Pointwise and Grand Maximal Function Characterizations of Besov-Type and Triebel–Lizorkin-Type Spaces

Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 41, 2016, 103–117

POINTWISE AND GRAND MAXIMAL FUNCTION CHARACTERIZATIONS OF BESOV-TYPE AND TRIEBEL–LIZORKIN-TYPE SPACES

Tomás Soto

University of Helsinki, Department of Mathematics and Statistics P.O. Box 68, FI-00014 University of Helsinki, Finland; tomas.soto@helsinki.ﬁ

Abstract. In this note, we establish characterizations for the homogeneous Besov-type spaces ˙ s,τ Rn ˙ s,τ Rn Bp,q ( ) and Triebel–Lizorkin-type spaces Fp,q ( ), introduced by Yang and Yuan, through fractional Hajłasz-type gradients for suitable values of the parameters p, q and τ when 0

1. Introduction The main purpose of this note is to establish pointwise characterizations of the Besov-type and Triebel–Lizorkin-type function spaces, introduced by Yang and Yuan in [14] and [16], through Hajłasz-type gradients. Characterizations of this type go back to Hajłasz’s pointwise characterization of the classical Sobolev spaces [6], and they have found many applications in both the Euclidean setting as well as in the setting of more general metric measure spaces. The families of function spaces con- sidered in this paper include the standard Besov and Triebel–Lizorkin spaces as well as the fractional Morrey–Sobolev spaces for smoothness indices s ∈ (0, 1) as special cases. To begin with, we first recall the definitions of the standard (homogeneous) Triebel–Lizorkin and Besov spaces. For a dimension n ∈ N := {1, 2, 3, ···}, which shall be fixed throughout the paper, we denote by S(Rn) the class of Schwartz functions, i.e. the class of complex-valued C∞(Rn) functions φ for which

m γ kφkSk,m := sup (1 + |x|) |∂ φ(x)| |γ|≤k, x∈Rn

N N γ γ1 γn is ﬁnite for all k, m ∈ 0 := ∪{0}; here |γ| = |γ1| + ··· + |γn| and ∂ = ∂x1 ··· ∂xn Nn for all multi-indices γ =(γ1, ··· ,γn) ∈ 0 . The seminorms k·kSk,m induce a locally convex topology on S(Rn). We denote by S′(Rn) the class of tempered distributions, i.e. the class of continuous complex-valued linear functionals on S(Rn), and equip S′(Rn) with the topology induced by the mappings f 7→ hf,φi, φ ∈ S(Rn). For standard facts about the Schwartz space and tempered distributions, particularly their Fourier-transforms and convolutions, we refer to e.g. [5].

doi:10.5186/aasfm.2016.4101 2010 Mathematics Subject Classiﬁcation: Primary 42B35; Secondary 46E35. Key words: Besov-type space, function space, grand maximal function, Hajłasz gradient, Triebel–Lizorkin-type space. The author was supported by the Finnish CoE in Analysis and Dynamics Research. 104 Tomás Soto

Following [3], ϕ and ψ shall throughout the paper be ﬁxed elements of S(Rn) satisfying supp ϕ, supp ψ ⊂ ξ ∈ Rn : 2−1 ≤|ξ| ≤ 2 , |ϕ(ξ)|, |ψ(ξ)| ≥ c> 0 when 3/5 ≤|ξ| ≤ 5/3 b b and b b ϕ(2jξ)ψ(2jξ)=1 when ξ =06 . Z Xj∈ n b jn j For φ ∈ S(R ) and j ∈ Z, web write φj for the Schwartz function x 7→ 2 φ(2 x). For s ∈ R, 0

1/p q p/q js kfk ˙ s Rn = 2 |ϕ ∗ f(z)| dz Fp,q( ) j ˆ n R Z ! Xj∈ is finite, with the obvious modification made when q = ∞. For s ∈ R, 0 < p ≤ ∞ ˙ s Rn and 0 < q ≤ ∞, the homogeneous Besov space Bp,q( ) is defined as the class of tempered distributions f for which

1/q q js kfk ˙ s Rn = 2 kϕ ∗ fk p Rn Bp,q( ) j L ( ) j∈Z ! X is ﬁnite, with again the obvious modiﬁcation made when q = ∞. It is well known that after quotienting out the tempered distributions whose Fourier-transforms are ˙ s Rn ˙ s Rn supported at the origin, i.e. the polynomials, Fp,q( ) and Bp,q( ) become quasi- Banach spaces, independent of the choice of ϕ in the sense that two admissible choices induce equivalent quasinorms; see for instance [13]. The following Triebel–Lizorkin-type and Besov-type spaces were introduced by Yang and Yuan in [14] and [16]. For s ∈ R, 0

