A Characterization of Hajłasz-Sobolev and Triebel-Lizorkin Spaces via Grand Littlewood-Paley Functions
aa r X i v : . [ m a t h . C A ] N ov J. Funct. Anal., to appear
A Characterization of Haj lasz-Sobolev and Triebel-LizorkinSpaces via Grand Littlewood-Paley Functions
Pekka Koskela, Dachun Yang and Yuan Zhou
Abstract
In this paper, we establish the equivalence between the Haj lasz-Sobolevspaces or classical Triebel-Lizorkin spaces and a class of grand Triebel-Lizorkin spaces onEuclidean spaces and also on metric spaces that are both doubling and reverse doubling.In particular, when p ∈ ( n/ ( n + 1) , ∞ ), we give a new characterization of the Haj lasz-Sobolev spaces ˙ M , p ( R n ) via a grand Littlewood-Paley function. Recently, analogs of the theory of first order Sobolev spaces on doubling metric spaceshave been established both based on upper gradients [23, 9, 32] and on pointwise inequal-ities [18]. For surveys on this see [19, 24]. These different approaches result in the samefunction class if the underlying space supports a suitable Poincar´e inequality [25]. In thispaper we further investigate the spaces introduced by Haj lasz [18] (also see [39]) that aredefined via pointwise inequalities.
Definition 1.1.
Let ( X , d ) be a metric space equipped with a regular Borel measure µ suchthat all balls defined by d have finite and positive measures. Let p ∈ (0 , ∞ ) and s ∈ (0 , .The homogeneous fractional Haj lasz-Sobolev space ˙ M s,p ( X ) is the set of all measurablefunctions f ∈ L p loc ( X ) for which there exists a non-negative function g ∈ L p ( X ) and a set E ⊂ X of measure zero such that (1.1) | f ( x ) − f ( y ) | ≤ [ d ( x, y )] s [ g ( x ) + g ( y )] for all x, y ∈ X \ E . Denote by D ( f ) the class of all nonnegative Borel functions g satisfying (1.1) . Moreover, define k f k ˙ M s,p ( X ) ≡ inf g ∈D ( f ) {k g k L p ( X ) } , where the infimumis taken over all functions g as above. In the Euclidean setting, ˙ M ,p coincides with the usual homogeneous first order Sobolevspace ˙ W , p , [18], provided 1 < p < ∞ . For p ∈ ( n/ ( n + 1) , Mathematics Subject Classification . Primary 42B35; Secondary 42B30, 46E35, 42B25.
Key words and phrases . Sobolev spaces, Triebel-Lizorkin space, Calder´on reproducing formula.Dachun Yang was supported by the National Natural Science Foundation (Grant No. 10871025) ofChina. Pekka Koskela and Yuan Zhou were supported by the Academy of Finland grant 120972 A Characterization of Haj lasz-Sobolev and Triebel-Lizorkin Spaces
Consequently, ˙ M , p ( R n ) = ˙ F p, ( R n ), for n/ ( n + 1) < p < ∞ , where ˙ F p, q ( R n ) refersto a homogeneous Triebel-Lizorkin space (see Theorem 5.2.3/1 in [34] and [30]). In thefractional order, s ∈ (0 , , case it was shown in [39] that ˙ M s, p ( R n ) = ˙ F sp, ∞ ( R n ), provided1 < p < ∞ . Notice the jump in the index q when s crosses 1 and that the result in thefractional order case does not allow for values of p below 1.We will next introduce a class of grand Triebel-Lizorkin spaces that allow us to char-acterize conveniently the fractional Haj lasz-Sobolev spaces for n/ ( n + s ) < p < ∞ . Thedefinition is based on grand Littlewood-Paley functions and we later extend it to the metricspace setting, establishing an analogous characterization.Let Z + ≡ N ∪ { } . Let S ( R n ) be the Schwartz space, namely, the space of rapidlydecreasing functions endowed with a family of seminorms {k · k S k, m ( R n ) } k, m ∈ Z + , where forany k ∈ Z + and m ∈ (0 , ∞ ), we set k ϕ k S k, m ( R n ) ≡ sup α ∈ Z n + , | α |≤ k sup x ∈ R n (1 + | x | ) m | ∂ α ϕ ( x ) | . Here we recall that for any α ≡ ( α , · · · , α n ) ∈ Z n + , | α | = α + · · · + α n and ∂ α ≡ ( ∂∂x ) α · · · ( ∂∂x n ) α n . It is known that S ( R n ) forms a locally convex topology vector space.Denote by S ′ ( R n ) the dual space of S ( R n ) endowed with the weak ∗ -topology. Moreover,for each N ∈ Z + , denote by S N ( R n ) the space of all functions f ∈ S ( R n ) satisfying that R R n x α f ( x ) dx = 0 for all α ∈ Z n + with | α | ≤ N . For the convenience, we also write S − ( R n ) ≡ S ( R n ). For any ϕ ∈ S ( R n ), t > x ∈ R n , set ϕ t ( x ) ≡ t − n ϕ ( t − x ).For each N ∈ Z + ∪ {− } , m ∈ (0 , ∞ ) and ℓ ∈ Z + , our class of test functions is(1.2) A ℓN, m ≡ { φ ∈ S N ( R n ) : k φ k S N + ℓ +1 , m ( R n ) ≤ } . Definition 1.2.
Let s ∈ R , p ∈ (0 , ∞ ) and q ∈ (0 , ∞ ] . Let A be a class of test functionsas in (1.2) . The homogeneous grand Triebel-Lizorkin space A ˙ F sp, q ( R n ) is defined as thecollection of all f ∈ S ′ ( R n ) such that when q ∈ (0 , ∞ ) , k f k A ˙ F sp, q ( R n ) ≡ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X k ∈ Z ksq sup φ ∈A | φ − k ∗ f | q ! /q (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( R n ) < ∞ , and when q = ∞ , k f k A ˙ F sp, ∞ ( R n ) ≡ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) sup k ∈ Z ks sup φ ∈A | φ − k ∗ f | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( R n ) < ∞ . For
A ≡ A ℓN, m , we also write A ˙ F sp, q ( R n ) as A ℓN, m ˙ F sp, q ( R n ). Moreover, if N ∈ Z + and k f k A ˙ F sp, q ( R n ) = 0, then it is easy to see that f ∈ P N , where P N is the space of polynomialswith degree no more than N . So the quotient space A ˙ F sp, q ( R n ) / P N is a quasi-Banachspace. As usual, an element [ f ] = f + P N ∈ A ˙ F sp, q ( R n ) / P N with f ∈ A ˙ F sp, q ( R n ), is simplyreferred to by f. By abuse of the notation, we always write the space A ˙ F sp, q ( R n ) / P N as A ˙ F sp, q ( R n ).The grand Triebel-Lizorkin spaces are closely connected with Haj lasz-Sobolev spacesand (consequently) with the classical Triebel-Lizorkin spaces. oskela, Yang and Zhou Theorem 1.1.
Let s ∈ (0 , and p ∈ ( n/ ( n + s ) , ∞ ) . If A = A ℓ , m with ℓ ∈ Z + and m ∈ ( n + 1 , ∞ ) , then ˙ M s, p ( R n ) = A ˙ F sp, ∞ ( R n ) with equivalent norms. To prove Theorem 1.1, for any f ∈ L p ( R n ), we introduce a special g ∈ D ( f ) via avariant of the grand maximal function; see (2.1) below. When s = 1, comparing this withthe proof of Theorem 1 of [30], we see that the gradient on f appearing there is transferredto the vanishing moments of the test functions and the size conditions of the test functionsand their first-order derivatives (see A ) here. We point out that the choice of the set A isvery subtle. This is the key point which allows us to extend Theorem 1.1 to certain metricmeasure spaces. Moreover, to prove Theorem 1.1, an imbedding theorem established byHaj lasz [19] is also employed.Theorem 1.1 also has a higher-order version. Definition 1.3.
Let p ∈ (0 , ∞ ) and s ∈ ( k, k + 1] with k ∈ N . The homogeneous Haj lasz-Sobolev space ˙ M s, p ( R n ) is defined to be the set of all measurable functions f ∈ L p loc ( R n ) such that for all α ∈ Z n + with | α | = k , ∂ α f ∈ ˙ M s − k, p ( R n ) , and normed by k f k ˙ M s, p ( R n ) ≡ P | α | = k k ∂ α f k ˙ M s − k, p ( R n ) . Corollary 1.1.
Let N ∈ Z + , s ∈ ( N, N + 1] and p ∈ ( n/ ( n + N − s ) , ∞ ) . If A = A ℓN, m with ℓ ∈ Z + and m ∈ ( n + N + 2 , ∞ ) when s = N + 1 or m ∈ ( n + N + 1 , ∞ ) when s ∈ ( N, N + 1) , then ˙ M s, p ( R n ) = A ˙ F sp, ∞ ( R n ) with equivalent norms. The essential point in the proof of Corollary 1.1 is to establish a lifting propertyfor A ˙ F sp, ∞ ( R n ) via Theorem 1.1. This is done with the aid of auxiliary lemmas (seeLemmas 2.4 and 2.5 below), where in Lemma 2.4, we decompose a test function in S N ( R n )into a sum of test functions in S k ( R n ) with subtle controls on their semi-norms for all − ≤ k ≤ N −
1. The decomposition of a test function in S ( R n ) into functions in S ( R n )already plays a key role in [30]. The proof of Corollary 1.1 also uses Theorem 1.2 below.Now we recall the definition of Triebel-Lizorkin spaces on R n . Definition 1.4.
Let s ∈ R , p ∈ (0 , ∞ ) and q ∈ (0 , ∞ ] . Let ϕ ∈ S ( R n ) satisfy that (1.3) supp b ϕ ⊂ { ξ ∈ R n : 1 / ≤ | ξ | ≤ } and | b ϕ ( ξ ) | ≥ constant > if / ≤ | ξ | ≤ / . The homogeneous Triebel-Lizorkin space ˙ F sp, q ( R n ) is defined as the collection of all f ∈S ′ ( R n ) such that k f k ˙ F sp, q ( R n ) ≡ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X k ∈ Z ksq | ϕ − k ∗ f | q ! /q (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( R n ) < ∞ with the usual modification made when q = ∞ . Notice that if k f k ˙ F sp, q ( R n ) = 0, then it is easy to see that f ∈ P ≡ ∪ N ∈ N P N . Sosimilarly to above, we write an element [ f ] = f + P in the quotient space ˙ F sp, q ( R n ) / P with f ∈ ˙ F sp, q ( R n ) as f , and also the space ˙ F sp, q ( R n ) / P as ˙ F sp, q ( R n ). A Characterization of Haj lasz-Sobolev and Triebel-Lizorkin Spaces
Theorem 1.2.
