# On the Baire Generic Validity of thet-Multifractal Formalism in Besov and Sobolev Spaces

Authors: Moez Ben Abid; Mourad Ben Slimane; Ines Ben Omrane; Borhen Halouani
Journal: Journal of Function Spaces
 (2019)

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## Abstract

Thet-multifractal formalism is a formula introduced by Jaffard and Mélot in order to deduce the t-spectrum of a function f from the knowledge of the (p,t)-oscillation exponent of f. The t-spectrum is the Hausdorff dimension of the set of points where f has a given value of pointwise Lt regularity. The (p,t)-oscillation exponent is measured by determining to which oscillation spaces Op,ts (defined in terms of wavelet coefficients) f belongs. In this paper, we first prove embeddings between oscillation and Besov-Sobolev spaces. We deduce a general lower bound for the (p,t)-oscillation exponent. We then show that this lower bound is actually equality generically, in the sense of Baire’s categories, in a given Sobolev or Besov space. We finally investigate the Baire generic validity of the t-multifractal formalism.


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## Body

## 1. Introduction

Multifractal analysis is concerned with the pointwise regularity and the scaling behavior of functions. It gives a powerful classification tool in various domains. In the setting of the classical Hölder pointwise regularity, it has been successfully used theoretically in (see [1–18] and references therein) and practically for signal and image processing (see [18–24] and references therein). Several related generic results, in the sense of Baire’s categories [15, 25–34] and in the prevalence sense [10, 11, 35, 36], were proved. Recall that a prevalent result is large in a measure sense, whereas a Baire generic result holds in a residual set (any countable intersection of open dense sets) in a well chosen topological vector space. Both Baire and prevalent generic sets are dense and stable by translation, dilation, and countable intersection. However, prevalence and Baire genericity usually differ widely. In Rd, prevalence coincides with Lebesgue almost everywhere, and there exist subsets of Rd with vanishing Lebesgue measure, but Baire generic. In infinite-dimensional spaces, there are stronger results of this type in [37, 38]. Nevertheless, in [39], Kolàr proved that the so-called HP-notion of genericity yields both Baire and prevalent results (see also [40]).The Hölder regularity has some limitations (see [41, 42] and references therein). Hölder regularity is only defined for locally bounded functions. It can not take negative values. It is not stable under some pseudodifferential and integral operators. It is not significant in fractal boundaries where it takes only two values 0 and ∞. For instance, in fully developed turbulence, velocity is not bounded near vorticity filaments and yields negative singularities [20]. The same holds for microcalcifications in mammography [20]. In order to overcome these weaknesses, Hölder regularity was replaced by the pointwise Lt regularity introduced by Calderón and Zygmund in [43] for functions that belong locally to Lt to better study elliptic partial differential equations. This notion has recently been put forward in the mathematical literature in [42, 44–46].Definition 1.
Lett≥1. Let u be a real number and x∈Rd. A function f in Lloct(Rd) belongs to Tu,t(x) if there exists R>0 and a polynomial P of degree less than u (with P=0 if u<0), such that(1)∀r≤Rfy-Py-xLtBx,r≤Cru+d/tfor some constant C independent of r.
The pointwiseLt regularity of f at x is (2)utx=sup⁡u;f∈Tu,tx.Definition1 written for t=∞ corresponds to the Hölder regularity.Definition1 can be also extended to the case t∈(0,1), where we consider the Hardy space Ht instead of Lt [47]. In [48] Theorem 1 p. 4, it is proved that ut(x)∈[-d/t,∞], t↦ut(x) is decreasing, and v↦u1/v(x) is concave on [0,1]. In [43], it is shown that, contrary to the Hölder regularity, the pointwise Lt regularity for 1≤t<∞ is invariant under pseudodifferential operators of order 0.Thet-sets of f are given by (3)Eth=x;utx=h.The t-spectrum of f is defined by the function(4)h↦dth=dimEthwhere dim denotes the Hausdorff dimension. By convention dim∅=-∞.In [11, 49], Fraysse has computed the t-spectrum for almost every function, in the prevalence setting, in a given Sobolev or Besov space.In [45, 46], Jaffard and Mélot have shown that the pointwise Lt regularity is well adapted for fractal interfaces. They have also proved that if f belongs to the Besov space Btε,∞(Rd) for an ε>0, then the pointwise Lt regularity is characterized by some conditions bearing on the moduli of the wavelet coefficients [47] (the definition of Besov is recalled in the next section). Note that (see [50]) (5)Btε,∞Rd↪LtRd↪Bt0,tRd.Let us recall the result obtained in [46]; let (ψi)i=1,…,2d-1 be either the Daubechies [51] compactly supported wavelets in Crψ(Rd) (where rψ is the uniform Hölder regularity of ψ) or the Lemarié-Meyer [50, 52] wavelets in the Schwartz class S(Rd) of rapidly decreasing C∞ functions (we will write rψ=∞), such that the family ψi(2jx-k), for j∈Z and k∈Zd, form an orthogonal basis of L2(Rd) (note that we choose the L∞ normalization, not L2). We will omit the letter i and the summation with respect to i. This will not affect the results of this work. The wavelets will be indexed in terms of dyadic cubes (6)λ=λj,k=k2-j+0,2-jd,so that we can write (7)ψλx=ψ2jx-k.For j∈N0, set (8)Λj=λ=λj,kwithk∈Zd.Put (9)Λ=⋃j∈N0Λj=λ=λj,kwithj,k∈N0×Zd.Let S′(Rd) be the space of tempered distributions (i.e., the dual of S(Rd); let f∈S′(Rd). Using the notation ψ˘(x)=ψ¯(-x), the wavelet coefficient Cλ of f is given by (10)Cλ=f.2j∗ψ˘k=2djf,ψλ.If f∈L2(Rd) then (11)Cλ=2dj∫Rdfyψλ¯ydy.And (12)f=∑j∈Z∑λ∈ΛjCλψλ.Recall that, for 0<t<∞, the t-exponent of f is given by (13)ηt=sup⁡τ:f∈Btτ/t,∞Rd=d+liminfj→∞⁡log⁡∑λ∈ΛjCλtlog⁡2-j.Remark 2.
In [48] Section 3.4, it is proved that if t≥1, then η(t) does not depend on the chosen wavelet basis as long as rψ>|hmin| (where hmin=liminfj→∞log⁡(supλ∈Λj|Cλ|)/log(2-j)). When t≥1, if η(t)>0, then f∈Lloct(Rd) and if η(t)<0 then f∉Lloct(Rd).For0<t<∞, the t-wavelet leader of f∈Lloct(Rd) at λ∈Λj was introduced in [46] (14)lt,λ=∑j′≥j,λ′⊂λCλ′2d/tj-j′t1/t,where the sum is over all λ′∈Λj′ such that λ′⊂λ with j′≥j. The sum (14) is finite if η(t)>0.Forx∈Rd, denote by λj(x) the unique cube at the scale j that contains x and 3λj(x) the set formed by the cube λj(x) and all its 3d-1 adjacent cubes at scale j. Let f∈S′(Rd). Suppose that 1<t<∞ and rψ>ut(x) (resp., 1≤t<∞ and rψ=∞). If η(t)>0, then (15)utx=liminfj→∞⁡log⁡lt,3λjxlog⁡2-j,and see [48], Section 3.3 (resp., [46] Corollary 1 p. 553).Remark 3.
Formula (15) was used in [48] to extend the definition of the pointwise Lt regularity to the case 0<t≤1 under the sole condition f∈Bt0,∞(Rd) (see [48], Section 3.4). From now on, the pointwise Lt regularity is defined by formula (15) for t>0.Using characterization (15), a t-multifractal formalism associated with pointwise Lt regularities was conjectured by Jaffard and Mélot in [46]. Let us recall it.Definition 4.
Fors∈R, p,t>0, the oscillation space Op,ts(Rd) (see [46]) is the space of tempered distributions f that satisfy (16)supj∈N0⁡2s-d/pj∑λ∈Λjlt,λp1/p<∞if0<p<∞supj∈N0,λ∈Λj⁡2sjlt,λifp=∞.The left-hand term defines the Op,ts(Rd)-seminorm.It is independent of the choice of the smooth enough wavelet basis or in the Schwartz class (see [53], Section 3.2).This space together with global notions (3) and (4) can also be defined locally; let (17)O=Ω⊂Rd:ΩisanonemptyboundedopensubsetofRd.For Ω∈O, set (18)EtΩh=x∈Ω;utx=hand (19)dtΩh=dimEtΩh.Clearly (20)dth=supΩ∈O⁡dtΩh.For Ω∈O and j∈N0, set (21)ΛjΩ=λ∈Λj:λ∩Ω≠∅.Definition 5.
Fors∈R, p,t>0, the oscillation space Op,ts(Ω) is the space of tempered distributions f that satisfy (22)supj∈N0⁡2s-d/pj∑λ∈ΛjΩlt,λp1/p<∞if0<p<∞supj∈N0,λ∈ΛjΩ⁡2sjlt,λifp=∞.Remark 6.
For this space, we can assume that functions and wavelets are compactly supported, with supportK that contains Ω. Then, there exists C>0, such that, at each scale j, there are at most C2dj dyadic cubes λ∈Λj(Ω) for which lt,λ does not vanish (see [17] in the proof of Proposition 10 p.36).For0<t,p<∞, the local (p,t)-oscillation exponent of f on Ω is given by (23)ζtΩp=sup⁡τ:f∈Op,tτ/pΩ=d+liminfj→∞⁡log⁡∑λ∈ΛjΩlt,λplog⁡2-j.The (p,t)-oscillation exponent of f is defined as (24)ζtp=infΩ∈O⁡ζtΩp.The t-multifractal formalism in [46] states that (25)dth=inf0<p<∞⁡d+hp-ζtp.The local t-multifractal formalism on Ω states that (26)dtΩh=inf0<p<∞⁡d+hp-ζtΩp.These formalisms yield upper bounds valid for any function (see either [46], Theorem 2, p. 561, or [53], Section 3.2); for t>0, define the upper t-set Bt(h) of f by (27)Bth=x;utx≤hand the upper t-spectrum by (28)h↦Dth=dimBth.Define the local upper t-set BtΩ(h) of f by (29)BtΩh=Bth∩Ωand the local upper t-spectrum on Ω by (30)h↦DtΩh=dimBtΩh.Then(31)∀t>0,ηt>0∀hdth≤Dth≤inf0<p<∞⁡d+hp-ζtpand (32)∀t>0,ηΩt>0∀hdtΩh≤DtΩh≤inf0<p<∞d+hp-ζtΩpwhere (33)ηΩt=d+liminfj→∞⁡log⁡∑λ∈ΛjΩCλtlog⁡2-j.As in (26), we state the t-multifractal formalism for upper t-sets as (34)Dth=inf0<p<∞⁡d+hp-ζtp.We also state the local t-multifractal formalism for upper t-sets on Ω as (35)DtΩh=inf0<p<∞⁡d+hp-ζtΩp.In [53], Leonarduzzi et al. have validated the t-multifractal formalism for some synthetic images and signals that include independent realizations of random processes.This paper is devoted to the study of the Baire generic validity of both local and globalt-multifractal formalisms in a given Sobolev or Besov space. Recall that Besov and Sobolev spaces are complete metrizable spaces [54]. Note that, if s>d/p then Bps,q(Rd)⊂Cs-d/p(Rd)⊂L∞(Rd) for all q. On the contrary, if s=d/p and q>1 (resp., s<d/p), Jaffard and Meyer [55] (resp., [28]) have proved that a function in Bps,q(Rd) can be infinite on a dense set and thus nowhere Hölder regular. Similarly, if s≤d/p, there exist functions in the usual Sobolev space (36)Lp,sRd=f∈LpRd;-Δs/2f∈LpRdwhich are everywhere locally unbounded [55].In this paper, we are interested in the Baire generic validity of thet-multifractal formalisms in Besov spaces Bp0s0,q0(Rd) for s0,q0,p0>0 and Sobolev spaces Lp0,s0(Rd) for s0>0 and 1≤p0<∞, under the condition s0-d/p0>-d/t. In the next section, we recall some tools from the theory of Besov spaces. We also add some embeddings between oscillation spaces and a relationship with the space BMO of functions of bounded mean oscillation. In the third section, we prove embeddings between Besov and oscillation spaces. We deduce a general lower bound for the (p,t)-oscillation exponent. In the fourth section, we show that the obtained lower bound is actually equality generically, in the sense of Baire categories, in a given Sobolev or Besov space. In the fifth section, we investigate the Baire generic validity of the t-multifractal formalisms. Finally, in the sixth section, we deduce a conclusion on the range of Baire validity of both the t-multifractal formalism and the t-multifractal formalism for upper t-sets.All generic results are studied locally onΩL=L+(0,1)d (L∈Zd) and also globally on Rd.Remark 7.
Since onlyt-wavelet leaders for j≥0 are needed in the values of pointwise Lt regularity and the (p,t)-exponent, then, from now on, we will identify functions that have the same wavelet coefficients Cλ for j≥0.


## 2. Besov, Sobolev, and Oscillation Spaces

### 2.1. Besov and Sobolev Spaces

Let us first recall some properties of Besov spaces (for details, see, for example, [54, 56, 57]). Given a function f∈S(Rd), its Fourier transform is denoted by f^. Let Ψ0∈S(Rd) with(37)Ψ0^ξ=1ifξ≤1,Ψ0^ξ=0ifξ≥2.For j∈N, let(38)Ψj^ξ=Ψ0^2-jξ-Ψ0^2-j-1ξ.Then the support of Ψj^ is included in the annulus {ξ;2j-1<ξ≤2j+1} and(39)∑j=0∞Ψj^=1is a partition of the unity. The Littlewood Paley definition of Besov spaces is the following.Definition 8.
Let0<p,q≤∞, s∈R. Then(40)Bps,qRd=f∈S′Rd;fBps,qRd<∞where(41)fBps,qRd=∑j=0∞2jsqf∗ΨjLpRdq1/qif0<q<∞supj∈N0⁡2jsf∗ΨjLpRdifq=∞.Note that .Bps,q(Rd) is a norm (quasinorm when p<1 or q<1) on Bps,q(Rd).Besov spaces do not depend on the choice ofΨ0. They are separable when both p and q are finite.Let us recall their wavelet characterizations; for a given sequence of scalar numbers(aλ)λ∈Λ (where Λ is as in (9)), let (42)aλλ∈Λbps,q=∑j=0∞2s-d/pj∑λ∈Λjaλp1/pq1/q,with the usual modification when p=∞ or q=∞ (i.e., 2(s-d/∞)j=2sj and the sums of pth or qth powers replaced by suprema over the same sets of indices), and (43)bps,q=aλλ∈Λ:aλλ∈Λbps,q<∞.Then, for the above wavelets if either rψ=∞ or rψ is large enough (rψ>max⁡s,2d/p+d/2-s in [57], Theorem 1.64, p. 48, and (rψ>s if p≥1, and rψ>max⁡{s,2d/p-d-s} if 0<p<1) in [50, 58]), then (44)I:f→Cλfλ∈Λis an isomorphic map from Bps,q(Rd) onto bps,q.Under the same condition onrψ, it is also proved that, for 1≤p<∞ and s>0, the Sobolev space Lp,s(Rd) given in (36) is characterized by (45)fLp,sRd≔∑j=0∞22sj∑λ∈ΛjCλ2χλ1/2LpRd<∞,where χλ denotes the characteristic function of the cube λ.Note that characterizations (44) and (45) do not depend on choice of the wavelet basis (see [50]).Besov and usual Sobolev spaces are closely related (see [12, 59]): (46)∀p≥1∀s>0Bps,1Rd↪Lp,sRd↪Bps,∞Rdand (47)∀p>1∀s>ε>0∀q>0Bps-ε,qRd↪Lp,sRd↪Bps+ε,qRd.The following embeddings hold (for example, see [60], Proposition 2.6, p. 245, Proposition 2.8, p. 245, and Theorem 2.14, p. 248, respectively):(48)∀0<q2≤q1bps,q2↪bps,q1,(49)∀s1≤s2bps2,q2↪bps1,q1,(50)∀0<p2≤p1bp2s,q↪bp1s+d/p1-d/p2,q.The following interpolation property holds:(51)∀0≤θ≤1bp1s1,q∩bp2s2,q↪bpθs1+1-θs2,q,where1p=θp1+1-θp2.Using the isomorphic (44), we obtain similar embeddings between Besov spaces.


### 2.2. Oscillation Spaces

Proposition 9.
The following embeddings between oscillation spaces hold:(52)∀p,t>0∀s1≤s2Op,ts2Rd↪Op,ts1Rd,(53)∀t>0∀0<p2≤p1Op2,tsRd↪Op1,ts+d/p1-d/p2Rd,and (54)∀0≤θ≤1Op1,ts1Rd∩Op2,ts2Rd↪Op,tθs1+1-θs2Rd,where1p=θp1+1-θp2.We also have (55)∀0<θ<1Op,t1s1Rd∩Op,t2s2Rd↪Op,θt1+1-θt2θt1/θt1+1-θt2s1+1-θt2/θt1+1-θt2s2Rd.And (56)∀p>0∀sOp,tsRd↪Bps,∞Rdand (57)O∞,20Rd=BMO.If O is as in (17), then the following local embedding holds: (58)∀Ω∈O∀t>0∀0<p1≤p2Op2,tsΩ↪Op1,tsΩ.Proof.
Embeddings (52) (resp., (53)) follows directly from (49) (resp., (50)). Embedding (54) follows from (51) and the equivalence (59)f∈Op,tsRd⇔lt,λλ∈Λ∈bps,∞.Let us now prove embedding (55). Write the t-wavelet leader of f at a cube λ∈Λj (given in (14)) as (60)lt,λ=2d/tjat,λ1/twhere (61)at,λ=∑j′=j∞2-dj′∑λ′⊂λCλ′t.If θ∈(0,1), then (62)aθt1+1-θt2,λ=∑j′=j∞∑λ′⊂λCλ′t12-dj′θCλ′t22-dj′1-θ.We know that, by Cauchy-Schwartz inequality, if P≥1 and 1/P+1/Q=1, then (63)∀Ai,Bi≥0∀∑AiBi≤∑AiP1/P∑BiQ1/Q.If P=1/θ then Q=1/(1-θ); then relation (63) applied to (62) with Ai=(|Cλ′|t12-dj′)θ and Bi=(|Cλ′|t22-dj′)1-θ (with i=(j′,λ′)) yields (64)aθt1+1-θt2,λ≤at1,λθat2,λ1-θ.Then(65)2djaθt1+1-θt2,λ≤2djat1,λθ2djat2,λ1-θ.It follows from (60) that (66)lθt1+1-θt2,λ≤lt1,λθt1/θt1+1-θt2lt2,λ1-θt2/θt1+1-θt2.Put (67)Stp,j≔2-dj∑λ∈ΛjΩlt,λp=2dp/t-1j∑λ∈ΛjΩat,λp/t.Then, as in (64), relation (66) yields(68)Sθt1+1-θt2p,j≤St1p,jθt1/θt1+1-θt2St2p,j1-θt2/θt1+1-θt2.This achieves the proof of embedding (55).
Embedding (56) follows from the fact that |Cλ|≤lt,λ.
With regard to (57), for p=∞, s=0, and t=2, (69)fO∞,20Rd=supj∈N0,λ∈Λj⁡l2,λ=supj∈N0,λ∈Λj⁡∑λ′⊂λCλ′2d/2j-j′21/2.It follows that (70)f∈O∞,20Rd⇔∃C>0:∀j∈N0∀λ∈Λj∑λ′⊂λCλ′22-dj′≤C2-dj.Thanks to our L∞ (resp., Meyer L2) normalization (7) for wavelets and (10), the right-hand term in (70) coincides with the Carleson condition for the wavelet characterization of the BMO space given in Theorem 4, p. 150-151 in [50].
Finally result (58) follows from Remark 6 and the Hölder inequality: (71)∑λ∈ΛjΩlt,λp1≤C2dj1-p1/p2∑λ∈ΛjΩlt,λp2p1/p2.Oscillation spacesOp,ts(Rd) were defined in terms of wavelet coefficients. They can be characterized by some Littlewood Paley conditions; moreover t-wavelet leaders can also be replaced by Lt local norms [61].Whent=∞, the t-wavelet leaders of f∈Lloc∞(Rd) given in (14) boil down to the classical wavelet leaders (72)l∞,λ=supλ′⊂λ⁡Cλ′used for the characterization of the Hölder exponent. The corresponding oscillation spaces Op,∞s(Rd) have been studied in [17, 62, 63] and denoted by Ops(Rd). Their characterizations by differences were obtained in [62]. Their independence of the chosen wavelet basis in the Schwartz class was obtained in [17]. For either s≥0 or s≤-d/p, it is proved in [63] that these spaces are a variation on the definition of Besov spaces. On the contrary, the spaces Ops(Rd) for -d/p<s<0 cannot be sharply imbedded between Besov spaces and thus are new spaces of really different nature.Generalized oscillation spacesOp,ts,s′(Rd) were also introduced in [46]; if s′>0 (resp., s′<0), then f belongs to Op,ts,s′(Rd) if its fractional derivative (resp., primitive) or order s′ which we denote by (-Δ)s′/2f (i.e., such that (-Δ)s′/2f^(ξ)=ξs′f^(ξ)) belongs to Op,ts(Rd). Spaces Op,∞s,s′(Rd) allow the computation of fractal dimension of graphs and yield a multifractal formalism for chirp-type Hölder singularities that behave like |x-x0|hsin⁡(1/|x-x0|β) (see [62]).


## 2.1. Besov and Sobolev Spaces

Let us first recall some properties of Besov spaces (for details, see, for example, [54, 56, 57]). Given a function f∈S(Rd), its Fourier transform is denoted by f^. Let Ψ0∈S(Rd) with(37)Ψ0^ξ=1ifξ≤1,Ψ0^ξ=0ifξ≥2.For j∈N, let(38)Ψj^ξ=Ψ0^2-jξ-Ψ0^2-j-1ξ.Then the support of Ψj^ is included in the annulus {ξ;2j-1<ξ≤2j+1} and(39)∑j=0∞Ψj^=1is a partition of the unity. The Littlewood Paley definition of Besov spaces is the following.Definition 8.
Let0<p,q≤∞, s∈R. Then(40)Bps,qRd=f∈S′Rd;fBps,qRd<∞where(41)fBps,qRd=∑j=0∞2jsqf∗ΨjLpRdq1/qif0<q<∞supj∈N0⁡2jsf∗ΨjLpRdifq=∞.Note that .Bps,q(Rd) is a norm (quasinorm when p<1 or q<1) on Bps,q(Rd).Besov spaces do not depend on the choice ofΨ0. They are separable when both p and q are finite.Let us recall their wavelet characterizations; for a given sequence of scalar numbers(aλ)λ∈Λ (where Λ is as in (9)), let (42)aλλ∈Λbps,q=∑j=0∞2s-d/pj∑λ∈Λjaλp1/pq1/q,with the usual modification when p=∞ or q=∞ (i.e., 2(s-d/∞)j=2sj and the sums of pth or qth powers replaced by suprema over the same sets of indices), and (43)bps,q=aλλ∈Λ:aλλ∈Λbps,q<∞.Then, for the above wavelets if either rψ=∞ or rψ is large enough (rψ>max⁡s,2d/p+d/2-s in [57], Theorem 1.64, p. 48, and (rψ>s if p≥1, and rψ>max⁡{s,2d/p-d-s} if 0<p<1) in [50, 58]), then (44)I:f→Cλfλ∈Λis an isomorphic map from Bps,q(Rd) onto bps,q.Under the same condition onrψ, it is also proved that, for 1≤p<∞ and s>0, the Sobolev space Lp,s(Rd) given in (36) is characterized by (45)fLp,sRd≔∑j=0∞22sj∑λ∈ΛjCλ2χλ1/2LpRd<∞,where χλ denotes the characteristic function of the cube λ.Note that characterizations (44) and (45) do not depend on choice of the wavelet basis (see [50]).Besov and usual Sobolev spaces are closely related (see [12, 59]): (46)∀p≥1∀s>0Bps,1Rd↪Lp,sRd↪Bps,∞Rdand (47)∀p>1∀s>ε>0∀q>0Bps-ε,qRd↪Lp,sRd↪Bps+ε,qRd.The following embeddings hold (for example, see [60], Proposition 2.6, p. 245, Proposition 2.8, p. 245, and Theorem 2.14, p. 248, respectively):(48)∀0<q2≤q1bps,q2↪bps,q1,(49)∀s1≤s2bps2,q2↪bps1,q1,(50)∀0<p2≤p1bp2s,q↪bp1s+d/p1-d/p2,q.The following interpolation property holds:(51)∀0≤θ≤1bp1s1,q∩bp2s2,q↪bpθs1+1-θs2,q,where1p=θp1+1-θp2.Using the isomorphic (44), we obtain similar embeddings between Besov spaces.


## 2.2. Oscillation Spaces

Proposition 9.
The following embeddings between oscillation spaces hold:(52)∀p,t>0∀s1≤s2Op,ts2Rd↪Op,ts1Rd,(53)∀t>0∀0<p2≤p1Op2,tsRd↪Op1,ts+d/p1-d/p2Rd,and (54)∀0≤θ≤1Op1,ts1Rd∩Op2,ts2Rd↪Op,tθs1+1-θs2Rd,where1p=θp1+1-θp2.We also have (55)∀0<θ<1Op,t1s1Rd∩Op,t2s2Rd↪Op,θt1+1-θt2θt1/θt1+1-θt2s1+1-θt2/θt1+1-θt2s2Rd.And (56)∀p>0∀sOp,tsRd↪Bps,∞Rdand (57)O∞,20Rd=BMO.If O is as in (17), then the following local embedding holds: (58)∀Ω∈O∀t>0∀0<p1≤p2Op2,tsΩ↪Op1,tsΩ.Proof.
Embeddings (52) (resp., (53)) follows directly from (49) (resp., (50)). Embedding (54) follows from (51) and the equivalence (59)f∈Op,tsRd⇔lt,λλ∈Λ∈bps,∞.Let us now prove embedding (55). Write the t-wavelet leader of f at a cube λ∈Λj (given in (14)) as (60)lt,λ=2d/tjat,λ1/twhere (61)at,λ=∑j′=j∞2-dj′∑λ′⊂λCλ′t.If θ∈(0,1), then (62)aθt1+1-θt2,λ=∑j′=j∞∑λ′⊂λCλ′t12-dj′θCλ′t22-dj′1-θ.We know that, by Cauchy-Schwartz inequality, if P≥1 and 1/P+1/Q=1, then (63)∀Ai,Bi≥0∀∑AiBi≤∑AiP1/P∑BiQ1/Q.If P=1/θ then Q=1/(1-θ); then relation (63) applied to (62) with Ai=(|Cλ′|t12-dj′)θ and Bi=(|Cλ′|t22-dj′)1-θ (with i=(j′,λ′)) yields (64)aθt1+1-θt2,λ≤at1,λθat2,λ1-θ.Then(65)2djaθt1+1-θt2,λ≤2djat1,λθ2djat2,λ1-θ.It follows from (60) that (66)lθt1+1-θt2,λ≤lt1,λθt1/θt1+1-θt2lt2,λ1-θt2/θt1+1-θt2.Put (67)Stp,j≔2-dj∑λ∈ΛjΩlt,λp=2dp/t-1j∑λ∈ΛjΩat,λp/t.Then, as in (64), relation (66) yields(68)Sθt1+1-θt2p,j≤St1p,jθt1/θt1+1-θt2St2p,j1-θt2/θt1+1-θt2.This achieves the proof of embedding (55).
Embedding (56) follows from the fact that |Cλ|≤lt,λ.
With regard to (57), for p=∞, s=0, and t=2, (69)fO∞,20Rd=supj∈N0,λ∈Λj⁡l2,λ=supj∈N0,λ∈Λj⁡∑λ′⊂λCλ′2d/2j-j′21/2.It follows that (70)f∈O∞,20Rd⇔∃C>0:∀j∈N0∀λ∈Λj∑λ′⊂λCλ′22-dj′≤C2-dj.Thanks to our L∞ (resp., Meyer L2) normalization (7) for wavelets and (10), the right-hand term in (70) coincides with the Carleson condition for the wavelet characterization of the BMO space given in Theorem 4, p. 150-151 in [50].
Finally result (58) follows from Remark 6 and the Hölder inequality: (71)∑λ∈ΛjΩlt,λp1≤C2dj1-p1/p2∑λ∈ΛjΩlt,λp2p1/p2.Oscillation spacesOp,ts(Rd) were defined in terms of wavelet coefficients. They can be characterized by some Littlewood Paley conditions; moreover t-wavelet leaders can also be replaced by Lt local norms [61].Whent=∞, the t-wavelet leaders of f∈Lloc∞(Rd) given in (14) boil down to the classical wavelet leaders (72)l∞,λ=supλ′⊂λ⁡Cλ′used for the characterization of the Hölder exponent. The corresponding oscillation spaces Op,∞s(Rd) have been studied in [17, 62, 63] and denoted by Ops(Rd). Their characterizations by differences were obtained in [62]. Their independence of the chosen wavelet basis in the Schwartz class was obtained in [17]. For either s≥0 or s≤-d/p, it is proved in [63] that these spaces are a variation on the definition of Besov spaces. On the contrary, the spaces Ops(Rd) for -d/p<s<0 cannot be sharply imbedded between Besov spaces and thus are new spaces of really different nature.Generalized oscillation spacesOp,ts,s′(Rd) were also introduced in [46]; if s′>0 (resp., s′<0), then f belongs to Op,ts,s′(Rd) if its fractional derivative (resp., primitive) or order s′ which we denote by (-Δ)s′/2f (i.e., such that (-Δ)s′/2f^(ξ)=ξs′f^(ξ)) belongs to Op,ts(Rd). Spaces Op,∞s,s′(Rd) allow the computation of fractal dimension of graphs and yield a multifractal formalism for chirp-type Hölder singularities that behave like |x-x0|hsin⁡(1/|x-x0|β) (see [62]).


## 3. General Lower Bound on the(p,t)-Oscillation Exponent

The following general lower bounds of both local and global(p,t)-oscillation exponents hold.Theorem 10.
Letf be any function in either the Besov space Bp0s0,q0(Rd) for s0,q0,p0>0 or the Sobolev space Lp0,s0(Rd) for s0>0 and p0≥1. Then, for all t>0 such that s0-d/p0>-d/t, (73)∀Ω∈OζtΩp≥ps0-dp0+d∀p≥p0ps0∀p≤p0and (74)ζtp≥ps0-dp0+d∀p≥p0ps0∀p≤p0.Proof.For Ω∈O, define the local Besov space Bps,q(Ω) (resp., Sobolev space Lps(Ω)) as in (44) (resp., (45)) with Λ replaced by ⋃j∈N0Λj(Ω) (resp., Λj replaced by Λj(Ω)).Result (73) will follow from (23) and the following proposition. Result (74) will be deduced from (24).Proposition 11.
LetΩ∈O. The following embeddings hold for all s0,q0,p0,t>0 such that s0-d/p0>-d/t:(75)Ift≥p0thenBp0s0,q0Ω↪Op,ts0-d/p0+d/pΩifp≥p0Bp0s0,q0Ω↪Op,ts0Ω∀p≤p0and (76)ift<p0then∀ε>0Bp0s0,q0Ω↪Op,ts0-d/p0+d/p-εΩ∀p≥p0Bp0s0,q0Ω↪Op,ts0-εΩ∀p≤p0.Embeddings between similar local Sobolev and oscillation spaces also hold (using (46) and (47)).Proof.
Letf∈Bp0s0,q0(Ω) for s0,q0,p0>0 and t>0 such that s0-d/p0>-d/t.
( 1 ) Assume that t≥p0. If at,λ is as in (61), then using the property(77)∀xn≥0∀p≥1∑xnp≤∑xnp
we get(78)at,λ1/t≤∑j′=j∞2-dp0/tj′∑λ′⊂λCλ′p01/p0.(a)
Ifp≥p0, then by Remark 6, the quantity St(p,j) given in (67) satisfies(79)Stp,j≤C2dp/t-1j∑λ∈ΛjΩ∑j′=j∞∑λ′⊂λCλ′p02-dp0/tj′p/p0.
Clearly(80)∑λ∈ΛjΩ∑j′=j∞∑λ′⊂λCλ′p02-dp0/tj′=∑j′=j∞∑λ′∈Λj′ΩCλ′p02-dp0/tj′.
Sincef∈Bp0s0,q0(Ω), then f∈Bp0s0,∞(Ω). There exists j0 such that (81)∀j′≥j02-dj′∑λ′∈Λj′ΩCλ′p0≤C2-s0p0j′.
It follows that for allj≥j0(82)Stp,j≤C2dp/t-1j∑j′=j∞2d1-p0/tj′2-s0p0j′p/p0.
Sinces0-d/p0>-d/t, then(83)Stp,j≤C2-d+d-s0p0p/p0j.Thus(84)f∈Op,ts0-d/p0+d/pΩ.(b)
Let nowp≤p0.(85)∀0≤r≤1∑λ∈ΛjΩlt,λrp≥C2dj1-r∑λ∈ΛjΩlt,λpr.
It follows that(86)Strp,j≥CStp,jr.
From 1.(a) sinces0>0, then f∈Ot,p0s0(Ω). Then(87)Stp0,j≤C2-s0p0j.
By takingr=p0/p, we deduce that(88)Stp,j≤C2-s0pj.
Thus(89)f∈Op,ts0Ω.
( 2 ) Assume that t<p0. Since s0-d/p0>-d/t, then there exists t′>t such that s0-d/p0=-d/t′. It follows from the fact that s0>0 that t′>p0. Let t1∈(p0,t′) and t2<t. Since t<p0 write t=θt1+(1-θ)t2.Sincet1<p′, then s0-d/p0>-d/t1. Since t1>p0, then from above 1.(a) yields(90)f∈Op,t1s0-d/p0+d/pΩifp≥p0f∈Op,t1s0Ωifp<p0.
On the other hand, sincet2<p0 then f∈Op,t2s0-d/p0(Ω). In fact (91)at2,λ≤C∑j′=j∞2d1-t2/p0j′-j∑λ′⊂λCλ′p0t2/p02-dj′=C2-d1-t2/p0j∑j′=j∞2-dj′∑λ′⊂λCλ′p0t2/p0.
Sincef∈Bp0s0,q0(Ω), then from (81) we get(92)∀j≥j0∀λ∈Λjat2,λ≤C2-d1-t2/p0j∑j′=j∞2-t2s0j′.
Sinces0>0 then(93)∀j≥j0∀λ∈Λjat2,λ≤C2-d1-t2/p0j2-t2s0j.It follows from (60) that(94)∀j≥j0∀λ∈Λjlt2,λ≤C2-s0-d/p0j.Then(95)St2p,j≤C2-ps0-d/p0j.Therefore (96)f∈Op,t2s0-d/p0Ω.Using (54), both (90) and (96) yield (97)f∈Op,tθt1/ts0-d/p0+d/p+1-θt2/ts0-d/p0Ω=Op,tθt1/td/p+s0-d/p0Ωifp≥p0f∈Op,tθt1/ts0+1-θt2/ts0-d/p0Ω=Op,ts0-1-θt2/td/p0Ωifp≤p0.When t2 tends to 0, fraction (1-θ)t2/t tends to 0 too, and θt1/t tends to 1. Hence (76) holds.


## 4. Generic(p,t)-Oscillation Exponent

For a given Besov or Sobolev space, the residual set that we will construct will be generated from a saturating functionF (i.e., for which the lower bounds obtained in Theorem 10 become equality). Thanks to embedding (46), we can choose for saturating function for the Sobolev space the one obtained for the Besov space Bp0s0,1(Rd).


### 4.1. Saturating Function

Forλ=k2-j+[0,2-j)d, let J≤j be the unique integer given by the irreducible representation(98)k2-j=K2-JwithK∈Zd-2Zd.For L=(l1,…,ld)∈Zd, put (99)L=l1+⋯+ld,ΩL=L+0,1d.The following wavelet series will be called a saturating function:(100)F=∑L∈Zd2-LFLwhere(101)FL=∑j=1∞∑λ∈ΛjΩLj-a2d/p0-s0j2-d/p0Jψλand a=2/p0+2/q0+1.Remark 12.
Ifλ=k2-j+[0,2-j)d∈Λj(Ω0) and L∈Zd then L+λ≔L+k2-j+[0,2-j)d∈Λj(ΩL). Both λ and L+λ share the same J. The previous function F satisfies(102)CL+λ=2-LCλ.This yields(103)lt,L+λ=2-Llt,λ.It follows that (104)ζtΩLp=ζtΩ0p.It is easy to show thatF0∈Bp0s0,q0(Ω0) (see [15], Proposition 2, p. 532). Relation (102) yields that FLBp0s0,q0(ΩL)=2-|L|F0Bp0s0,q0(Ω0). Note that the norm of the local Besov space is as the global one but with restriction on (105)ΛΩ=⋃j∈N0ΛjΩ.Therefore F∈Bp0s0,q0(Rd) and FBp0s0,q0(Rd)=F0Bp0s0,q0(Ω0).Theorem 13.
Lets0,q0,p0>0 and t>0 be such that s0-d/p0>-d/t. Then the saturating function F satisfies (106)∀L∈Zdζtp=ζtΩLp=ps0-dp0+d∀p≥p0ps0∀p≤p0.Proof. Thanks to both (104) and Theorem 10, it suffices to show that (107)ζtΩ0p≤ps0-dp0+d∀p≥p0ps0∀p≤p0.We know that, for j∈N, (108)ΛjΩ0=λ=k2-j+0,2-jdwithk∈0,…,2j-1d.It is easy to show that (109)∀p>0ηΩ0p=d+ps0-dp0ifp≥p0s0pifp≤p0(the proof is the same as in [15], Proposition 4, p. 540-541, and we do not need the assumption s0>d/p0).The following result estimates thet-wavelet leaders of F0 and allows the computation of its local (p,t)- oscillation exponent on Ω0.Proposition 14.
Lets0,p0,q0>0 and t>0 such that s0-d/p0>-d/t. Let F0 be as in (101). Then there exist two positive constants C1(t) and C2(t) such that (110)∀ε>0∃jε∀j≥jε∀λ∈ΛjΩ0C1t2-εjjaCλ≤lt,λ≤C2tjaCλ.Proof.
Forλ′⊂λ, either J′=J or J′>j. There exists C>0 such that, at each scale J′>j, there are C2d(J′-j) subdyadic cubes λ′ of λ that have the irreducible representation K′2-J′. Then (111)at,λ=∑j′=j∞∑λ′⊂λj′-at2d/p0-s0tj′2-dt/p0J′2-dj′=∑j′=j∞j′-at2d/p0-s0t-dj′2-dt/p0J+∑J′=j+1∞2dJ′-j∑j′=J′∞j′-at2d/p0-s0t-dj′2-dt/p0J′.The left-hand series corresponds to λ′=k2-j+[0,2-j′)d if λ=k2-j+[0,2-j)d.
Consequently, there existshj such that(112)at,λ=hj∑j′=j∞2d/p0-s0t-dj′2-dt/p0J+∑J′=j+1∞2dJ′-j∑j′=J′∞2d/p0-s0t-dj′2-dt/p0J′with(113)∀jhj≤1and(114)∀ε>0∃jε∀j≥jεhj≥2-tεj.Since s0-d/p0>-d/t, we get(115)at,λ≈hj2d/p0-s0t-dj2-dt/p0J+2-s0t+dj≈hj2-s0t+dj2dt/p0j-Jwhere ≈ means that the left quantity is bounded from below and above by positive constants times the right quantity.
It follows that(116)lt,λ=2d/tjaλ1/t≈hj1/t2d/p0-s0j2-d/p0J.Hence (110) holds.Relation (110) together with (23) yields (107).


### 4.2. The Residual Set

We will need the following lemma.Lemma 15.
Lets0,p0,q0>0 and t>0 such that s0-d/p0>-d/t. Then, for all Ω∈O, for all f∈Bp0s0,q0(Ω), (117)∀j∀λ∈ΛjΩlt,λ≤Ct2d/p0-s0jfBp0s0,q0Ωwhere C(t)=max⁡1/1-2s0t,1/1-2-d(1-t/p0)-s0t1/t.Proof.
LetΩ∈O and f∈Bp0s0,q0(Ω). Write f instead of fBp0s0,q0(Ω). Clearly (118)∀j∑λ∈ΛjΩCλp0≤2d-s0p0jfp0.Let λ∈Λj(Ω) and t>0.(i)
Suppose thatt≤p0. Then(119)∑λ′∈Λj′,λ′⊂λCλ′t2-dj′≤2-dj′2d1-t/p0j′-j∑λ′∈Λj′,λ′⊂λCλ′p0t/p0.Using (118) (120)∑λ′∈Λj′,λ′⊂λCλ′t2-dj′≤2-d1-t/p0j2-dt/p0j′2d-s0p0j′fp0t/p0≤2-d1-t/p0j2-s0tj′ft.Then(121)at,λ≤2-d1-t/p0j∑j′=j∞2-s0tj′ft≤C1t2-d1-t/p0-s0tjftwhereC1(t)=1/1-2s0t.We therefore get(122)lt,λ≤C1t1/t2d/p0-s0jf.(ii)
Ift>p0 and s0-d/p0>-d/t, then (123)∑λ′∈Λj′,λ′⊂λCλ′t2-dj′≤2-dj′∑λ′∈Λj′,λ′⊂λCλ′p0t/p0≤2-dj′2d-s0p0j′fp0t/p0=2-d1-t/p0-s0tj′ft.Sinces0-d/p0>-d/t, we get(124)at,λ≤∑j′=j∞2-d1-t/p0-s0tj′ft≤C2t2-d1-t/p0-s0tjftwhereC2(t)=1/1-2-d(1-t/p0)-s0t.Then(125)lt,λ≤C2t1/t2d/p0-s0jf.Therefore (117) holds with C(t)=maxC1(t),C2(t)1/t.
Hence Lemma15 holds.Theorem 16.
Lets0,p0,q0>0 (resp., s0>0 and p0≥1) and t>0 such that s0-d/p0>-d/t. (1)
For allL∈Zd, there exists a residual set A(ΩL) of Bp0s0,q0(ΩL) (resp., Lp0,s0(ΩL)) such that for all f∈A(ΩL)(126)∀p>0ζtΩLp=d+ps0-dp0ifp≥p0s0pifp≤p0
and(127)inf0<p<∞⁡d+hp-ζtΩLp=-∞ifh<s0-dp0p0h-s0+difs0-dp0≤h≤s0difh>s0.(2)
There exists a residual setA of Bp0s0,q0(Rd) (resp., Lp0,s0(Rd)) such that for all f∈A(128)∀p>0ζtp=d+ps0-dp0ifp≥p0s0pifp≤p0and(129)inf0<p<∞⁡d+hp-ζtp=-∞ifh<s0-dp0p0h-s0+difs0-dp0≤h≤s0difh>s0.Relations (126) and (127) also hold on A.Proof.
(1)
(i) First considerL=0. By Theorem 10, it suffices to prove the upper bound in (126). Thanks to embedding (46), it suffices to write the proof in Besov spaces. We will follow the idea of [64] in the construction of the residual set since it does not depend on the separability of the space Bp0s0,q0(Ω0). Let F0 be the saturating function defined in (101). From now on, when it is necessary, we will make the dependency on the function in the previous notations (for example, we write Cλ(f) instead of Cλ in (10), lt,λ(f) instead of lt,λ, and ζt,fΩ0(p) instead of ζtΩ0(p)). Ifn∈N, set (130)EnΩ0=gn∈Bp0s0,q0Ω0:∀λ∈ΛΩ0∃M∈Z∖0:Cλgn=MnCλF0, whereΛ(Ω) is as in (105). Clearly(131)∀n∀gn∈EnΩ0∀λ∈ΛΩ0lt,λgn≥1nlt,λF0. Writef instead of fBp0s0,q0(Ω0). In [64] (Proof of Lemma 2.9, p. 1520-1521), it is proved that there exists C>0 such that (132)∀f∈Bp0s0,q0Ω0∀n∈N∃gn∈EnΩ0f-gn≤Cn, and so, ifN∈N, then the set DN(Ω0)≔⋃n≥NEn(Ω0) is dense in Bp0s0,q0(Ω0). Let(rn)n be a sequence of positive numbers which converges to 0. Put (133)AnΩ0=f∈Bp0s0,q0Ω0:∃gn∈EnΩ0f-gn<rn. Then(134)AΩ0≔⋂N∈N⋃n≥NAnΩ0 is a residual set ofBp0s0,q0(Ω0). IfC~(t)=max⁡(21/t-1,1), then the following trivial relation holds: (135)∀f,g∀λlt,λf+g≤C~tlt,λf+lt,λg.Choose(136)rn=C1t2n2aC~tCt2-dn/p0 whereC(t) is as in Lemma 15 and C1(t) is any constant satisfying (110). Let A(Ω0) be the residual set given by (141) associated with the sequence (rn). Letf∈A(Ω0). Then for infinitely many integers n, there exists gn∈En(Ω0) such that f-gn<rn. By (135)(137)∀λ∈Λlt,λf≥1C~tlt,λgn-lt,λgn-f. By Lemma15 and Proposition 14,(138)∀λ∈ΛnΩ0lt,λgn-f≤Ct2d/p0-s0nf-gn≤Ct2d/p0-s0nrn≤12naC~tlt,λF0. It follows from (131) that (139)∀λ∈ΛnΩ0lt,λf≥12nC~tlt,λF0. Since (139) holds for infinitely many scales n and for all λ∈Λn, and thanks to the fact that the liminf in ζt,F0(p) is actually a limit, we deduce that(140)∀p>0ζt,fΩ0p≤ζt,F0Ωp. Result (106) allows achieving the proof of Theorem 16. (ii)
Now ifL≠0, it suffices to replace F0 by FL defined in (101) and rn by 2-|L|rn. This means that the residual set of Bp0s0,q0(Ω0) on which (126) will hold is (141)AΩL≔⋂N∈N⋃n≥NAnΩL, where(142)AnΩL=f∈Bp0s0,q0ΩL:∃gn∈EnΩLf-gnBp0s0,q0ΩL<2-Lrn and(143)EnΩL=gn∈Bp0s0,q0ΩL:∀λ∈ΛΩL∃M∈Z∖0:Cλgn=MnCλFL.(2)
Clearly(144)A=⋂L∈ZdAΩL is a residual set ofBp0s0,q0(Rd) on which (128) and (126) hold (using (24) and the first result in this theorem).


## 4.1. Saturating Function

Forλ=k2-j+[0,2-j)d, let J≤j be the unique integer given by the irreducible representation(98)k2-j=K2-JwithK∈Zd-2Zd.For L=(l1,…,ld)∈Zd, put (99)L=l1+⋯+ld,ΩL=L+0,1d.The following wavelet series will be called a saturating function:(100)F=∑L∈Zd2-LFLwhere(101)FL=∑j=1∞∑λ∈ΛjΩLj-a2d/p0-s0j2-d/p0Jψλand a=2/p0+2/q0+1.Remark 12.
Ifλ=k2-j+[0,2-j)d∈Λj(Ω0) and L∈Zd then L+λ≔L+k2-j+[0,2-j)d∈Λj(ΩL). Both λ and L+λ share the same J. The previous function F satisfies(102)CL+λ=2-LCλ.This yields(103)lt,L+λ=2-Llt,λ.It follows that (104)ζtΩLp=ζtΩ0p.It is easy to show thatF0∈Bp0s0,q0(Ω0) (see [15], Proposition 2, p. 532). Relation (102) yields that FLBp0s0,q0(ΩL)=2-|L|F0Bp0s0,q0(Ω0). Note that the norm of the local Besov space is as the global one but with restriction on (105)ΛΩ=⋃j∈N0ΛjΩ.Therefore F∈Bp0s0,q0(Rd) and FBp0s0,q0(Rd)=F0Bp0s0,q0(Ω0).Theorem 13.
Lets0,q0,p0>0 and t>0 be such that s0-d/p0>-d/t. Then the saturating function F satisfies (106)∀L∈Zdζtp=ζtΩLp=ps0-dp0+d∀p≥p0ps0∀p≤p0.Proof. Thanks to both (104) and Theorem 10, it suffices to show that (107)ζtΩ0p≤ps0-dp0+d∀p≥p0ps0∀p≤p0.We know that, for j∈N, (108)ΛjΩ0=λ=k2-j+0,2-jdwithk∈0,…,2j-1d.It is easy to show that (109)∀p>0ηΩ0p=d+ps0-dp0ifp≥p0s0pifp≤p0(the proof is the same as in [15], Proposition 4, p. 540-541, and we do not need the assumption s0>d/p0).The following result estimates thet-wavelet leaders of F0 and allows the computation of its local (p,t)- oscillation exponent on Ω0.Proposition 14.
Lets0,p0,q0>0 and t>0 such that s0-d/p0>-d/t. Let F0 be as in (101). Then there exist two positive constants C1(t) and C2(t) such that (110)∀ε>0∃jε∀j≥jε∀λ∈ΛjΩ0C1t2-εjjaCλ≤lt,λ≤C2tjaCλ.Proof.
Forλ′⊂λ, either J′=J or J′>j. There exists C>0 such that, at each scale J′>j, there are C2d(J′-j) subdyadic cubes λ′ of λ that have the irreducible representation K′2-J′. Then (111)at,λ=∑j′=j∞∑λ′⊂λj′-at2d/p0-s0tj′2-dt/p0J′2-dj′=∑j′=j∞j′-at2d/p0-s0t-dj′2-dt/p0J+∑J′=j+1∞2dJ′-j∑j′=J′∞j′-at2d/p0-s0t-dj′2-dt/p0J′.The left-hand series corresponds to λ′=k2-j+[0,2-j′)d if λ=k2-j+[0,2-j)d.
Consequently, there existshj such that(112)at,λ=hj∑j′=j∞2d/p0-s0t-dj′2-dt/p0J+∑J′=j+1∞2dJ′-j∑j′=J′∞2d/p0-s0t-dj′2-dt/p0J′with(113)∀jhj≤1and(114)∀ε>0∃jε∀j≥jεhj≥2-tεj.Since s0-d/p0>-d/t, we get(115)at,λ≈hj2d/p0-s0t-dj2-dt/p0J+2-s0t+dj≈hj2-s0t+dj2dt/p0j-Jwhere ≈ means that the left quantity is bounded from below and above by positive constants times the right quantity.
It follows that(116)lt,λ=2d/tjaλ1/t≈hj1/t2d/p0-s0j2-d/p0J.Hence (110) holds.Relation (110) together with (23) yields (107).


## 4.2. The Residual Set

We will need the following lemma.Lemma 15.
Lets0,p0,q0>0 and t>0 such that s0-d/p0>-d/t. Then, for all Ω∈O, for all f∈Bp0s0,q0(Ω), (117)∀j∀λ∈ΛjΩlt,λ≤Ct2d/p0-s0jfBp0s0,q0Ωwhere C(t)=max⁡1/1-2s0t,1/1-2-d(1-t/p0)-s0t1/t.Proof.
LetΩ∈O and f∈Bp0s0,q0(Ω). Write f instead of fBp0s0,q0(Ω). Clearly (118)∀j∑λ∈ΛjΩCλp0≤2d-s0p0jfp0.Let λ∈Λj(Ω) and t>0.(i)
Suppose thatt≤p0. Then(119)∑λ′∈Λj′,λ′⊂λCλ′t2-dj′≤2-dj′2d1-t/p0j′-j∑λ′∈Λj′,λ′⊂λCλ′p0t/p0.Using (118) (120)∑λ′∈Λj′,λ′⊂λCλ′t2-dj′≤2-d1-t/p0j2-dt/p0j′2d-s0p0j′fp0t/p0≤2-d1-t/p0j2-s0tj′ft.Then(121)at,λ≤2-d1-t/p0j∑j′=j∞2-s0tj′ft≤C1t2-d1-t/p0-s0tjftwhereC1(t)=1/1-2s0t.We therefore get(122)lt,λ≤C1t1/t2d/p0-s0jf.(ii)
Ift>p0 and s0-d/p0>-d/t, then (123)∑λ′∈Λj′,λ′⊂λCλ′t2-dj′≤2-dj′∑λ′∈Λj′,λ′⊂λCλ′p0t/p0≤2-dj′2d-s0p0j′fp0t/p0=2-d1-t/p0-s0tj′ft.Sinces0-d/p0>-d/t, we get(124)at,λ≤∑j′=j∞2-d1-t/p0-s0tj′ft≤C2t2-d1-t/p0-s0tjftwhereC2(t)=1/1-2-d(1-t/p0)-s0t.Then(125)lt,λ≤C2t1/t2d/p0-s0jf.Therefore (117) holds with C(t)=maxC1(t),C2(t)1/t.
Hence Lemma15 holds.Theorem 16.
Lets0,p0,q0>0 (resp., s0>0 and p0≥1) and t>0 such that s0-d/p0>-d/t. (1)
For allL∈Zd, there exists a residual set A(ΩL) of Bp0s0,q0(ΩL) (resp., Lp0,s0(ΩL)) such that for all f∈A(ΩL)(126)∀p>0ζtΩLp=d+ps0-dp0ifp≥p0s0pifp≤p0
and(127)inf0<p<∞⁡d+hp-ζtΩLp=-∞ifh<s0-dp0p0h-s0+difs0-dp0≤h≤s0difh>s0.(2)
There exists a residual setA of Bp0s0,q0(Rd) (resp., Lp0,s0(Rd)) such that for all f∈A(128)∀p>0ζtp=d+ps0-dp0ifp≥p0s0pifp≤p0and(129)inf0<p<∞⁡d+hp-ζtp=-∞ifh<s0-dp0p0h-s0+difs0-dp0≤h≤s0difh>s0.Relations (126) and (127) also hold on A.Proof.
(1)
(i) First considerL=0. By Theorem 10, it suffices to prove the upper bound in (126). Thanks to embedding (46), it suffices to write the proof in Besov spaces. We will follow the idea of [64] in the construction of the residual set since it does not depend on the separability of the space Bp0s0,q0(Ω0). Let F0 be the saturating function defined in (101). From now on, when it is necessary, we will make the dependency on the function in the previous notations (for example, we write Cλ(f) instead of Cλ in (10), lt,λ(f) instead of lt,λ, and ζt,fΩ0(p) instead of ζtΩ0(p)). Ifn∈N, set (130)EnΩ0=gn∈Bp0s0,q0Ω0:∀λ∈ΛΩ0∃M∈Z∖0:Cλgn=MnCλF0, whereΛ(Ω) is as in (105). Clearly(131)∀n∀gn∈EnΩ0∀λ∈ΛΩ0lt,λgn≥1nlt,λF0. Writef instead of fBp0s0,q0(Ω0). In [64] (Proof of Lemma 2.9, p. 1520-1521), it is proved that there exists C>0 such that (132)∀f∈Bp0s0,q0Ω0∀n∈N∃gn∈EnΩ0f-gn≤Cn, and so, ifN∈N, then the set DN(Ω0)≔⋃n≥NEn(Ω0) is dense in Bp0s0,q0(Ω0). Let(rn)n be a sequence of positive numbers which converges to 0. Put (133)AnΩ0=f∈Bp0s0,q0Ω0:∃gn∈EnΩ0f-gn<rn. Then(134)AΩ0≔⋂N∈N⋃n≥NAnΩ0 is a residual set ofBp0s0,q0(Ω0). IfC~(t)=max⁡(21/t-1,1), then the following trivial relation holds: (135)∀f,g∀λlt,λf+g≤C~tlt,λf+lt,λg.Choose(136)rn=C1t2n2aC~tCt2-dn/p0 whereC(t) is as in Lemma 15 and C1(t) is any constant satisfying (110). Let A(Ω0) be the residual set given by (141) associated with the sequence (rn). Letf∈A(Ω0). Then for infinitely many integers n, there exists gn∈En(Ω0) such that f-gn<rn. By (135)(137)∀λ∈Λlt,λf≥1C~tlt,λgn-lt,λgn-f. By Lemma15 and Proposition 14,(138)∀λ∈ΛnΩ0lt,λgn-f≤Ct2d/p0-s0nf-gn≤Ct2d/p0-s0nrn≤12naC~tlt,λF0. It follows from (131) that (139)∀λ∈ΛnΩ0lt,λf≥12nC~tlt,λF0. Since (139) holds for infinitely many scales n and for all λ∈Λn, and thanks to the fact that the liminf in ζt,F0(p) is actually a limit, we deduce that(140)∀p>0ζt,fΩ0p≤ζt,F0Ωp. Result (106) allows achieving the proof of Theorem 16. (ii)
Now ifL≠0, it suffices to replace F0 by FL defined in (101) and rn by 2-|L|rn. This means that the residual set of Bp0s0,q0(Ω0) on which (126) will hold is (141)AΩL≔⋂N∈N⋃n≥NAnΩL, where(142)AnΩL=f∈Bp0s0,q0ΩL:∃gn∈EnΩLf-gnBp0s0,q0ΩL<2-Lrn and(143)EnΩL=gn∈Bp0s0,q0ΩL:∀λ∈ΛΩL∃M∈Z∖0:Cλgn=MnCλFL.(2)
Clearly(144)A=⋂L∈ZdAΩL is a residual set ofBp0s0,q0(Rd) on which (128) and (126) hold (using (24) and the first result in this theorem).


## 5. Generic Validity of thet-Multifractal Formalisms

We first show that conditionη(t)>0 in both result (15) and upper bound (31) holds for all functions in Besov spaces Bp0s0,q0(Rd) for s0,q0,p0>0 and Sobolev spaces Lp0,s0(Rd) for s0>0 and p0≥1, for all t>0 such that s0-d/p0>-d/t.Lemma 17.
Letf be any function in either the Besov space Bp0s0,q0(Rd) for s0,q0,p0>0 or the Sobolev space Lp0,s0(Rd) for s0>0 and p0≥1. Then, for all t>0 such that s0-d/p0>-d/t,(145)ηt>0.Proof.
Embedding (56) implies that η(p)≥ζt(p). It follows that η(t)≥ζt(t). Since s0-d/p0>-d/t, then Theorem 10 yields η(t)>0.Lemma17 together with both Theorem 10 and upper bound (31) yields the following corollary.Corollary 18.
Letf be any function in either the Besov space Bp0s0,q0(Rd) for s0,q0,p0>0 or the Sobolev space Lp0,s0(Rd) for s0>0 and p0≥1. Then, for all t>0 such that s0-d/p0>-d/t(146)∀hdth≤Dth≤inf0<p<∞d+hq-ζtp=-∞ifh<s0-dp0≤p0h-s0+difs0-dp0≤h≤s0≤difh>s0.The following result shows that the Baire generict-spectrum is the same as the prevalent generic one obtained in [49].Theorem 19.
Lets0,p0,q0>0 (resp., s0>0 and p0≥1) and t>0 such that s0-d/p0>-d/t. (1)
For allL∈Zd, for all functions f in the residual set A(ΩL) of Bp0s0,q0(ΩL) (resp., Lp0,s0(ΩL)) constructed in Theorem 16, (147)dtΩLh=-∞ifh<s0-dp0orh>s0p0h-s0+difs0-dp0≤h≤s0and(148)DtΩLh=-∞ifh<s0-dp0p0h-s0+difs0-dp0≤h≤s0difh>s0.(2)
For all functionsf in the residual set A of Bp0s0,q0(Rd) (resp., Lp0,s0(Rd)) constructed in Theorem 16, (149)dth=-∞ifh<s0-dp0orh>s0p0h-s0+difs0-dp0≤h≤s0and(150)Dth=-∞ifh<s0-dp0p0h-s0+difs0-dp0≤h≤s0difh>s0.Both (147) and (148) also hold on A.Proof. As in the proof of Theorem 16, it suffices to give the proof for L=0; let f∈A(Ω0). Relation (139) is satisfied for infinitely many scales n. Let α≥1. For each such scale n, write λ∈Γn(α) if λ∈Λn satisfies (98) with J=[n/α]. Let K(α) be the set of points x∈Ω0 that belong to λ∈Γn(α), for an infinite number of values of the above n’s. In [15], p. 535-536, we can find the following result (see also [65]).Proposition 20.
Forα≥1, the Hausdorff dimension of K(α) is d/α. Moreover, there exists a σ-finite measure mα carried by K(α) such that mα(Ω)=0 for any Ω⊂K(α) with dim⁡Ω<d/α.The following proposition bounds the pointwiseLt regularity of f∈A(Ω0) on K(α).Proposition 21.
Lets0,p0,q0>0 (resp., s0>0 and p0≥1) and t>0 such that s0-d/p0>-d/t. For all f∈A(Ω0), for all x∈Kα,(151)utx≤s0-dp01-1α.Proof.
Letx∈K(α). Result (15), relation (139), and result (145) imply that (152)ut,fx≤liminfn→∞log⁡lt,λnxflog⁡2-n≤liminfn→∞log⁡lt,λnxF0log⁡2-n=s0-dp01-1α.PutH(α)=s0-d/p0(1-1/α). Take α=d/p0(h-s0)+d if h∈[s0-d/p0,s0] (resp., α=1 if h>s0). Then H(α)=h if h∈[s0-d/p0,s0] (resp., H(α)=s0<h if h>s0). Using Proposition 21, for all x∈K(α), ut(x)≤h. It follows that K(α)⊂Bt(h). Therefore, for all h≥s0-d/p0,(153)min⁡p0h-s0+d,d≤DtΩ0h.Thus, by (20) and (146), for all h≥s0-d/p0,(154)DtΩ0h=min⁡p0h-s0+d,d.For h∈[s0-d/p0,s0], consider the set J(h)=x∈Ω0;ut,f(x)<H(α)=h, where α=d/p0(h-s0)+d. Clearly J(α)=⋃n≥1‍BtΩ0(h-1/n). By (150), for all n≥1, dim⁡BtΩ0h-1/n<min⁡p0(h-s0)+d,d. Then, by Proposition 20, mα(BtΩ0(h-1/n))=0. Using the σ-additivity of the measure mα, we get mα(J(h))=0.SinceK(α)⊂BtΩ0(h)=EtΩ0(h)∪J(h), it follows that mα(EtΩ0(h))>0. Thus,(155)p0h-s0+d≤dth.By (150), for h∈[s0-d/p0,s0], we get(156)dtΩ0h=p0h-s0+d.For α=1, K(1)=Ω0. Then for all x∈Ω0, ut(x)≤s0. Thus, for h>s0, EtΩ0(h)=∅.Both (147) and (148) also hold on A. This achieves the proof of Theorem 19.


## 6. Conclusion

The following theorem summarizes the range of Baire validity of botht-multifractal formalism and t-multifractal formalism for upper t-sets, locally on the ΩL and also globally on Rd. It follows directly from both Theorems 16 and 19.Theorem 22.
Lets0,p0,q0>0 (resp., s0>0 and p0≥1) and p>0 such that s0-d/p0>-d/t. Baire generically in Bp0s0,q0(Rd) (resp., Lp0,s0(Rd)), Et(h)=∅ for all h>s0, the t-multifractal formalism is valid for h≤s0, and the t-multifractal formalism for upper t-sets is valid for all h.
The same results hold for the localt-multifractal formalism on ΩL.



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*Source: 4358261-2019-08-29.xml*