1/p q p/q 1 js kfk ˙ s,τ n = sup 2 |ϕ ∗ f(z)| dz Fp,q (R ) −ℓ τ j x∈Rn,ℓ∈Z |B(x, 2 )| ˆB(x,2−ℓ) ! Xj≥ℓ is finite, with the obvious modification when q = ∞. For s ∈ R, 0 < p ≤ ∞, ˙ s,τ Rn 0 < q ≤∞ and 0 ≤ τ < ∞, the homogeneous Besov-type space Bp,q ( ) is defined as the class of tempered distributions f for which

1/q q 1 js kfk ˙ s,τ n = sup 2 kϕ ∗ fk p −ℓ Bp,q (R ) −ℓ τ j L (B(x,2 )) x∈Rn,ℓ∈Z |B(x, 2 )| ! Xj≥ℓ is finite, with again the obvious modification when q = ∞. Again, these spaces become quasinormed spaces after quotienting out the polynomials. Actually, in the definitions of [14] and [16], the supremum is taken over dyadic cubes instead of balls with dyadic radii, but it is quite easy to see that the above definitions yield equivalent quasinorms. It is also known that these spaces are independent of the choice of ϕ ([16, Corollary 3.1]) and that they are quasi-Banach spaces ([11, Proposition 2.2] and the Characterizations of Besov-type and Triebel–Lizorkin-type spaces 105 references therein). Here are a couple examples of how these spaces coincide1 with other well-known function spaces: ˙ s,0 Rn ˙ s Rn ˙ s,0 Rn ˙ s Rn • Fp,q ( )= Fp,q( ) and Bp,q ( )= Bp,q( ) for all admissible parameters. ˙ s,1/p Rn ˙ s Rn ˙ 0,1/p Rn • Fp,q ( )= F∞,q( ) for all admissible parameters; in particular Fp,2 ( ) = BMO(Rn). s, 1 − 1 ˙ u p Rn ˙ s Rn R ˙ s Rn • Bu,∞ ( ) = Np,u,∞( ) for 0 < u ≤ p < ∞ and s ∈ , where Np,u,q( ) stands for the homogeneous Besov-Morrey space, i.e. the Besov-type space p Rn p Rn based on the Morrey space Mu ( ) instead of L ( ), introduced in [9] and [10]. s, 1 − 1 ˙ u p Rn ˙s Rn R • Fu,q ( ) = Ep,u,q( ) for 0 < u ≤ p < ∞, 0 < q ≤ ∞ and s ∈ , where ˙s Rn Ep,u,q( ) stands for the homogeneous Triebel–Lizorkin-Morrey space, i.e. the p Rn Triebel–Lizorkin-type space based on the Morrey space Mu ( ) instead of Lp(Rn). α, 1 − α n ˙ 2 n Rn • For 0 <α < min(1, 2 ), the space F2,2 ( ) coincides with the space n Qα(R ) introduced in [2]. 1 • For p <τ < ∞ and all admissible values of p, q and s,

s+n(τ− 1 ) s+n(τ− 1 ) ˙ s,τ Rn ˙ p Rn ˙ s,τ Rn ˙ p Rn (1) Fp,q ( )= F∞,∞ ( ) and Bp,q ( )= B∞,∞ ( ).

˙ s Rn ˙ s Rn Recall that for s > 0, F∞,∞( ) and B∞,∞( ) both coincide with the homogeneous Hölder–Zygmund space of order s; see e.g. [5, Theorem 6.3.6] and [3, Section 5]. ˙ s Rn Rn We refer to e.g. [11] and [17] for the deﬁnitions of the spaces F∞,q( ), BMO( ), ˙s Rn ˙ s Rn Rn Ep,u,q( ), Np,u,q( ) and Qα( ) (which we shall not need in the sequel) and to [17, Proposition 1 and Theorem 2] as well as the references therein for details about the above coincidences. We also refer to [11] for a detailed discussion of the history of the spaces in question. ˙ s,τ Rn ˙ s,τ Rn Inspired by [8], we now deﬁne function spaces analogous to Fp,q ( ) and Bp,q ( ) through Hajłasz-type gradients. If u : Rn → C is a (Lebesgue-)measurable function and 0 set E ⊂ Rn of measure zero such that, for all k ∈ Z,

s |u(x) − u(y)|≤|x − y| (gk(x)+ gk(y)) whenever x, y ∈ Rn\E and 2−k−1 ≤|x − y| < 2−k. ˙ s,τ Rn For 0

p/q 1/p 1 q kuk ˙ s,τ Rn := inf sup gk(z) dz Mp,q ( ) −→ Ds −ℓ τ g ∈ (u) x∈Rn,ℓ∈Z |B(x, 2 )| ˆB(x,2−ℓ) ! Xk≥ℓ

