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 2 (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 1 × 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 2 1 n 2 2 f = sup `(I) − f(x) f dx < : k kBMO − I 1 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 on2 R−∞n obeying1 1 2 2 2α n f(x) f(y) f = sup `(I) − j − j dxdy < : k kQα x y n+2α 1 I ZI ZI j − j This Qα is a natural extension of BMO according to the following result (proved in [11] and [30]): BMO; α ( ; 0); 1;n 2 −∞ Qα = New space between W and BMO; α [0; 1); 8 2 <>C; α [1; ): 2 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 jr j 1 Z More importantly, Qα, α (0; 1), can be regarded as the homothetically invariant 2 _ 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 = j − n+2αj dxdy < : k k α Rn Rn x y 1 Z Z j − j 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,2 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 C1 real-valued function on R with 1 (n+1) n x L ; (x) . (1 + x )− ; (x)dx = 0 and t(x) = t− ( ): 2 j j j j 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 t(y) t− dydt < : 2 () x Rn;r (0; ) 0 y x <r j ∗ j 1 2 2 1 Z Zj − j Here and henceforth, Lp = Lp(Rn) represents the complex Lebesgue space equipped with p-norm Lp ; stands for the convolution operating on the space variable; and U . V meansk · k that∗ there exists a constant c > 0 such that U cV . ≤ Two particular choices of in (1.1) yield two characterizations of Qα involving tp ∆ the Poisson and heat semi-groups. As for this aspect, denote by e− − ( ; ) and et∆( ; ) are the Poisson and heat kernels respectively; that is, · · · · tp ∆ n + 1 n+1 2 2 n+1 e− − (x; y) = Γ π− 2 t( x y + t )− 2 2 j − j and 2 t∆ n x y e (x; y) = (4πt)− 2 exp j − j : − 4t β Of course, for β R, the notation ( ∆) 2 is the β=2-th power of the Laplacian operator 2 − n n @2 ∆ = @2 = − − j − @x2 j=1 j=1 j X X determined by the partial derivatives @ = @=@x n and the Fourier transform f j jgj=1 ( ) · \β β ( ∆) 2 f(x) = (2π x ) f^(x): c − j j On the one hand, if n+1 2 n+1 2 1 + x (n + 1)Γ 2 π− 0(x) = j j − n+3 ; (1 + x 2) 2 j j then tp ∆ ( 0)t(x) = t@te− − (x; 0) and hence for α (0; 1) and a measurable complex-valued function f on Rn, (1.2) 2 r 2 2α n tp ∆ 1 2α f Qα sup r − @te− − f(y) t − dydt < : 2 () x Rn;r (0; ) 0 y x <r 1 2 2 1 Z Zj − j On the other hand, if 2 n xj x (x) = (4π)− 2 exp j j for j = 1; :::; n; j − 2 − 4 then t2∆ ( j )t(x) = t@j e (x; 0) and so, for α (0; 1) and a measurable complex-valued function f on Rn, 2 r 2α n t2∆ 2 1 2α (1.3) f Qα sup r − e f(y) t − dydt < : 2 () x Rn;r (0; ) 0 y x <r jr j 1 2 2 1 Z Zj − j With the help of the above-mentioned facts, we can establish the following result. Theorem 1.2. Let α (0; 1). Then 2 230 JIE XIAO 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 j − j 1 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 j n+2α − j dy : f 2 Rn y f L_ 2 >0 L_ Γ 2 ! Γ 2 ! k k α k k α Z j j 1 n Rn 1 (iii) Qα−; = (Qα) , where a tempered distribution f on belongs to Qα−; if and only1if r · 1 1 r2 2 2α n t∆ 2 α f −1 = sup r − e f(y) t− dydt < : Qα;1 k k x Rn;r (0; ) 0 y x <r j j 1 2 2 1 Z Zj − j ! 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) − j − j dxdy < : 2 − () x y n+2α 1 I ZI ZI j − j 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 thatr · f = n @ f .