ON CERTAIN COMMUTATOR ESTIMATES FOR VECTOR FIELDS

Transcription

1 ON CERTAIN COMMUTATOR ESTIMATES FOR VECTOR FIELDS JAROD HART AND VIRGINIA NAIBO Abstract. A unifying approach for proving certain commutator estimates involving smooth, not-necessarily divergence-free vector fields is introduced and implemented in the scales of weighted Triebel-Lizorkin and Besov spaces and certain variable exponent Triebel-Lizorkin and Besov spaces. Such commutator estimates are motivated by the study of well-posedness results for some models in incompressible fluid mechanics.. Introduction and main results Let V = (V,..., V n ) be a smooth vector field in, and let { j } denote a family of Littlewood-Paley operators associated to a smooth function with Fourier transform supported in an annulus centered at the origin. In the PDE literature, and usually under the assumption that V is divergence-free, there can be found a number of bilinear estimates for the commutator n [V, j ](f) := V k k ( j f) j (V k k f), k= taking the forms (or variants of them) ( ) 2 js [V, j ](f) f X V Y + V X f Y Z or ( 2 js [V, j ](f) Z ) f X V Y + V X f Y, for suitable function spaces X, Y, Z. For instance, in the study of well-posedness results for the ideal magneto hydrodynamics system in the scale of homogeneous Triebel-Lizorkin spaces, it is used that if s > 0, < p <, and <, then ( ) 2 js [V, j ](f) L p f L V F p s, + V L f F p s,. (.) Date: April 25, Mathematics Subject Classification. Primary: 42B25, 42B35; Secondary: 42B37. Key words and phrases. Commutator estimates, vector fields, Littlewood-Paley operators. The second author is supported by the NSF under grant DMS

2 2 JAROD HART AND VIRGINIA NAIBO See [7, Section 4] and [5, (3.5)] for such ineuality and other versions as well as their applications. As a key tool to prove local existence and uniueness of solutions for the incompressible Euler euations in the scale of Besov spaces of weak type, [26, Lemma 3.3] states, always under the vanishing assumption div(v ) = 0, that if s > 0, < p <, and it holds that ( 2 jsr [V, j ](f) L p, ) V L f Ḃs,, p + f L V Ḃs,, p. (.2) Similar estimates, also used in the settings of models for incompressible fluid mechanics, such as the Boussines system and others, can be found in [3] (see p.982 and its appendix) and in [28, Lemmas 2.6 and 2.7] in the scales of Besov spaces. Other examples of such estimates are given in [3, Lemma 2.00] (see also [6,, 2]); for instance, for the inhomogeneous Besov spaces, p p,, n min(/p, /p ) < s < + n/p, and under no vanishing assumptions on div(v ) one has ( j 0 2 js [V, j ](f) L p ) V L B n p, p f B s, p. (.3) (Here 0 corresponds to a multiplier supported in a ball centered at the origin, not in an annulus as in all other previous cases.) Moreover, variants of the estimate (.3) can be proved in the scale of homogeneous Besov spaces under certain conditions on the indices involved (see [3, Remark 2.02]). All the ineualities cited above are proved by means of paraproducts. The purpose of this article is to present and develop an alternative unifying approach for proving a variety of estimates in the spirit of the ones presented above under no vanishing assumptions on the divergence of the vector field V and in more general settings that include weighted and variable exponent Lebesgue, Triebel-Lizorkin and Besov spaces. Instead of using paraproducts, our approach relies on an elementary real-analysis lemma (Lemma 3.) and duality arguments, and it is inspired by techniues for bilinear suare functions as developed in [2] and [22]. Through the use of Lemma 3. we prove Theorems.,.2 and.3 stated below. For the sake of exposition, the estimates in these theorems are presented in the scale of homogeneous weighted Triebel-Lizorkin spaces. Analogous estimates can also be proved in the settings of inhomogeneous weighted Triebel-Lizorkin spaces, homogeneous and inhomogeneous weighted Besov spaces and certain variable exponent Triebel-Lizorkin and Besov spaces; these estimates are discussed in Section 6. We refer the reader to Section 2 for details regarding the notation in the statements of the theorems. Briefly, S( ) stands for the Schwartz class of smooth rapidly decreasing functions defined on, j is the Littlewood-Paley operator associated to a function ψ S( ) with Fourier transform supported in an annulus ( ψ j in Section 2), A p is the Muckenhoupt class of weights, and F p s, (w) denotes a homogeneous weighted Triebel- Lizorkin space. The notation means C for some constant C that may only

3 COMMUTATOR ESTIMATES FOR VECTOR FIELDS 3 depend on some of the parameters or on the A p -characteristic of the weights used, but not on the functions involved. Theorem.. Let s > 0, assume < p, p 2, < and < p satisfy /p = /p + /p 2, and let (w, w 2 ) A p A p2 be such that w := w p/p w p/p 2 2 A p. For every vector field V = (V,..., V n ) with V j S( ) for j =,, n, and for all f S( ) we have ( ) 2 sj [V, j ](f) (.4) f L p (w ) V F p s, 2 (w 2 ) + V L p (w ) f ( ) 2 sj [V, j ](f) and ( ) 2 sj [V, j ](f) f div(v ) f L p (w ) V F p s, 2 (w 2 ) + V L p (w ) f Fp s, (w) + V L p (w ) f F s, p 2 (w 2 ) + f L p (w ) V Fp s, 2 (w 2 ), Fp s, 2 (w 2 ), Fp s+, 2 (w 2 ). (.5) (.6) We note that if p = then L p (w ) = L and w 2 = w. Such choice for (.5) and (.6) corresponds to a weighted version of the estimates in [7, Section 4] (where div(v ) = 0 is assumed). In particular, those are recovered by taking w =. The next theorem shows an improvement of (.4) and (.5) when the regularity parameter s belongs to the interval (0, ), and a stronger version of (.6) when s is in the interval (, 0). Theorem.2. Let < p, p 2, < and < p satisfy /p = /p + /p 2, and let (w, w 2 ) A p A p2 be such that w := w p/p w p/p 2 2 A p. For every vector field V = (V,..., V n ) with V j S( ) for j =,, n, and for all f S( ) we have: if 0 < s <, then ( ) 2 sj [V, j ](f) f L p (w ) V F p s, 2 (w 2 ) ; (.7) if < s < 0, then ( ) 2 sj [V, j ](f) f div(v ) Fp s, (w) + f L p (w ) V F s+,p p 2 (w 2 ). (.8)