1 Here and in the sequel, X = Y for function spaces X and Y means that they embed continuously into each other. 106 Tomás Soto is ﬁnite, and for 0 < p ≤ ∞, 0 < q ≤ ∞, 0

l Rn γ AN,m = φ ∈ S( ): x φ(x) dx =0 when |γ| ≤ N and kφkSN+l+1,m ≤ 1 , ˆ n R where the moment condition is interpreted to be void when N = −1. For s ∈ R, 0

kfk l ˙ s,τ Rn AN,mFp,q ( ) 1 p p q q 1 js := sup −ℓ τ 2 sup |φj ∗ f(z)| dz x∈Rn,ℓ∈Z |B(x, 2 )| ˆB(x,2−ℓ) φ∈Al ! Xj≥ℓ N,m is finite, with the obvious modification for q = ∞. For s ∈ R, 0 < p ≤ ∞, 0 < q ≤∞, 0 ≤ τ < ∞ and N, m and l as above, we define the grand Besov-type space l ˙ s,τ Rn AN,mBp,q ( ) as the class of tempered distributions f for which 1 q q 1 js kfk l ˙ s,τ Rn := sup 2 sup |φj ∗ f| AN,mBp,q ( ) −ℓ τ p −ℓ x∈Rn,ℓ∈Z |B(x, 2 )| φ∈Al L (B(x,2 )) ! j≥ℓ N,m X is finite, where the supremum is taken pointwise and the obvious modification is made for q = ∞. The two quantities defined above are quasinorms when N = −1, and when N ∈ N0, they become quasinorms after quotienting out the polynomials l ˙ s,0 Rn l ˙ s Rn with degree at most N. We obviously have AN,mFp,q ( ) = AN,mFp,q( ) and Characterizations of Besov-type and Triebel–Lizorkin-type spaces 107

l ˙ s,0 Rn l ˙ s Rn l ˙ s Rn AN,mBp,q( ) = AN,mBp,q( ) for all admissible parameters, where AN,mFp,q( ) l ˙ s Rn and AN,mBp,q( ) are the grand Triebel–Lizorkin and grand Besov spaces deﬁned in [7] and [8]. The following result extends [8, Theorem 3.1]. Theorem 1.2. (i) Let 0 integers N ≥ −1, m ≥ 0 and l ≥ 0 satisfy

(2) N +1 > max(s,J − n − s) and m> max(J, n + N + 1),

˙ s,τ Rn l ˙ s,τ Rn 1 then Fp,q ( )= AN,mFp,q ( ) for all τ ∈ [0, p ]. (ii) Let 0 < p ≤ ∞, 0 < q ≤ ∞, s ∈ R and J = n/ min(1,p). If the integers ˙ s,τ Rn l ˙ s,τ Rn N ≥ −1, m ≥ 0 and l ≥ 0 are related by (2), then Bp,q ( )= AN,mBp,q ( ) 1 for all τ ∈ [0, p ]. 1 ǫ (iii) The results of (i) and (ii) actually hold for all τ ∈ [0, p + 2n ), where

ǫ = min 2(N +1 − s), m − J, 2(N +1+ n + s − J) > 0. The proof of the theorem is presented in section 2. We also have the following result which generalizes the analogous results for τ =0 obtained in [8, Theorem 3.2]. Theorem 1.3. Suppose that s ∈ (0, 1) and m ≥ n +1 or s =1 and m ≥ n +2. n n 1 1−s N (i) For p ∈ ( n+s , ∞), q ∈ ( n+s , ∞], τ ∈ [0, p + n ) and l ∈ 0, we have ˙ s,τ Rn l ˙ s,τ Rn Mp,q ( )= A0,mFp,q ( ) with equivalent quasinorms. n 1 1−s N ˙ s,τ Rn (ii) For p ∈ ( n+s , ∞], q ∈ (0, ∞], τ ∈ [0, p + n ) and l ∈ 0, we have Np,q ( )= l ˙ s,τ Rn A0,mBp,q ( ) with equivalent quasinorms. The proof of this is presented in section 3. Previously, the Besov-type and Triebel–Lizorkin type spaces have been charac- terized through difference integrals (see e.g. [1] where the non-homogeneous versions of the spaces are treated) and through Peetre-type maximal functions as well as local means in [15]. We refer to [11] for a variety of other characterizations. We end this section with some notation conventions. We write |A| for the n-di- − −1 mensional Lebesgue measure of a measurable set A, and A f or fA for |A| A f(x) dx whenever the latter quantity is well-defined. For two non-ne´ gative functions´ f and g with the same domain, we write f . g if f ≤ Cg for some positive and finite constant C, usually independent of some paramters; f ≈ g means that f . g and g . f. For real numbers x and y, we may write x ∧ y for min(x, y). For a dyadic −j n n −j −j cube Q := 2 (k + [0, 1] ), j ∈ Z, k ∈ Z , we let ℓ(Q)=2 and xQ = 2 k. For a ball B of Rn, we denote by λB, λ> 0, the ball cocentric with B with radius λ times the radius of B. n n n We write P(R ) for the vector space of polynomials in R and PN (R ) for the vector space of polynomials with degree at most N when N ∈ N0; when N < 0, Rn ˙ s,τ Rn we let PN ( ) := {0}. We shall frequently abuse notation by writing Fp,q ( ) for s,τ n both the class of tempered distributions f such that kfkF˙p,q (R ) is finite as well as ′ n n R R s,τ n the function space {f ∈ S ( )/P( ): kfkF˙p,q (R ) < ∞}, and similarly for other ˙ s,τ Rn ˙ s,τ Rn spaces defined in this section as well. When talking about Fp,q ( ) and Bp,q ( ) in the same sentence with some indicated parameter range, it is understood that the ˙ s,τ Rn possibility p = ∞ is excluded in the case of Fp,q ( ), and similarly for the other pairs of spaces defined in this section. 108 Tomás Soto 2. Proof of Theorem 1.2 The argument below modifies the proof of [8, Theorem 3.1]; we have left out some details that carry over unchanged. Proof of Theorem 1.2. (i) First, if a tempered distribution f belongs to l s,τ n A F˙ (R ), then kfk ˙ s,τ Rn . kfk l ˙ s,τ Rn < ∞ since our fixed ϕ is a constant N,m p,q Fp,q ( ) AN,mFp,q ( ) l l ˙ s,τ Rn ˙ s,τ Rn multiple of an element of AN,m. In this sense we have AN,mFp,q ( ) ⊂ Fp,q ( ). ˙ s,τ Rn Conversely, if a tempered distribution f belongs to Fp,q ( ), it is shown in the proof of [16, Lemma 4.2] that