Let s ∈ R , p ∈ (0 , ∞ ) , q ∈ (0 , ∞ ] and J ≡ n/ min { , p, q } . If A = A ℓN, m with ℓ ∈ Z + , (1.4) N + 1 > max { s, J − n − s } and m > max { J, n + N + 1 } , then A ˙ F sp, q ( R n ) = ˙ F sp, q ( R n ) with equivalent norms. To prove Theorem 1.2, we use the Calder´on reproducing formula in [31, 12] and theboundedness of almost diagonal operators on the sequence spaces corresponding to theTriebel-Lizorkin spaces. The almost diagonal operators were introduced by Frazier andJawerth [13] and proved to be a very powerful tool therein (see also [8]). It is perhapsworthwhile to point out that the proof of Theorem 1.1 does not rely on Theorem 1.2.Recall that ˙ M , p ( R n ) = ˙ F p, ( R n ) when p ∈ ( n/ ( n + 1) , ∞ ) by [30], A ˙ F p, ∞ ( R n ) = L p ( R n ) when p ∈ (1 , ∞ ) and A ˙ F p, ∞ ( R n ) = H p ( R n ) when p ∈ ( n/ ( n + 1) , A = A ℓ − , m with ℓ ≥ m ∈ ( n + 1 , ∞ ), and H p ( R n ) is the classical real Hardy space(see [33, 16]). Combining these facts with Theorem 1.1, we have the following result. Corollary 1.2. (i) If p ∈ ( n/ ( n + 1) , ∞ ) and A = A ℓ , m with ℓ ∈ Z + and m ∈ ( n + 1 , ∞ ) ,then A ˙ F p, ∞ ( R n ) = ˙ M , p ( R n ) = ˙ F p, ( R n ) with equivalent norms.(ii) If s ∈ (0 , , p ∈ ( n/ ( n + s ) , ∞ ) and A = A ℓ , m with ℓ ∈ Z + and m ∈ ( n + 1 , ∞ ) ,then A ˙ F sp, ∞ ( R n ) = ˙ M s, p ( R n ) = ˙ F sp, ∞ ( R n ) with equivalent norms.(iii) Let A ≡ A ℓ − , m with ℓ ≥ and m ∈ ( n + 1 , ∞ ) . If p ∈ ( n/ ( n + 1) , ,then A ˙ F p, ∞ ( R n ) = H p ( R n ) = ˙ F p, ( R n ) with equivalent norms; if p ∈ (1 , ∞ ) , then A ˙ F p, ∞ ( R n ) = L p ( R n ) = ˙ F p, ( R n ) with equivalent norms. Moreover, for N ∈ Z + , s ∈ ( N, N + 1] and p ∈ ( n/ ( n + N − s ) , ∞ ), by Corollary 1.1,Theorem 1.2 and the lifting property of homogeneous Triebel-Lizorkin spaces, we have that˙ M s, p ( R n ) = ˙ F sp, ∞ ( R n ) with equivalent norms when s ∈ ( N, N + 1), and ˙ M N +1 , p ( R n ) =˙ F N +1 p, ( R n ) with equivalent norms. Remark 1.1. (i) In a sense, Corollary 1.2 (i) gives a grand maximal characterizationsof Hardy-Sobolev spaces ˙ H , p ( R n ) = ˙ F p, ( R n ) with p ∈ ( n/ ( n + 1) , , where ˙ H , p ( R n ) is defined as the space of all f ∈ S ′ ( R n ) such that ∇ f ∈ H p ( R n ) . We point out theadvantage of this grand maximal characterization is that it only depends on the first-orderderivatives of test functions, which can be replaced by Lipschitz regularity (see Definition1.5). In fact, our approach transfers the derivatives on f to vanishing moments, sizeconditions and Lipschitz regularity of test functions. This is a key observation, whichallows us to extend this characterization to certain metric measure spaces without anydifferential structure; see Theorems 1.3 and 1.4 below.(ii) We point out that Auchser, Russ and Tchamitchian [3] characterized the Hardy-Sobolev space ˙ F p, ( R n ) via a maximal function which is obtained by transferring the gra-dient on f to a size condition on the divergence of the vectors formed by certain test func-tions; see Theorem 6 of [3]. However, this characterization still depends on the derivatives.(iii) We also point out that Cho [11] characterized Hardy-Sobolev spaces ˙ H k, p ( R n ) =˙ F kp, ( R n ) with k ∈ N via a nontangential maximal function by transferring the derivativeson the distribution to a fixed specially chosen Schwartz function; see Theorem I of [11]. oskela, Yang and Zhou (iv) We finally remark that a continuous version of the grand Littlewood-Paley function ( P k ∈ Z sup φ ∈A | φ − k ∗ f | ) / with a different choice of A was used by Wilson [38] to solvea conjecture of R. Fefferman and E. M. Stein on the weighted boundedness of the classicalLittlewood-Paley S -function. Finally, we discuss the metric space setting. Let ( X , d, µ ) be a metric measure space.For any x ∈ X and r >
0, let B ( x, r ) ≡ { y ∈ X : d ( x, y ) < r } . Recall that ( X , d, µ ) iscalled an RD-space if there exist constants 0 < C ≤ ≤ C and 0 < κ ≤ n such that forall x ∈ X , 0 < r < X ) and 1 ≤ λ < X ) /r ,(1.5) C λ κ µ ( B ( x, r )) ≤ µ ( B ( x, λr )) ≤ C λ n µ ( B ( x, r )) , where and in what follows, diam X ≡ sup x, y ∈X d ( x, y ); see [21].We point out that (1.5) implies the doubling property, there exists a constant C ∈ [1 , ∞ ) such that for all x ∈ X and r > µ ( B ( x, r )) ≤ C µ ( B ( x, r )), and the reversedoubling property: there exists a constant a ∈ (1 , ∞ ) such that for all x ∈ X and0 < r < diam X /a , µ ( B ( x, ar )) ≥ µ ( B ( x, r )). For more equivalent characterizationsof RD-spaces and the fact that each connected doubling space is an RD-space, see [40].In what follows, we always assume that ( X , d, µ ) is an RD-space. We also assumethat µ ( X ) = ∞ in this section and in Section 4. In the remaining part of this section, let˚ G (1 , G ( x, − k , , G (1 , ′ and (˚ G ǫ ( β, γ )) ′ be as in Section 4. Definition 1.5.
Let s ∈ (0 , , p ∈ (0 , ∞ ) and q ∈ (0 , ∞ ] . Let A := {A k ( x ) } x ∈X , k ∈ Z and A k ( x ) = { φ ∈ ˚ G (1 , , k φ k ˚ G ( x, − k , , ≤ } for all x ∈ X . The homogeneous grandTriebel-Lizorkin space A ˙ F sp, q ( X ) is defined to be the set of all f ∈ ( G (1 , ′ that satisfy k f k A ˙ F sp, q ( X ) ≡ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)( ∞ X k = −∞ ksq sup φ ∈A k ( · ) |h f, φ i| q ) /q (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( X ) < ∞ with the usual modification made when q = ∞ . Here we also point out that k f k A ˙ F sp, q ( X ) = 0 implies that f = constant. Similarlyto the above, we write the element [ f ] = f + C in the quotient space A ˙ F sp, q ( X ) / C with f ∈ A ˙ F sp, q ( X ) as f , and also the space A ˙ F sp, q ( X ) / C as A ˙ F sp, q ( X ).We have the following analog of Theorem 1.1. Theorem 1.3.
Let s ∈ (0 , and p ∈ ( n/ ( n + s ) , ∞ ) . Then ˙ M s,p ( X ) = A ˙ F sp, ∞ ( X ) withequivalent norms. The proof of Theorem 1.3 uses essentially the same ideas as those used in the proof ofTheorem 1.1. For further characterizations of ˙ M , ( X ) when X is a doubling Riemannianmanifold see [6].We recall the definition of homogeneous Triebel-Lizorkin spaces ˙ F sp, q ( X ) in [21]. A Characterization of Haj lasz-Sobolev and Triebel-Lizorkin Spaces
Definition 1.6.
Let ǫ ∈ (0 , , s ∈ (0 , ǫ ) , p ∈ ( n/ ( n + ǫ ) , ∞ ) and q ∈ ( n/ ( n + ǫ ) , ∞ ] . Let β, γ ∈ (0 , ǫ ) satisfy (1.6) β ∈ ( s, ǫ ) and γ ∈ (max { s − κ/p, n (1 /p − + } , ǫ ) . Let { S k } k ∈ Z be an approximation of the identity of order ǫ with bounded support as inDefinition 4.2. For k ∈ Z , set D k ≡ S k − S k − . The homogeneous grand Triebel-Lizorkinspace ˙ F sp, q ( X ) is defined to be the set of all f ∈ (˚ G ǫ ( β, γ )) ′ that satisfy k f k ˙ F sp, q ( X ) ≡ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)( ∞ X k = −∞ ksq | D k ( f ) | q ) /q (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( X ) < ∞ with the usual modification made when q = ∞ . As shown in [40], the definition of ˙ F sp, q ( X ) is independent of the choices of ǫ , β , γ andthe approximation of the identity as in Definition 4.2. Theorem 1.4.
