# Directional Multifractal Analysis in theLp Setting

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

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## Abstract

The classical Hölder regularity is restricted to locally bounded functions and takes only positive values. The localLp regularity covers unbounded functions and negative values. Nevertheless, it has the same apparent regularity in all directions. In the present work, we study a recent notion of directional local Lp regularity introduced by Jaffard. We provide its characterization by a supremum of a wide range oriented anisotropic Triebel wavelet coefficients and leaders. In addition, we deduce estimates on the Hausdorff dimension of the set of points where the directional local Lp regularity does not exceed a given value. The obtained results are illustrated by some examples of self-affine cascade functions.


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## Body

## 1. Introduction

Multifractal analysis describes geometrically and statistically the distribution of pointwise regularities (or singularities) of irregular functionsf on Rm. It was first introduced in the context of the statistical study of fully developed turbulence in the mid 80’s [1]. The classical notion of pointwise regularity most commonly used is Hölder regularity. Let y∈Rm and α>0. Let B(y,r) denote the open ball of radius r, centered at y. Recall that a function f on Rm belongs to Cα(y) if there exist constants C and 0<R<1 and a polynomial P of degree at most the integer part [α] of α, such that(1)∀x∈By,Rfx-Px-y≤Cx-yα.Clearly, this condition makes sense only if f is locally bounded. It is equivalent to(2)∀r≤Rf·-P·-yL∞By,r≤Crα.It describes how the L∞ norm of f, properly renormalized by substracting a polynomial, behaves in small neighborhoods of y.The Hölder exponent off at y is defined as(3)αy=sup⁡α;f∈Cαy.The multifractal formalism relates some function spaces norms of f to the Hölder spectrum (which is the Hausdorff dimension of the set of points where f has a given Hölder exponent). The idea of using wavelets in multifractal analysis has been worked out first by Arneodo et al. [2]. Wavelets characterize Hölder regularity and many function spaces.In many PDE’s, the natural function space setting isLp for 1≤p<∞ or a Sobolev space which includes unbounded functions. In such cases, one replaces the L∞ norm in (2) by the Lp norm. This corresponds to the following weaker condition introduced by Calderón and Zygmund [3].Definition 1 (letp≥1).
Letf be a function which belongs locally to Lp(Rm).
Ifα≥0, one says that f∈Tαp(y) if there exist constants C and 0<R<1 and a polynomial P of degree at most [α] such that(4)∀r≤Rf·-P·-yLpBy,r≤Crα+m/p.If -m/p≤α<0, one says that f∈Tαp(y) if there exist constants C and 0<R<1 such that(5)∀r≤RfLpBy,r≤Crα+m/p.The p-exponent of f at y is defined as(6)αpy=supα;f∈Tαpy.Clearly iff∈Llocp(Rm) then f∈T-m/pp(y). The usual Hölder regularity Cα(y), for α>0, corresponds to p=∞. If f∈Cα(y), then f∈Tαp(y) for all p≥1. If f∈Tαp(y) then f∈Tαp′(y) for all p′≤p.In [4, 5], Jaffard and Mélot proved that if Ω is a domain of Rm and y is on the boundary ∂Ω of Ω, then by taking P=0 or P=1, the Tα/pp(y) condition for the characteristic function χΩ of Ω coincides with the bilaterally weak α accessibility of Ω at y. Let α≥0. Recall that Ω is called bilaterally weak α accessible at y if there exist two constants C and 0<R<1 such that(7)∀r≤Rmin⁡measΩc∩By,r,measΩ∩By,r≤Crα+m,where meas is the Lebesgue measure. This remark allows performing a multifractal analysis of fractal boundaries [5, 6]. This analysis has many applications in physics, mechanics, or chemistry where many phenomena involve fractal interfaces. More pointwise exponents for classification of the geometry of fractal boundaries were studied by Jaffard and Heurteaux [7]. The relationship of these exponents to local dimension computation was proved by Tricot [8].In [4, 5], a wavelet characterization of the Tαp(y) regularity was obtained. Moreover, an associated multifractal formalism was conjectured: the Hausdorff dimension dp(H) of the set of points y, where the p-exponent αp(y) of a function f is equal to H, may be derived from some global functional norms extracted from f.Unfortunately, ifm≥2 then Definition 1 has the same apparent value in all coordinate directions. However many signals belong to classes of functions with various directional regularity behaviors (see, for instance, [9–17] and the references therein). These behaviors are important for detection of edges, efficient image compression, etc. (see, for instance, [12] and the references therein). Classical (isotropic) wavelets are not optimal for analyzing directional or anisotropic features like edges. Extensions of wavelet bases which can be elongated in particular directions were considered. Candes and Donoho [18] and Mallat [19] (resp., Donoho [20] and Guo and Labate [21]) have used ridgelets and bandelets (resp., wedgelets and shearlets) to detect singularities along lines and hyperplanes (resp., discontinuities along smooth edges). Sampo and Sumetkijakan [16, 22, 23] (resp., Jaffard [24]) have used curvelets and Hart-Smith transform (resp., the anisotropic Gabor-wavelet transform) to detect directional pointwise Hölder singularities.Many signals present anisotropies quantified through regularity characteristics and features that strongly differ when measured in different directions [13, 14, 24–29]. Several authors were concerned with the problem of obtaining useful decompositions of isotropic and anisotropic function spaces in simple building blocks: atoms, quarks, wavelets, splines (see, for example, Farkas [30], Garrigós et al. [31, 32], Hochmuth [33], and Kamont [34]). Triebel family of anisotropic wavelets yield characterizations of anisotropic Besov spaces [35, 36] and anisotropic Hölder regularity (see Ben Slimane et al. [37–39]).Lete∈Rm be a unit vector. To take into account pointwise directional Lp behavior in direction e, it is natural to define the local Lp regularity αp,e(y) at a point y in direction e as the p-exponent at 0 of the one variable function fy,e:t↦f(y+te), that is,(8)αp,ey=sup⁡α;fy,e∈Tαp0.We wish to derive or estimate the Hausdorff dimension dp,e(H) of the set of points y where αp,e(y)=H from some global quantities extracted from the function f itself and not its traces fy,e (actually these traces are unknown since corresponding points y are unknown). One cannot expect directional local Lp regularity αp,e(y) to be characterized in terms of the size of the wavelet coefficients by taking wavelets on Rm because fy,e is defined as the trace of f on a line, which is a set of vanishing measure and wavelets on Rm have a support of nonempty interior. Thus one should take into account the values of f around the line considered. Therefore the definition of directional local Lp regularity should include such information. However, in the asymptotic of small scales, the values taken into account should be localized more and more sharply around this line. These considerations motivated the following definition of Jaffard [24].Definition 2.
Letα→=(α1,…,αm) be an m-uple of nonnegative real numbers. The average regularity α~ is the harmonic mean of the αn, i.e.,(9)1α~=1m∑n=1m1αn.The anisotropy indices are(10)vn=α~αn.Let b=(e1,…,em) be an orthonormal basis of Rm. Denote by (x1,…,xm) the coordinates of x on the basis b.
Set(11)v=v1,…,vmandxv=max⁡x11/v1,…,xm1/vm.Then ·v is a quasi-norm on Rm; i.e., it satisfies the requirements of a norm except for the triangular inequality which is replaced by the weaker condition(12)∃C>0∀x,y∈Rmx+yv≤Cxv+yv.It is also continuous and homogeneous in the sense that(13)∀r>0∀x∈Rmrv1x1,…,rvmxmv=rxv.The corresponding v-ball Bv,b(x,r)≔y∈Rm:|x-y|v<r of v-radius r centered on x is a rectangle with sides parallel to the axes of coordinates, centered at x and with side-length 2rvn in the xn-direction.
Letf:Rm→C be a function which belongs locally to Lp. One says that f∈Tp,α→(y,b) if there exist constants C,0<R<1 and a polynomial P(x)=∑I=(i1,…,im)∈N0m‍aIxI=∑I=(i1,…,im)∈N0m‍aIx1i1…xmim of degree(14)dJP<α→in the sense that(15)max∑n=1minαn:aI≠0<1such that(16)∀r≤Rf·-P·-yLpBv,by,r≤Crα~+m/p.Remark 3.
Actually, hereBv,b(x,r) replaces the elongated ellipsoid Nb,α→(x,r) of axis of lengths 2rv1,…,2rvm, centered on x, considered in [24]. This ellipsoid corresponds to the ball centered at x with ρv radius r, where ρv is the quasi-norm on Rm defined by ρv(0)=0, and for x≠0, ρv(x) is the unique value of r for which ∑n=1m(xn/rvn)2=1. Both ·v and ρv are equivalent. Nevertheless, ·v has the advantage of being easier in the computations of distances.
In the isotropic case (vn=1 for all 1≤n≤m), ρv coincides with the Euclidean norm.We first extend the definition ofTp,α→(y,b) for an m-uple α→=(α1,…,αm) of nonpositive real numbers as follows.Definition 4.
Letα→=(α1,…,αm) be an m-uple of nonpositive real numbers. We say that f∈Tp,α→(y,b) if there exist constants C and 0<R<1 such that(17)∀r≤RfLpBv,by,r≤Crα~+m/p.Contrary to the casep=∞ studied in [24], there is no partial ordering property for 1≤p<∞; i.e., if either 0<α1≤β1,…,0<αm≤βm or α1≤β1<0,…,αm≤βm<0 then f∈Tp,β→(y,b) does not imply that f∈Tp,α→(y,b). Indeed, the harmonic mean values α~ of α→ and β~ of β→ satisfy α~≤β~, but the corresponding anisotropy indices v=(v1,…,vm) and u=(u1,…,um) are not necessarily the same. Nevertheless, we will show that the following substitute for directional local Lp regularity αp,e(y) given in (8) makes sense (see next section).Definition 5.
Lete∈Rm be a unit vector and b be any orthonormal basis starting with the vector e. Let f:Rm→C be a function. Set(18)E+p,y,e≔α1>0:∃0<ε≤α1f∈Tp,α1,ε,…,ε→y,band(19)E-p,y,e≔α1<0:∃0<ε≤-α1f∈Tp,α1,-ε,…,-ε→y,b.If E+(p,y,e)≠∅, define the p-exponent of f at y in direction e by(20)αp,y,e=sup⁡E+p,y,e.If E+(p,y,e)=∅, define the p-exponent of f at y in direction e by(21)αp,y,e=sup⁡E-p,y,e.Remark 6.
Clearly we can choose any orthonormal basisb starting with the vector e, in fact the component x1-y1 is the same in any b and is equal to the inner product of x-y with e, and all norms of Rm-1 are equivalent.Definition 7.
Lete∈Rm be a unit vector. Let f:Rm→C be a function. The directional Lp spectrum (resp., upper Lp spectrum) of f in direction e is the map which associates with each H the Hausdorff dimension dp(H,e) (resp., Dp(H,e)) of the set of points y where α(p,y,e)=H (resp., α(p,y,e)≤H).The paper is organized as follows. In Section2, we first show that Definition 5 makes sense. Then we prove a criterion of directional pointwise p-exponent in terms of a supremum on a wide range of oriented anisotropic Lp regularities. We therefore deduce a first upper bound for the directional Lp spectra. In Section 3, we establish criteria of directional pointwise p-exponent in terms of decay condition on Triebel anisotropic wavelet coefficients and wavelet leaders. In Section 4, we obtain an alternative upper bound for the directional Lp spectra expressed as an infimum of Legendre transforms of some p-anisotropic scaling functions. Finally, in Section 5, the obtained results are illustrated by some examples of self-affine cascade functions.


## 2. DirectionalLp Regularity Criterion

### 2.1. Approval of Definition5

We first show that Definition5 makes sense.(i)
Suppose thatf∈Tp,(β1,ε,…,ε)→(y,b) with 0<ε<β1. Let α1<β1. Set ε′=α1/β1ε. Then ε′<ε and ε′<α1. Set α→≔(α1,ε′,…,ε′)→ and β→≔(β1,ε,…,ε)→. We will prove that f∈Tp,α→(y,b); clearly, the harmonic mean values α~ of α→ and β~ of β→ satisfy α~≤β~. The corresponding anisotropy indices v=(v1,…,vm) are the same.Since f∈Tp,β→(y,b) then there exist constants C,0<R<1 and a polynomial P(x)=∑I=(i1,…,im)∈N0m‍aIxI of degree dJ(P) less than β→ (see (14) and (15)), i.e.,(22)max⁡i1β1+i2ε+⋯+imε:aI≠0<1,such that(23)∀r≤Rf·-P·-yLpBv,by,r≤Crβ~+m/p.Split P as the sum of two polynomials Q and S, where the indices I of the nonvanishing coefficients aI of Q satisfy i1/α1+i2/ε′+⋯+im/ε′<1. Therefore the indices I of the nonvanishing coefficients aI of S satisfy both i1/β1+i2/ε+⋯+im/ε<1 and i1/α1+i2/ε′+⋯+im/ε′≥1. Clearly dJ(Q)<α→ and if R<1 then rβ~+m/p≤Crα~+m/p. For each index I of the nonvanishing coefficients aI of S(24)∫Bvy,ry-xIpdx=∫Bv0,rxIpdx=∏n=1m∫-rvnrvnxninpdxn≤C∏n=1mrvninp+1≤Crα~p+m.It follows that f∈Tp,α→(y,b).(ii)
Suppose thatf∈Tp,(β1,-ε,…,-ε)→(y,b) with 0<ε<-β1. Let α1<β1. Set ε′=α1/β1ε. Then -ε′<-ε and 0<ε′<-α1. Clearly the harmonic mean values of α→≔(α1,-ε′,…,-ε′)→ and β→≔(β1,-ε,…,-ε)→ satisfy α~≤β~<0. The corresponding anisotropy indices v=(v1,…,vm) are the same. Since f∈Tp,β→(y,b) then there exist constants C and 0<R<1 such that(25)∀r≤Rf·LpBvy,r≤Crβ~+m/p≤Crα~+m/p.It follows that f∈Tp,α→(y,b).


### 2.2. Criterion of Directional Pointwisep-Exponent

We will establish a criterion of directional pointwisep-exponent in terms of a supremum on a wide range of oriented anisotropic Lp regularities. The latest regularity is reminiscent of [40, 41] where Calderón and Torchinsky on one side and Folland and Stein on the other side have developed a theory of anisotropic Hup(Rm) spaces using the ρu quasi-norm of Remark 3.Letu=(u1,…,um)∈Rm be such that(26)∀nun>0and∑n=1mun=m.We will say that u is an anisotropy vector.IfI=(i1,…,im)∈N0m, set du(I)=∑n=1munin. If P=∑I‍aIxI, aI∈C is a polynomial, define its u-homogeneous degree by(27)duP=max⁡duI:aI≠0.Definition 8.
Letf:Rm→C be a function which belongs locally to Lp. Let b be an orthonormal basis of Rm.
Ifα≥0, we say that f belongs to Tu,αp(y,b) if there exist constants C,0<R<1 and a polynomial P of u-homogeneous degree less than α such that(28)∀r≤Rf·-P·-yLpBu,by,r≤Crα+m/p.If -m/p≤α<0, we say that f belongs to Tu,αp(y,b) if there exist constants C and 0<R<1 such that(29)∀r≤RfLpBu,by,r≤Crα+m/p.Definitions2 and 4 are related to the previous anisotropic regularity in Lp setting. Let α→=(α1,…,αm) be an m-uple of either nonpositive or nonnegative real numbers. Let α~ (resp., v=(v1,…,vm)) be the corresponding harmonic mean value as in (9) (resp., anisotropy indices as in (10)).Ifα→ is an m-uple of nonpositive real numbers then it follows from (29) and Definition 4 that(30)Tp,α→y,b=Tv,α~py,b.If α→ is an m-uple of nonnegative real numbers then(31)dvI=∑n=1mvnin<α~⇔∑n=1minαn<1.It follows from (14) and (15) that(32)dvP<α~⇔dJP<α→.We deduce that(33)Tp,α→y,b=Tv,α~py,b.The following theorem extends the result obtained for the directional Hölder regularity in [38] to the Lp setting. It has the advantage to cover unbounded functions and negative values.Theorem 9 (letm≥2).
Lete∈Rm be a unit vector and b be any orthonormal basis starting with the vector e. Let E be the set of all anisotropy vectors u with u2=⋯=um.
Letf∈Llocp(Rm). Then the p-exponent of f at y in direction e is given by(34)αp,y,e=supu∈Eαp,uy,bu1,where αp,u(y,b) is the so-called anisotropic p-u regularity of f at y oriented in basis b, defined as(35)αp,uy,b=sup⁡α~;f∈Tu,α~py,b.Proof (letu∈E).
Assume thatf∈Tu,α~p(y,b). Take α1=α~/u1 and for all 2≤n≤m take αn=α~/u2. Set α→=(α1,…,αm). Clearly, the anisotropy indices v=(v1,…,vm) of α→ coincide with u.(i)
Ifα~<0 then(36)f∈Tu,α~py,b⇔f∈Tp,α→y,b.Since α→=(α1,…,αm) with 0>αm=⋯=α2≥α1, then it follows from Definition 5 that(37)αp,y,e≥α1=α~u1.Then(38)αp,y,e≥1u1sup⁡α~;f∈Tu,α~py,b.Taking at the right the supremum over E yields the lower bound in (34).(ii)
Ifα~>0 then(39)duI=∑n=1munin<α~⇔∑n=1minαn<1.So(40)duP<α~⇔dJP<α→.We deduce that (36) holds too.Since α→=(α1,…,αm) with 0<α2=⋯=αm≤α1, then it follows from Definition 5 that(41)αp,y,e≥α1=α~u1.Then(42)αp,y,e≥1u1sup⁡α~;f∈Tu,α~py,b.Taking at the right the supremum over E yields the lower bound in (34).
Let us now prove the upper bound.(i)
Assume thatf∈Tp,α→(y,b) with α→=(α1,…,αm), α2=⋯=αm=ε and 0<ε≤α1. It follows from (30) that(43)α~≤αp,vy,b.Then(44)α1=α~v1≤supu∈Eαp,uy,bu1.Thus(45)αp,y,e≤supu∈Eαp,uy,bu1.Whence Theorem 9 holds.(ii)
The case wheref∈Tp,α→(y,b) with α→=(α1,…,αm) with α2=⋯=αm=-ε and 0<ε≤-α1 is similar.


### 2.3. General Upper Bound for the DirectionalLp Spectra

LetE be the set of anisotropy vectors u given in Theorem 9. Let b be an orthonormal basis of Rm. Define the p-u-sets of f in basis b by(46)Ep,uh,b=x;αp,ux,b=hand the upper p-u-sets of f in basis b by(47)Ep,uh,b=x;αp,ux,b≤h.Theorem 9 yields the following general upper bound for the directional upper Lp spectrum given in Definition 7.Corollary 10.
LetE be as in Theorem 9. Let e∈Rm be a unit vector and b be any orthonormal basis starting with the vector e. If f∈Llocp(Rm) then the directional Lp spectra of f in direction e given in Definition 7 satisfy(48)∀HdpH,e≤DpH,e≤infu∈Edim⁡Ep,uu1H,b,where dim denotes the Hausdorff dimension.


## 2.1. Approval of Definition5

We first show that Definition5 makes sense.(i)
Suppose thatf∈Tp,(β1,ε,…,ε)→(y,b) with 0<ε<β1. Let α1<β1. Set ε′=α1/β1ε. Then ε′<ε and ε′<α1. Set α→≔(α1,ε′,…,ε′)→ and β→≔(β1,ε,…,ε)→. We will prove that f∈Tp,α→(y,b); clearly, the harmonic mean values α~ of α→ and β~ of β→ satisfy α~≤β~. The corresponding anisotropy indices v=(v1,…,vm) are the same.Since f∈Tp,β→(y,b) then there exist constants C,0<R<1 and a polynomial P(x)=∑I=(i1,…,im)∈N0m‍aIxI of degree dJ(P) less than β→ (see (14) and (15)), i.e.,(22)max⁡i1β1+i2ε+⋯+imε:aI≠0<1,such that(23)∀r≤Rf·-P·-yLpBv,by,r≤Crβ~+m/p.Split P as the sum of two polynomials Q and S, where the indices I of the nonvanishing coefficients aI of Q satisfy i1/α1+i2/ε′+⋯+im/ε′<1. Therefore the indices I of the nonvanishing coefficients aI of S satisfy both i1/β1+i2/ε+⋯+im/ε<1 and i1/α1+i2/ε′+⋯+im/ε′≥1. Clearly dJ(Q)<α→ and if R<1 then rβ~+m/p≤Crα~+m/p. For each index I of the nonvanishing coefficients aI of S(24)∫Bvy,ry-xIpdx=∫Bv0,rxIpdx=∏n=1m∫-rvnrvnxninpdxn≤C∏n=1mrvninp+1≤Crα~p+m.It follows that f∈Tp,α→(y,b).(ii)
Suppose thatf∈Tp,(β1,-ε,…,-ε)→(y,b) with 0<ε<-β1. Let α1<β1. Set ε′=α1/β1ε. Then -ε′<-ε and 0<ε′<-α1. Clearly the harmonic mean values of α→≔(α1,-ε′,…,-ε′)→ and β→≔(β1,-ε,…,-ε)→ satisfy α~≤β~<0. The corresponding anisotropy indices v=(v1,…,vm) are the same. Since f∈Tp,β→(y,b) then there exist constants C and 0<R<1 such that(25)∀r≤Rf·LpBvy,r≤Crβ~+m/p≤Crα~+m/p.It follows that f∈Tp,α→(y,b).


## 2.2. Criterion of Directional Pointwisep-Exponent

We will establish a criterion of directional pointwisep-exponent in terms of a supremum on a wide range of oriented anisotropic Lp regularities. The latest regularity is reminiscent of [40, 41] where Calderón and Torchinsky on one side and Folland and Stein on the other side have developed a theory of anisotropic Hup(Rm) spaces using the ρu quasi-norm of Remark 3.Letu=(u1,…,um)∈Rm be such that(26)∀nun>0and∑n=1mun=m.We will say that u is an anisotropy vector.IfI=(i1,…,im)∈N0m, set du(I)=∑n=1munin. If P=∑I‍aIxI, aI∈C is a polynomial, define its u-homogeneous degree by(27)duP=max⁡duI:aI≠0.Definition 8.
Letf:Rm→C be a function which belongs locally to Lp. Let b be an orthonormal basis of Rm.
Ifα≥0, we say that f belongs to Tu,αp(y,b) if there exist constants C,0<R<1 and a polynomial P of u-homogeneous degree less than α such that(28)∀r≤Rf·-P·-yLpBu,by,r≤Crα+m/p.If -m/p≤α<0, we say that f belongs to Tu,αp(y,b) if there exist constants C and 0<R<1 such that(29)∀r≤RfLpBu,by,r≤Crα+m/p.Definitions2 and 4 are related to the previous anisotropic regularity in Lp setting. Let α→=(α1,…,αm) be an m-uple of either nonpositive or nonnegative real numbers. Let α~ (resp., v=(v1,…,vm)) be the corresponding harmonic mean value as in (9) (resp., anisotropy indices as in (10)).Ifα→ is an m-uple of nonpositive real numbers then it follows from (29) and Definition 4 that(30)Tp,α→y,b=Tv,α~py,b.If α→ is an m-uple of nonnegative real numbers then(31)dvI=∑n=1mvnin<α~⇔∑n=1minαn<1.It follows from (14) and (15) that(32)dvP<α~⇔dJP<α→.We deduce that(33)Tp,α→y,b=Tv,α~py,b.The following theorem extends the result obtained for the directional Hölder regularity in [38] to the Lp setting. It has the advantage to cover unbounded functions and negative values.Theorem 9 (letm≥2).
Lete∈Rm be a unit vector and b be any orthonormal basis starting with the vector e. Let E be the set of all anisotropy vectors u with u2=⋯=um.
Letf∈Llocp(Rm). Then the p-exponent of f at y in direction e is given by(34)αp,y,e=supu∈Eαp,uy,bu1,where αp,u(y,b) is the so-called anisotropic p-u regularity of f at y oriented in basis b, defined as(35)αp,uy,b=sup⁡α~;f∈Tu,α~py,b.Proof (letu∈E).
Assume thatf∈Tu,α~p(y,b). Take α1=α~/u1 and for all 2≤n≤m take αn=α~/u2. Set α→=(α1,…,αm). Clearly, the anisotropy indices v=(v1,…,vm) of α→ coincide with u.(i)
Ifα~<0 then(36)f∈Tu,α~py,b⇔f∈Tp,α→y,b.Since α→=(α1,…,αm) with 0>αm=⋯=α2≥α1, then it follows from Definition 5 that(37)αp,y,e≥α1=α~u1.Then(38)αp,y,e≥1u1sup⁡α~;f∈Tu,α~py,b.Taking at the right the supremum over E yields the lower bound in (34).(ii)
Ifα~>0 then(39)duI=∑n=1munin<α~⇔∑n=1minαn<1.So(40)duP<α~⇔dJP<α→.We deduce that (36) holds too.Since α→=(α1,…,αm) with 0<α2=⋯=αm≤α1, then it follows from Definition 5 that(41)αp,y,e≥α1=α~u1.Then(42)αp,y,e≥1u1sup⁡α~;f∈Tu,α~py,b.Taking at the right the supremum over E yields the lower bound in (34).
Let us now prove the upper bound.(i)
Assume thatf∈Tp,α→(y,b) with α→=(α1,…,αm), α2=⋯=αm=ε and 0<ε≤α1. It follows from (30) that(43)α~≤αp,vy,b.Then(44)α1=α~v1≤supu∈Eαp,uy,bu1.Thus(45)αp,y,e≤supu∈Eαp,uy,bu1.Whence Theorem 9 holds.(ii)
The case wheref∈Tp,α→(y,b) with α→=(α1,…,αm) with α2=⋯=αm=-ε and 0<ε≤-α1 is similar.


## 2.3. General Upper Bound for the DirectionalLp Spectra

LetE be the set of anisotropy vectors u given in Theorem 9. Let b be an orthonormal basis of Rm. Define the p-u-sets of f in basis b by(46)Ep,uh,b=x;αp,ux,b=hand the upper p-u-sets of f in basis b by(47)Ep,uh,b=x;αp,ux,b≤h.Theorem 9 yields the following general upper bound for the directional upper Lp spectrum given in Definition 7.Corollary 10.
LetE be as in Theorem 9. Let e∈Rm be a unit vector and b be any orthonormal basis starting with the vector e. If f∈Llocp(Rm) then the directional Lp spectra of f in direction e given in Definition 7 satisfy(48)∀HdpH,e≤DpH,e≤infu∈Edim⁡Ep,uu1H,b,where dim denotes the Hausdorff dimension.


## 3. Criteria of DirectionalLp Regularity in Anisotropic Triebel Wavelet Bases

For eachβ∈N there are Daubechies [42] real valued compactly supported father and mother wavelets ψF and ψM in Cβ(R) (in the sense that they have classical continuous derivatives up to order β) such that ∫R‍ψF(x)dx=1, the moments ∫R‍xnψM(x)dx=0 for n=0,…,β, and the collection (ψF(·-k))k∈Z and (2j/2ψM(2j·-k))j∈N0,k∈Z is an orthonormal basis of L2(R).Letu be an anisotropic vector. For r>0, define the u-anisotropy map as(49)rux=ru1x1,…,rumxm.In [35, 36], Triebel has considered anisotropic multiresolution analysis; consider, for any j∈N0, the closed subspace Vj,u of L2(Rm) spanned by the orthonormal basis (2∑n=1m‍[jun]/2Φj,k,u)k∈Zm, where(50)Φj,k,ux=∏n=1mψF2junxn-kn.The sequence (Vj,u)j∈N0 is an anisotropic multiresolution analysis of L2(Rm) in the sense that(i)∀j∈N0,Vj,u⊂Vj+1,u.(ii)f(x)∈Vj,u⇔f(2ux)∈Vj+1,u.(iii)⋃j∈N0Vj,u¯=L2(Rm).Forj∈Z, let Ij,u be the set of pairs (G,l) where G=(G1,…,Gm)∈{F,M}m such that at least one component Gn is M and l=(l1,…,lm)∈N0m where(51)ln=junifGn=F,(52)jun≤ln<j+1unifGn=Mandj+1un>jun,and(53)ln=junifGn=Mandj+1un=jun.Clearly the cardinality ♯Ij,u of Ij,u is bounded independently of j, more precisely(54)1≤♯Ij,u≤2m-1∏n=1m2+un.The following proposition is given in [35, 36] in the case where b is the canonical basis of Rm. It remains valid in the case where b is any orthonormal basis of Rm.Proposition 11.
Letb be an orthonormal basis of Rm.Let(x1,…,xm)be the coordinates ofxinb. Set(55)Φk,bx≔∏i=1mψFxi-ki,Ψj,k,u,bG,lx=∏i=1mψGi2lixi-kiandl≔∑i=1mli.The collection of the union of (Φk,b) for k∈Zm and (2|l|/2Ψj,k,u,b(G,l)) for j∈N0, (G,l)∈Ij,u, and k∈Zm is then an orthonormal basis of L2(Rm). Thus any function f∈L2(Rm) can be written as(56)fx=∑k∈ZmCk,bΦk,bx+∑j=0∞∑k∈Zm∑G,l∈Ij,ucj,k,u,bG,lΨj,k,u,bG,lx,with(57)Ck,b=∫RmfxΦk,bxdxand(58)cj,k,u,bG,l=2l∫RmfxΨj,k,u,bG,lxdx.A straightforward extension is given by the following result.Proposition 12.
The collection of(2|l|/2Ψj,k,u,B(G,l)) for j∈Z, (G,l)∈Ij,u, and k∈Zm is an orthonormal basis of L2(Rm). Thus any function f∈L2(Rm) can be written as(59)fx=∑j∈Z∑k∈Zm∑G,l∈Ij,ucj,k,u,bG,lΨj,k,u,bG,lx.


### 3.1. Wavelet Characterization of theTu,αp(y) Spaces

Letu be an anisotropic vector. One of the fundamental properties of the anisotropic wavelet bases is that it characterizes Tu,αp(y,b) spaces. The results are reminiscent of [5] where the authors have considered the isotropic case u1=⋯=um=1.Without any loss of generality, we will present the results in the canonical basis and drop the letterb. The proofs adapt all steps and arguments of [5] to the anisotropic setting, using(i)
the properties of the homogeneous norm·u,(ii)
anisotropic versions of mean value theorem and Taylor’s theorem with remainder (see [40, 41]),(iii)
anisotropic Triebel wavelet characterization of anisotropic Besov spacesBp,us,q(Rm) (see [35, 36]).Proposition 13 (theu-mean value theorem).
Letu be an anisotropic vector. There exist two positive constants C and ν such that for all functions f of class C1 on Rm and all x,y∈Rm,(60)fy-fx≤C∑i=1my-xuuisuphu≤νy-xu∂xifx+h.We denote by Δu the additive subsemigroup of R generated by 0,u1,u2,…  and um. In other words, Δu is the set of all numbers du(I) as I ranges over N0m.Proposition 14 (theu-Taylor inequality).
Letu be an anisotropic vector. Put umin=min1≤n≤mun. Suppose δu∈Δu, δu>0, and k=[δu/umin]. There are two constants Cδ>0 and ν>0 such that for all functions f of class C(k+1) on Rm and all x,y∈Rm,(61)fy-Py-x≤Cδ∑I≤k+1,duI>δuy-xuduIsuphu≤νk+1y-xu∂Ifx+hwhere P is the so-called u-Taylor polynomial of f at x of u-homogeneous degree δu, given by(62)Py-x=∑I:duI≤δu∂IfxI!y-xI.Proposition 15.
Letu be an anisotropic vector. Let 0<p,q≤∞ and s∈R. Then f∈Bp,us,q(Rm) if and only if its norm(63)fBp,us,qRm≔∑k∈ZmCkp1/p+∑j∈N0∑k∈Zm∑G,l∈Ij,u2s-m/pjcj,k,uG,lpq/p1/qis finite, with the usual modification if p=∞ and/or q=∞. This characterization does not depend on the chosen (smooth enough) wavelets ψF and ψM.In [35] (section 5.3), a transference method has been proposed with the aim of showing that isotropic and anisotropic Besov spaces are isomorphic together and with the space bp,qs of all union of sequences (Ck)k∈Zm and (cj,k)j∈N0,k∈Zm such that(64)∑k∈ZmCkp1/p+∑j∈N0∑k∈Zm2s-m/pjcj,kpq/p1/q<∞,with the usual modification if p=∞ and/or q=∞. In particular, any embedding theorem for isotropic Besov spaces can be carried over to anisotropic Besov spaces.Definition 16.
Letu be an anisotropic vector. Let s,s′ be real number and let p and q be positive real numbers. One says that a function f belongs to X˙us,s′,p,q(y) if there exists A>0 such that(65)εj≔2js-m/p∑y-2-lku≤A∑G,l∈Ij,ucj,k,uG,lp1+2jy-2-lkus′p1/p∈lq,where(66)2-lk=k12l1,…,km2lm.Ifu1=⋯=um=1 then this space was already introduced by Meyer and Xu [43] in order to study local oscillating behaviors.Proposition 17.
Letu be an anisotropic vector. Let 1≤p<∞, 0≤s≤β where β is the regularity of the Daubechies wavelets. Let y∈Rm and f∈Llocp(Rm).(i)
Iff∈X˙us,-s,p,1(y), then f∈Tu,s-m/pp(y).(ii)
Iff∈Tu,s-m/pp(y), then there exist A and C>0 such that(67)∀j2jsp-m∑y-2-lku≤A∑G,l∈Ij,ucj,k,uG,lp1+2jy-2-lku-sp≤Cj.Letp≥1 and f be a function which belongs locally to Lp. For A>0 small enough let(68)Σjpu,s,A,y=2jsp-m∑y-2-lku≤A∑G,l∈Ij,ucj,k,uG,lp1+2jy-2-lku-sp,and(69)iu,py≔sup⁡s:liminfj↦∞log⁡Σjpu,s,A,y1/p-jlog⁡2≥0.Proposition 18.
Letu be an anisotropic vector. Let p≥1 and f be a function which belongs locally to Lp(Rm). Then(1)
iu,p(y) is positive and independent of A,(2)
we have always(70)αp,uy≤iu,py-mp,(3)
iff∈Bp,uδ,p(Rm) for some δ>0 then(71)αp,uy=iu,py-mp.Both Theorem9 and Proposition 18 yield the following characterization of the pointwise directional p-exponent.Corollary 19 (letm≥2).
Lete∈Rm be a unit vector and b be any orthonormal basis starting with the vector e. Let E be the set of all anisotropy vectors u such that u2=⋯=um.
Letf∈Llocp(Rm). If f∈Bp,uδ,p(Rm) for some δ>0 then the p-exponent of f at y in direction e is given by(72)αp,y,e=supu∈Eiu,py-m/pu1.


### 3.2. Wavelet Leaders Characterization

Without any loss of generality, we will present the results in the canonical basis. Letu be an anisotropic vector. By λu=λj,k,ul we denote a u-dyadic rectangle in Rm of scale j, which has the form(73)λu=λj,k,ul=2-lk+∏i=1m0,2-li,where 2-lk was defined in (66).Denote byΛj,u all u-dyadic rectangles in Rm of scale j.Forλu∈Λj,u set(74)cλu=max⁡cj,k,uG,l,where the maximum is taken over all indices G giving the same l at scale j.Letf be a function on Rm. Define its u wavelet scaling function as(75)ηup=liminfj→∞log⁡2-mj∑λ∈Λj,ucλuplog⁡2-j.By Proposition 15, this scaling function does not depend on the chosen (smooth enough) wavelets, and(76)∀0<q≤∞ηup=sup⁡τ:f∈Bp,uτ/p,qRm.For p>0 such that ηu(p)>0, the p-u wavelet leader of f at a rectangle λu∈Λj,u is defined as(77)lλup=∑j′=j∞∑λu′∈Λj′,u,λu′⊂3λucλu′p2-mj′-j1/pwhere 3λu is the set formed by the rectangle λu and all its adjacent rectangles at scale j.Remark 20.
Note that ifηu(p)>0 then f∈Llocp and if ηu(p)<0 then f∉Llocp (see [44]).Ifx∈Rm, denote by λj,u(x) the unique rectangle at the scale j that contains x.As in [45], the third point in Proposition 18 yields the following result.Proposition 21.
Letu be an anisotropic vector. Let p≥1. If ηu(p)>0 then(78)αp,ux=liminfj→∞log⁡lλj,uxplog⁡2-j.Both Theorem9 and Proposition 21 yield the following characterization of the pointwise directional p-exponent.Corollary 22 (letm≥2).
Lete∈Rm be a unit vector and b any orthonormal basis starting with the vector e. Let E be the set of all anisotropy vectors u such that u2=⋯=um.
Letp≥1. If ηu(p)>0 then the p-exponent of f at y in direction e is given by(79)αp,y,e=supu∈Eliminfj→∞log⁡lλj,uxplog⁡2-ju1.


## 3.1. Wavelet Characterization of theTu,αp(y) Spaces

Letu be an anisotropic vector. One of the fundamental properties of the anisotropic wavelet bases is that it characterizes Tu,αp(y,b) spaces. The results are reminiscent of [5] where the authors have considered the isotropic case u1=⋯=um=1.Without any loss of generality, we will present the results in the canonical basis and drop the letterb. The proofs adapt all steps and arguments of [5] to the anisotropic setting, using(i)
the properties of the homogeneous norm·u,(ii)
anisotropic versions of mean value theorem and Taylor’s theorem with remainder (see [40, 41]),(iii)
anisotropic Triebel wavelet characterization of anisotropic Besov spacesBp,us,q(Rm) (see [35, 36]).Proposition 13 (theu-mean value theorem).
Letu be an anisotropic vector. There exist two positive constants C and ν such that for all functions f of class C1 on Rm and all x,y∈Rm,(60)fy-fx≤C∑i=1my-xuuisuphu≤νy-xu∂xifx+h.We denote by Δu the additive subsemigroup of R generated by 0,u1,u2,…  and um. In other words, Δu is the set of all numbers du(I) as I ranges over N0m.Proposition 14 (theu-Taylor inequality).
Letu be an anisotropic vector. Put umin=min1≤n≤mun. Suppose δu∈Δu, δu>0, and k=[δu/umin]. There are two constants Cδ>0 and ν>0 such that for all functions f of class C(k+1) on Rm and all x,y∈Rm,(61)fy-Py-x≤Cδ∑I≤k+1,duI>δuy-xuduIsuphu≤νk+1y-xu∂Ifx+hwhere P is the so-called u-Taylor polynomial of f at x of u-homogeneous degree δu, given by(62)Py-x=∑I:duI≤δu∂IfxI!y-xI.Proposition 15.
Letu be an anisotropic vector. Let 0<p,q≤∞ and s∈R. Then f∈Bp,us,q(Rm) if and only if its norm(63)fBp,us,qRm≔∑k∈ZmCkp1/p+∑j∈N0∑k∈Zm∑G,l∈Ij,u2s-m/pjcj,k,uG,lpq/p1/qis finite, with the usual modification if p=∞ and/or q=∞. This characterization does not depend on the chosen (smooth enough) wavelets ψF and ψM.In [35] (section 5.3), a transference method has been proposed with the aim of showing that isotropic and anisotropic Besov spaces are isomorphic together and with the space bp,qs of all union of sequences (Ck)k∈Zm and (cj,k)j∈N0,k∈Zm such that(64)∑k∈ZmCkp1/p+∑j∈N0∑k∈Zm2s-m/pjcj,kpq/p1/q<∞,with the usual modification if p=∞ and/or q=∞. In particular, any embedding theorem for isotropic Besov spaces can be carried over to anisotropic Besov spaces.Definition 16.
Letu be an anisotropic vector. Let s,s′ be real number and let p and q be positive real numbers. One says that a function f belongs to X˙us,s′,p,q(y) if there exists A>0 such that(65)εj≔2js-m/p∑y-2-lku≤A∑G,l∈Ij,ucj,k,uG,lp1+2jy-2-lkus′p1/p∈lq,where(66)2-lk=k12l1,…,km2lm.Ifu1=⋯=um=1 then this space was already introduced by Meyer and Xu [43] in order to study local oscillating behaviors.Proposition 17.
Letu be an anisotropic vector. Let 1≤p<∞, 0≤s≤β where β is the regularity of the Daubechies wavelets. Let y∈Rm and f∈Llocp(Rm).(i)
Iff∈X˙us,-s,p,1(y), then f∈Tu,s-m/pp(y).(ii)
Iff∈Tu,s-m/pp(y), then there exist A and C>0 such that(67)∀j2jsp-m∑y-2-lku≤A∑G,l∈Ij,ucj,k,uG,lp1+2jy-2-lku-sp≤Cj.Letp≥1 and f be a function which belongs locally to Lp. For A>0 small enough let(68)Σjpu,s,A,y=2jsp-m∑y-2-lku≤A∑G,l∈Ij,ucj,k,uG,lp1+2jy-2-lku-sp,and(69)iu,py≔sup⁡s:liminfj↦∞log⁡Σjpu,s,A,y1/p-jlog⁡2≥0.Proposition 18.
Letu be an anisotropic vector. Let p≥1 and f be a function which belongs locally to Lp(Rm). Then(1)
iu,p(y) is positive and independent of A,(2)
we have always(70)αp,uy≤iu,py-mp,(3)
iff∈Bp,uδ,p(Rm) for some δ>0 then(71)αp,uy=iu,py-mp.Both Theorem9 and Proposition 18 yield the following characterization of the pointwise directional p-exponent.Corollary 19 (letm≥2).
Lete∈Rm be a unit vector and b be any orthonormal basis starting with the vector e. Let E be the set of all anisotropy vectors u such that u2=⋯=um.
Letf∈Llocp(Rm). If f∈Bp,uδ,p(Rm) for some δ>0 then the p-exponent of f at y in direction e is given by(72)αp,y,e=supu∈Eiu,py-m/pu1.


## 3.2. Wavelet Leaders Characterization

Without any loss of generality, we will present the results in the canonical basis. Letu be an anisotropic vector. By λu=λj,k,ul we denote a u-dyadic rectangle in Rm of scale j, which has the form(73)λu=λj,k,ul=2-lk+∏i=1m0,2-li,where 2-lk was defined in (66).Denote byΛj,u all u-dyadic rectangles in Rm of scale j.Forλu∈Λj,u set(74)cλu=max⁡cj,k,uG,l,where the maximum is taken over all indices G giving the same l at scale j.Letf be a function on Rm. Define its u wavelet scaling function as(75)ηup=liminfj→∞log⁡2-mj∑λ∈Λj,ucλuplog⁡2-j.By Proposition 15, this scaling function does not depend on the chosen (smooth enough) wavelets, and(76)∀0<q≤∞ηup=sup⁡τ:f∈Bp,uτ/p,qRm.For p>0 such that ηu(p)>0, the p-u wavelet leader of f at a rectangle λu∈Λj,u is defined as(77)lλup=∑j′=j∞∑λu′∈Λj′,u,λu′⊂3λucλu′p2-mj′-j1/pwhere 3λu is the set formed by the rectangle λu and all its adjacent rectangles at scale j.Remark 20.
Note that ifηu(p)>0 then f∈Llocp and if ηu(p)<0 then f∉Llocp (see [44]).Ifx∈Rm, denote by λj,u(x) the unique rectangle at the scale j that contains x.As in [45], the third point in Proposition 18 yields the following result.Proposition 21.
Letu be an anisotropic vector. Let p≥1. If ηu(p)>0 then(78)αp,ux=liminfj→∞log⁡lλj,uxplog⁡2-j.Both Theorem9 and Proposition 21 yield the following characterization of the pointwise directional p-exponent.Corollary 22 (letm≥2).
Lete∈Rm be a unit vector and b any orthonormal basis starting with the vector e. Let E be the set of all anisotropy vectors u such that u2=⋯=um.
Letp≥1. If ηu(p)>0 then the p-exponent of f at y in direction e is given by(79)αp,y,e=supu∈Eliminfj→∞log⁡lλj,uxplog⁡2-ju1.


## 4. Alternative Upper Bound of the DirectionalLp Spectra

In Corollary10, if u is an anisotropic vector different from (1,…,1), then the sets Ep,u(u1H,b) are anisotropic but dim is isotropic. Computing dim⁡Ep,u(u1H,b) is a very difficult task (see the example of Sierpinski carpet in [46] pages 118–119). To overcome this problem, in [47, 48], we adapted the notion of Hausdorff dimension to the anisotropy u; if Ω⊂Rd, define its u-diameter to be |Ω|u≔supx,y∈Ω|x-y|u. By replacing in the definition of Hausdorff measure, the usual notion of diameter by the u-diameter, we easily check (see [49]) that we get the following notion of anisotropic dimension.Definition 23.
LetΩ⊂Rd, ε>0 and let Rε be the set of all coverings R=(Ωn)n∈N of Ω by sets Ωn of u-diameter |Ωn|u at most ε. Let(80)Mε,uδΩ=infR∈Rε∑n∈NΩnuδ.The δ-dimensional u-Hausdorff measure of Ω is(81)MuδΩ=limsupε→0Mε,uδΩ.The u-Hausdorff dimension of Ω is(82)dimuΩ=inf⁡δ:MuδΩ=0=sup⁡δ:MuδΩ=∞.Note that we get the same value ofdimu(Ω) if we use coverings R=(Ωn)n∈N of Ω by rectangles Ωn with sides parallel to the axes of coordinates and with side-length 2εui in the xi-direction.In the isotropic case,·(1,…,1) is equivalent to the Euclidean norm · on Rd and dim(1,…,1)(Ω) coincides with dim⁡Ω. But if u≠(1,…,1), then dimu(Ω) does not necessarily coincide with dim⁡Ω. Actually, if umin=min1≤n≤mun and umax=max1≤n≤mun there exists C≥1 such that(83)∀x∈Rd1Cmin⁡x1/umin,x1/umax≤xu≤Cmax⁡x1/umin,x1/umax.and(84)umindim⁡Ω≤dimuΩ≤umaxdim⁡Ω.Definition 24.
Thep-u-spectrum is defined as(85)dp,uH=dimux∈Rm:αp,ux=H.The p-u-upper-spectrum is defined as(86)Dp,uH=dimux∈Rm:αp,ux≤H.The restrictedp-u wavelet leader of f at a rectangle λu∈Λj,u is defined as(87)l~λup=∑j′=j∞∑λu′∈Λj′,u,λu′⊂λucλu′p2-mj′-j1/p.The p-u structure function is defined as(88)S~upq,j=2-mj∑λu∈Λj,ul~λupqThe p-u wavelet leader scaling function is given by(89)ζupq=liminfj→∞log⁡S~upq,jlog⁡2-j.We can easily adapt the (isotropic) general upper bound obtained in [45] or [50] to the anisotropic setting.Proposition 25 (letp≥1).
Ifηu(p)>0 then(90)∀hDp,uh≤infq∈Rd+hq-ζupq.Consequently, we obtain the following result.Theorem 26 (letm≥2).
Lete∈Rm be a unit vector and b any orthonormal basis starting with the vector e. Let E be the set of all anisotropy vectors u such that u2=⋯=um.
Letp≥1. Then the directional Lp spectra of f in direction e given in Definition 7 satisfy(91)∀HdpH,e≤DpH,e≤infu∈E,ηup>01umininfq∈R⁡u1Hq-ζupq+m.Proof.
By (84)(92)DpH,e≤infu∈E⁡1uminDp,uu1H.Thus (90) yields (91).


## 5. Examples of Affine Cascade Functions

We will apply Theorem26 for some examples of self-affine cascade functions. Let s and t be two nonnegative integers. We divide the unit square R=[0,1]2 into a uniform grid of rectangles of sides 1/t and 1/s. Choose A⊂{0,1,…,s-1}×{0,1,…,t-1}. For ω=(a,b)∈A, the contraction Sω(x1,x2)=x1/s+a/s,x2/t+b/t maps the unit square R into the rectangle(93)Rω=as,a+1s×bt,b+1t.If G is a subset of Rm, we define the mapping S by(94)SG=⋃ω∈ASωG.The Sierpinski carpet K (see [28, 29, 48]) and references therein) is the unique nonempty compact set satisfying(95)K=⋂n∈NSnR.Let (γω)ω∈A be nonnegative scalars. Put g(x)=ψM(x1)ψM(x2) where ψM is a mother (smooth enough) wavelet. The Sierpinski cascade function adapted to the subdivision A satisfies the self-affine equation(96)∀x∈RFx=∑ω∈AγωFSω-1x+gx.Proposition 27 (let1≤p<∞).
If(∑ω∈A‍γω)p<st, then the series(97)Fx=gx+∑n=1∞∑ω1,…,ωn∈Anγω1⋯γωngSωn-1⋯Sω1-1xis the unique solution in Lp(R) for equation (96).
If moreover,s=2S and t=2T with S and T two nonnegative integers such that S<T and S+T is even and (e1,e2) is the canonical basis in R2, then the directional Lp spectra of F in directions e1 and e2, respectively, satisfy(98)∀HdpH,e1≤DpH,e1≤infq∈RHq-log⁡∑ω∈Aγωqlog⁡sand(99)∀HdpH,e2≤DpH,e2≤infq∈RTSHq-log⁡∑ω∈Aγωqlog⁡s.Proof.
Clearly series (97) satisfies (96). Since(100)∀ω∈AF∘Sω-1Lp=st1/pFLp,then assumption (∑ω∈A‍γω)p<st implies that F is the unique solution in Lp(R).
Clearly ifωl=(al,bl) then(101)gSωn-1⋯Sω1-1x=ψMsnx1-sn-1a1-⋯-san-1-anψMtnx2-tn-1b1-⋯-tbn-1-bn.Let S and T be two positive integers such that S<T and S+T is even. Assume that s=2S and t=2T. Put(102)u1=2SS+Tandu2=2TS+T.Clearly u=(u1,u2) is an anisotropic vector with u1<u2,(103)sn=2nS=2ju1andtn=2nT=2ju2with(104)j=nS+T2.By a straightforward computation, the u wavelet scaling function defined in (75) is given by(105)ηup=2-2log2⁡∑ω∈AγωpS+T.Since (∑ω∈A‍γω)p<st, then ∑ω∈A‍γωp<st. It follows that ηu(p)>0.
The restrictedp-u wavelet leader of F at a rectangle λu∈Λj,u (defined in (87)) satisfies(106)l~λupp=∑n′≥nstn-n′∑ω∈Aγωpn′-nγω1⋯γωnp.Since ∑ω∈A‍γωp<st, then(107)l~λup~γω1⋯γωnand(108)ζupq=2-2log2⁡∑ω∈AγωqS+T.If (e1,e2) is the canonical basis in R2 then Theorem 26 yields(109)∀HdpH,e1≤DpH,e1≤1u1infq∈R⁡u1Hq-ζupq+2and(110)∀HdpH,e2≤DpH,e2≤1u1infq∈R⁡u2Hq-ζupq+2.Using (102) and (108), we deduce results (98) and (99).



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*Source: 1691903-2019-07-16.xml*