γ j(|γ|+n/p−s−nτ) ∞ Rn . s,τ n k∂ (ψj ∗ ϕj ∗ f)kL ( ) 2 kfkF˙p,q (R ), for all integers j andf multi-indices γ, so the discussion in [3, pp. 153–155] applies: if γ ∞ |γ| >s+nτ −n/p, then j≤1 k∂ (ψj ∗ϕj ∗f)kL (Rn) < ∞, so there exists polynomials n (P ) N in P (R ) with L = ⌊s + nτ − n/p⌋ and a polynomial P such that K K∈ L P f f ∞

f + Pf = lim ψj ∗ ϕj ∗ f + PK K→∞ j=−K ! X ′ n f n with convergence in S (R ); here the polynomial Pf is unique modulo PL(R ) in the (i) (i) i i sense that if ψ , ϕ , (PK)K∈N and Pf are, for i ∈{1, 2}, two choices of admissible (1) (2) Rn ǫ functions as above, then Pf −Pf ∈ PL( ). Furthermore, L n + N +1, by the proof of [7, Theorem 1.2] we have

−1/2 (3) |φj ∗ f(z)| . aQR|tR| |Q| χQ(z) −j ℓ(QX)=2 XR e l Rn Z for all φ ∈AN,m, z ∈ and j ∈ , with the implied constant independent of these parameters. Here the outer sum is taken over all dyadic cubes Q of Rn with side −j n length 2 and the inner sum over all dyadic cubes R of R ; tR denotes hf, ψRi with −in/2 −i ψR(x)=2 ψi(x − xR) for all dyadic cubes R with side length 2 ; and f −|i−j|(n/2+N+1) min(i,j) −m (4) aQR =2 1+2 |xQ − xR| for all cubes Q and R with side lengths2−j and 2−i respectively. The cubes Q in (3) are pairwise disjoint, so for any dyadic cube P with side length 2−ℓ we have

1/p q p/q 1 js τ 2 sup |φj ∗ f(z)| dz |P | ˆP φ∈A ! Xj≥ℓ e 1/p 1 q p/q . 2jsq a |t | |Q|−q/2χ (z) dz |P |τ ˆ QR R Q  P −j R Xj≥ℓ ℓ(QX)=2 X   Characterizations of Besov-type and Triebel–Lizorkin-type spaces 109

q p/q 1/p 1 −sq/n−q/2 ≈ τ |Q| aQR|tR| χQ(z) dz |P | ˆP ! QX⊂P XR

. aQR|tR| , ˙s,τ Rn R Q fp,q ( ) X ˙s,τ Rn where fp,q ( ) stands for the space of sequences (bQ)Q of complex numbers, indexed by the dyadic cubes of Rn, such that

1/p q p/q 1 −s/n−1/2 k(b ) k ˙s,τ n = sup |Q| |b |χ (x) dx , Q Q fp,q (R ) τ Q Q P |P | ˆP ! QX⊂P where the supremum is taken over all dyadic cubes P of Rn, is ﬁnite. From (4) it is easy to check that for ǫ> 0 as in the statement of part (iii), we have

n+ǫ ǫ−n ℓ(Q) s |x − x | −J−ǫ ℓ(Q) 2 ℓ(R) J+ 2 a ≤ 1+ Q R min , QR ℓ(R) max(ℓ(Q),ℓ(R)) ℓ(R) ℓ(Q) " # for all dyadic cubes Q and R, i.e. that the operator