Let all the assumptions be as in Definition 1.6. Then ˙ F sp, q ( X ) = A ˙ F sp, q ( X ) with equivalent norms. To prove Theorem 1.4, we employ the discrete Calder´on reproducing formula estab-lished in [21], which was already proved to be very useful therein.This paper is organized as follows. In Section 2, we present the proofs of Theorems 1.1and 1.2 and Corollary 1.1. In Section 3, we generalize these results to the inhomogeneouscase; see Theorems 3.1 and 3.2 and Corollary 3.2 below. In Section 4, we present theproofs of Theorems 1.3 and 1.4. Finally, in Section 5, we generalize Theorems 1.3 and 1.4to the inhomogeneous case; see Theorems 5.1 and 5.2 below.We point out that Theorems 1.3, 1.4, 5.1 and 5.2 apply in a wide range of settings,for instance, to Ahlfors n -regular metric measure spaces (see [22]), d -spaces (see [35]), Liegroups of the polynomial volume growth (see [36, 37, 29, 2]), the complete connected non-compact manifolds with a doubling measure (see [4, 5]), compact Carnot-Carath´eodory(also called sub-Riemannian) manifolds (see [29, 26, 27]) and to boundaries of certainunbounded model domains of polynomial type in C N appearing in the work of Nagel andStein (see [28, 29, 26, 27]).Finally, we state some conventions. Throughout the paper, we denote by C a positiveconstant which is independent of the main parameters, but which may vary from line toline. Constants with subscripts, such as C , do not change in different occurrences. Thesymbol A . B or B & A means that A ≤ CB . If A . B and B . A , we then write A ∼ B .For any a, b ∈ R , we denote min { a, b } , max { a, b } , and max { a, } by a ∧ b , a ∨ b and a + ,respectively. If E is a subset of a metric space ( X , d ), we denote by χ E the characteristicfunction of E . For any locally integrable function f , we denote by – R E f dµ (or m E ( f ))the average of f on E , namely, – R E f dµ ≡ µ ( E ) R E f dµ . oskela, Yang and Zhou To prove Theorem 1.1, we need a Sobolev embedding theorem, which for s = 1 is dueto Haj lasz [19, Theorem 8.7], and for s ∈ (0 ,
1) can be proved by a slight modification ofthe proof of [19, Theorem 8.7]. We omit the details.
Lemma 2.1.
Let s ∈ (0 , , p ∈ (0 , n/s ) and p ∗ = np/ ( n − sp ) . Then there exists apositive constant C such that for all u ∈ ˙ M s, p ( B ( x, r )) and g ∈ D ( u ) , inf c ∈ R – Z B ( x,r ) | u ( y ) − c | p ∗ dy ! /p ∗ ≤ Cr s – Z B ( x, r ) [ g ( y )] p dy ! /p . The following result follows from Lemma 2.1. We omit the details.
Lemma 2.2.
Let s ∈ (0 , , p ∈ [ n/ ( n + s ) , n/s ) and p ∗ = np/ ( n − sp ) . Then for each u ∈ ˙ M s, p ( R n ) , there exists a constant C such that u − C ∈ L p ∗ ( R n ) and k u − C k L p ∗ ( R n ) ≤ e C k u k ˙ M s, p ( R n ) , where e C is a positive constant independent of u .Proof of Theorem 1.1. Let A = A ℓ , m with ℓ ∈ Z + and m ∈ ( n + 1 , ∞ ). We first provethat if f ∈ A ˙ F sp, ∞ ( R n ), then f ∈ ˙ M s, p ( R n ) and k f k ˙ M s, p ( R n ) . k f k A ˙ F sp, ∞ ( R n ) .To see this, we first assume that f is a locally integrable function. Fix ϕ ∈ S ( R n ) withcompact support and R R n ϕ ( x ) dx = 1. Notice that ϕ − k ∗ f ( x ) → f ( x ) as k → ∞ for almostall x ∈ R n . Then for almost all x, y ∈ R n , taking k ∈ Z such that 2 − k − < | x − y | ≤ − k ,we have | f ( x ) − f ( y ) | ≤ | ϕ − k ∗ f ( x ) − ϕ − k ∗ f ( y ) | + X k ≥ k ( | ϕ − k − ∗ f ( x ) − ϕ − k ∗ f ( x ) | + | ϕ − k − ∗ f ( y ) − ϕ − k ∗ f ( y ) | ) . Write ϕ − k ∗ f ( x ) − ϕ − k ∗ f ( y ) = ( φ ( x, y ) ) − k ∗ f ( x ) with φ ( x, y ) ( z ) ≡ ϕ ( z − k [ x − y ]) − ϕ ( z )and ϕ − k − ∗ f ( x ) − ϕ − k ∗ f ( x ) = ( ϕ − − ϕ ) − k ∗ f ( x ). Notice that ϕ − − ϕ and φ ( x, y ) are fixed constant multiples of elements of A ℓ , m . For all x ∈ R n , set(2.1) g ( x ) ≡ sup k ∈ Z ks sup φ ∈A ℓ , m | φ − k ∗ f ( x ) | . Since f ∈ A ˙ F sp, ∞ ( R n ) and s ∈ (0 , g ∈ L p ( R n ) and | f ( x ) − f ( y ) | . | ϕ − k ∗ f ( x ) − ϕ − k ∗ f ( y ) | + X k ≥ k sup φ ∈A ℓ , m ( | φ − k ∗ f ( x ) | + | φ − k ∗ f ( y ) | ) . X k ≥ k − ks [ g ( x ) + g ( y )] . | x − y | s [ g ( x ) + g ( y )] . Thus, f ∈ ˙ M s, p ( R n ) and k f k ˙ M s, p ( R n ) . k f k A ˙ F sp, ∞ ( R n ) . A Characterization of Haj lasz-Sobolev and Triebel-Lizorkin Spaces
Generally, if f ∈ A ˙ F sp, ∞ ( R n ) is only known to be an element in S ′ ( R n ) at first, thenshow that we may identify f with a locally integrable function e f in S ′ ( R n ). Indeed, let ϕ be as above. Notice that for all x ∈ R n , k ∈ Z and i ∈ N ,(2.2) | ϕ − k ∗ f ( x ) − ϕ − ( k + i ) ∗ f ( x ) | ≤ i − X j =0 | ϕ − k − j ∗ f ( x ) − ϕ − k − j − ∗ f ( x ) | . − ks g ( x ) . If p ∈ (1 , ∞ ), then { ϕ − k ∗ f − ϕ − ( k + i ) ∗ f } i ∈ N is a Cauchy sequence in L p ( R n ), whichtogether with the completeness of L p ( R n ) implies that there exists an f k ∈ L p ( R n ) suchthat ϕ − k ∗ f − ϕ − ( k + i ) ∗ f → f k in L p ( R n ) and thus almost everywhere as i → ∞ . Observethat for any k, k ′ ∈ Z , we have f k = lim i →∞ [ ϕ − k ∗ f − ϕ − k − i ∗ f ]= [ ϕ − k ∗ f − ϕ − k ′ ∗ f ] + lim i →∞ [ ϕ − k ′ ∗ f − ϕ − k − i ∗ f ]= [ ϕ − k ∗ f − ϕ − k ′ ∗ f ] + f k ′ in L p ( R n ) and almost everywhere. Set e f ≡ ϕ ∗ f − f . Then e f ∈ L ( R n ) and e f = ϕ − k ∗ f − f k almost everywhere. Since { ϕ − k ∗ f } k ∈ Z is a sequence of continuous functionsthat converges to f in S ′ ( R n ) as k → ∞ (see, for example, Lemma 3.8 of [7]), then for any ψ ∈ S ( R n ), we have Z R n e f ( x ) ψ ( x ) dx = Z R n ϕ ∗ f ( x ) ψ ( x ) dx − lim i →∞ Z R n [ ϕ ∗ f ( x ) − ϕ − i ∗ f ( x )] ψ ( x ) dx = lim i →∞ Z R n ϕ − i ∗ f ( x ) ψ ( x ) dx = h f, ψ i , which implies that f coincides with e f in S ′ ( R n ). Now we identify f with the locallyintegrable function e f in S ′ ( R n ). Therefore, by the above proof, e f ∈ ˙ M s, p ( R n ) and k e f k ˙ M s, p ( R n ) . k e f k A ˙ F sp, ∞ ( R n ) ∼ k f k A ˙ F sp, ∞ ( R n ) . In this sense, we have that f ∈ ˙ M s, p ( R n )and k f k ˙ M s, p ( R n ) . k f k A ˙ F sp, ∞ ( R n ) . Now assume that p ∈ ( n/ ( n + s ) , x, y ∈ R n , let k ∈ Z such that 2 − k − < | x − y | ≤ − k . If k > k , then | ϕ − k ∗ f ( x ) − ϕ − k ∗ f ( y ) | . | ϕ − k ∗ f ( x ) − ϕ − k ∗ f ( x ) | + | ϕ − k ∗ f ( y ) − ϕ − k ∗ f ( y ) | + | ϕ − k ∗ f ( x ) − ϕ − k ∗ f ( y ) | . | x − y | s [ g ( x ) + g ( y )] , where g is as in (2.1). If k ≤ k , then 2 k | x − y | ≤ | ϕ − k ∗ f ( x ) − ϕ − k ∗ f ( y ) | = 2 k | x − y || ( φ x, y ) − k ∗ f ( x ) | . | x − y | s [ g ( x ) + g ( y )] , where g is as in (2.1) and for all z ∈ R n , φ x, y ( z ) = 2 − k | x − y | − [ ϕ ( z ) − ϕ ( z − k ( x − y ))] . oskela, Yang and Zhou ϕ − k ∗ f ∈ ˙ M s, p ( R n ) and k ϕ − k ∗ f k ˙ M s, p ( R n ) . k f k A ℓ , m ˙ F sp, ∞ ( R n ) uniformly in k ∈ Z .By Lemma 2.2, for each k ∈ Z , there exists a constant C k such that ϕ − k ∗ f − C k ∈ L p ∗ ( R n ) with uniform bounded norms. By the weak compactness property of L p ∗ ( R n ),there exists a subsequence which we still denote by the full sequence such that ϕ − k ∗ f − C k converges weakly in L p ∗ ( R n ) and thus almost everywhere to a certain function e f ∈ L p ∗ ( R n ). Moreover, for all k, k ′ ∈ Z , since ϕ − k ∗ f − ϕ − k ′ ∗ f ∈ L p ( R n ) (see (2.2))and ϕ − k ∗ f − ϕ − k ′ ∗ f + ( C ′ k − C k ) ∈ L p ∗ ( R n ), we know that C ′ k = C k . This, togetherwith the fact that ϕ − k ∗ f → f ∈ S ′ ( R n ) as k → ∞ , implies that f coincides with e f + C in S ′ ( R n ) and hence with e f in S ′ ( R n ) / C . Now, we identify f with the locally integrablefunction e f . As in the case p ∈ (1 , ∞ ), in this case, we also have that f ∈ ˙ M s, p ( R n ) and k f k ˙ M s, p ( R n ) . k f k A ˙ F sp, ∞ ( R n ) . Now we show that if f ∈ ˙ M s, p ( R n ), then f ∈ A ˙ F sp, ∞ ( R n ) and k f k A ˙ F sp, ∞ ( R n ) . k f k ˙ M s, p ( R n ) .Let φ ∈ A ℓ , m ( R n ) and g ∈ D ( f ). Then for all k ∈ Z and i ≥
0, by Lemma 2.1, wehave that f ∈ L ( R n ) and – Z B ( x, − k + i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f ( y ) − – Z B ( x, − k ) f ( z ) dz (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dy . i X j =0 – Z B ( x, − k + j ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f ( y ) − – Z B ( x, − k + j ) f ( z ) dz (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dy . i X j =0 − ks + js – Z B ( x, − k + j ) [ g ( y )] n/ ( n + s ) dy ! ( n + s ) /n . − ks is h M (cid:16) g n/ ( n + s ) (cid:17) ( x ) i ( n + s ) /n . From this, m > n + 1 ≥ n + s and R R n φ ( x ) dx = 0, it follows that for all k ∈ Z and x ∈ X , | φ − k ∗ f ( x ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z X φ − k ( x − y ) " f ( y ) − – Z B ( x, − k ) f ( z ) dz dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (2.3) ≤ ∞ X i =0 − ( m − n ) i – Z B ( x, − k + i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f ( y ) − – Z B ( x, − k ) f ( z ) dz (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dy . − ks h M (cid:16) g n/ ( n + s ) (cid:17) ( x ) i ( n + s ) /n , which together with the L p ( n + s ) /n -boundedness of M implies that if p ∈ ( n/ ( n + s ) , ∞ ),then (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) sup k ∈ Z sup φ ∈A ℓ , m ( R n ) ks | φ − k ∗ ( f ) | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( X ) . (cid:13)(cid:13)(cid:13)(cid:13)h M (cid:16) g n/ ( n + s ) (cid:17)i ( n + s ) /n (cid:13)(cid:13)(cid:13)(cid:13) L p ( X ) . k g k L p ( X ) . A Characterization of Haj lasz-Sobolev and Triebel-Lizorkin Spaces
Moreover, without loss of generality, we may assume that M ( g n/ ( n + s ) )(0) < ∞ . Then forany ψ ∈ S ( R n ), by an argument similar to that of (2.3), we have that (cid:12)(cid:12)(cid:12)(cid:12)Z X f ( x ) ψ ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ k ψ k L ( X ) – Z B (0 , | f ( z ) | dz + Z R n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f ( x ) − – Z B (0 , f ( z ) dz (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | ψ ( x ) | dx . k ψ k L ( X ) – Z B (0 , | f ( z ) | dz + k ψ k S , m ( R n ) [ M ( g n/ ( n + s ) )(0)] ( n + s ) /n . C ( f ) k ψ k S , m ( R n ) , which implies that f ∈ S ′ ( R n ). Thus, f ∈ A ˙ F sp, ∞ ( R n ) and k f k A ˙ F sp, ∞ ( R n ) . k f k ˙ M s, p ( R n ) , which completes the proof of Theorem 1.1.To prove Theorem 1.2, we need the following estimate. Lemma 2.3.