4 4 JAROD HART AND VIRGINIA NAIBO In Corollary 5., we show that if V is such that (.8) holds without the term f div(v ) F p s, (w) for all f S(Rn ) then div(v ) = 0, when < p < n and w n+s satisfies an extra condition (see Remark 5.; the class of weights satisfying such condition includes for instance w = and power weights). The next theorem presents commutator estimates with regularity s > 0 on their left-hand sides and various parameters for Lebesgue spaces and Triebel-Lizorkin spaces on their right-hand sides. Theorem.3. Let s > 0 and s, s 2 R be such that s 2 < and s = s + s 2 ; consider < p, p, p 2,,, 2 < with /p = /p + /p 2 and / = / + / 2 ; assume (w, w 2 ) A p A p2 with w := w p/p w p/p 2 2 A p. For every vector field V = (V,..., V n ) such that V j S( ) for j =,, n, and for all f S( ) we have ( ) 2 sj [V, j ](f) f L p (w ) V F p s, 2 (w 2 ) + f F s, p (w ) V F s 2, 2 p 2 (w 2 ). (.9) The remainder of this paper is organized as follows. In Section 2 we set some notation and present definitions and basic results about weights and the scale of weighted Triebel-Lizorkin spaces. In Section 3, we state and prove Lemma 3.. Section 4 contains statements and proofs of preliminary results used in the proofs of Theorems.,.2 and.3, which are presented in Section 5 along with a remark and corollary concerning (.8). The extensions of the main theorems to the settings of weighted Besov spaces and variable exponent Triebel-Lizorkin and Besov spaces are discussed in Section 6. The latter extension uses a duality result in the context of variable Lebesgue spaces (Lemma 6.), which is proved in Appendix A. The appendix also contains some calculations regarding Remark Preliminaries In this section we set some notation and present definitions and basic results about weights and the scale of weighted Triebel-Lizorkin spaces. The notation S( ) and S ( ) will be used for the Schwartz class of smooth rapidly decreasing functions defined on and its dual, the class of tempered distributions on, respectively. The class of tempered distributions modulo polynomials will be denoted by S ( )/P( ). Throughout the article, all functions will be defined on and therefore we omit in the notation of the function spaces defined below. A weight on is a nonnegative, locally integrable function defined on. Given < p <, the Muckenhoupt class A p consists of all weights w on such that [w] Ap := sup Q ( Q Q ) ( w(x) dx Q Q w(x) p dx ) p <,

5 COMMUTATOR ESTIMATES FOR VECTOR FIELDS 5 where the supremum is taken over all cubes Q and Q means the Lebesgue measure of Q. The uantity [w] Ap is called the A p -characteristic of w. We set A = p> A p. Given w A and 0 < p, we denote by the space of measurable functions defined on such that ) f (R = f(x) p p w(x) dx <, n with the corresponding change when p =. When w =, we will just write L p ; note that L (w) = L for all w. For a locally integrable function f defined on, M(f) will denote the Hardy- Littlewood maximal function of f, that is M(f)(x) = sup f(y) dy x, x B B B where the supremum is taken over all euclidean balls B containing x. We recall that if < p <, then M is bounded on if and only if w A p (see, for instance, [6]). We will freuently use the following vector-valued version of such result, the weighted Fefferman-Stein ineuality (see [2]): If < p, < and w A p, then for all seuences {f j } of locally integrable functions defined on, we have ( ) ( ) M(f j ) f j, (2.0) where the implicit constant depends on p,, and w. The Fourier transform of a tempered distribution φ S ( ) will be denoted by φ; in particular, we use the formula φ(ξ) = φ(x)e 2πiξ x dx ξ for φ L. Given a function φ on and j Z, φ j is defined by φ j (x) = 2 jn φ(2 j x) for x. This subscripted j notation conflicts with some other notation we use later, though changing this notation may cause more confusion. So we make the following convention for the meaning of subscripts for the remainder of this article: for any function (or its derivative) represented by a Greek letter, an integer subscript represents the dilation φ j (x) = 2 jn φ(2 j x). All other integer subscripts for functions merely represent an index label for that function, not necessarily for a dilation; for example w, w 2 represents two different weights, V,, V n, represent the components of V, and {h j } and {f j } represent collections of functions indexed by j. If φ S( ) we denote by φ j, j Z, the Littlewood-Paley operators associated to φ; that is, φ j f(x) = (φ j f)(x) for x and f S ( ).

6 6 JAROD HART AND VIRGINIA NAIBO Let ψ and ψ denote functions in S( ) satisfying the following conditions: supp( ψ) {ξ : 2 ψ(ξ) > c for all ξ such that 3 5 < ξ < 5 3 supp( ψ) {ξ : ξ < 2}, ψ(ξ) > c for all ξ such that ξ < 5 3 < ξ < 2}, (2.) and some c > 0, (2.2) and some c > 0. Let s R, 0 < p <, 0 < and w A. The homogeneous weighted Triebel-Lizorkin space F p s, (w) consists of all f S ( )/P( ) such that ( ) f F p s, (w) = (2 sj ψ j f ) <. Similarly, the inhomogeneous weighted Triebel-Lizorkin space Fp s, (w) is the class of all f S ( ) such that ( ) S f F s, p (w) = ψ 0 f + L (2 sj ψ p j (w) f ) <, j N where Ŝ ψ 0 f(ξ) = ψ(ξ) f(ξ) for ξ. In both cases, the summation in j is replaced by the supremum in j if =. These definitions are independent of the choice of the functions ψ and ψ (see [8, Proposition 0.4, Remark 2.6 and paragraph preceding Section 3] or [25, Theorem 2.2]) and F p s, (w) and Fp s, (w) are uasi-banach spaces (Banach spaces for p, ). Also, H p (w) = F p 0,2 (w) for 0 < p <, where H p (w) denote the weighted Hardy spaces, and H p (w) = F p 0,2 (w) = Fp 0,2 (w) = if < p < and w A p (see [25, Theorem.4 and Remark 4.5]). The class S( ) is contained in Fp s, (w) for all p, s, w as in the definition and is dense for finite p and (see [25, Theorem 2.4]); S( ) is also contained in F p s, (w) for 0 <, < p <, s > 0, and w A p. We refer the reader to classical references such as [8, 9, 23, 25, 27] for the theory of Triebel-Lizorkin spaces. Given p, and a weight w we denote by (l ) the Banach space of seuences f = {f j } of measurable functions defined on such that ( ) f (l ) = {f j} (l ) := f j <. The duality arguments in the proofs of our theorems will make use of the fact that for any seuence {f j } of measurable functions defined on, it holds that {f j } (l ) = f j (x)h j (x) w(x) dx, (2.3) sup {h j } U p,

