y) F2(t, y ) t− dydt dx ≤ J R1+n S(2J) | − || | Z Z + \ ! 2 ∞ 1 (ψ0)t(x y) F2(t, y) t− dydt dx ≤ J S(2k+1J) S(2kJ) | − || | Z k=1 Z \ ! X 2 ∞ k 1 . 2 `(J) − (ψ0)t(x y) F2(t, y) dydt dx J S(2k+1J) S(2k J) | − || | Z k=1 Z \ ! X  2 ∞ k (n+1) . 2 `(J) − F2(t, y) dydt dx J S(2k+1J) S(2k J) | | ! Z Xk=1 Z \  1 2 2 ∞ n k 2 2 1 . 2 `(J) − F2(t, y) t− dydt dx J  S(2k+1J) S(2k J) | |  Z k=1 Z \ ! X  n 2α 2 −  . F α `(J) . k kC Thus 

2 2 Π F (x) Π F dx . Π F (x) dx ψ0 − ψ0 J ψ0 ZJ ZJ  2 2 . Πψ0 F1(x) dx + Πψ0 F2(x) dx J J Z n 2α Z 2 − . F α `(J) , k kC namely, (2.3) holds. Applying (2.3) to Π (ψ ) f which equals f under (2.2), ψ0 0 t ∗ we achieve f 2,n 2α.  ∈ L −  PROOF OF THEOREM 1.2 (i). Since the imbedding with that constant can be α derived from a routine computation, it suffices to show Qα = ( ∆)− 2 2,n 2α. − L − 236 JIE XIAO

1+α t√ ∆ For f 2,n 2α, let F (t, y) = t ∂te− − f(y). Then by Lemma 2.1 we get ∈ L − r 2α n 2 1 2α 2 sup r − F (t, y) t− − dydt . f 2,n−2α , x Rn,r (0, ) 0 y x

PROOF OF THEOREM 1.2 (ii). According to [27, p. 175], we use Fubini’s theo- 1 rem, Plancherel’s formula, the change of variables y = x − z and an orthonormal transform to obtain | |

2 2 (n+2α) f L˙ 2 = f(x + y) f(x) dx y − dy k k α Rn Rn | − | | | Z Z  2πy x 2 (n+2α) 2 = e− · 1 y − dy fˆ(x) dx Rn Rn | − | | | | | Z Z  2πz (1,0,...,0) 2 (n+2α) 2 2α = e− · 1 z − dz fˆ(x) x dx. Rn | − | | | Rn | | | | Z  Z α ˙ 2 2 2 ˆ Accordingly, L = ( ∆)− L . Note that f(x) = n f(y) exp( 2πix y)dy and α − R − · t√ ∆ R e− − f(x) = fˆ(y) exp 2π(iy x + y t) dy. Rn − · | | Z So, by differentiation and integration (cf. [25 , p. 83]), we find 

t√ ∆ 2 2 2 ˆ 2 e− − f L2 = 8π x f(x) exp( 4π x t)dx. k∇ k Rn | | | | − | | Z Consequently, 2 ∞ t√ ∆ 2 1 2α 8π Γ 2(1 α) 2α 2 2 − ˆ (2.4) e− − f L t − dt = 2(1 α) x f(x) dx. k∇ k (4π) Rn | | | | Z0 −  Z Moreover, the preceding consideration actually tells us that proving the desired sharp imbedding amounts to verifying the following best inequality

2 ∞ t√ ∆ 2 1 2α (2.5) f 2n τn,α e− − f L2 t − dt, k kL n−2α ≤ k∇ k Z0 where 2α 1 4α n 2α n 2 − Γ −2 Γ(n) τn,α = .  n+2α n παΓ 2(1 α) Γ Γ 2 ! − 2     HOMOTHETIC FRACTIONAL SOBOLEV SPACE WITH APPLICATION TO NS SYSTEM237

To this end, we use f, g as f(x)g(x)dx, and then get h i Rn R 2 2 2α 2 2α 2 f, g = fˆ, gˆ x fˆ(x) dx x − gˆ(x) dx . |h i| |h i| ≤ Rn | | | | Rn | | | | Z  Z  Because (cf. [19, Corollary 5.10])

n 2α 2 n 2α 2α 2 π − Γ −2 g(x)g(y) x − gˆ(x) dx = n 2α dxdy, Rn | | | | Γ(α) Rn Rn x y Z  Z Z | − | − it follows from Lieb’s sharp version [18] of the Hardy-Littlewood-Sobolev inequality that