(bQ)Q 7→ aQRbR R Q X ˙s,τ Rn 1 ǫ is ǫ-almost diagonal on fp,q ( ) [16, Deﬁnition 4.1]. Now τ < p + 2n , so according ˙s,τ Rn to [16, Theorem 4.1], the operator described above is bounded on fp,q ( ). Also, ˙ s,τ Rn by [16, Theorem 3.1], the operator u 7→ hu, ψQi Q is bounded from Fp,q ( ) to ˙s,τ Rn fp,q ( ). Combining these results with the estimates above yields f

kfk l ˙ s,τ Rn . aQR|tR| . k(tQ)Qk ˙s,τ Rn . kfk ˙ s,τ Rn , AN,mFp,q ( ) fp,q ( ) Fp,q ( ) ˙s,τ Rn R Q fp,q ( ) X e ˙ s,τ Rn i.e. the mapping described above is a continuous embedding of Fp,q ( ) into l ˙ s,τ Rn AN,mFp,q ( ). l ˙ s,τ Rn ˙ s,τ Rn (ii) We have AN,mBp,q ( ) ⊂ Bp,q ( ) in the same sense as above. Conversely, if ˙ s,τ Rn a tempered distribution f belongs to Bp,q ( ), again by the proof of [16, Lemma 4.2] and [3, pp. 153–155] it suﬃces to show that kfk l ˙ s,τ Rn , where f is as in part AN,mBp,q ( )

s,τ n (i), is controlled by a constant times kfkB˙ p,q (R ). Using (3), one can check that e e

kfk l ˙ s,τ Rn . aQR|tR| , AN,mBp,q ( ) ˙s,τ Rn R Q bp,q ( ) X ˙s,τ Rn e where bp,q( ) stands for the space of sequences (bQ)Q of complex numbers, indexed by the dyadic cubes of Rn, such that

∞ 1/p q p/q 1 −s/n−1/2+1/p k(b ) k˙s,τ n = sup |Q| |b | , Q Q bp,q (R ) τ Q P |P | j=j −j ! XP Q⊂P,X ℓ(Q)=2 110 Tomás Soto

Rn where the supremum is taken over all dyadic cubes P of and jP = − log2 ℓ(P ), is finite. Combining this estimate with [16, Theorems 4.1 and 3.1] as above then yields kfk l ˙ s,τ Rn . kfk ˙ s,τ Rn . AN,mBp,q ( ) Bp,q ( ) (iii) This is contained in the arguments above. e 3. Proof of Theorem 1.3 We now turn to the identification of the Hajłasz-type spaces with the spaces defined through grand maximal functions. We shall need the following Sobolev-type embedding, which is a special case of [8, Lemma 2.3]. Lemma 3.1. Let s ∈ (0, 1] and 0 <ǫ<ǫ′ < s. Then there exists a positive constant C such that for all x ∈ Rn, k ∈ Z, measurable functions u and −→g ∈ Ds(u),

n+ǫ ′ ′ n n inf − |u(y) − c| dy ≤ C2−kǫ 2−j(s−ǫ ) − g (y) n+ǫ dy . C j c∈ ˆB(x,2−k) ˆB(x,2−k+1) j≥Xk−2 To talk about the identification of the spaces of measurable functions defined through Hajłasz gradients and the spaces of tempered distributions defined through grand maximal functions, we need the following basic lemma. The techniques of the proof are similar to the ones employed in [7, Theorem 1.1] and [8, Theorem 3.2]. Lemma 3.2. ˙ s,τ Rn ˙ s,τ Rn (i) Let u ∈ Mp,q ( ) or u ∈ Np,q ( ) with s ∈ (0, 1], p ∈ n ( n+s , ∞], q ∈ (0, ∞] and τ ≥ 0. Then u defines a tempered distribution in the sense that for all functions φ ∈ S(Rn), uφ is integrable and

uφ ≤ C(u)kφkS0,N ˆRn

for some integer N depending on n, s, p and τ. ′ Rn l ˙ s,τ Rn l ˙ s,τ Rn (ii) Suppose that f ∈ S ( ) belongs to A0,mFp,q ( ) or A0,mBp,q ( ) with n N s ∈ (0, ∞), p ∈ ( n+s , ∞], q ∈ (0, ∞], τ ∈ [0, ∞) and l, m ∈ 0. Then f coincides with a locally integrable function in the sense that there exists a ˜ 1 Rn function f ∈ Lloc( ) such that

hf,φi = fφ˜ ˆRn for all φ ∈ S(Rn) with compact support.

s,τ n s,τ n Proof. (i) Let kuk denote either kukM˙ p,q (R ) or kukN˙ p,q (R ), whichever is ﬁnite. ′ ′ n Rn Fix ǫ and ǫ such that 0 <ǫ<ǫ < s and n+ǫ < p. For any x ∈ , Lemma 3.1 yields

n+ǫ ′ n n −j(s−ǫ ) + inf − |u(y) − c| dy . inf 2 − gj(y) n ǫ dy c∈Cˆ −→g ∈Ds(u) ˆ B(x,1) j≥−2 B(x,2) X −j(s−ǫ′) . inf 2 kgjkLp(B(x,2)) −→g ∈Ds(u) j≥−2 X′ . 2−j(s−ǫ )kuk < ∞, j≥−2 X so u is locally integrable. Characterizations of Besov-type and Triebel–Lizorkin-type spaces 111

n Now write N for the smallest integer strictly greater than max(n, n+s+nτ − p ). For any φ ∈ S(Rn) we then have

|u(x)φ(x)| dx . |u|B(0,1)kφkS0,N + u(x) − uB(0,1) |φ(x)| dx ˆRn ˆ k k−1 k≥1 B(0,2 )\B(0,2 ) X

