On some characterizations of Besov spaces of generalized smoothness

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On some characterizations of Besov spaces of generalized smoothness

Susana D. Moura∗1 1

CMUC, Department of Mathematics, University of Coimbra, 3001-454 Coimbra, Portugal Received 29 November 2005, accepted 8 October 2006

Published online 6 June 2007

Key words Function spaces of generalized smoothness, characterizations, approximation, differences MSC (2000) 46E35

Dedicated to Professor Frank-Olme Speck on the occasion of his 60th birthday

We present characterizations of the Besov spaces of generalized smoothnessB_p,qσ,N(Rn) via approximation and by means of differences.

c

2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction

Function spaces of generalized smoothness have been introduced and considered, in particular since the middle of the seventies of the last century, by several authors from different points of view and different degrees of generality. We refer, for instance, to Goldman [9, 10], Kalyabin [13, 14], Merucci [16], Cobos and Fernandez [4], the survey by Kalyabin and Lizorkin [12], Leopold [15], Edmunds and Triebel [6, 7] and Moura [17, 18].

From the standpoint of a Fourier analytic characterization, the recent work by Farkas and Leopold [8] repre-sents a unified and general approach on this topic. Following their approach, we see that, under certain conditions, the Besov spaces of generalized smoothness can be analogously characterized via approximation and by means of differences.

2 Preliminaries

2.1 General notation

In this paper we shall adopt the following general notation:N denotes the set of all natural numbers, N0=N∪{0},

Rn_,_n_{∈ N, denotes the n-dimensional real euclidean space and R = R}1_{. We always use the equivalence “}_{∼” in}

ak∼ bk or φ(r)∼ ψ(r)

to mean that there are two positive numbersc1andc2such that

c1ak ≤ bk≤ c2ak or c1φ(x)≤ ψ(x) ≤ c2φ(x)

for all admitted values of the discrete variablek or the continuous variable x, where (ak)k,(bk)kare nonnegative

sequences andφ, ψ are nonnegative functions. If a∈ R then a₊ := max(a, 0). We consider here only function spaces defined onRn; so for convenience we shall usually omit the “Rn” from their notation.

2.2 Sequences

In this subsection we explain the class of sequences we shall be interested in and some related basic results.

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A sequenceγ = (γ_j)_j∈N₀of positive real numbers is said to be admissible if there exist two positive constants

d0andd1such that

d₀γ_j≤ γ_j+1≤ d₁γ_j, j∈ N₀. (2.1)

For an admissible sequenceγ = (γj)j∈N0, let

γ_j := inf k≥0 γ_j+k γk and γj:= supk≥0 γ_j+k γk , j∈ N0. (2.2)

Then clearlyγ_jγk ≤ γj+k≤ γkγj for anyj, k∈ N0. In particular,γ₁andγ1are the best possible constantsd0

andd1in (2.1), respectively. Then, the lower and upper Boyd indices of the sequenceγ are defined, respectively,

by s(γ) := lim j→∞ log γ_j j and s(γ) := limj→∞ log γ_j j . (2.3)

The above definition is well-posed: the sequencelog γ_j_j∈N is sub-additive and hence the right-hand side limit in (2.3) exists, it is finite(since γ is an admissible sequence) and it coincides with infj>0log γj/j. The

corresponding assertions for the lower counterparts(γ) can be read off observing that log γ_j =− logγ−1_j. The Boyd indexs(γ) of an admissible sequence γ describes the asymptotic behaviour of the γ_j’s and provides more information than simplyγ₁and, what is more, is stable under the equivalence of sequences: ifγ∼ τ, then s(γ) = s(τ ) as one readily verifies. The same applies to the lower counterpart.

Observe also that for eachε > 0 there are two positive constants c1= c1(ε) and c2= c2(ε) such that

c₁2(s(γ)−ε)j≤ γ_j≤ γ_j ≤ c₂2(s(γ)+ε)j, j∈ N₀. (2.4)

Example 2.1 We consider some examples of admissible sequences.