2α παΓ n 2α n 2 −2 Γ(n) 2 2α ˆ 2 f, g n+2α n g 2n x f(x) dx. |h i| ≤ Γ Γ k kL n+2α Rn | | | | 2  2 ! Z

  4α n−2α ˙ 2 In the last inequality we take g = f f for f Lα, and use (2.4) to achieve | | ∈ 2 (2α n)/2 (2.5) whose equality can be checked for f(x) = (1 + x ) − through a direct calculation. | | The proof of Theorem 1.2 (iii) depends on the following lemma.

1 Lemma 2.2. Given α (0, 1). For j, k = 1, ..., n let fj,k = ∂j ∂k( ∆)− f. If 1 1 ∈ − f Qα−; then fj,k Qα−; . ∈ ∞ ∈ ∞ n Proof. Assume that φ is a C∞ real-valued function on R with compact Rn support suppφ B(0, 1) = x : x < 1 and Rn φ(x)dx = 1. Recall n ⊂ { ∈ | | } 1 t∆ φ (x) = r− φ(x/r), and write g (t, x) = φ ∂ ∂ ( ∆)− e f(x). Then r r r ∗ j k − R

t∆ 1 t∆ e f (x) = ∂ ∂ ( ∆)− e f(x) = f (t, x) + g (t, x). j,k j k − r r ˙ 1,1 ˙ 1, If B1 stands for the predual of the homogeneous Besov space B− ∞, then f 1 1 ˙ 1, ∞ ∈ Qα−; yields f BMO− B− ∞ (see also [Le, p. 160, Lemma 16.1]) and ∞ ∈ ⊆ ∞

1 t∆ 1 ∞ 1,1 . −1,∞ gr(t, ) L φr B˙ ∂j ∂k( ∆)− e f −1,∞ r− f B˙ ∞ . k · k ≤ k k 1 − B˙ ∞ k k

Consequently,

r2 2 α n 2α 2 n 2α 2 (2.6) gr(t, y) t− dydt . r − f −1,∞ . r − f −1 . B˙ ∞ Qα;∞ 0 y x

Next, we estimate fr. Take another C∞ real-valued function ψ with compact support in Rn such that ψ = 1 on B(0, 10) = x Rn : x < 10 . Define y x { ∈ | | } ψr,x = ψ −r and write fr = Fr,x + Gr,x where

 1 t∆ 1 t∆ G = ∂ ∂ ( ∆)− ψ e f φ ∂ ∂ ( ∆)− ψ e f. r,x j k − r,x − r ∗ j k − r,x 238 JIE XIAO

Thus, we employ the Plancherel formula for the space variable to get

r2 1 t∆ 2 dt ∂ ∂ ( ∆)− ψ e f j k − r,x L2 tα Z0 r2 \ 2 dt . (∂ ∂ ( ∆) 1ψ et∆f j k − − r,x L2 tα Z0 r2  2 \t∆ 2 dt . yj yk y − ψr,xe f (y) dy α Rn | | t Z0 Z r2    \ 2 dt . ψ et∆f r,x L2 tα Z0 r2 2 dt . ψ et∆f . r,x L2 tα Z0

And, by Minkowski’s inequalit y (for φ r) as well as the Plancherel formula again, we have

r2 r2 1 t∆ 2 dt t∆ 2 dt φ ∂ ∂ ( ∆)− ψ e f . ψ e f 2 . r ∗ j k − r,x L2 tα r,x kL tα Z0 Z0

The last two estimates imply

r2 r2 2 dt t∆ 2 dt Gr, (t, ) 2 . ψr,xe f 2 . · · L tα L tα Z0 Z0

To control Fr,x, we bring the follo wing estimate (proved in [17, p. 161])