7 COMMUTATOR ESTIMATES FOR VECTOR FIELDS 7 where U p, is the unit ball centered at zero in Lp (w)(l ). See [4, Theorems and 2] for a proof. Finally, for a vector-valued function f = (f, f 2,..., f n ) and a normed space X such that f j X for all j =,..., n, we set f X = n j= f j X. 3. Main lemma In this section we state and prove a lemma that plays a key role in the proof of the main results. For N > 0 and j Z set Φ N j (x) := 2 jn ( + 2 j x ) N x. (3.4) If α N n 0 is a multiindex, α denotes the sum of the components of α. Lemma 3.. Fix ϕ, ζ S( ) and K N. For every f, g S( ), x and j, m Z we have ϕ j (x y)(ζ m (x z) ζ m (y z))f(y)g(z)dy dz (3.5) R 2n = 2 α (m j) (ϕ α j f)(x)(( α ζ) m g)(x) + E j,m,k (f, g)(x), where α <K ϕ α j (x) = ( ) α (2 j x) α ϕ j (x), α N n α! 0, x (3.6) and E j,m,k (f, g)(x) 2 (m j)(k ) 2 min(m j,0) M(M(f)M(g))(x), (3.7) with the implicit constant depending only on the dimension n, K, and Schwartz seminorms of ϕ and ζ. When K =, the sum on the right hand side of (3.5) is interpreted as zero. Proof. By Taylor s theorem up to order K we have ζ m (y z) ζ m (x z) = α! ( α ζ m )(x z)(y x) α (3.8) α <K + α =K α! ( α ζ m )(ξ α )(y x) α, where ξ α is on the line segment joining y z and x z. For each multiindex α with α < K it follows that R 2n ϕ j (x y)( α ζ m )(x z)(y x) α f(y)g(z)dy dz = 2 α m R 2n ϕ j (x y)( α ζ) m (x z)(y x) α f(y)g(z)dy dz = 2 α (m j) R 2n ϕ j (x y)( α ζ) m (x z)(2 j y 2 j x) α f(y)g(z)dy dz = α! 2 α (m j) (ϕ α j f)(x)(( α ζ) m g)(x),

8 8 JAROD HART AND VIRGINIA NAIBO with ϕ α j as in (3.6). Next, let N > 0 and consider Φ N j estimate: α =K α! ( α ζ m )(ξ α )(y x) α Suppose first that for j Z as in (3.4). We will prove the following min(2 m(k ) x y K, 2 mk x y K ) ( Φ N m(x z) + Φ N m(y z) ). (3.9) min(2 m(k ) x y K, 2 mk x y K ) = 2 m(k ) x y K, which is euivalent to 2 m x y. Using that ζ S( ) and (3.8) gives α =K α! ( α ζ m )(ξ α )(y x) α ζ m (x z) ζ m (y z) + ζ m (x z) + ζ m (y z) + α <K α <K ( Φ N m(x z) + Φ N m(y z) ) + α! ( α ζ m )(x z) y x α α! 2m α ( α ζ) m (x z) y x α α <K 2 m(k ) x y K ( Φ N m(x z) + Φ N m(y z) ), 2 m α y x α where in the last ineuality we used that 2 m x y. Thus, (3.9) is true in this case. Suppose now that min(2 m(k ) x y K, 2 mk x y K ) = 2 mk x y K, that is 2 m x y <, and consider two cases: 2 m min( x z, y z ) 2 and 2 m min( x z, y z ) > 2. Case : 2 m min( x z, y z ) 2: For each multiindex α with α = K we have α! ( α ζ m )(ξ α ) sup α =K α! α ζ m L C K,ζ 2 m(n+k).

9 COMMUTATOR ESTIMATES FOR VECTOR FIELDS 9 Using the above estimate and the assumption 2 m min( x z, y z ) 2 we obtain α =K α! ( α ζ m )(ξ α )(y x) α 2m(n+K) x y K 2 m(n+k) x y K ( + 2 m min( x z, y z )) ( N 2 m(n+k) x y K ( + 2 m x z ) + N ( + 2 m y z ) N = 2 mk x y K ( Φ N m(x z) + Φ N m(y z) ), ) as desired. Case 2: 2 m min( x z, y z ) > 2: For each multiindex α with α = K we have ξ α = δ α (x z) + ( δ α )(y z) for some δ α (0, ). Since x y 2 m and min( x z, y z ) > 2 m+, it then follows that ξ α = (y z) δ α (y x) y z δ α x y y z 2 m y z /2. Using that ζ S( ) and the latter ineuality, we get α =K α! ( α ζ m )(ξ α )(y x) α = 2 mk ( α ζ) m (ξ α )(y x) α α! α =K 2 mk y x K Φ N m(ξ α ) α =K 2 mk y x K Φ N m(y z) 2 mk y x K ( Φ N m(x z) + Φ N m(y z) ), which gives (3.9) for this case. Finally, set E j,m,k (f, g)(x) = ϕ j (x y) R 2n α =K α! ( α ζ m )(ξ α )(y x) α f(y)g(z)dy dz.

10 0 JAROD HART AND VIRGINIA NAIBO Choose N > 0 such that N K > n. Recalling that ϕ S( ) and applying (3.9), we then obtain E j,m,k (f, g)(x) Φ N j (x y) min(2 m(k ) x y K, 2 mk x y K ) R 2n (Φ N m(x z) + Φ N m(y z)) f(y)g(z) dy dz 2 (m j)(k ) Φ N j (x y) min( 2 j (x y) K, 2 (m j) 2 j (x y) K ) R 2n (Φ N m(x z) + Φ N m(y z)) f(y)g(z) dy dz 2 (m j)(k ) min(, 2 (m j) ) (x y)(φ N m(x z) + Φ N m(y z)) 2 (m j)(k ) 2 min(m j,0) Φ N K j R 2n Φ N K j R 2n + 2 (m j)(k ) 2 min(m j,0) f(y)g(z) dy dz (x y)φ N m(x z) f(y)g(z) dy dz Φ N K j R 2n (x y)φ N m(y z) f(y)g(z) dy dz 2 (m j)(k ) 2 min(m j,0) (M(f)(x)M(g)(x) + M(f M(g))(x)) 2 (m j)(k ) 2 min(m j,0) M(M(f)M(g))(x), where in the last two estimates we have used the pointwise ineualities Φ L k h M(h) (uniformly in k for any L > n), and h M(h), respectively. This concludes the proof of the lemma. 4. Preparations for the proofs of the main theorems In this section we introduce more notation and prove some preliminary results that will be used in the proofs of the main theorems Let ψ be as in Section 2 with its Fourier transform supported in {ξ : 2 + ε < ξ < 2 ε} for some small ε > 0. Consider Ψ, θ S( ) such that Ψ and θ satisfy (2.) and (2.2) along with the following conditions: Ψ(2 j ξ) ψ(2 j ξ) = ξ 0, (4.20) θ(ξ) ψ(ξ) = ψ(ξ) ξ. (4.2) For the construction of Ψ see, for instance, [9, Lemma 6.9]. Note that the assumption given above for the support of ψ allows for the construction of θ. Throughout the proofs, we will reason by duality and use the following notation. Let p,, s and w be as in the hypotheses of the theorems and consider an arbitrary