Let N ∈ Z + ∪ {− } and m ∈ ( n + N + 1 , ∞ ) . Then there exists a positiveconstant C such that for all x ∈ R n and i, j ∈ Z with i ≥ j , φ ∈ S N ( R n ) and ψ ∈ S ( R n ) , | φ − i ∗ ψ − j ( x ) | ≤ C k φ k S N +1 , m ( R n ) k ψ k S , m ( R n ) − ( i − j )( N +1) jn (1 + 2 j | x | ) − m . Proof.
Without loss of generality, we may assume that k φ k S N +1 , m ( R n ) = k ψ k S N +1 , m ( R n ) =1. For simplicty, we only consider the case N ≥
0. If j = 0, then by φ ∈ S N ( R n ) and theTaylor formula, we have | φ − i ∗ ψ ( x ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z R n φ − i ( y ) ψ ( x − y ) − X | α |≤ N α ! y α ∂ α ψ ( x ) dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Z | y |≤ (1+ | x | ) / | φ − i ( y ) | X | α | = N +1 | y | N +1 | ∂ α ψ ( x − θy ) | dy + Z | y | > (1+ | x | ) / | φ − i ( y ) ψ ( x − y ) | dy + Z | y | > (1+ | x | ) / | φ − i ( y ) | X | α |≤ N | y | | α | | ∂ α ψ ( x ) | dy ≡ I + I + I , where θ ∈ [0 , | y | ≤ (1 + | x | ) /
2, then for any θ ∈ [0 , | x | ≤ | x − θy | + | y | and hence, 1 + | x | ≤ | x − θy | ). By this and m ∈ ( n + N + 1 , ∞ ),we obtain I . Z | y |≤ (1+ | x | ) / in | y | N +1 (1 + 2 i | y | ) m | x − θy | ) m dy . − i ( N +1) | x | ) m Z R n in | i y | N +1 (1 + 2 i | y | ) m dy . − i ( N +1) (1 + | x | ) − m . For I and I , we also have I . Z | y | > (1+ | x | ) / in (1 + 2 i | y | ) m | x − y | ) m dy . − i ( N +1) (1 + | x | ) − m oskela, Yang and Zhou I . Z | y | > (1+ | x | ) / in (1 + 2 i | y | ) m | y | N | x | ) m dy . − i ( N +1) (1 + | x | ) − m . For j = 0, we obtain | φ − i ∗ ψ ( x ) | = 2 jn | φ − ( i − j ) ∗ ψ (2 j x ) | . −| i − j | ( N +1) jn (1 + 2 j | x | ) − m , which completes the proof of Lemma 2.3.Now we turn to the proof of Theorem 1.2. Proof of Theorem 1.2.
Let A = A ℓN, m with ℓ ∈ Z + , N and m satisfying (1.4). Obviously, A ˙ F sp, q ( R n ) is continuously imbedded into ˙ F sp, q ( R n ). We now prove that if f ∈ ˙ F sp, q ( R n ),then f ∈ A ˙ F sp, q ( R n ) and k f k A ˙ F sp, q ( R n ) . k f k ˙ F sp, q ( R n ) . This proof is similar to the proofthat the definition of ˙ F sp, q ( R n ) is independent of the choice of ϕ satisfying (1.3), but a bitmore complicated. In fact, we need to use the boundedness of almost diagonal operatorsin sequences spaces. For reader’s convenience, we sketch the argument.Recall that there exists a function ψ ∈ S ( R n ) satisfying the same conditions as ϕ suchthat P k ∈ Z b ϕ (2 − k ξ ) b ψ (2 − k ξ ) = 1 for all ξ ∈ R n \ { } ; see [14, Lemma (6.9)]. Then theCalder´on reproducing formula says that for all f ∈ S ′ ( R n ), there exist polynomials P f and { P i } i ∈ N depending on f such that for all x ∈ R n ,(2.4) f ( x ) + P f ( x ) = lim i →−∞ ∞ X j = i ϕ − j ∗ ψ − j ∗ f ( x ) + P i ( x ) , where the series converges in S ′ ( R n ); see, for example, [31, 12]. When f ∈ ˙ F sp, q ( R n ), itis known that the degrees of the polynomials { P i } i ∈ N here are no more than ⌊ s − n/p ⌋ ;see [13, pp. 153-155], and also [31, p. 53] and [34, pp. 17-18]. Recall that ⌊ α ⌋ for α ∈ R denotes the maximal integer no more than α . Moreover, as shown in [13, pp. 153-155], f + P f is the canonical representative of f in the sense that if ϕ ( i ) , ψ ( i ) satisfy (1.3) and P k ∈ Z d ϕ ( i ) (2 − k ξ ) d ψ ( i ) (2 − k ξ ) = 1 for all ξ ∈ R n \ { } for i = 1 ,
2, then P (1) f − P (2) f is apolynomial of degree no more than ⌊ s − n/p ⌋ , where P ( i ) f is as in (2.4) corresponding to ϕ ( i ) , ψ ( i ) for i = 1 ,
2. So in this sense, we identify f with e f ≡ f + P f .Let e ϕ ( x ) = ϕ ( − x ) for all x ∈ R n . Denote by Q the collection of the dyadic cubeson R n . For any dyadic cube Q ≡ − j k + 2 − j [0 , n ∈ Q with certain k ∈ Z n , we set x Q ≡ − j k , denote by ℓ ( Q ) ≡ − j the side length of Q and write ϕ Q ( x ) ≡ jn/ ϕ (2 j x − k ) =2 − jn/ ϕ − j ( x − x Q ) for all x ∈ R n . It is known that for all x ∈ R n ,(2.5) ϕ − j ∗ ψ − j ∗ f ( x ) = X ℓ ( Q )=2 − j h f, e ϕ Q i ψ Q ( x )2 A Characterization of Haj lasz-Sobolev and Triebel-Lizorkin Spaces in S ′ ( R n ) and pointwise; see [12, 14] and also [8, Lemma 2.8]. Notice also that N + 1 > s implies that N ≥ ⌊ s − n/p ⌋ . Then for all f ∈ ˙ F sp, q ( R n ), φ ∈ S N ( R n ) with N ≥ ⌊ s − n/p ⌋ , i ∈ Z and x ∈ R n , by (2.4) and (2.5), we have e f ∗ φ − i ( x ) = X Q ∈Q h f, e ϕ Q i ψ Q ∗ φ − i ( x ) = X Q ∈Q t Q ψ Q ∗ φ − i ( x ) , where t Q = h f, e ϕ Q i , and by [13, Theorem 2.2] or [14, Theorem (6.16)],(2.6) k f k ˙ F sp, q ( R n ) ∼ k{ t Q } Q ∈Q k ˙ f sp, q ( R n ) ≡ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X Q ∈Q [ | Q | − s/n − / | t Q | χ Q ] q /q (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( R n ) . Moreover, by Lemma 2.3, for all R ∈ Q with ℓ ( R ) = 2 − i and x ∈ R , we have | e f ∗ φ − i ( x ) | . X j ∈ Z −| i − j | ( N +1) ( i ∧ j ) n − jn/ X ℓ ( Q )=2 − j | t Q | (1 + 2 i ∧ j | x − x Q | ) m . X j ∈ Z X ℓ ( Q )=2 − j −| i − j | ( n/ N +1) | t Q | (1 + 2 i ∧ j | x R − x Q | ) m | R | − / . For
R, Q ∈ Q with ℓ ( R ) = 2 − i and ℓ ( Q ) = 2 − j , setting a RQ = 2 −| i − j | ( n/ N +1) (1 + 2 i ∧ j | x R − x Q | ) − m , by (1.4), we have a RQ ≤ (cid:20) ℓ ( R ) ℓ ( Q ) (cid:21) s (cid:20) | x R − x Q | max { ℓ ( R ) , ℓ ( Q ) } (cid:21) − J − ǫ min ((cid:20) ℓ ( R ) ℓ ( Q ) (cid:21) n + ǫ , (cid:20) ℓ ( Q ) ℓ ( R ) (cid:21) J + ǫ − n ) for certain ǫ >
0. Thus { a RQ } R, Q ∈Q forms an almost diagonal operator on ˙ f sp, q ( R n ), whichis known to be bounded on ˙ f sp, q ( R n ); see [13, Theorem 3.3] and also [14, Theorem (6.20)].Therefore, by (2.6), we have k e f k A ˙ F sp, q ( R n ) . (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X R ∈Q | R | − s/n − / X Q ∈Q a RQ t Q χ R q /q (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( R n ) . k{ t Q } Q ∈Q k ˙ f sp, q ( R n ) ∼ k f k ˙ F sp, q ( R n ) , which completes the proof of Theorem 1.2.To prove Corollary 1.1, we need to establish a lifting property of A ˙ F sp, q ( R n ), whichheavily depends on the following result. oskela, Yang and Zhou Lemma 2.4.
Let N ∈ Z + , ϕ ∈ S N ( R n ) and ≤ k ≤ N + 1 . Then there exist functions { ϕ α } α ∈ Z n + , | α | = k ⊂ S N − k ( R n ) such that ϕ = P | α | = k ∂ α ϕ α ; moreover, for any ℓ ∈ Z + , thereexists a positive constant C , depending on N, k, ℓ and m , but not on ϕ and ϕ α , such that (2.7) X | α | = k k ϕ α k S N + ℓ +1 , m − kn ( R n ) ≤ C k ϕ k S N + ℓ +1 , m ( R n ) . Proof.
We begin by proving Lemma 2.4 for k = 1. We point out that when N = 0, thisproof is essentially given by [30, Lemma 6] and [1, Lemma 3.29] except for checking theestimate (2.7). Now assume N ≥
0. We decompose ϕ by using the idea appearing in theproof of [30, Lemma 6] and then verify (2.7).Let ϕ ∈ S N ( R n ). We apply induction on n . For n = 1, set ψ ( x ) ≡ R x −∞ ϕ ( y ) dy for all x ∈ R . Then ϕ ( x ) = ddx ψ ( x ) for x ∈ R . Moreover, for any 0 ≤ j ≤ N −
1, by integrationby parts, we have R R ψ ( x ) x j dx = − j +1 R R ϕ ( x ) x j +1 dx = 0, which means ψ ∈ S N − ( R ).Moreover, for all x ∈ R , since ϕ ∈ S N ( R ), | ψ ( x ) | ≤ k ϕ k S N + ℓ +1 , m ( R ) Z ∞| x | | y | ) m dy . k ϕ k S N + ℓ +1 , m ( R ) (1 + | x | ) − ( m − , and for all 1 ≤ j ≤ N + ℓ + 1, (cid:12)(cid:12)(cid:12)(cid:12) d j dx j ψ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) d j − dx j − ϕ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ k ϕ k S N + ℓ +1 , m ( R ) (1 + | x | ) − ( m − . Thus Lemma 2.4 holds for n = 1.Suppose that Lemma 2.4 holds true for a fixed n ≥
1. Let ϕ ∈ S N ( R n +1 ). Withoutloss of generality, we may assume that k ϕ k S N + ℓ +1 , m ( R n +1 ) = 1. For any x ∈ R n +1 , we write x = ( x ′ , x n +1 ) and define h ( x ′ ) ≡ R R ϕ ( x ′ , u ) du, where x ′ = ( x , · · · , x n ) ∈ R n . Then h ∈ S N ( R n ). Moreover, for all x ′ ∈ R n and α ′ ∈ Z n + with | α ′ | ≤ N + ℓ + 1, we have | ∂ α ′ h ( x ′ ) | ≤ Z R | | x ′ | + | u | ) m du . | | x ′ | ) m − , which implies that k h k S N + ℓ +1 ,m − ( R n ) .
1. By induction hypothesis, we write h ( x ′ ) = P ni =1 ∂∂x i h i ( x ′ ) with h i ∈ S N − ( R n ) and k h i k S N + ℓ +1 , m − n − ( R n ) . i = 1 , · · · , n . Let a ∈ S ( R ) be fixed and R R a ( u ) du = 1. For all x ∈ R n +1 , set ϕ n +1 ( x ) ≡ R x n +1 −∞ [ ϕ ( x ′ , u ) − a ( u ) h ( x ′ )] du and ϕ i ( x ) ≡ a ( x n +1 ) h i ( x ′ ) with i = 1 , · · · , n . Then ϕ i ∈ S N − ( R n +1 ) for i = 1 , · · · , n . For any ℓ ≤ N − | α | ≤ ℓ with α = ( α ′ , α n +1 ) ∈ Z n +1+ , by integrationby parts again, we have Z R n +1 ϕ n +1 ( x ) x α dx = Z R n Z R Z x n +1 −∞ ϕ ( x ′ , u )( x ′ ) α ′ x α n +1 n +1 du dx n +1 dx ′ = − α n +1 + 1 Z R n +1 ϕ ( x )( x ′ ) α ′ x α n +1 +1 n +1 dx = 0 . For any α ∈ Z n +1+ , for i = 1 , · · · , n , we have | ∂ α ϕ i ( x ) | . k h i k S N + ℓ +1 , m − n − ( R n ) (1 + | x n +1 | ) m − n − (1 + | x ′ | ) m − n − . (1 + | x | ) − ( m − n − , A Characterization of Haj lasz-Sobolev and Triebel-Lizorkin Spaces which implies that k ϕ i k S N + ℓ +1 , m − n − ( R n +1 ) . . For any α ∈ Z n +1+ with | α | ≤ N + ℓ + 1, if α n +1 = 0, then by k h k S N + ℓ +1 , m − ( R n ) .
1, we have that | ∂ α ϕ n +1 ( x ) | . (1 + | x | ) − ( m − n − for all x ∈ R n ; if α n +1 = 0, then by R R [ ψ ( x ′ , u ) − a ( u ) h ( x ′ )] du = 0, we have that for all x ∈ R n , | ∂ α ϕ n +1 ( x ) | . Z ∞| x n +1 | | x ′ | + u ) m du + 1(1 + | x ′ | ) m − n Z ∞| x n +1 | | u | ) m du . k ϕ k S N + ℓ +1 , m ( R n +1 ) (1 + | x ′ | ) − ( m − n − . Thus, k ϕ n +1 k S N + ℓ +1 , m − n − ( R n +1 ) .
1, which completes the proof of Lemma 2.4.
Lemma 2.5.
For any N , ℓ ∈ Z + and m, m ′ ∈ ( n + N + 2 , ∞ ) , A N, m ′ ˙ F N +1 p, ∞ ( R n ) = A ℓN, m ˙ F N +1 p, ∞ ( R n ) with equivalent norms.Proof. It suffices to prove that if f ∈ A ℓN, m ˙ F N +1 p, ∞ ( R n ), then f ∈ A N, m ′ ˙ F N +1 p, ∞ ( R n ) and k f k A N, m ′ ˙ F N +1 p, ∞ ( R n ) . k f k A ℓN, m ˙ F N +1 p, ∞ ( R n ) . Without loss of generality, we may assume that m ≥ m ′ . To this end, fix ψ ∈ S ( R n ) such that R R n ψ ( x ) dx = 1. Obviously, for any α ∈ Z n + with | α | = N + 1, if β ∈ Z n + , | β | ≤ N + 1 and β = α , then R R n ∂ α ψ ( x ) x β dx = 0; if α = β ,then R R n ∂ α ψ ( x ) x α dx = ( − N +1 . For any φ ∈ A N, m ′ , let(2.8) φ = φ − ( − N +1 X | α | = N +1 (cid:18)Z R n φ ( x ) x α dx (cid:19) ∂ α ψ. Then φ ∈ S N +1 ( R n ). Moreover, for | α | = N +1, since φ ∈ A N, m ′ with m ′ ∈ ( n + N +2 , ∞ ),we have Z R n | φ ( x ) x α | dx ≤ Z R n | x | N +1 (1 + | x | ) m ′ dx . , which implies that φ is a fixed constant multiple of an element of A N +1 , m ′ . Notice that { ∂ α ψ } | α | = N +1 are also fixed constant multiples of elements of A ℓN, m . Then, by (2.8), wehave sup φ ∈A N, m ′ | φ − k ∗ f ( x ) | . sup φ ∈A ℓN, m | φ − k ∗ f ( x ) | + sup φ ∈A N +1 , m ′ | φ − k ∗ f ( x ) | , which implies that k f k A N, m ′ ˙ F N +1 p, ∞ ( R n ) . k f k A ℓN, m ˙ F N +1 p, ∞ ( R n ) + k f k A N +1 , m ′ ˙ F N +1 p, ∞ ( R n ) . By The-orem 1.2 together with m ′ ∈ ( n + N + 2 , ∞ ), we have that k f k A N +1 , m ′ ˙ F N +1 p, ∞ ( R n ) ∼k f k ˙ F N +1 p, ∞ ( R n ) . k f k A ℓN, m ˙ F N +1 p, ∞ ( R n ) , which yields that k f k A N, m ′ ˙ F N +1 p, ∞ ( R n ) . k f k A ℓN, m ˙ F N +1 p, ∞ ( R n ) .This finishes the proof of Lemma 2.5. Proof of Corollary 1.1.
First, let f ∈ ˙ M s, p ( R n ). Then by Theorem 1.1, for any ℓ ∈ Z + and m ∈ (( n + 2) N + 1 , ∞ ), and all α ∈ Z n + with | α | = N , ∂ α f ∈ ˙ M s − N, p ( R n ) = A ℓ , m − nN ˙ F s − Np, ∞ ( R n ). Moreover, for any φ ∈ A ℓN, m , by Lemma 2.4, there exist { φ α } | α | = N oskela, Yang and Zhou C independent of φ such that { C φ α } | α | = N ⊂ A ℓ , m − nN and φ = P | α | = N ∂ α φ α . This implies that for all x ∈ R n , φ − k ∗ f ( x ) = X | α | = N ( ∂ α φ α ) − k ∗ f ( x ) = 2 kN ( − N X | α | = N ( φ α ) − k ∗ ( ∂ α f )( x ) , and thus, sup φ ∈A ℓN, m | φ − k ∗ f ( x ) | . kN sup | α | = N sup φ ∈A ℓ , m − nN | φ − k ∗ ( ∂ α f )( x ) | . From this and ∂ α f ∈ A ℓ , m − nN ˙ F s − Np, ∞ ( R n ) for all α ∈ Z n + with | α | = N together withTheorem 1.1, it follows that f ∈ A ℓN, m ˙ F sp, ∞ ( R n ) and k f k A ℓN, m ˙ F sp, ∞ ( R n ) . X | α | = N k ∂ α f k A ℓ , m − nN ˙ F s − Np, ∞ ( R n ) ∼ X | α | = N k ∂ α f k ˙ M s − N, p ( R n ) ∼ k f k ˙ M s, p ( R n ) . On the other hand, let f ∈ A ℓN, m ˙ F sp, ∞ ( R n ). Let ℓ ≥ N and m ∈ ( n + N + 1 , ∞ ).Observe that for any φ ∈ A ℓ , m and α ∈ Z n + with | α | = N , ∂ α φ ∈ A ℓ − NN, m . Thus for all k ∈ Z , sup φ ∈A ℓN, m | φ − k ∗ ( ∂ α f ) | ≤ sup φ ∈A ℓ − N , m kN | φ − k ∗ ( f ) | , which implies that { ∂ α f } | α | = N ⊂ A ℓ − N , m ˙ F s − Np, ∞ ( R n ) = ˙ M s − N, p ( R n ) and thus, f ∈ M s, p ( R n )and k f k ˙ M s, p ( R n ) ∼ X | α | = N k ∂ α f k ˙ M s − N, p ( R n ) ∼ X | α | = N k ∂ α f k A ℓ − N , m ˙ F s − Np, ∞ ( R n ) . k f k A ℓN, m ˙ F sp, ∞ ( R n ) . Finally, combining the above results with Lemma 2.5 and Theorem 1.2, for all ℓ ∈ Z + and m ∈ ( n + N + 2 , ∞ ) when s = N + 1 or m ∈ ( n + N + 1 , ∞ ) when s ∈ ( N, N + 1),we have that ˙ M s, p ( R n ) = A ℓN, m ˙ F sp, ∞ ( R n ). This finishes the proof of Corollary 1.1. We first recall the definitions of inhomogeneous Triebel-Lizorkin spaces; see [34].
Definition 3.1.
Let s ∈ R , p ∈ (0 , ∞ ) and q ∈ (0 , ∞ ] . Let ϕ ∈ S ( R n ) satisfy (1.3) and Φ ∈ S ( R n ) be such that supp b Φ ⊂ B (0 , and | Φ( ξ ) | ≥ constant > for all | ξ | ≤ / .The inhomogeneous Triebel-Lizorkin space F sp, q ( R n ) is defined as the collection of all f ∈S ′ ( R n ) such that k f k F sp, q ( R n ) ≡ k Φ ∗ f k L p ( R n ) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X k =1 ksq | ϕ − k ∗ f | q ! /q (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( R n ) < ∞ with the usual modification made when q = ∞ . A Characterization of Haj lasz-Sobolev and Triebel-Lizorkin Spaces
Recall that the local Hardy space h p ( R n ) of Goldberg is just F p, ( R n ) (see [34, Theorem2.5.8/1]). A variant of inhomogeneous Haj lasz-Sobolev spaces is defined as follows. Definition 3.2.
Let p ∈ (0 , ∞ ) and s ∈ (0 , M s, p ( R n ) is the set of all measurable functions f ∈ L p loc ( R n ) such that f ∈ ˙ M s, p ( R n ) and f ∈ h p ( R n ). Moreover, define k f k M s, p ( R n ) ≡ k f k h p ( R n ) + k f k ˙ M s, p ( R n ) . Now we introduce the inhomogeneous grand Triebel-Lizorkin spaces.
Definition 3.3.
Let s ∈ R , p ∈ (0 , ∞ ) and q ∈ (0 , ∞ ] . Let A = A ℓN, m with ℓ ∈ Z + , N ∈ Z + ∪{− } and m ∈ (0 , ∞ ) be a class of test functions as in (1.2) . The inhomogeneousgrand Triebel-Lizorkin space A F sp, q ( R n ) is defined as the collection of all f ∈ S ′ ( R n ) suchthat k f k A F sp, q ( R n ) ≡ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) sup φ ∈A ℓ +1 − , m | φ ∗ f | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( R n ) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X k =1 ksq sup φ ∈A | φ − k ∗ f | q ! /q (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( R n ) < ∞ with the usual modification made when q = ∞ . Moreover, similarly to Theorems 1.1 and 1.2 and Corollary 1.1, we have the followingresults.
Theorem 3.1.
Let all the assumptions be the same as in Theorem 1.2. Then A F sp, q ( R n ) = F sp, q ( R n ) with equivalent norms. The proof of Theorem 3.1 is similar to that of Theorem 1.2. In fact, since the inho-mogeneous Calder´on reproducing formula is available (see [13, p. 131]), then by using theargument of Theorem 1.2 and the estimates in Lemma 2.3, we have Theorem 3.1. Herewe omit the details.
Theorem 3.2.
Let s ∈ (0 , , p ∈ ( n/ ( n + s ) , ∞ ) , ℓ ∈ Z + and m ∈ ( n + 1 , ∞ ) . Then M s, p ( R n ) = A ℓ , m F sp, ∞ ( R n ) with equivalent norms.Proof. The proof of Theorem 3.2 is similar to that of Theorem 1.1 and much easier. In fact,if f ∈ M s, p ( R n ), then f ∈ ˙ M s, p ( R n ) by Definition 3.1 and thus f ∈ A ℓ , m ˙ F sp, ∞ ( R n ). Noticethat k sup φ ∈A ℓ − , m | φ ∗ f |k . k f k h p ( R n ) (see [15]). Then we know that f ∈ A ℓ , m F sp, ∞ ( R n )and k f k A ℓ , m F sp, ∞ ( R n ) . k f k M s, p ( R n ) .Conversely, assume that f ∈ A ℓ , m F sp, ∞ ( R n ). Obviously, by h p ( R n ) = F p, ( R n ), weknow that f ∈ h p ( R n ) and k f k h p ( R n ) ∼ k f k F p, ( R n ) . k f k A ℓ , m F p, ( R n ) . Fix ϕ ∈ S ( R n )with compact support and R R n ϕ ( x ) dx = 1. For any k ∈ N , if | x − y | ≤
1, by an argumentsimilar to that for Theorem 1.1, we then know that | ϕ − k ∗ f ( x ) − ϕ − k ∗ f ( y ) | . | x − y | s sup k ≥ ks sup φ ∈A ℓ , m ( | φ − k ∗ f ( x ) | + | φ − k ∗ f ( y ) | ) . If | x − y | >
1, then, obviously, | ϕ − k ∗ f ( x ) − ϕ − k ∗ f ( y ) | . | x − y | s sup φ ∈A ℓ +1 − , m ( | φ − k ∗ f ( x ) | + | φ − k ∗ f ( y ) | ) . oskela, Yang and Zhou ϕ − k ∗ f ∈ ˙ M s, p ( R n ) and k ϕ − k ∗ f k ˙ M s, p ( R n ) . k f k A ℓ , m F sp, q ( R n ) . Then similarly tothe proof of Theorem 1.1, we can prove that f ∈ L p loc ( R n ) and k ϕ − k ∗ f k ˙ M s, p ( R n ) . k f k A ℓ , m F sp, q ( R n ) . Thus, k f k M s, p ( R n ) . k f k h p ( R n ) + k f k A ℓ , m F sp, q ( R n ) . k f k A ℓ , m F sp, q ( R n ) , which completes the proof of Theorem 1.3. Corollary 3.1.
Let s ∈ (0 , , p ∈ ( n/ ( n + s ) , ∞ ) , ℓ ∈ Z + and m ∈ ( n + 1 , ∞ ) .(i) If s = 1 , then A ℓ , m F p, ∞ ( R n ) = M , p ( R n ) = F p, ( R n ) with equivalent norms.(ii) If s ∈ (0 , , then A ℓ , m F sp, ∞ ( R n ) = M s, p ( R n ) = F sp, ∞ ( R n ) with equivalent norms. Define the inhomogeneous Haj lasz-Sobolev spaces M s, p ( R n ) of higher orders as inDefinition 1.3 by replacing ˙ M s − N, p ( R n ) with M s − N, p ( R n ). Then we have the followinginhomogeneous version of Corollary 1.1. We omit the details of its proof. Corollary 3.2.
Let N ∈ Z + , s ∈ ( N, N +1] and p ∈ ( n/ ( n + s − N ) , ∞ ) . If A = A ℓN, m with ℓ ∈ Z + and m ∈ ( n + N +2 , ∞ ) when s = N +1 or m ∈ ( n + N +1 , ∞ ) when s ∈ ( N, N +1) ,then M s, p ( R n ) = A F sp, ∞ ( R n ) with equivalent norms. Moreover, M s, p ( R n ) = F sp, ∞ ( R n ) when s ∈ ( N, N + 1) and M N +1 , p ( R n ) = F N +1 p, ( R n ) with equivalent norms. Remark 3.1.
Notice that when p ∈ (1 , ∞ ) , h p ( R n ) = L p ( R n ) , and when p ∈ (0 , , h p ( R n ) ( L p ( R n ) . Another way to define the inhomogeneous Haj lasz-Sobolev space denotedby f M s, p ( R n ) is to replace k f k h p ( R n ) by k f k L p ( R n ) in the Definition 3.1. Recall that itwas proved in [30] that f M , p ( R n ) = F p, ( R n ) ∩ L p ( R n ) for p ∈ ( n/ ( n + 1) , ∞ ) . Anargument similar to that used in the proof of Theorem 3.2 can show that f M s, p ( R n ) = A ℓ , m F sp, ∞ ( R n ) ∩ L p ( R n ) for p ∈ ( n/ ( n + s ) , ∞ ) and s ∈ (0 , . Similar results for Corollary3.2 also hold true. We omit the details. The following spaces of test functions play a key role in the theory of function spaceson RD-spaces; see [21]. In what follows, for any x, y ∈ X and r >
0, set V ( x, y ) ≡ µ ( B ( x, d ( x, y ))) and V r ( x ) ≡ µ ( B ( x, r )). It is easy to see that V ( x, y ) ∼ V ( y, x ) for all x, y ∈ X . Definition 4.1.
Let x ∈ X , r ∈ (0 , ∞ ) , β ∈ (0 , and γ ∈ (0 , ∞ ) . A function ϕ on X issaid to be in the space G ( x , r, β, γ ) if there exists a nonnegative constant C such that(i) | ϕ ( x ) | ≤ C V r ( x )+ V ( x ,x ) (cid:16) rr + d ( x ,x ) (cid:17) γ for all x ∈ X ;(ii) | ϕ ( x ) − ϕ ( y ) | ≤ C (cid:16) d ( x,y ) r + d ( x ,x ) (cid:17) β V r ( x )+ V ( x ,x ) (cid:16) rr + d ( x ,x ) (cid:17) γ for all x , y ∈ X satisfyingthat d ( x, y ) ≤ ( r + d ( x , x )) / .Moreover, for any ϕ ∈ G ( x , r, β, γ ) , its norm is defined by k ϕ k G ( x , r, β, γ ) ≡ inf { C :( i ) and ( ii ) hold } . A Characterization of Haj lasz-Sobolev and Triebel-Lizorkin Spaces
Throughout the whole paper, we fix x ∈ X and let G ( β, γ ) ≡ G ( x , , β, γ ) . Then G ( β, γ ) is a Banach space. We also let ˚ G ( β, γ ) = (cid:8) f ∈ G ( β, γ ) : R X f ( x ) dµ ( x ) = 0 (cid:9) . De-note by ( G ( β, γ )) ′ and (˚ G ( β, γ )) ′ the dual spaces of G ( β, γ ) and ˚ G ( β, γ ), respectively. Ob-viously, (˚ G ( β, γ )) ′ = ( G ( β, γ )) ′ / C .For any given ǫ ∈ (0 , G ǫ ( β, γ ) be the completion of the set G ( ǫ, ǫ ) in the space G ( β, γ ) when β , γ ∈ (0 , ǫ ]. Obviously, G ǫ ( ǫ, ǫ ) = G ( ǫ, ǫ ). If ϕ ∈ G ǫ ( β, γ ), define k ϕ k G ǫ ( β,γ ) ≡k ϕ k G ( β,γ ) . Obviously, G ǫ ( β, γ ) is a Banach space. The space ˚ G ǫ ( β, γ ) is defined to bethe completion of the space ˚ G ( ǫ, ǫ ) in ˚ G ( β, γ ) when β, γ ∈ (0 , ǫ ]. Let ( G ǫ ( β, γ )) ′ and( G ǫ ( β, γ )) ′ be the dual space of G ǫ ( β, γ ) and G ǫ ( β, γ ), respectively. Also we have that(˚ G ǫ ( β, γ )) ′ = ( G ǫ ( β, γ )) ′ / C . Remark 4.1.
Because (˚ G ǫ ( β, γ )) ′ = ( G ǫ ( β, γ )) ′ / C , if we replace (˚ G ǫ ( β, γ )) ′ with ( G ǫ ( β, γ )) ′ / C or ( G ǫ ( β, γ )) ′ in Definition 1.6, then we obtain a new Triebel-Lizorkin space which, moduloconstants, is equivalent to the original Triebel-Lizorkin space. So we can replace (˚ G ǫ ( β, γ )) ′ with ( G ǫ ( β, γ )) ′ / C or ( G ǫ ( β, γ )) ′ in the Definition 1.6 if need be, in what follows. Now we recall the notion of approximations of the identity on RD-spaces, which werefirst introduced in [21].
Definition 4.2.
Let ǫ ∈ (0 , . A sequence { S k } k ∈ Z of bounded linear integral operatorson L ( X ) is called an approximation of the identity of order ǫ (for short, ǫ - AOTI ) withbounded support, if there exist constants C , C > such that for all k ∈ Z and all x , x ′ , y and y ′ ∈ X , S k ( x, y ) , the integral kernel of S k is a measurable function from X × X into C satisfying(i) S k ( x, y ) = 0 if d ( x, y ) > C − k and | S k ( x, y ) | ≤ C V − k ( x )+ V − k ( y ) ; (ii) | S k ( x, y ) − S k ( x ′ , y ) | ≤ C kǫ [ d ( x, x ′ )] ǫ V − k ( x )+ V − k ( y ) for d ( x, x ′ ) ≤ max { C , } − k ;(iii) Property (ii) holds with x and y interchanged;(iv) | [ S k ( x, y ) − S k ( x, y ′ )] − [ S k ( x ′ , y ) − S k ( x ′ , y ′ )] | ≤ C kǫ [ d ( x,x ′ )] ǫ [ d ( y,y ′ )] ǫ V − k ( x )+ V − k ( y ) for d ( x, x ′ ) ≤ max { C , } − k and d ( y, y ′ ) ≤ max { C , } − k ;(v) R X S k ( x, y ) dµ ( y ) = 1 = R X S k ( x, y ) dµ ( x ) . It was proved in [21, Theorem 2.6] that there always exists a 1- AOTI with boundedsupport on an RD-space.To prove Theorem 1.3, we need a Sobolev embedding theorem, which for s = 1 is dueto Haj lasz [19, Theorem 8.7], and for s ∈ (0 ,
1) can be proved by a slight modification ofthe proof of [19, Theorem 8.7]. We omit the details.
Lemma 4.1.
Let s ∈ (0 , , p ∈ (0 , n/s ) and p ∗ = np/ ( n − sp ) . Then there exists apositive constant C such that for all u ∈ ˙ M s,p ( X ) , g ∈ D ( u ) and all balls B with radius r , u ∈ L p ∗ ( B ) and inf c ∈ R (cid:18) – Z B | u − c | p ∗ dµ (cid:19) /p ∗ ≤ Cr s (cid:18) – Z B g p dµ (cid:19) /p . By Lemma 4.1, we have the following version of Lemma 2.2. oskela, Yang and Zhou Lemma 4.2.
Let s ∈ (0 , , p ∈ [ n/ ( n + s ) , n/s ) and p ∗ = np/ ( n − sp ) . Then for each u ∈ ˙ M s, p ( X ) , there exists constant C such that u − C ∈ L p ∗ ( X ) and k u − C k L p ∗ ( X ) ≤ e C k u k ˙ M s, p ( X ) , where e C is a positive constant independent of u and C . With the aid of Lemmas 4.1 and 4.2, we can prove Theorem 1.3 by following the ideasused in the proof of Theorem 1.1. For reader’s convenience, we sketch the argument.
Proof of Theorem 1.3.
We first prove that if f ∈ A ˙ F sp, ∞ ( X ), then f ∈ ˙ M s, p ( X ) and k f k ˙ M s, p ( X ) . k f k A ˙ F sp, ∞ ( X ) . Let { S k } k ∈ Z be a 1- AOTI with bounded support. If f is alocally integrable function, using S k ( f ) to replace ϕ − k ∗ f and following the procedure asin the proof of Theorem 1.1, we know that f ∈ ˙ M s, p ( X ) and g ( · ) = sup k ∈ Z sup φ ∈A k ( · ) ks |h f, φ i| ∈ L p ( X )and k f k ˙ M s, p ( X ) . k f k A ˙ F sp, ∞ ( X ) . If f ∈ A ˙ F sp, ∞ ( X ) is only known to be an element in( G (1 , ′ , we may also identify f with a locally integrable function e f in ( G (1 , ′ and k e f k ˙ M s, p ( X ) . k f k A ˙ F sp, ∞ ( X ) by using Lemma 4.2 and an argument used in that of Theorem1.1. In this sense, we have that f ∈ ˙ M s, p ( X ) and k f k ˙ M s, p ( X ) . k f k A ˙ M s, p ( X ) .Conversely, let f ∈ ˙ M s, p ( X ). Choose g ∈ D ( f ) such that k g k L p ( X ) ≤ k f k ˙ M s, p ( X ) . Then for all x ∈ X , k ∈ Z and φ ∈ A k ( x ), similarly to the proof of (2.3) and using Lemma4.1, we have that |h f, φ i| . − ks h M (cid:16) g n/ ( n + s ) (cid:17) ( x ) i ( n + s ) /n , which together with the L p ( n + s ) /n ( X )-boundedness of M implies that k f k A ˙ F sp, q ( X ) . k g k L p ( X ) for all p ∈ ( n/ ( n + s ) , ∞ ). Moreover, without loss of generality, we may as-sume that M (cid:0) g n/ ( n + s ) (cid:1) ( x ) < ∞ . Then for all ψ ∈ G (1 , σ := R X ψ ( y ) dµ ( y ),by Lemma 4.2 and an argument similar to the proof of (2.3), we have that f ∈ L ( X )and (cid:12)(cid:12)(cid:12)(cid:12)Z X f ( x ) ψ ( x ) dµ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z X f ( x )[ ψ ( x ) − σS ( x , x )] dµ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) + | σ || S ( f )( x ) | . k ψ k L ( X ) – Z B (0 , C ) | f ( z ) | dµ ( z ) + k ψ k G (1 , [ M ( g n/ ( n + s ) )] s/n ( x ) . C ( f ) k ψ k G (1 , , which implies that f ∈ ( G (1 , ′ . Thus f ∈ A ˙ F sp, ∞ ( X ) and k f k A ˙ F sp, ∞ ( X ) . k f k ˙ M s, p ( X ) , which completes the proof of Theorem 1.3. Remark 4.2.
By the above proof, if we replace the space G (1 , of test functions by G ( β, γ ) with β ∈ [ s, and γ ∈ ( s, ∞ ) in Definition 1.5, then Theorem 1.3 still holds true.Thus the definition of the space A ˙ F sp, ∞ ( X ) is independent of the choice of the space of testfunctions G ( β, γ ) with β ∈ [ s, and γ ∈ ( s, ∞ ) . A Characterization of Haj lasz-Sobolev and Triebel-Lizorkin Spaces
To prove Theorem 1.4, we need the following homogeneous Calder´on reproducing for-mula established in [21]. We first recall the following construction given by Christ in [10],which provides an analogue of the set of Euclidean dyadic cubes on spaces of homogeneoustype.
Lemma 4.3.