−kN . kφkS0,N |u|B(0,1) + 2 |u(x) − uB(0,1)| dx ˆB(0,2k) ! Xk≥1 −k(N−n) ≈kφkS0,N |u|B(0,1) + 2 − |u(x) − uB(0,1)| dx ˆB(0,2k) ! Xk≥1 k −k(N−n) i . kφkS0,N |u|B(0,1) + 2 − |u(x) − uB(0,2 )| dx ˆ i i=0 B(0,2 ) ! Xk≥1 X −i(N−n) i . kφkS0,N |u|B(0,1) + 2 − |u(x) − uB(0,2 )| dx . ˆ i i≥0 B(0,2 ) ! X As above, the integral in the ith term of the latter sum can be estimated by n+ǫ n iǫ′ −j(s−ǫ′) n i . n+ǫ − |u(x) − uB(0,2 )| dx −→ inf 2 2 − gj(y) dy ˆ i g ∈Ds(u) ˆ i+1 B(0,2 ) j≥−i−2 B(0,2 ) ′ X ′ 1 iǫ −j(s−ǫ ) i+1 − p . inf 2 2 |B(0, 2 )| kgjkLp(B(0,2i+1)) −→g ∈Ds(u) j≥−i−2 X i(ǫ′+nτ− n ′ i(s+nτ− n ) . 2 p ) 2−j(s−ǫ )kuk . 2 p kuk. j≥−i−2 X Thus,

n i(n+s+nτ− p −N) |u(x)φ(x)| dx . kφkS0,N |u|B(0,1) + 2 kuk , ˆRn i≥0 ! X n where the quantity inside the latter parentheses is ﬁnite because n+s+nτ − p −N < 0. (ii) Let kfk denote either kfk l ˙ s,τ Rn or kfk l ˙ s,τ Rn , whichever is ﬁnite. A0,mFp,q ( ) A0,mBp,q ( ) Rn Fix a compactly supported ϑ ∈ S( ) such that Rn ϑ =1. It is well known that ∞ ´

f = ϑ ∗ f + (ϑj+1 − ϑj) ∗ f j=0 X with convergence in S′(Rn), so it suﬃces to show that

k(ϑj+1 − ϑj) ∗ fkL1(B) < ∞ j≥0 X n whenever B ⊂ R is a ball with radius 1. For p ≥ 1, this is almost immediate: ϑ1 −ϑ l is a constant multiple of an element of A0,m, so

−js js −js k(ϑj+1 − ϑj) ∗ fkL1(B) . 2 2 (ϑ1 − ϑ)j ∗ f Lp(B) . 2 kfk < ∞. j≥0 j≥0 j≥0 X X X n Rn −j N Suppose now that n+s

Using this estimate in an arbitrary ball B of radius 1 we thus get 1−p p 1 k(ϑ1 − ϑ)j ∗ fkL (B) . k(ϑ1 − ϑ)j ∗ fkL∞(B)k(ϑ1 − ϑ)j ∗ fkLp(B) j≥0 j≥0 X X 1−p jn −js−jnτ p . 2 p kfk 2−jskfk j≥0 X j n (1−p)−s . 2 ( p )kfk < ∞, j≥0 X n n since p (1 − p) − s< (n + s)(1 − n+s ) − s =0. We are now ready to give the proof of Theorem 1.3. The methods are based on the case p = ∞ of the proof of [8, Theorem 3.2]. Proof of Theorem 1.3. We shall ﬁrst prove (i) and (ii) under the assumption that s ∈ (0, 1) and m ≥ n +1. ˙ s,τ Rn ℓ ˙ s,τ Rn (i) We start by establishing the embedding Mp,q ( ) ⊂ A0,mFp,q ( ) with the assumption that q < ∞. To this direction, ﬁx ǫ and ǫ′ so that 0 <ǫ<ǫ′ < s and n n+ǫ < min(p, q). ˙ s,τ Rn −→ Ds Let u ∈ Mp,q ( ), and choose g ∈ (u) so that