(i) The sequenceγ =γ_j_j∈N

0,

γ_j= 2sj(1 + j)b(1 + log(1 + j))c

with arbitrary fixed real numberss, b and c is a standard example of an admissible sequence with s(γ) = s(γ) = s.

(ii) LetΦ : (0, 1] → R be a slowly varying function (or equivalent to a slowly varying one) in the sense of [1]. Then, for eachs∈ R the sequence γ =2sjΦ2−j_j∈N

0is an admissible sequence. Also here we have

s(γ) = s(γ) = s.

(iii) In view of [2, Proposition 1.9.7], the caseγ =2sjΨ2−j_j∈N

0, where nowΨ is an admissible function

in the sense of [6]i.e., a positive monotone function defined on(0, 1] such that Ψ2−2j∼ Ψ2−j,j ∈ N0, can be regarded as a special case of (ii).

Remark 2.2 The examples above have in common the fact that their upper and lower Boyd indices coincide.

However, this is not in general the case. Examples 2.2.6 and 2.2.7 in [8], due to Kalyabin, show that an admissible sequence has not necessarily a fixed main order.

2.3 Function spaces of generalized smoothness

LetN =Nj_j∈N₀ be an admissible sequence withN₁> 1 (recall (2.2)). In particular N is a so-called strongly

increasing sequence (cf. [8, Def. 2.2.1]), which guarantees the existence of a numberk0∈ N0such that

N_k ≥ 2N_j for any k, j such that k≥ j + k₀. (2.5) It should be noted that the sequenceN = Nj_j∈N₀ plays the same role as the sequence2j_j∈N

0 in the

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For a fixed sequenceN =N_j_j∈N

0, as above, we define the associated coveringΩ

N ₌_ΩN j j∈N0 ofR n_by ΩN_j ={ξ ∈ Rn:|ξ| ≤ N_j+k₀}, j = 0, . . . , k₀− 1, and ΩN_j ={ξ ∈ Rn: Nj−k0 ≤ |ξ| ≤ Nj+k0}, j ≥ k0, withk0according to (2.5).

Definition 2.3 For a fixed admissible sequenceN =Nj_j∈N₀ withN₁> 1, and for the associated covering

ΩN =ΩN_j _j∈N

0ofR

n_{, a system}_ϕN ₌_ϕN j

j∈N0will be called a (generalized) partition of unity subordinated

toΩN if: (i)

ϕN_j ∈ C₀∞ and ϕ_jN(ξ)≥ 0 if ξ ∈ Rn for any j∈ N₀; (ii)

supp ϕN_j ⊂ ΩN_j for any j∈ N0;

(iii) for anyα∈ Nn₀ there exists a constantcα> 0 such that for any j∈ N0

DαϕN_j (ξ) ≤ c_α1 +|ξ|2−|α|/2 for any ξ∈ Rn; (iv) there exists a constantcϕ> 0 such that

0 <

∞

j=0

ϕN_j (ξ) = c_ϕ<∞ for any ξ ∈ Rn.

Before turning to the definition of the Besov spaces of generalized smoothness let us recall thatS denotes the Schwartz space of all complex-valued rapidly decreasing infinitely differentiable functions onRnequipped with the usual topology and byS we denote its topological dual, the space of all tempered distributions onRn. If

f ∈ S, thenFf and F−1f denote the Fourier and the inverse Fourier transform of f , respectively. For ϕ∈ S

andf ∈ Swe will use the notationϕ(D)f = [F−1(ϕFf)]. Furthermore, if 0 < p ≤ ∞ and 0 < q ≤ ∞ then

L_p and _q have the standard meaning and, iff_j_j∈N

0 is a sequence of complex-valued Lebesgue measurable

functions onRn, then (f_j)_j∈N₀| _qL_p= _∞ j=0 fj|Lp q 1/q

with the appropriate modification ifq =∞.