11 COMMUTATOR ESTIMATES FOR VECTOR FIELDS seuence {h j } S( ) such that ( ) 2 js h j Given V = (V,, V n ) and f in S( ), we have and L p (w) n [V, ψ j ](f)(x) = [V k k, ψ j ](f)(x) k=. (4.22) [V k k, ψ j R ](f)(x) = ψ j (x y)(v k (x) V k (y)) k f(y)dy n = ψ j (x y)( Ψ m ψ mv k (x) Ψ m ψ mv k (y)) k f(y)dy (4.23) m Z = ψ j (x y)(ψ m (x z) Ψ m (y z)) k f(y)( ψ mv k )(z)dy dz, m Z R 2n where we have used (4.20). It can be shown that the above series converges uniformly to [V k k, ψ j ](f) in Rn. For each j, m Z, introduce the bilinear operator Θ j,m defined for F, G S( ) as Θ j,m (F, G)(x) = ψ j (x y)(ψ m (x z) Ψ m (y z))f (y)( ψ mg)(z)dy dz (4.24) R 2n and consider the sums Θ j,m (F, G)(x)h j (x) Θ j,m (F, G)(x)h j (x) m Z j m + Θ j,m (F, G)(x)h j (x). By duality arguments, as explained in Section 2, the proofs of the theorems will follow from appropriately estimating the integrals over with respect to w(x)dx of each of the sums on the right hand side above with F = k f and G = V k. 4.. Sum corresponding to j m. For the estimates involving the sum corresponding to j m, we will prove the following lemma. Lemma 4.. Let s, a R be such that s + a > 0; consider < p, p 2, < and < p with /p = /p + /p 2 ; assume (w, w 2 ) A p A p2 with w := w p/p w p/p 2 2 A p. With the notation given above and {h j } satisfying (4.22), we have j m 2 ja Θ j,m (F, G)(x)h j (x) w(x) dx F L p (w ) G F s+a, p 2 (w 2 ),

12 2 JAROD HART AND VIRGINIA NAIBO where the implicit constant is independent of F, G and {h j }. Proof. Fix the indices and weights as in the hypothesis of the lemma and let {h j } be as in (4.22). By Lemma 3. with K = applied to Θ j,m (F, G), we have j m = j m 2 ja Θ j,m (F, G)(x)h j (x) w(x) dx j m 2 ja M(M(F )M( ψ mg))(x) h j (x) w(x) dx 2 (s+a)(m j) M(M(F )M(2 m(s+a) ψ mg))(x)2 js h j (x) w(x) dx ( ( M(M(F )M(2 m(s+a) ψ mg))(x) ) m Z ) ( 2 js h j (x) ) w(x) dx, where in the last ineuality it was used that s + a > 0. Using Hölder s ineuality, (4.22) and the weighted Fefferman-Stein ineuality (2.0), it follows that 2 ja Θ j,m (F, G)(x)h j (x) w(x) dx j m ( ( M(M(F )M(2 m(s+a) ψ mg)) ) ) m Z ( M(F ) ( M(2 m(s+a) ψ mg) ) ) m Z ( ( M(F ) L p (w ) M(2 m(s+a) ψ mg) ) m Z ( ) F L p (w ) 2 m(s+a) ψ mg m Z ) L p 2 (w2 ) L p 2 (w2 ) = F L p (w ) G F s+a, p 2 (w 2 ), which proves the lemma Sum corresponding to m < j. For the estimate involving the sum corresponding to m < j we will use the following ineuality. Given a R, the decomposition

13 COMMUTATOR ESTIMATES FOR VECTOR FIELDS 3 (3.5) and the bound (3.7) in Lemma 3., applied to Θ j,m (F, G), give 2 ja Θ j,m (F, G)(x)h j (x) 2 ja E j,m,k (F, ψ mg)(x)h j (x) (4.25) + α <K 2 ja 2 α (m j) (ψj α θ jf )(x)(( α Ψ) m ψ mg)(x)h j (x), where the second term on the right hand side is zero if K =, and we have used that ψ θ = ψ to insert the operator θ j. We also note that the series in m < j in the second term is absolutely and uniformly convergent in. For the first term in (4.25) we have the following lemma. Lemma 4.2. Let K N and s, a R be such that K > s + a; consider < p, p 2, < and < p with /p = /p + /p 2 ; assume (w, w 2 ) A p A p2 with w := w p/p w p/p 2 2 A p. With the notation given above and {h j } satisfying (4.22), we have 2 ja E j,m,k (F, ψ mg)(x)h j (x) w(x) dx F L p (w ) G F s+a, p 2 (w 2 ), where the implicit constant is independent of F, G and {h j }. Proof. Fix the indices and weights as in the hypothesis of the lemma and let {h j } be as in (4.22). By (3.7), we have 2 ja E j,m,k (F, ψ mg)(x)h j (x) w(x) dx (4.26) 2 (K s a)(m j) M(M(F )M(2 m(s+a) ψ mg))(x)2 js h j (x) w(x) dx, and then one can proceed as in Lemma 4., using that (K s a) > 0, to obtain the desired ineuality. The following corollary follows from Lemmas 4. and 4.2 and the ineuality (4.25). Corollary 4.3. Let K N and s, a R be such that 0 < s + a < K; consider < p, p 2, < and < p with /p = /p +/p 2 ; assume (w, w 2 ) A p A p2 with w := w p/p w p/p 2 2 A p. With the notation given above and {h j } satisfying

14 4 JAROD HART AND VIRGINIA NAIBO (4.22), we have 2 ja Θ j,m (F, G)(x)h j (x) R w(x) dx F L p n (w ) G F p s+a, 2 (w 2 ) (4.27) m Z + 2 ja 2 α (m j) (ψj α θ jf )(x)(( α Ψ) m ψ mg)(x)h j (x) w(x) dx, α <K where the implicit constant is independent of F, G and {h j }. For K =, the second term on the right hand side of the above ineuality is interpreted as zero. 5. Proofs of Theorems.,.2, and.3 Proof of Theorem.. Let < p, p 2, < and < p satisfy /p = /p + /p 2, and let (w, w 2 ) A p A p2 be such that w = w p/p w p/p 2 2 A p. Proof of (.4) and (.5): Let s > 0. In view of Corollary 4.3 applied to F = k f and G = V k for k =,, n, with a = 0 and K > s, (.4) and (.5) will follow from controlling the second term on the right hand side of (4.27) (with a=0) by V k L p (w ) kf F p s, 2 (w 2 ) and V k L p (w ) f F p s, 2 (w 2 ), respectively. Regarding (.4), it will be enough to estimate 2 (m j) M( θ j k f)(x)m( ψ mv k )(x) h j (x) w(x) dx 2 (m j) M( θ j k f)(x)m(m(v k ))(x) h j (x) w(x) dx. Now, using Hölder s ineuality, (4.22) and the Fefferman-Stein ineuality (2.0), we have 2 (m j) M( θ j k f)(x)m(m(v k ))(x) h j (x) w(x) dx = 2 (m j) M(2 sj θ j k f)(x)m(m(v k ))(x)2 sj h j (x) w(x) dx ( M(M(V ( k)) (M(2 sj θ j k f) ) as desired. ) V k L p (w ) kf F p s, 2 (w 2 ),