2 n+1 t∆ 2 (n+1) Fr,x(t, y) dy . r e f(w) x w − dw y x

r2 2 dt Fr,x(t, y) dy α 0 y x

The integral estimates on Fr,x and Gr,x give

r2 2 α n 2α 2 (2.7) fr(t, y) t− dydt . r − f −1 . Qα;∞ 0 y x

n PROOF OF THEOREM 1.2 (iii). If f (Qα) , then there are f1, ..., fn Qα n ∈ ∇ · ∈ such that f = j=1 ∂j fj. So the Minkowski inequality derives n n P −1 − f ∂j fj 1 . fj Qα . k kQα;∞ ≤ Qα;∞ k k j=1 j=1 X X 1 This means f Qα−; . ∈ ∞ 1 1 Conversely, let f Qα−; . If fj,k = ∂j ∂k( ∆)− f, j, k = 1, ..., n, then fj,k 1 ∈ ∞ − 1 ∈ Qα−; by Lemma 2.2, and hence fk = ∂k( ∆)− f Qα. This leads to ∞ − − ∈ n\ n ∂ f = f = fˆ, k k − k,k Xk=1 Xk=1 completing the proof. d 3. Proof of Theorem 1.4 To prove Theorem 1.4 we need two lemmas. Lemma 3.1. Let α (0, 1). Given a number T (0, ] and a function f( , ) R1+n ∈ t (t s)∆ ∈ ∞ · · on + . Let Af(t, x) = 0 e − ∆f(s, x)ds. Then T T R 2 dt 2 dt (3.1) Af(t, ) . f(t, ) . · L2 tα · L2 tα Z0 Z0

Proof. It suffices to justify (3.1) for T = . This is because: If T < , then one may extend f by putting f = 0 on (T, ),∞since Af counts only on the∞values of f on (0, t) Rn. Moreover, we may define∞ f = 0 = Af for t ( , 0). 2 t∆× n x ∈ −∞ Recall e (x, 0) = (4πt)− 2 exp( | | ). Define − 4t ∆et∆(x, 0), t > 0 Ω(t, x) = 0, t 0. ( ≤ Then we read that Af(t, x) = Ω(t s, x y)f(s, y)dyds, R Rn − − Z Z and hence A becomes a convolution operator over R1+n = R Rn. Since × \ Ω(t, )(ζ) = Ω(t, x) exp( 2πix ζ)dx = (2π)2 ζ 2 exp (2π)2t ζ 2 , t > 0, · Rn − · − | | − | | Z we conclude 

\ Af(t, )(ζ) · = Ω(t s, x y)f(s, y) exp( 2πix ζ)dx dyds R1+n Rn − − − · Z Z  = f(s, y) Ω(t s, u) exp( 2πi(u + y) ζ)du dyds R1+n Rn − − · Z Z  t = f(s, y) exp( 2πiy ζ) (2π)2 ζ 2 exp (2π)2(t s) ζ 2 dyds 0 Rn − · − | | − − | | Z Z t   \  = (2π)2 ζ 2 exp (2π)2(t s) ζ 2 f(s, )(ζ)ds. − | | − − | | · Z0  240 JIE XIAO

This formula, together with the Fubini theorem and the Plancherel formula, implies

∞ 2 dt Af(t, ) · L2 tα Z0

∞ \ 2 dt = Af(t, )(x) dx α Rn | · | t Z0 Z  t 2 ∞ ζ \ 2 dt f(s, )(ζ) ds dζ |2 | 2 α ≤ 0 Rn 0 exp (2π) (t s) ζ | · | ! t Z Z  Z − | |  t 2 ∞ ζ  \ 2 dt = (2π)2 f(s, )(ζ) ds dζ |2 | 2 α Rn 0 0 exp (2π) (t s) ζ | · | t ! Z Z  Z − | |  2 \ ∞ ∞ ζ f(s, )(ζ) 2 dt = (2π)2 1 ds dζ. 0 s t | | | 2 · | 2 α Rn 0 0 { ≤ ≤ } exp (2π) (t s) ζ t ! Z Z  Z   − | |  This tells us that if one can prove  2 2 \ ∞ ∞ ζ f(s, )(ζ) dt ∞ \ dt (3.2) 1 ds . f(t, )(ζ) 2 , 0 s t | | | · 2| α α 0 0 { ≤ ≤ } exp (t s) ζ ! t 0 | · | t Z Z   − | | Z then the Plancherel formula can be used againto yield