p/q 1/p 1 q sup g (y) dy ≤ 2kuk ˙ s,τ n . −ℓ τ k Mp,q (R ) x∈Rn,ℓ∈Z |B(x, 2 )| ˆB(x,2−ℓ) ! Xk≥ℓ Recall that by Lemma 3.2 above, u deﬁnes a tempered distribution. Since m ≥ n+1, we have n+ǫ ′ ′ n n −k j(1−ǫ ) −i(s−ǫ ) + (5) sup |φk ∗ u(z)| . 2 2 2 − gi(y) n ǫ dy l ˆ −j+1 φ∈A0,m i≥j−2 B(z,2 ) Xj≤k X for all k ∈ Z and z ∈ Rn, where the implied constant does not depend on these two parameters; a similar estimate is established in [8, pp. 15–16], but (5) can also be deduced in a manner similar to the proof of part (i) of Lemma 3.2. We now consider a ball B := B(x, 2−ℓ) with ℓ ∈ Z and use the estimate (5) for z ∈ B and k ≥ ℓ. The terms of the sum with j ≤ ℓ can be estimated as in the proof of Lemma 3.2: n+ǫ n n jn jn n+ǫ p p −jnτ . p −j+2 . s,τ n (6) − gi(y) dy 2 kgikL (B(z,2 )) 2 kukM˙ p,q (R ), ˆ −j+1 B(z,2 ) n and since 1 − s + p − nτ > 0, we get n+ǫ ′ ′ n n jn j(1−ǫ ) −i(s−ǫ ) n+ǫ j−js+ p −jnτ s,τ n 2 2 − gi(y) dy . 2 kukM˙ p,q (R ) ˆ −j+1 j≤ℓ i≥j−2 B(z,2 ) j≤ℓ X X X n ℓ(1−s+ p −nτ) s,τ n . 2 kukM˙ p,q (R ). Characterizations of Besov-type and Triebel–Lizorkin-type spaces 113

For the terms with j>ℓ, we have B(z, 2−j+1) ⊂ 2B for all z ∈ B, so that

n+ǫ n n n n+ǫ n+ǫ n+ǫ n (7) − gi(y) dy ≤M gi χ2B (z) , ˆB(z,2−j+1) where M is the Hardy–Littlewood maximal function. Combining these estimates, we have

p q q 1 ks τp 2 sup |φk ∗ u(z)| dz |B| ˆB l k≥ℓ φ∈A0,m ! X p q |B| n q −k(1−s)q ℓ(1−s+ p −nτ) (8) . 2 2 kuk ˙ s,τ Rn |B|τp Mp,q ( ) k≥ℓ ! X p q n q ′ ′ n+ǫ 1 −k(1−s) j(1−ǫ ) −i(s−ǫ ) n+ǫ + 2 2 2 M g χ (z) n dz. |B|τp ˆ i 2B B i≥j−2 ! Xk>ℓ ℓ 0, the ﬁrst quantity can be estimated easily:

p q |B| n q −k(1−s)q ℓ(1−s+ p −nτ) p 2 2 kuk s,τ n . kuk s,τ . τp M˙ p,q (R ) ˙ Rn |B| ! Mp,q ( ) Xk≥ℓ For the second quantity, we exchange the order of summation in the integrand as follows: q n ′ ′ n+ǫ n+ǫ −k(1−s) j(1−ǫ ) −i(s−ǫ ) n 2 2 2 M gi χ2B (z) " # k>ℓ ℓℓ i≥Xℓ−1 ℓℓ ℓ−1≤i≤k−2 X X q n ′ ′ n+ǫ n+ǫ k(s−ǫ ) −i(s−ǫ ) n +2 2 M gi χ2B (z) # i≥Xk−1 Now, using Hölder’s inequality when q > 1 and the sub-additivity of t 7→ tq when n n+s < q ≤ 1, the latter quantity can be estimated from above by a constant times

n n+ǫ n+ǫ −k(1−s)(q∧1) i(1−s)(q∧1) n q 2 2 M gi χ2B (z) " Xk>ℓ ℓ−1≤Xi≤k−2 n ′ ′ n+ǫ n+ǫ k(s−ǫ )(q∧1) −i(s−ǫ )(q∧1) n q +2 2 M gi χ2B (z) i≥k−1 # n X n+ǫ n+ǫ n q . M gi χ2B (z) . i≥Xℓ−1 114 Tomás Soto

n+ǫ n+ǫ Since n p> 1 and n q > 1, the Feﬀerman-Stein vector valued maximal inequality thus yields p q n q ′ ′ n+ǫ 1 n+ǫ −k(1−s) j(1−ǫ ) −i(s−ǫ ) n τp 2 2 2 M gi χ2B (z) dz |B| ˆB ! k>ℓ ℓ

kuk l ˙ s,τ Rn . kuk ˙ s,τ Rn , A0,mFp,q ( ) Mp,q ( ) ˙ s,τ Rn l ˙ s,τ Rn i.e. Mp,q ( ) ⊂ A0,mFp,q ( ). The case q = ∞ can be handled in a similar but easier manner. l ˙ s,τ Rn Now suppose that f ∈A0,mFp,q ( ). According to Lemma 3.2, f coincides with a locally integrable function, so ﬁxing a compactly supported ϑ ∈ S(Rn) such that