15 COMMUTATOR ESTIMATES FOR VECTOR FIELDS 5 As for (.5), we consider 2 α (m j) (ψj α θ j k f)(x)(( α Ψ) m ψ mv k )(x)h j (x) w(x) dx α <K = 2 ( α )(m j) (2 j ψj α θ j k f)(x) α <K (2 m ( α Ψ) m ψ mv k )(x)h j (x) w(x) dx. We have (2 j ψ α j θ j k f)(x) = (( k ψ α 0 ) j θ jf)(x) M( θ jf)(x) and 2 m ( α Ψ) m ψ mv k = 2 m ( α Ψ) m ψ m V k = 2 m ( α (Ψ ψ)) m V k. Also, since the multiindex α satisfies α there exists l α {,..., n} such that α e lα 0 componentwise, so that Then, Hence, = 2 m ( α (Ψ ψ)) m (x) = e lα (( α e lα(ψ ψ))m )(x). (2 m ( α Ψ) m ψ mv k )(x) = ( e lα (( α e lα(ψ ψ))m ) V k )(x) = (( α e lα (Ψ ψ))m e lα Vk )(x). 2 α (m j) (ψj α θ j k f)(x)(( α Ψ) m ψ mv k )(x)h j (x) w(x) dx α <K (( k ψ α ) j θ jf)(x) α <K ( ) 2 ( α )(m j) (( α e lα (Ψ ψ))m lα V k )(x) h j (x) w(x) dx. Now, we note that 2 ( α )(m j) (( α e lα (Ψ ψ))m lα V k )(x) M( V k)(x). uniformly in j. Indeed, if α = 0 then α = e lα and conseuently ( α elα (Ψ ψ))m (x) = (Ψ ψ) m (x) = Φ j (x),

16 6 JAROD HART AND VIRGINIA NAIBO where Φ j (x) = 2 nj Φ(2 j x) for some Φ S( ) such that Φ is supported in a ball centered at the origin (recall that m Z Ψ(2 m ξ) ψ(2 m ξ) = for all ξ 0); then the claim follows from the fact that (Φ j lα V k )(x) M( lα V k )(x) uniformly in j. If α > 0 we have (( α e lα (Ψ ψ))m lα V k )(x) M( lα V k )(x) uniformly in m while the series 2 ( α )(m j) converges to a number independent of j. It then follows that, 2 α (m j) (ψj α θ j k f)(x)(( α Ψ) m ψ mv k )(x)h j (x) w(x) dx α <K M( θ jf)(x)m( V k )(x) h j (x) w(x) dx = M(2 sj θ jf)(x)m( V k )(x)2 sj h j (x) w(x) dx V k L p (w ) f F p s, 2 (w 2 ), where the last ineuality is obtained by using Hölder s ineuality, (4.22), and the weighted Fefferman-Stein ineuality (2.0). This gives the desired estimate. Proof of (.6): Let s > 0. In order to prove (.6), we first integrate by parts to get Θ j,m ( k f, V k )(x) = 2 j ( k ψ) j (x y)(ψ m (x z) Ψ m (y z))f(y)( ψ mv k )(z)dy dz R 2n + ψ j (x y)( k Ψ m )(y z)f(y)( ψ mv k )(z)dy dz. (5.28) R 2n For the second term on the right hand side of (5.28) we have ψ j (x y)( k Ψ m )(y z)f(y)( ψ mv k )(z)dy dz R 2n = ψ j (x y)f(y)(( k Ψ m ) ψ mv k )(y)dy R n = ψ j (x y)f(y)( Ψ m ψ m k V k )(y)dy = ψ j (f Ψ m ψ m k V k )(x). Then, since m Z Ψ m ψ m is the identity operator, n ψ j (f Ψ m ψ m k V k ) = k= m Z n ψ j (f kv k ) = ψ j (f div(v )). k= It then follows that ( n ) 2js ψ j (f Ψ m ψ m k V k ) k= m Z which leads to the first term in (.6). = f div(v ) F p s, (w),

17 COMMUTATOR ESTIMATES FOR VECTOR FIELDS 7 The first term in (5.28) is of the form 2 j Θj,m (f, V k ), where Θ j,m (f, V k ) is the same as Θ j,m (f, V k ) but with ( k ψ) j instead of ψ j (in particular, Lemmas 4. and 4.2 hold true with Θ j,m instead of Θ j,m ). Given {h j } as in (4.22), we consider 2 j Θj,m (f, V k )(x)h j (x) 2 j Θj,m (f, V k )(x)h j (x) (5.29) m Z j m + 2 j Θj,m (f, V k )(x)h j (x). For the first term in (5.29), we use Lemma 4. with a = and obtain that, since s + > 0, j m 2 j Θ j,m (f, V k )(x)h j (x) w(x) dx f L p (w ) V k F s+, p 2 (w 2 ). For the second term in (5.29), we apply Lemma 3. with K > s + and get 2 j Θj,m (f, V k )(x)h j (x) 2 j 2 α (m j) (( k ψ) α j θ jf)(x)(( α Ψ) m ψ mv k )(x)h j (x) α <K + 2 j E j,m,k (f, ψ mv k )(x)h j (x). (5.30) By Lemma 4.2 with a = and K > s +, for the second term in (5.30) we have 2 j E j,m,k (f, ψ mv k )(x)h j (x) w(x) dx f L p (w ) V k F s+, p 2 (w 2 ). Finally, for the first term in (5.30), we obtain 2 j 2 α (m j) (( k ψ) α j θ jf)(x)(( α Ψ) m ψ mv k )(x)h j (x) w(x) dx α <K = 2 ( α )(m j) (( k ψ) α j θ jf)(x) α <K (2 m ( α Ψ) m ψ mv k )(x)h j (x) w(x) dx V L p (w ) f F s, p 2 (w 2 ),