∞ 2 dt ∞ 2 dt Af(t, ) . f(t, ) 2 , · L2 tα · kL tα Z0 Z0 as desired. To verify (3.2), we rewrite its left side as 2 ∞ ∞ K(s, t)F (s, ζ)ds dt, Z0 Z0  where α 2 α \ s 2 ζ F (s, ζ) = s− 2 f(s, )(ζ) and K(s, t) = 1 0 s t | | . | · | { ≤ ≤ } t exp((t s) ζ 2)   − | | Clearly,  t α ∞ s 2 K(s, t)ds = ζ 2 exp( (t s) ζ 2)ds 1 exp( t ζ 2) 1 0 | | 0 t − − | | ≤ − − | | ≤ Z Z   and α ∞ ∞ s 2 K(s, t)dt = ζ 2 exp( (t s) ζ 2)dt 1. | | t − − | | ≤ Z0 Zs Therefore, by Schur’s lemma we get   2 ∞ ∞ ∞ 2 K(s, t)F (s, ζ)ds dt . F (t, ζ) dt, Z0 Z0  Z0 reaching (3.2).   Lemma 3.2. Given α (0, 1). For a function f on (0, 1) Rn let ∈ × r2 2α n α C(f; α) = sup r − f(t, y) t− dtdy. x Rn,r (0,1) 0 y x

Then 1 t 2 1 dt 2 dt √ t∆ . (3.3) ∆e f(s, )ds α C(f; α) f(t, ) L2 α . − · 2 t · t Z0 Z0 L Z0

Proof. In the sequel, , stands for the inner product in L 2 with respect to the space variable x Rn. Soh· ·i ∈

t 2 2 t∆ L2 = √ ∆e f(s, y)ds dy k · · · k Rn − Z Z0 t t t∆ t∆ = √ ∆e f(s, y)ds, √ ∆e f(s, y)ds − −  Z0 Z0  t t = √ ∆et∆f(s, ), √ ∆et∆f(h, ) dsdh. 0 0 − · − · Z Z D E This gives 1 1 2 dt 2t∆ α 2 = 2 f(s, ), ( ∆)e f(h, )t− dt dsdh k · · · kL tα < · − · Z0 ZZ0

As estimated in [17, p. 163], it follows that

s r2 2s∆ n sup e f(h, z) dh . sup r− f(s, y) dyds. z Rn,s (0,1] 0 | | x Rn,r (0,1) 0 y x

PROOF OF THEOREM 1.4 (i)-(ii). In accordance with the Picard contraction principle (cf. [17, p. 145, Theorem 15.1]), we find that proving Theorem 1.4 amounts to demonstrating that the bilinear operator t (t s)∆ B(u, v) = e − P (u v)ds ∇ · ⊗ Z0 n n n n is bounded from (Xα;T ) (Xα;T ) to (Xα;T ) . Naturally, u (Xα;T ) and a 1 n × ∈ ∈ (Qα−;T ) are respectively equipped with the norms n n

n −1 −1 u (Xα;T ) = uj Xα;T and a (Q )n = aj Q . k k k k k k α;T k k α;T j=1 j=1 X X 242 JIE XIAO

Step 1. L∞-bound. We are about to prove that if t (0, T ) then ∈ 1 (3.4) B(u, v) . t− 2 u n v n . | | k k(Xα;T ) k k(Xα;T ) If t s < t then 2 ≤ ∞ ∞ (t s)∆ u L v L 1 1 e − P (u v) L∞ . k k k k . (t s)− 2 s− u n v n . k ∇ · ⊗ k √t s − k k(Xα;T ) k k(Xα;T ) − t If 0 < s < 2 then

(t s)∆ e − P (u v) | ∇ · ⊗ | u(s, y) v(s, y) . | || | dy Rn (√t + x y )n+1 Z | − | (n+1) . (√t(1 + k )− u(s, y) v(s, y) dy. | | √ n | || | k Zn x y t(k+[0,1] ) X∈ Z − ∈ An application of the Cauchy-Schwarz inequality yields

t u(s, y) v(s, y) dyds 0 x y √t(k+[0,1]n) | || | Z Z − ∈ 1 t 2 2 α u(s, y) . t | α | dyds 0 x y √t(k+[0,1]n) s Z Z − ∈ ! 1 t v(s, y) 2 2 | α | dyds × 0 x y √t(k+[0,1]n) s Z Z − ∈ ! n . t 2 u n v n . k k(Xα;T ) k k(Xα;T ) From the foregoing inequalities it follows that