Definition 2.4 LetN =Nj_j∈N₀ be an admissible sequence withN1> 1 and ϕN a system of functions as

in Definition 2.3. Let0 < p, q ≤ ∞ and σ =σj_j∈N₀ be an admissible sequence. Then the Besov space of generalized smoothnessBσ,N_p,q is the set of all tempered distributionsf such that the quasi-norm

_f_|Bσ,N p,q :=σjϕNj (D)f j∈N0| q Lp is finite.

Remark 2.5 Note that if0 < p < ∞ then the Triebel–Lizorkin space of generalized smoothness F_p,qσ,N is defined in an analogous way, by interchanging the roles of theLpand qquasi-norms.

Ifσ = (2sj)j∈N0, withs a real number, and N = (2j)j∈N0then the spacesBp,qσ,Ncoincide with the usual Besov

spacesB_p,qs andF_p,qσ,N coincide with the usual Triebel–Lizorkin spacesF_p,qs . If we letσ = (2sjΨ(2−j))_j∈N₀, whereΨ is an admissible function in the sense of [6, 7] (see Example 2.1 (iii)), the corresponding Besov space coincide with the space Bp,q(s,Ψ) introduced by Edmunds and Triebel in [6, 7] and also considered by Moura

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3 Characterization by approximation

Let N = N_j_j∈N

0 be an admissible sequence with N1 > 1 (recall (2.2)) and k0 ∈ N0 as in (2.5). Let

0 < p≤ ∞ and UN p := a =aj_j∈N₀ : aj∈ S∩ Lp, suppFaj⊂ {y : |y| ≤ Nj+k0}, j ∈ N0 .

Theorem 3.1 Let σ = σj_j∈N₀ be an admissible sequence,0 < p, q ≤ ∞, and N = Nj_j∈N₀ be an admissible sequence withN₁> 1, as above.

Suppose thats(σ) s(N )−1> n(1/p− 1)₊. Then B_p,qσ,N =f∈ S:∃ a =a_j_j∈N

0 ∈ U

N

p such that f = lim_k→∞ak inSand

where the infimum is taken over all admissible systemsa∈ UN_p, is an equivalent quasi-norm inBσ,N_p,q .

P r o o f. First of all let us remark that, taking into account the inequalities in (2.4), ifs(σ) s(N )−1 > δ ≥ 0

then

σ−1_j Nδ_j

j∈N∈ u for any u∈ (0, ∞].

Step 1. In this step we prove that ·|Bσ,N_p,q Xcan be estimated from above by ·|B_p,qσ,N. Letf ∈ B_p,qσ,Nand letϕN_j ∞_j=0be a generalized partition of unity, according to Definition 2.3. Then

a_k:=

k

j=0

ϕN_j (D)f −→ f in S as k−→ ∞.

Moreover,ak ∈ S∩ LpandsuppFak ⊂ {y : |y| ≤ Nk+k0}, k ∈ N0. Hence,a = (ak)k∈N0 ∈ UNp. Let

¯

p := min(1, p) and put ϕN_l ≡ 0 if −l ∈ N. Then σ_k f − a_k|L_p ≤ σ_k _∞ j=k+1 ϕN_j (D)f|L_pp¯1/¯p ≤ _∞ m=1 σ−¯_mpσ_k+mp¯ ϕN_k+m(D)f|Lpp¯ 1/¯p . (3.1)

If 0 < q≤ ¯p , from (3.1) and using the fact thatσ−1_j _j∈N

If q > ¯p, from (3.1) by using the triangle inequality with respect to _q/¯_pandσ−1_j _j∈N

0 ∈ p¯, we obtain _∞ k=1 σ_kq f − ak|Lp q _1/q ≤ ⎧ ⎨ ⎩ ∞ m=1 σ−¯_mp _∞ k=1 σq_k+mϕN_k+m(D)f|Lpq _p/q_¯ ⎫_⎬ ⎭ 1/¯p ≤

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≤σ−1_j _j∈N

0| p¯f|B

σ,N

p,q  ≤ cf|Bp,qσ,N

(with the usual modification if q =∞). Since we also have

a0|Lp =ϕ0N(D)f|Lp ≤ cf|Bσ,Np,q ,

then

_f_|Bσ,N

p,q a ≤ cf|Bp,qσ,N, f ∈ Bp,qσ,N. Step 2. Letf ∈ Sanda = (ak)k∈N0 ∈ UNp be such that

f = lim

k→∞ak in S

_and _f_|Bσ,N

p,q a<∞.

Letl₀= l₀(N )∈ N be such that Nl0

1 > N1and letµ0, µ∈ S be positive functions such that

µ₀(ξ) = 1 if |ξ| ≤ N₁, supp µ₀⊂ {ξ ∈ Rn :|ξ| ≤ N₁N₁} and µ(ξ) = 1 if N−1₁ ≤ |ξ| ≤ N1, supp µ⊂ ξ∈ Rn : N−l0 1 ≤ |ξ| ≤ Nl10 .

For eachj∈ N define µj(ξ) := µNj−1ξ

,ξ∈ Rn. Puttingal≡ 0 if −l ∈ N, we have µk(D)f = ∞ j=1−l0−k0 F−1_[µ kF(ak+j− ak+j−1)], k∈ N0,

and hence, withp as in step 1, we get¯ _µ_k_(D)f_L_p_≤ ∞ j=−1 F−1_[µ kF(ak+j− ak+j−1)]|Lpp¯ _1/¯_p , k∈ N0. (3.2)

Recall thatsuppF(ak+j− ak+j−1)⊂ {ξ : |ξ| ≤ Nk+j+k0}. By hypothesis s(σ) s(N)−1 > n(1/p− 1)+, so

that we can choose a real numberλ satisfying s(σ) s(N )−1> λ− n/2 > n(1/p − 1)+. Then the inequality (13)

of [19, 1.5.2, p. 28] yields F−1_{[ µ}

kF(ak+j− ak+j−1)]|Lp

≤ cµk(Nk+j+k0·)|H2λ ak+j− ak+j−1|Lp , k ∈ N0,

(3.3)

wherec is a positive constant independent of k and j. Using [5, Proposition 1, p. 33] we obtain, for k∈ N,

µ_k(N_k+j+k₀·)|H₂λ=µN_k+j+k₀N_k−1·|H₂λ ≤ c1Nk+j+k0Nk−1 λ−n 2_µ(a·)|Hλ 2 ≤ c2Nλ− n 2 j (3.4)

wherea = a(N ) and c1, c2are positive constants independent ofj and k. In an analogous way, for k = 0 we

also have _µ₀_(N_j+k₀_·)|Hλ 2 ≤ cNλ− n 2 j . (3.5) Inserting (3.3)–(3.5) in (3.2), leads us to µk(D)f|Lp ≤ c _∞ j=1−l0−k0 N(λ−n2)¯p j ak+j− ak+j−1|Lp p¯ 1/¯p

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with the understanding thatN_j≡ 1 if −j ∈ N. If0 < q≤ ¯p, it follows

where we have used the fact that

N(λ−n2)

j σj−1

j∈N0 ∈ q

, which is a consequence of the choice ofλ. On the

other hand, ifq > ¯p, reasoning as in step 1, we obtain ∞ k=0 σ_kq µk(D)f|Lp q ≤ c1 N(λ−n2) j σ−1j j∈N0 p¯ q ∞ m=0 σ_mq am− am−1|Lp q ≤ c2 a0|Lp + c3 ∞ m=0 σ_mq f − am|Lp q (3.7)

(with the usual modification ifq = ∞). Using now the [8, Theorem 4.2.2], by (3.6) and (3.7) we get, for any q∈ (0, ∞],

_f_|Bσ,N

p,q  ≤ cf|Bp,qσ,Na, f ∈ Bp,qσ,N,

wherec is a positive constant independent of the chosen system a = (ak)k∈N0. Therefore,

_f_|Bσ,N

p,q  ≤ cf|Bp,qσ,NX, f ∈ Bp,qσ,N.

Remark 3.2 ForN = (2j)j∈N0 andσ = (2sj)j∈N0 orσ = (2sjΦ(2−j))j∈N0 (cf. Examples 2.1 (i) and (ii)),

the above theorem coincide with [19, Theorem 2.5.3 (i)] and [11, Theorem 2.4], respectively.

A characterization as above is valid for the spaceF_p,qσ,N, replacing the conditions(σ) s(N )−1> n(1/p− 1)+

bys(σ) s(N )−1> n(1/ min(p, q)− 1)₊and interchanging the roles of theL_pand _q quasi-norms.

4 Characterization by differences

Next we recall the definition of differences of functions. Iff is an arbitrary function onRn,h∈ Rnandk∈ N,

then ∆k_hf(x) := k j=0 k j (−1)k−jf (x + jh), x∈ Rn.

Note that∆k_hcan also be defined iteratively via

∆1_hf(x) = f (x + h)− f(x) and ∆k+1_h f(x) = ∆1_h∆k_hf(x), k∈ N.

Furthermore, thek-th modulus of smoothness of a function f ∈ Lp,1≤ p ≤ ∞, k ∈ N, is defined by ω_k(f, t)_p= sup

|h|≤t

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Theorem 4.1 Let0 < p, q ≤ ∞, σ = (σ_j)_j∈N₀ andN = N_j_j∈N

0 be admissible sequences, the latter

satisfyingN₁> 1, such that s(σ) s(N )−1> n(1/p− 1)+. LetM ∈ N, M > s(σ) s(N)−1. Then

B_p,qσ,N = ⎧ ⎨ ⎩f ∈ Lmax(1,p):f|Bp,qσ,N(M):= f|Lp + _∞ j=0 σq_j ωMf, Nj−1 _q p 1/q <∞ ⎫ ⎬ ⎭ (with the usual modification if q =∞) in the sense of equivalent quasi-norms.

Remark 4.2 It can be easily verified that, for anya∈ (0, +∞),

_f_|Bσ,N p,q (M)∼ f|Lp + _a 0 Λt−1ω_M(f, t)_pq dt t 1/q

whereΛ : (0,∞) → (0, ∞) is, according to [3, Def. 2.2.1], an admissible function such that Λ(z) ∼ Λ(Nj) = σj,z∈ [Nj, Nj+1], j∈ N0.

P r o o f. Step 1. In this step we shall prove that there exists a positive constantc such that

_f_|Bσ,N

p,q (M)≤ cf|Bσ,Np,q (4.1)

holds for allf ∈ B_p,qσ,N. Iff ∈ Bσ,N_p,q withs(σ) s(N )−1 > n(1/p− 1)₊, thenf ∈ L_max(1,p)and

f|Lp ≤ cf|Bσ,Npq , (4.2)

for some positive constantc which does not depend on f . This is a consequence of [3, Corollary 3.18], as

σ−1_j N_jn(1p−1)+

∈ q in this case.

WithϕN_j _j∈No being a generalized partition of unity according to Definition 2.3 we have, with convergence inLp, f = ∞ j=0 ϕ_j(D)f. (4.3)

Leta > n/p. If k∈ N0then for each∈ Rnwith|| ≤ Nk−1andj∈ {0, . . . , k} we have

_∆M ϕN_j (D)f(x)= M =0 M (−1)M− ϕN_j (D)f(x + ) ≤ M =0 M (−1)M− |β|≤M−1 DβϕN_j (D)f(x) β! ( ) β + M =0 M (−1)M− |α|=M DαϕN_j (D)f(x_j,) α! ( ) α ≤ c1Nk−M sup |y−x|≤N−1 k |α|=M DαϕN_j (D)f(y) ≤ c2NjMNk−M sup |y−x|≤N−1 k sup z∈Rn _ϕN j (D)f (y− z) 1 +|Nj+1z|a ≤ c3N−Mk−j ϕN∗_j f_a(x), (4.4)

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_M

=0

_M

(−1)M− m= 0 for any m∈ N₀withm < M , and [19, Theorem 1.3.1, p. 16] for the third inequality.

Hence, forj∈ N0, sup ||≤N−1 k _∆M ϕN_j (D)f|Lp ≤ c min1,N−Mk−j ϕN∗j f a|Lp, (4.5)

wherec > 0 is a constant independent of f .

Suppose that0 < p ≤ 1 (otherwise, similarly to the proof of the previous theorem, one has to modify the calculations below in an obvious way), then, by (4.3) and (4.5), we get

sup ||≤N−1 k _∆M f|Lpp≤ ∞ j=0 sup ||≤N−1 k _∆M ϕM_j (D)f|Lpp ≤ c k j=0 N−Mp_k−j ϕN∗_j f_a|Lpp+ c ∞ j=k+1 _ϕN∗ j f a|Lp p . (4.6)

WithΛ as in Remark 4.2, it follows _f_|Bσ,N p,q (M)q≤ c1 f|Lp q+ c1 _N−1 0 0 Λt−1ωM(f, t)pq dt_t ≤ c1 f|Lp q+ c1 ∞ k=0 _N−1 k N−1 k+1 Λt−1ωM(f, t)pq dt_t ≤ c1 f|Lp q+ c2 ∞ k=0 σq_kωMf, N_k−1q_p ≤ c1 f|Lp q+ c3 ∞ k=0 σ_kq _k j=0 N−Mp_k−j ϕN∗_j f_a|Lpp q/p + c₃ ∞ k=0 σ_kq _∞ j=k+1 _ϕN∗ j f a|Lp p q/p (4.7)

(with the usual modification ifq =∞). By hypothesis M > s(σ) s(N)−1, so that (2.4) entailsσ_jN−M_j _j∈N

0 ∈

u, for anyu∈ (0, ∞]. Then we obtain ∞ k=0 σq_k _k j=0 N−Mp_k−j ϕN∗_j f_a|Lpp _q/p ≤ cσjN−Mj j∈N0| min(1,p,q) q ∞ j=0 σ_jqϕN∗_j f_a|Lpq (4.8) and also ∞ k=0 σ_kq _∞ j=k+1 ϕN∗_j f_a|L_pp q/p ≤ cσ−1_j _j∈N 0| min(1,p,q) q∞ j=0 σ_jqϕN∗_j f_a|Lpq. (4.9)

We insert (4.8) and (4.9) in (4.7). Then (4.1) is a consequence of [3, Theorem 4.3.4] and (4.2).

Step 2. Letf ∈ Bσ,N_p,q . By Theorem 3.1, _f_|Bσ,N p,q  ≤ c f|Lp + c _∞ j=1 σq_jEp(Nj+k0, f )q _1/q (4.10)

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for some positive constantc and with Ep(b, f ) := inf f − g|Lp

where the infimum is taken over allg∈ S∩ L_psuch thatsuppg ⊂ {y : |y| ≤ b}. As there is a positive constant

c (which depends on p and k) such that Ep(b, f )≤ c sup

|h|≤b−1

_∆M h f|Lp

for allb≥ 1 and f ∈ Lp(cf. [19, Prop. 2.5.12, p. 109]) from (4.10) we get

_f_|Bσ,N p,q  ≤ c1 f|Lp + c2 _∞ j=1 σq_j ωMf, N_j+k−1₀q_p _1/q ≤ c3f|Bσ,Np,q (M),

which completes the proof.

Remark 4.3 ForN = (2j)_j∈N₀ andσ = (2sj)_j∈N₀ orσ = (2sjΦ(2−j))_j∈N₀ (cf. Examples 2.1 (i) and (ii)), the above theorem is covered by [19, Theorem 2.5.12] and [11, Theorem 2.5], respectively. For1 < p < ∞, 1 < q≤ ∞ this type of characterization has been obtained by Kalyabin in [14].

Acknowledgements The author is grateful to Professor H.-G. Leopold for valuable remarks concerning the subject. This work was partially supported by Centro de Matem´atica da Universidade de Coimbra and by a project under the agreement GRICES/DAAD.

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