18 8 JAROD HART AND VIRGINIA NAIBO where the last ineuality follows as in the treatment of the second term in (4.25) used to prove (.5). This finishes the proof of (.6). Proof of Theorem.2. Estimate (.7) follows from Corollary 4.3 with K =, a = 0, F = k f and G = V k, k =,, n. As for (.8), we note that the proof of (.6) works well as long as s + > 0 and that the proof of (.8) follows from it. Indeed, the first term in (5.30) is the only one responsible for the appearance of V L p (w ) f F p s, 2 (w 2 ) in (.6). But, for the second term in (5.29), when applying Lemma 3., one can choose K = for < s < 0. Then the first term in (5.30) can be taken eual to zero for such range of s. Proof of Theorem.3. Consider indices and weights as in the hypothesis, that is, s > 0, s, s 2 R with s 2 < and s = s + s 2 ; < p, p 2, p,,, 2 < with /p = /p +/p 2 and / = / +/ 2 ; and (w, w 2 ) A p A p2 with w = w p/p w p/p 2 2 A p. In view of Corollary 4.3 applied to F = k f and G = V k for k =,, n, with a = 0 and K > s, it is enough to estimate the second term on the right hand side of (4.27) (with a = 0) by k f F s, p (w ) V k F s 2, 2 p 2 (w 2 ). Let 3 to be chosen later and write 2 α (m j) (ψj α θ j k f)(x)(( α Ψ) m ψ mv k )(x)h j (x) w(x) dx α <K 2 (m j) M( θ j k f)(x)m( ψ mv k ) h j (x) w(x) dx = 2 (m j)( s 2) M(2 s j θ j k f)(x)m(2 s 2m ψ mv k )2 sj h j (x) w(x) dx ( ( M(2 s j θ j k f)(x) ) 2 2 sj 2 hj (x) 2 ( ( M(2 s j θ j k f)(x) ) 2 3 ( m Z ) ( m Z ) 2 ( M(2 s 2 m ψ mv k )(x) ) 2 ) 2 w(x) dx ( sj 2 3 hj (x) 2 3 ) 2 3 ( M(2 s 2 m ψ mv k )(x) ) 2 ) 2 w(x) dx, where in the second ineuality we have used that s 2 > 0. Note that since / = / +/ 2 and > then / 2 > ; set 3 := / 2, which gives 2 3 = and 2 3 =. From Hölder s ineuality, the weighted Fefferman-Stein ineuality (2.0) and

19 COMMUTATOR ESTIMATES FOR VECTOR FIELDS 9 (4.22), we then obtain that the above is controlled by k f from which the desired ineuality follows. F s, p (w ) V k F s 2, 2 p 2 (w 2 ), We finish this section with a remark and corollary regarding the estimate (.8). Remark 5.. Under the assumptions < s < 0, < p < w A p such that lim R n, 0 <, and n+s w(x) x pn( p n+s n ) dx =, (5.3) B(0,R) it follows that any f F p s, (w) S( ) verifies f(x)dx = 0. Examples of weights in A p satisfying (5.3) include w(x) = x τ with n + p(n + s) < τ < n(p ) and any weight w A p such that w u ɛ for some u A and 0 < ɛ < p n+s. (A n is the class of weights in such that M(w)(x) w(x) for almost every x.) For a proof of this remark see Appendix A. Remark 5. and Theorem.2 imply the following result. Corollary 5.. Let < s < 0, assume < p, p 2, < and < p satisfy /p = /p + /p 2, and let (w, w 2 ) A p A p2 be such that w := w p/p w p/p 2 2 A p and w verifies (5.3). Fix a vector field V = (V,..., V n ) with V j S( ) for j =,, n. If ( ) 2 sj [V, j ](f) f L p (w ) V F p s+, 2 (w 2 ). (5.32) for all f S( ), then div(v ) = 0. Proof of Corollary 5.. Consider indices and weights as in the hypothesis. By (4.23), (4.24), (5.28) and the computations that follow (5.28), we have n [V, ψ j ](f)(x) = Θ j,m ( k f, V k )(x) k= m Z n = ψ j (fdiv(v )) + 2 j Θj,m (f, V k )(x), k= m Z and, using the estimate for the second term on the right hand size along with (5.32), fdiv(v ) Fp s, (w) f L p (w ) V F s+, p 2 (w 2 ). Since f, V,, V n S( ) and s + > 0, then f L p (w ) V F p s+, 2 (w 2 ) <. Therefore fdiv(v ) F p s, (w) for all f S( ). By Remark 5., fdiv(v ) has zero integral for all f S( ). Taking f = div(v ) S( ), gives div(v ) L 2 = 0 and therefore div(v ) = 0.

20 20 JAROD HART AND VIRGINIA NAIBO 6. Commutator estimates on other function spaces The techniues used to prove Theorems.,.2 and.3 are flexible and transparent enough to readily apply to other function spaces. 6.. Inhomogeneous weighted Triebel-Lizorkin spaces. We recall that the definition of the inhomogeneous weighted Triebel-Lizorkin spaces Fp s, (w) was given in Section 2. Analogous estimates to those in Theorems.,.2 and.3 hold for these spaces with the following changes: The left hand side of each ineuality becomes ( [V, S 0 ](f) + ) 2 sj [V, j ](f) j N and, on the right hand side, F is replaced by F. The proofs are analogous Weighted Besov spaces. Let s R, 0 < p,, and w A. The homogeneous and inhomogeneous weighted Besov spaces are denoted by Ḃs, p (w) and Bp s, (w), respectively. They are defined analogously to the weighted Triebel-Lizorkin spaces by interchanging the order of the uasi-norms (norms if p, ) in l and. As in the case of the Triebel-Lizorkin spaces, these definitions are independent of the choice of the functions ψ and ψ and Ḃs, p (w) and Bp s, (w) are uasi-banach spaces (Banach spaces for p, ). The class S( ) is contained in Bp s, (w) for all p, s, w as in the definition and is dense for finite p and (see [25, Theorem 2.4]); S( ) is also contained in Ḃs, p (w) for 0 <, < p, s > 0, and w A p. We refer the reader to the classical references [7, 9, 24, 25, 27] for the theory of such spaces. Given p, and a weight w we denote by l () the Banach space of seuences f = {f j } of measurable functions defined on such that It holds that f l () = {f j} l () := ( {f j } l () = sup {h j } U p, f j ) <. f j (x)h j (x) w(x) dx, (6.33) where U,p is the unit ball centered at zero in l (L p (w)) ([4, Theorems and 2]). Almost identical statements of Theorems.,.2 and.3 apply to the setting of homogeneous weighted Besov spaces with the following changes: The left hand side of each of the ineualities becomes ( 2 sj [V, j ](f) and, on the right hand side, F is replaced by Ḃ. Moreover, any of the indices, p, p, p 2 can take the value infinity in the hypothesis (note that the case p = forces )