B(u, v) | t | 2 t (t s)∆ (t s)∆ . e − P (u v) ds + e − P (u v) ds 0 | ∇ · ⊗ | t | ∇ · ⊗ | Z Z 2 t 1 1 1 2 n n 2 n n . t− u (Xα;T ) v (Xα;T ) + s− (t s)− ds u (Xα;T ) v (Xα;T ) k k k k t − k k k k Z 2 ! 1 . t− 2 u n v n , k k(Xα;T ) k k(Xα;T ) establishing (3.4). Step 2. L2-bound. We are about to show that if x Rn and r2 (0, T ) then ∈ ∈ r2 2α n 2 α 2 2 n n (3.5) r − B(u, v) s− dyds . u (Xα;T ) v (Xα;T ) . 0 y x

s 1 (s h)∆ 1 h∆ B2(u, v) = ( ∆)− 2 P e − ∆ ( ∆)− 2 (I e )(1r,x)u v dh, − ∇ · 0 − − ⊗ Z   and s 1 1 s∆ B (u, v) = ( ∆)− 2 P ( ∆) 2 e 1 )u v dh . 3 − ∇ · − r,x ⊗ Z0 Here and henceafter, I stands for the identity operator.   When 0 < s < r2 and y x < r, the Cauchy-Schwarz inequality produces | − |

B (u, v) | 1 | s u(h, z) v(h, z) . | || | dzdh n+1 0 z x 10r (√s h + y z ) Z Z| − |≥ 2 − | − | r u(h, z) v(h, z) . | || n+1 |dzdh 0 z x 10r x z Z Z| − |≥ | − | 1 1 2 2 r u(h, z) 2 2 r v(h, z) 2 2 . | n+1| dzdh | n+1| dzdh 0 z x 10r x z ! 0 y x 10r x z ! Z Z| − |≥ | − | Z Z| − |≥ | − | 1 . r− u n v n . k k(Xα;T ) k k(Xα;T ) Therefore, r2 2 α n 2α 2 2 n n B1(u, v) t− dydt . r − u (Xα;T ) v (Xα;T ) . 0 y x

Similarly for B3( u, v), we obtain

r2 r2 t 2 2 dt 1 dt 2 t∆ B3(u, v) L2 α . ( ∆) e M(s, )ds α t − · 2 t Z0 Z0 Z0 L 1 τ 2 1 dτ 4+n 2α 2 τ∆ 2 . r − ( ∆) e M (r θ, r ) dθ α . − | · | 2 τ Z0 Z0 L

Now, making a use of Lemma 3.2 we achiev e

1 τ 2 1 2 1 τ∆ 2 dτ 2 dθ ( ∆) 2 e M(r θ, r ) dθ . D(M; α) M(r θ, r ) , α 2 α − | · | 2 τ · L θ Z0 Z0 L Z0

244 JIE XIAO where ρ2 n 2 α D(M; α) = sup ρ− M(r θ, rw) τ − dwdτ ρ (0,1) 0 w x <ρ | | ∈ Z Z| − | 2 . r− u n v n . k k(Xα;T ) k k(Xα;T ) Observe also that 1 2 2 dθ 2 M(r θ, r ) 2 . r− u n v n . k · kL θα k k(Xα;T ) k k(Xα;T ) Z0 So, it follows that r2 2 dt n 2α 2 2 B3(u, v) 2 . r − u n v n . k kL tα k k(Xα;T ) k k(Xα;T ) Z0 Adding the previous estimates on Bj (u, v), j = 1, 2, 3 together gives (3.5). n n n Clearly, the boundedness of B( , ) : (Xα;T ) (Xα;T ) (Xα;T ) follows from (3.4) and (3.5). Furthermore, ·the· case T = × produces→(i); and the other case T (0, ) yields (ii). The proof is complete. ∞ ∈ ∞ References

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Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, NL A1C 5S7, Canada E-mail address: [email protected]