Rn ϑ =1, we have f = limj→∞ ϑj ∗ f pointwise almost everywhere. In particular, if x´ and y are distinct Lebesgue points of f and k ∈ Z satisﬁes 2−k−1 ≤|x − y| < 2−k, we have

kuk ˙ s,τ Rn . kuk l ˙ s,τ Rn , Mp,q ( ) A0,mFp,q ( ) l ˙ s,τ Rn ˙ s,τ Rn i.e. A0,mFp,q ( ) ⊂ Mp,q ( ). Characterizations of Besov-type and Triebel–Lizorkin-type spaces 115

˙ s,τ Rn l ˙ s,τ Rn n (ii) That Np,q ( ) ⊂ A0,mBp,q ( ) for n+s n+ǫ , instead ˙ s,τ Rn of the Feﬀerman–Stein maximal inequality. Let u be a function in Np,q ( ) and −→g ∈ Ds(u) a fractional s-gradient that yields the quasinorm of u up to a constant multiple of at most two. Letting B := B(x, 2−ℓ) for arbitrary x ∈ Rn and ℓ ∈ Z, arguing as in (5), (6) and (7) we have

n ks −k(1−s) ℓ(1−s+ p −nτ) s,τ n 2 sup |φk ∗ u(z)| . 2 2 kukN˙ p,q (R ) l φ∈A0,m n ′ ′ n+ǫ n+ǫ −k(1−s) j(1−ǫ ) −i(s−ǫ ) n +2 2 2 M gi χ2B (z) ℓ

ks p 2 sup |φk ∗ f Lp(B) l φ∈A0,m + n n ǫ p τp (ℓ−k)(1−s) p p −k(1−s)p i(1−s) n+ǫ n . |B| 2 kuk s,τ n +2 2 M g χ2B N˙ p,q (R ) i Lp(B) ℓ−1≤i≤k−2 n+ǫ Xp ′ ′ n k(s−ǫ )p −i(s−ǫ ) n+ǫ n +2 2 M gi χ2B p i≥k−1 L (B) X τp (ℓ−k )(1−s)p p . |B| 2 kuk s,τ n N˙ p,q (R ) + n n ǫ p −k(1−s)(p∧1) i(1−s)(p∧1) n+ǫ n +2 2 M gi χ2B Lp(B) ℓ−1≤i≤k−2 X n+ǫ p ′ ′ n k(s−ǫ )(p∧1) −i(s−ǫ )(p∧1) n+ǫ n +2 2 M gi χ2B p i≥k−1 L (B) X τp (ℓ−k)(1−s)p p −k(1−s)(p∧1) i(1−s)(p∧1) p . |B| 2 kuk s,τ n +2 2 kgik p N˙ p,q (R ) L (2B) ℓ−1X≤i≤k−2 k(s−ǫ′)(p∧1) −i(s−ǫ′)(p∧1) p +2 2 kgikLp(2B). i≥Xk−1 Dividing out by |B|τp ≈|2B|τp, taking ﬁrst the ℓq/p norm over k ≥ ℓ and then taking the supremum over all admissible balls B yields

kuk l ˙ s,τ Rn . kuk ˙ s,τ Rn , A0,mBp,q ( ) Np,q ( )

˙ s,τ Rn l ˙ s,τ Rn i.e. Np,q ( ) ⊂ A0,mBp,q ( ). The case p = ∞ can be handled in a similar but easier manner. l ˙ s,τ Rn n On the other hand, if f ∈A0,mBp,q ( ) where n+s

−ℓ functions hk are as in part (i). For any ball B := B(x, 2 ), we have 1 1 −τ −τ ks(p∧1)( ∧1) −js(p∧1)( ∧1) js p p p |B| khkkL (B) . |B| 2 2 · 2 sup |φj ∗ f| Lp(B), l j≥k φ∈A0,m X so taking the ℓq norm over k ≥ ℓ and then the supremum over all admissible balls B yields

kfk ˙ s,τ Rn . kfk l ˙ s,τ Rn , Np,q ( ) A0,mBp,q ( ) l ˙ s,τ Rn ˙ s,τ Rn which means that A0,mBp,q ( ) ⊂ Np,q ( ). The case p = ∞ can be handled in a similar but easier manner. The case with s ∈ (0, 1) and m ≥ n +1 is thus proven. The case with s = 1 and m ≥ n +2 can be proven using exactly the same argument, except for the fact that 1 − s is not positive. To remedy this, we use the assumption that m ≥ n +2 to replace (5) with the estimate n+ǫ ′ ′ n n −2k j(2−ǫ ) −i(s−ǫ ) + sup |φk ∗ u(z)| . 2 2 2 − gi(y) n ǫ dy . l ˆ −j+1 φ∈A0,m i≥j−2 B(z,2 ) Xj≤k X The above proof can thus be carried out by replacing 1−s with 2−s where necessary. We omit the details. Acknowledgements. The author would like to thank Eero Saksman and Yuan Zhou for reading the manuscript and making several valuable remarks. The author also wishes to thank IPAM at UCLA for supporting his participation in the program “Interactions Between Analysis and Geometry” in Spring 2013, during which part of this note was written. References

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Received 23 February 2015 • Accepted 31 May 2015