21 COMMUTATOR ESTIMATES FOR VECTOR FIELDS 2 p = p 2 = and w = w = w 2 = ). The proofs follow analogously noting that the boundedness of M in weighted Lebesgue spaces is used rather than the weighted Fefferman-Stein ineuality (then = and p = are allowed). The finiteness of p, p, p 2 in the case of the Triebel-Lizorkin spaces (note that p is allowed to be infinity in Theorems. and.2) is due to the fact that the definitions of these spaces given in Section 2 do not apply when the integrability parameter euals infinity; in the case of the Besov spaces the definitions do apply in such instance and the computations given work as well. Analogous comments to those in Section 6. follow for the theorems corresponding to inhomogeneous Besov spaces Variable exponent Triebel-Lizorkin and Besov spaces. The main properties associated to the weighted Triebel-Lizorkin and Besov spaces that are used in the proofs of Theorems.,.2 and.3 and their respective Besov spaces counterparts are the following: (i) the norm duality (2.3) and (6.33) for (l ) and l (), respectively; (ii) Hölder s ineuality for weighted Lebesgue spaces; (iii) the weighted Fefferman-Stein ineuality (2.0) in the case of the Triebel-Lizorkin spaces and the boundedness of the maximal operator M in weighted Lebesgue spaces in the case of the Besov spaces. As we will see, the Triebel-Lizorkin and Besov spaces with variable integrability exponent also possess properties corresponding to those given in (i), (ii) and (iii) above. As a result, the same methods used in the proofs of Theorem.,.2 and.3 give rise to analogous results in the context of these variable integrability exponent spaces. We next give the precise definitions of these spaces and state the corresponding results. Let P be the collection of measurable functions p( ) : [, ) such that p = ess inf x p(x) > and p + = ess sup x p(x) <. For p( ) P we define the modular ρ p( ) (f) = f(x) p(x) dx, where f is a measurable function defined on. The variable Lebesgue space L p( ) consists of all measurable functions f defined on that satisfy ρ p( ) (f/λ) < for some λ > 0. This is a Banach space with the norm given by f L p( ) = inf { λ > 0 : ρ p( ) (f/λ) }. We note that if p( ) = p 0 is constant then L p( ) = L p 0 with euality of norms. We refer the reader to the books [8, 4] for more information about variable Lebesgue spaces. Let B be the family of all p( ) P such that M is bounded from L p( ) to L p( ). A sufficient condition for p( ) B is that p( ) be log-hölder continuous locally: there

22 22 JAROD HART AND VIRGINIA NAIBO exists C 0 > 0 such that p(x) p(y) C 0 log( x y ), x y < 2 ; and log-hölder continuous at infinity: there exists p, C > 0 such that C p(x) p log(e + x ), x Rn. Given s R, 0 < <, p( ) P, and functions ψ and ψ as in Section 2, the inhomogeneous Triebel-Lizorkin spaces F s, p( ) and Besov spaces Bs, p( ), with integrability variable exponent, are defined analogously to Fp s, (w) and Bp s, (w) with the norm in replaced by the norm in L p( ). If p( ) B, these definitions are independent of ψ and ψ (see [30]) and contain S( ). If p( ) B and s > 0, F s,2 p( ) coincides with the variable exponent Bessel potential space L s,p( ) and, for s N, with the variable exponent Sobolev space W s,p( ) (see [20, 29]). More general versions of these spaces, where s and are also allowed to be functions, were introduced in [] and [5]. We next check that appropriate versions of the properties (i), (ii) and (iii) hold in this setting. Property (ii) is given by the following version of Hölder s ineuality in the context of variable Lebesgue spaces. For a proof see, for instance, in [8, Corollary 2.28]. Lemma 6.A. Given p( ), ( ), r( ) P, suppose = +. Then for all r( ) p( ) ( ) f L p( ) and g L ( ), fg L r( ) f L p( ) g L ( ). The implicit constant depends only on p( ) and ( ). For the corresponding property (iii), the following version of the Fefferman-Stein ineuality was proved in [9, Corollary 2.]. Lemma 6.B. If p( ) B and < < then ( ) ( ) M(h j ) h j L p( ) L p( ) We next consider property (i). If p( ) P, p ( ) will denote the conjugate variable exponent of p( ) given by + =. We define Lp( ) (l ), l (L p( ) ), p( ) p ( ) L p( ) (l ), l (L p( ) ), U p ( ), and U,p ( ) as above by replacing with L p( ). The corresponding property (i) is the content of Lemma 6. below. To the best of our knowledge, Lemma 6. is not present in the current literature. Its proof, which follows standard techniues, is sketched in Appendix A. Lemma 6.. Let p( ) P,, and {f j } be a seuence of measurable functions defined on. Then {f j } L p( ) (l ) sup {h j } U p ( ), f j (x)h j (x) dx (6.34).

23 and COMMUTATOR ESTIMATES FOR VECTOR FIELDS 23 {f j } l (L p( ) ) sup {h j } U,p ( ) where the implicit constants are independent of f. f j (x)h j (x) dx, (6.35) We next state the version of Theorem. for variable exponent Triebel-Lizorkin spaces as a model result. The corresponding version for variable exponent Besov spaces and counterparts of Theorems.2 and.3 in both settings are obtained similarly. Theorem 6.2. Let s > 0, < < and assume p( ), p ( ), p 2 ( ) B are such that = +. For every vector field V = (V p( ) p ( ) p 2 ( ),..., V n ) with V j S( ) for j =,, n, and for all f S( ) we have ( [V, S 0 ](f) + ) 2 sj [V, j ](f) j N and ( [V, S 0 ](f) + ) 2 sj [V, j ](f) j N L p( ) f L p ( ) V F s, p 2 ( ) + V L p ( ) f F s, p 2 ( ), L p( ) f L p ( ) V F s, p 2 ( ) + V L p ( ) f F s, p 2 ( ), ( [V, S 0 ](f) + ) 2 sj [V, j ](f) j N L p( ) f div(v ) F s, + V p( ) L p ( ) f F s, + f p 2 ( ) L p ( ) V F s+,. Remark 6.. Versions of Theorem 6.2 and the counterparts of Theorems.2 and.3 for variable exponent Triebel-Lizorkin spaces may also be obtained through the use of extrapolation results. For instance, [0, Theorem 3.] says, roughly speaking, that if there are indices 0 < p, p 2, p < satisfying p = p + p 2, such that h f L p (w ) g L p 2 (w 2 ) for all (h, f, g) in a given family F of ordered triples of non-negative, measurable functions defined on and for all w, w 2 A and w = w p/p w p/p 2 2, then h L p( ) f L p ( ) g L p 2 ( ) for all (h, f, g) F and for appropriate variable exponents satisfying = +. p( ) p ( ) p 2 ( ) As an example, noting that w, w 2 A implies w p/p w p/p 2 2 A for = p p + p 2, Lemma 4. adapted to the inhomogeneous setting, (2.3) and the above mentioned p 2 ( )