Let X be a space of homogeneous type. Then there exists a collection { Q kα ⊂X : k ∈ Z , α ∈ I k } of open subsets, where I k is some index set, and constants δ ∈ (0 , and C , C > such that(i) µ ( X \ ∪ α Q kα ) = 0 for each fixed k and Q kα ∩ Q kβ = ∅ if α = β ;(ii) for any α, β, k, ℓ with ℓ ≥ k, either Q ℓβ ⊂ Q kα or Q ℓβ ∩ Q kα = ∅ ;(iii) for each ( k, α ) and each ℓ < k , there exists a unique β such that Q kα ⊂ Q ℓβ ;(iv) diam ( Q kα ) ≤ C δ k ;(v) each Q kα contains some ball B ( z kα , C δ k ) , where z kα ∈ X . In fact, we can think of Q kα as being a dyadic cube with diameter roughly δ k andcentered at z kα . In what follows, to simplify our presentation, we always suppose that δ = 1 /
2; otherwise, we need to replace 2 − k in the definition of approximations to theidentity by δ k and some other changes are also necessary; see [21] for more details.In the following, for k ∈ Z and τ ∈ I k , we denote by Q k,ντ , ν = 1 , , · · · , N ( k, τ ) , theset of all cubes Q k + jτ ′ ⊂ Q kτ , where Q kτ is the dyadic cube as in Lemma 4.3 and j is a fixedpositive large integer such that 2 − j C < / . Denote by z k,ντ the “center” of Q k,ντ as inLemma 4.3 and by y k,ντ a point in Q k,ντ . Lemma 4.4.
Let ǫ ∈ (0 , and { S k } k ∈ Z be a 1- AOTI with bounded support. For k ∈ Z ,set D k := S k − S k − . Then, for any fixed j ∈ N large enough, there exists a family { e D k } k ∈ Z of linear operators such that for any fixed y k,ντ ∈ Q k,ντ with k ∈ Z , τ ∈ I k and ν = 1 , · · · , N ( k, τ ) , x ∈ X , and all f ∈ (˚ G ǫ ( β, γ )) ′ with β, γ ∈ (0 , ǫ ) , f ( x ) = ∞ X k = −∞ X τ ∈ I k N ( k,τ ) X ν =1 µ ( Q k,ντ ) e D k ( x, y k,ντ ) D k ( f )( y k,ντ ) , where the series converge in (˚ G ǫ ( β, γ )) ′ . Moreover, for any ǫ ′ ∈ ( ǫ, , there exists a positiveconstant C , depending on ǫ ′ , such that the kernels, denoted by e D k ( x, y ) , of the operators e D k satisfy(i) for all x, y ∈ X , | e D k ( x, y ) | ≤ C V − k ( x )+ V ( x,y ) h − k − k + d ( x,y ) i ǫ ′ , (ii) for all x, x ′ , y ∈ X with d ( x, x ′ ) ≤ (2 − k + d ( x, y )) / , | e D k ( x, y ) − e D k ( x ′ , y ) | ≤ C (cid:20) d ( x, x ′ )2 − k + d ( x, y ) (cid:21) ǫ ′ V − k ( x ) + V ( x, y ) (cid:20) − k − k + d ( x, y ) (cid:21) ǫ ′ , (iii) for all k ∈ Z , R X e D k ( x, y ) dµ ( y ) = 0 = R X e D k ( x, y ) dµ ( x ) . oskela, Yang and Zhou Proof of Theorem 1.4. If f ∈ A ˙ F sp, q ( X ), by Remark 4.2 and the fact that A ˙ F sp, q ( X ) ⊂A ˙ F sp, ∞ ( X ), we know that f ∈ ( G ( β, γ )) ′ with β ∈ ( s,
1) and γ ∈ ( s, ∞ ) and thus, f ∈ ˙ F sp, q ( X ) and k f k ˙ F sp, q ( X ) . k f k A ˙ F sp, q ( X ) . Conversely, assume that f ∈ ˙ F sp, q ( X ). ByLemma 4.4, for all x ∈ X , ℓ ∈ Z and φ ∈ A ℓ ( x ), we have h f, φ i = ∞ X k = −∞ X τ ∈ I k N ( k,τ ) X ν =1 µ ( Q k,ντ ) D k ( f )( y k,ντ ) Z X e D k ( z, y k,ντ ) φ ( z ) dµ ( z ) , where we fix y k,ντ ∈ Q k, ντ such that(4.1) | D k ( f )( y k,ντ ) | ≤ z ∈ Q k, ντ | D k ( f )( z ) | . Recall that e D k depends on the choice of y k,ντ and thus on f , but they do have uniformestimates as in Lemma 4.4, which is enough for us. In fact, by these estimates and φ ∈ ˚ G ǫ ( β, γ ), we further know that for any fixed β ′ ∈ ( s, β ) and γ ′ ∈ ( s, γ ) satisfying(1.6), (cid:12)(cid:12)(cid:12)(cid:12)Z X e D k ( z, y k,ντ ) φ ( z ) dµ ( z ) (cid:12)(cid:12)(cid:12)(cid:12) . −| k − ℓ | β ′ V − ( k ∧ ℓ ) ( x ) + V ( x, y k,ντ ) " − ( k ∧ ℓ ) − ( k ∧ ℓ ) + d ( x, y k,ντ ) γ ′ ;see [21] for a detailed proof. Thus, choosing an r ∈ ( n/ ( n + [ β ′ ∧ γ ′ ]) , min { p, q } ), by (4.1),we have |h f, φ i| . ∞ X k = −∞ −| k − ℓ | β ′ X τ ∈ I k N ( k,τ ) X ν =1 µ ( Q k,ντ ) | D k ( f )( y k,ντ ) | V − ( k ∧ ℓ ) ( x ) + V ( x, y k,ντ ) " − ( k ∧ ℓ ) − ( k ∧ ℓ ) + d ( x, y k,ντ ) γ ′ . ∞ X k = −∞ −| k − ℓ | β ′ [( k ∧ ℓ ) − k ] n (1 − /r ) M X τ ∈ I k N ( k,τ ) X ν =1 | D k ( f )( y k,ντ ) | r χ Q k, ντ ( x ) /r . ∞ X k = −∞ −| k − ℓ | β ′ [( k ∧ ℓ ) − k ] n (1 − /r ) [ M ( | D k ( f ) | r ) ( x )] /r . This implies that k f k A ˙ F sp, q ( X ) . (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)( ∞ X ℓ = −∞ ( ℓ − k ) sq ∞ X k = −∞ −| k − ℓ | β ′ [( k ∧ ℓ ) − k ] n (1 − /r ) × h M (cid:16) ksr | D k ( f ) | r (cid:17)i /r (cid:19) q (cid:27) /q (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( X ) . Applying the H¨older inequality when q > P k | a k | ) q ≤ P k | a k | q when q ∈ (0 ,
1] for all { a k } k ∈ Z ⊂ C , and using the vector-valued inequality of the Hardy-Littlewood maximal operator (see [17]), we then have k f k A ˙ F sp, q ( X ) . (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)( ∞ X k = −∞ h M (cid:16) ksr | D k ( f ) | r (cid:17)i q/r ) /q (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( X ) . k f k ˙ F sp, q ( X ) . A Characterization of Haj lasz-Sobolev and Triebel-Lizorkin Spaces
This finishes the proof of Theorem 1.4.
We consider both cases µ ( X ) < ∞ and µ ( X ) = ∞ at the same time. We next recallthe notions of inhomogeneous Besov and Triebel-Lizorkin spaces from [21].We call { S k } k ∈ N to be an inhomogeneous approximation of the identity of order ǫ withbounded support if their kernels satisfy (i) through (v) of Definition 4.1. Definition 5.1.
Let ǫ , s, p, q, β, γ be as in Definition 1.6. Let { S k } k ∈ N be an inhomo-geneous approximation of the identity of order ǫ with bounded support. For k ∈ N , set D k ≡ S k − S k − . Let { Q ,ντ : τ ∈ I , ν = 1 , · · · , N (0 , τ ) } with a fixed large j ∈ N bedyadic cubes as in Section 4. Let s ∈ (0 , ǫ ) . The inhomogeneous Triebel-Lizorkin space F sp, q ( X ) is defined to be the set of all f ∈ ( G ǫ ( β, γ )) ′ that satisfy k f k F sp, q ( X ) ≡ X τ ∈ I N (0 ,τ ) X ν =1 µ ( Q ,ντ ) h m Q ,ντ ( | S ( f ) | ) i p /p + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)( ∞ X k =1 ksq | D k ( f ) | q ) /q (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( X ) < ∞ with the usual modification made when q = ∞ . As shown in [40], the definition of F sp, q ( X ) is independent of the choices of ǫ , β , γ andthe inhomogeneous approximation of the identity. Definition 5.2.
Let s ∈ (0 , , p ∈ (0 , ∞ ) and q ∈ (0 , ∞ ] . Let A : ≡ {A k ( x ) } k ∈ Z + , x ∈X with A ( x ) = { φ ∈ G (1 , , k φ k G ( x, , , ≤ } and for k ∈ N , A k ( x ) := { φ ∈ G (1 , , k φ k ˚ G ( x, − k , , ≤ } . The inhomogeneous grand Triebel-Lizorkin space A ˙ F sp, q ( X ) is defined to be the set of all f ∈ ( G (1 , ′ that satisfy k f k A F sp, q ( X ) ≡ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)( ∞ X k =0 ksq sup φ ∈A k ( · ) |h f, φ i| q ) /q (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( X ) < ∞ with the usual modification made when q = ∞ . Then we have the following result.
Theorem 5.1.
Let all the assumptions be as in Definition 5.1. Then F sp, q ( X ) = A F sp, q ( X ) with equivalent norms. oskela, Yang and Zhou Definition 5.3.
Let p ∈ (0 , ∞ ) and s ∈ (0 , . The inhomogeneous fractional Haj lasz-Sobolev space M s,p ( X ) is defined to be the set of all measurable functions f ∈ L p loc ( X ) that satisfy both f ∈ h p ( X ) = F p, ( X ) and f ∈ ˙ M s,p ( X ) ; moreover, define k f k M s,p ( X ) ≡k f k h p ( X ) + inf g ∈D ( f ) k f k ˙ M s,p ( X ) , Theorem 5.2.
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E-mail: [email protected]
Dachun Yang : School of Mathematical Sciences, Beijing Normal University, Laboratoryof Mathematics and Complex Systems, Ministry of Education, Beijing 100875, People’sRepublic of China
E-mail: [email protected]
Yuan Zhou : School of Mathematical Sciences, Beijing Normal University, Laboratoryof Mathematics and Complex Systems, Ministry of Education, Beijing 100875, People’sRepublic of ChinaandUniversity of Jyv¨askyl¨a, Department of Mathematics and Statistics, P.O. Box 35 (MaD),Fin-40014 University of Jyv¨askyl¨a, Finland