24 24 JAROD HART AND VIRGINIA NAIBO extrapolation result imply that { } 2 ja Θ j,m (F, G) m=j j L p( ) (l ) F L p ( ) G F s+a, p 2 ( ) for s + a > 0 and suitable variable exponents. The same type of thought process can be applied to the different pieces that make the commutator [V, j ](f), as defined in the proof of the weighted estimates in the setting of Triebel-Lizorkin spaces. Appendix A. Proof of Remark 5.. Let ψ j, j, and Φ N j be as defined in the previous sections; assume without loss of generality that ψ(0) 0. Let f F p s, (w) S( ). We have ( f F p s, (w) R [ f(y)dy 2 sj ψ j (x) ] ) p p w(x) dx n j<0 ( 2 R sj (ψ j (x y) ψ j (x))f(y)dy n j<0, ) p w(x) dx Note that for j < 0, (ψ j (x y) ψ j (x))f(y)dy 2j Φ n 2 j j (x) R (2 n j + x ) 2 j n ( + x ), n and hence we have ( ) p A := 2 R sj (ψ j (x y) ψ j (x))f(y)dy w(x)dx n j<0 ( j<0 2 (+s)j ) p w(x) dx <, ( + x ) pn where we have used that s + > 0 and w A p. Since η := ψ(0) > 0, there exists 0 < δ < such that ψ j (x) 2 jn η/2 for 2 j x < δ. Then ( [ 2 sj ψ j (x) ] ) p ( w(x)dx sup 2 sj ψ j (x) ) p w(x) dx j<0 j<0 (η/2) p sup 2 (n+s)pj w(x) dx j<0 = (η/2) p δ (n+s)p sup j<0 (η/2) p δ (n+s)p sup j<0 x <2 j δ (δ2 j ) n (δ2 j ) n x <2 j δ x <2 j δ w(x)(δ2 j ) ( p n+s n )pn dx w(x) x ( p n+s n )pn dx =, p.

25 COMMUTATOR ESTIMATES FOR VECTOR FIELDS 25 where we have used (5.3). It follows that p ( [ f(y)dy 2 sj ψ j (x) ] ) p w(x) dx f p j<0 Fp s, (w) + A <. (A.36) The term in the suare brackets on the left hand side of (A.36) is infinite. Hence the only way to avoid a contradiction is if f has integral zero. We next check that w u ɛ for some u A and 0 < ɛ < p n+s is sufficient n for (5.3). Since u A, it follows that u(x) M(u)(x) ( + x ) n. Then w(x) ( + x ) nɛ and w(x) x pn( p n+s n ) dx x nɛ x pn( p n+s n ) dx B(0,R) < x <R which tends to infinity as R tends to infinity since 0 < ɛ < p n+s n. Proof of Lemma 6.. The proof follows the ideas from [4, Theorems and 2]. We first prove (6.34). Let p( ), and f = {f j } be as in the assumptions. We will use the fact that ρ p( ) (F/ F L p( )) = if F L p( ) and F L p( ) 0. (See [8, Proposition 2.2] for a proof.) It is enough to show that f L p( ) (l ) sup {h j } U p ( ), since the reverse estimate is a conseuence of Hölder s ineuality. Suppose first that < and 0 < f L p( ) (l ) < ; define We have that ρ p ( ) h j (x) = sign(f j (x)) f j (x) ( k Z f j (x)h j (x) dx (A.37) ) p(x) f k (x) ( ( h j ) ) = ρ p( ) ( ( f j ) f L p( ) (l ) ) f p(x) L p( ) (l ). =, where corresponding changes in notation apply if =. By the definition of the L p ( ) norm, we obtain that {h j } L p ( ) (l ). Moreover, f j (x)h j (x) dx = f L p( ) (l ), from which (A.37) follows. When = and 0 < f L p( ) (l ) <, suppose first that f j = 0 for j N 0 for some N 0 N. For ε > 0 and x, set F ε (x) = { j Z : f j (x) > sup +ε k Z f k (x) } and ( ) p(x) h j,ε (x) = sign(f j (x))g j,ε (x) sup f k (x) f p(x) k Z L p( ) (l ),

26 26 JAROD HART AND VIRGINIA NAIBO where g j,ε (x) = χ Fε(x)(j) if #(F #(F ε(x)) ε(x)) 0 and g j,ε (x) = 0 otherwise. It follows that {h j,ε } L p ( ) (l ) and that f j (x)h j,ε (x) dx + ε f L p( ) (l ). Since ε is arbitrary, we get the desired result. We next remove the assumption that f j = 0 if j N 0 ; for N N, set f j,n = f j if j N and f j,n = 0 if j > N. By the previous case, we have {f j,n } L p( ) (l ) sup f j (x)h j (x) dx N N. {h j } U p ( ), Since sup f j,n (x) sup f j (x) as N and for all x, the Monotone Convergence Theorem in the setting of variable Lebesgue spaces (see [8, Theorem 2.59]) and the above estimate imply (A.37). Suppose now that f L p( ) (l ) = ; for each N N define I N = {j Z : j N} {x : x N} and fj N (x) = f j (x)χ IN (j, x) if f j (x)χ IN (j, x) < N and fj N (x) = N, otherwise. Since f N = {fj N } satisfies f N L <, then p( ) (l ) f N L p( ) (l ) sup {h j } U p ( ), f j (x)h j (x) dx. Since {f N j (x)} l {f j (x)} l as N for x, the Monotone Convergence Theorem for variable Lebesgue spaces and the last ineuality imply (A.37). The proof of (6.35) is similar; we briefly indicate the corresponding main changes. As in the proof of (6.34), it is enough to show the analogous ineuality to (A.37). In the case < and 0 < f l (L p( ) ) < define h j (x) = sign(f j (x)) f j (x) p(x) f j p(x) f. L p( ) l (L p( ) ) For the case = and 0 < f l (L p( ) ) <, assume first that f j = 0 for j N { } 0 for some N 0 N. Given ε > 0, define F ε = j Z : f j L p( ) > f +ε l (L p( ) ) and set h j,ε (x) = sign(f j (x)) χ F ε (j) #(F ε ) f j(x) p(x) f j p(x). L p( ) The limiting arguments used to remove the assumption f j = 0 for j N 0 in the case = and for the case f l (L p( ) ) = are analogous to the ones used for (A.37). References [] A. Almeida and P. Hästö. Besov spaces with variable smoothness and integrability. J. Funct. Anal., 258(5): , 200. [2] K. Andersen and R. John. Weighted ineualities for vector-valued maximal functions and singular integrals. Studia Math., 69():9 3, 980/8.

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28 28 JAROD HART AND VIRGINIA NAIBO [28] J. Wu, X. Xu, and Z. Ye. Global smooth solutions to the n-dimensional damped models of incompressible fluid mechanics with small initial datum. J. Nonlinear Sci., 25():57 92, 205. [29] J. Xu. The relation between variable Bessel potential spaces and Triebel-Lizorkin spaces. Integral Transforms Spec. Funct., 9(7-8): , [30] J. Xu. Variable Besov and Triebel-Lizorkin spaces. Ann. Acad. Sci. Fenn. Math., 33(2):5 522, Jarod Hart, Higuchi Biosciences Center, University of Kansas, Lawrence, KS 66045, USA. address: Virginia Naibo, Department of Mathematics, Kansas State University. 38 Cardwell Hall, 228 N. 7th Street, Manhattan, KS 66506, USA. address: