# Generalized Pointwise Hölder Spaces Defined via Admissible Sequences

Authors: Damien Kreit; Samuel Nicolay
Journal: Journal of Function Spaces
 (2018)

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## Abstract

We introduce in this paper a generalization of the pointwise Hölder spaces. We give alternative definitions of these spaces, look at their relationship with the wavelets, and introduce a notion of generalized Hölder exponent.


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## Body

## 1. Introduction

A real-valued functionf defined on Rd belongs to the uniform Hölder space Λα(Rd) (α>0) if there exist a constant C and a polynomial P of degree less than α such that(1)fx+h-Px≤Chα,for all x∈Rd and h small enough. The supremum of all these values α is called the global Hölder exponent of f. One can naturally define the pointwise version of these spaces: a continuous function f belongs to the pointwise Hölder space Λαx0 if and only if there exist C,J>0 and a polynomial P of degree less than α such that(2)suph≤2-j⁡fx0+h-Ph≤C2-jα,for all j≥J. Of course, the supremum of all these values α is called the Hölder exponent of f at x0. If f is differentiable at x0, the Hölder exponent of f at x0 is at least 1. The fact that the constant C appearing in (1) is uniform for Λα(Rd) implies that the uniform Hölder exponent is not necessarily the infimum of the pointwise Hölder exponents. A classical example is the function sinc⁡1/x=xsin⁡1/x, which is C1 at the origin, C∞ everywhere else, while its global Hölder exponent is 1/2.In [1, 2], the properties of generalized uniform Hölder spaces have been investigated. The idea underlying the definition is to replace the exponent α of the usual spaces Λα(Rd) [3, 4] with a sequence σ satisfying some conditions. The so-obtained spaces Λσ(Rd) generalize the spaces Λα(Rd); the spaces Λσ(Rd) are actually the spaces B∞,∞1/σ(Rd), but they present specific properties (induced by L∞-norms) when compared to the more general spaces Bp,q1/σ(Rd) studied in [5–10], for example. Indeed, it is shown in [1, 2] that most of the usual properties holding for the spaces Λα(Rd) can be transposed to the spaces Λσ(Rd).Here, we introduce the pointwise version of these spaces: the spacesΛσ,M(x0), with x0∈Rd. As in [1, 2], the idea is again to replace the sequence 2-jαj appearing in (2) with a positive sequence σjj such that σj+1/σj and σj/σj+1 are bounded (for any j); the number M stands for the maximal degree of the polynomial (this degree cannot be induced by a sequence σ). By doing so, one tries to get a better characterization of the regularity of the studied function f; a usual choice is to replace 2-jα with j2-jα (see, e.g., [11–13]). For example, it is well known that, for the Brownian motion B, there exist C,J>0 such that(3)suph≤2-j⁡Bx+h-Bx≤Cj2-j/2,for any j≥J almost surely (see, e.g., [14, 15]). More generally, the behavior of the oscillations of f(x+h)-f(x) as h tends to 0 can reveal specific local behaviors such as approximate similarities [11]. Generalizations of the pointwise Hölder spaces have already been proposed, but, to our knowledge, the definition we give here is the most general version and leads to the sharpest results (in particular, the regularity spaces introduced in [13] are a special case of the spaces Λσ,M(x0); see Definition 2 and Remark 4). These spaces could also be useful in the study of singularities of PDEs where the function is smooth near x0 except on small sets of points (such a situation could be met in the case of the Navier-Stokes equations in three dimensions [16]).As a first application, we introduce here the sufficient tools to develop a multifractal formalism based on the wavelet characterization (Theorem16), in the same spirit as in [12]. The theory presented here also contributes to enlightening the reasons why the customary proofs for the classical spaces Λα(x0) work.This paper is organized as follows. We first give the definitions leading to generalized pointwise Hölder spacesΛσ,M(x0) and prove that, under some general conditions, the polynomials appearing in the definition are independent of the scale, as is the case with the usual Hölder spaces. Next, we give some alternative definitions of the spaces Λσ,M(x0), mimicking the different possible definitions of Λα(x0). One of the nicest properties of the Hölder spaces is their relationship with the wavelet theory given in [12]; we show here that this result still holds in the general case. Finally, we give some conditions under which one gets embedded generalized pointwise Hölder spaces and define a generalized pointwise Hölder exponent.Throughout this paper,B denotes the open unit ball of Rd centered at the origin; moreover, we set Bj=2-jB. The floor function is denoted by · and Pα designates the set of polynomials of degree at most α. We use the letter C for a generic positive constant whose value may be different at each occurrence.


## 2. Pointwise Generalized Hölder Spaces

To present the generalized pointwise Hölder spaces, we first need to recall some notions concerning the admissible sequences. After having introduced the definitions, we point out a major difference between the usual spaces and the generalized ones: the polynomial arising in the definition depends on the scale. It is then natural to look under which conditions this constraint can be dropped.


### 2.1. Definition

The generalization of the Hölder spaces we propose here is based on the notion of admissible sequence [10].Definition 1.
A sequenceσ=(σj)j∈N of real positive numbers is called admissible if there exists a positive constant C such that(4)C-1σj≤σj+1≤Cσj,for any j∈N.Ifσ is such a sequence, we set(5)Θ_j=infk∈N⁡σj+kσk,Θ¯j=supk∈N⁡σj+kσk,and we define the lower and upper Boyd indices as follows:(6)s_σ=limj⁡log2⁡Θ_jj,s¯σ=limj⁡log2⁡Θ¯jj.Since (log⁡Θ_j)j∈N is a subadditive sequence, such limits always exist [17]. The following relations about such sequences are well known (see, e.g., [1]). If σ is an admissible sequence, let ε>0; there exists a positive constant C such that (7)C-12js_σ-ε≤Θ_j≤σj+kσk≤Θ¯j≤C2js¯σ+ε,for any j, k∈N. Let σ be an admissible sequence:(i)
ifs_σ>0, then there exists a positive constant C such that, for any J∈N,(8)∑j=J∞σj-1≤CσJ-1,(ii)
ifn∈N satisfies s¯σ<n, then there exists a positive constant C such that, for any J∈N,(9)∑j=1J2jnσj-1≤C2JnσJ-1.In this paper, σ will always stand for an admissible sequence and M for a natural number, possibly zero.Starting from the definitions of the pointwise Hölder spacesΛα(x0) (with α>0) and the generalized uniform Hölder spaces Λσ(Rd) introduced in [1], we are naturally led to the following definition.Definition 2.
Letx0∈Rd; a continuous function f belongs to Λσ,M(x0) if there exist C,J>0 such that(10)infP∈PMf-PL∞x0+Bj≤Cσj,for any j≥J.We trivially have the following alternative definition forΛσ,M(x0).Definition 3.
A continuous functionf belongs to Λσ,M(x0) if there exist C,J>0 such that, for any j≥J, there exists a polynomial Pj∈P[M] for which(11)suph∈Bj⁡fx0+h-Pjx0+h≤Cσj.Remark 4.
In [13], a generalization of the pointwise Hölder spaces is introduced by replacing the admissible sequence appearing in Definition 2 by a modulus of smoothness (more precisely, the definition is based on (34)). As shown in [1], given a modulus of smoothness ω, σj=ω(2-j) defines an admissible sequence. The converse is not necessarily true: only the decreasing admissible sequences converging to 0 give rise to a modulus of continuity [1].Sometimes, we will also need to impose a slightly stronger condition than continuity to the function.Definition 5.
A functionf is uniformly Hölder if there exists ε>0 such that f∈Λε(Rd).


### 2.2. Independence of the Polynomial from the Scale

It is important to remark that the polynomial occurring in inequality (11) is a function of the scale j. However, for the classical Hölder spaces, such polynomial is independent of j. Here, we look under which conditions the independence still holds in the generalized case, that is, under which conditions Pj=P∈P[M] for any j≥J. In this section, f will designate a continuous function of Lloc∞Rd (although the continuity hypothesis is very often dropped when dealing with L∞ norms, we keep it here to ensure the equivalence with Definition 3).We will need the following Markov inequality (see, e.g., [18]): Let p∈0,∞, k∈1,…,d, and S⊂Rd be a bounded convex set with nonempty interior; one has(12)DkPLpS≤Cn2PLpS,for any P∈P[n], where DkP denotes the partial derivative of P following the kth variable and where C depends only on S and p. If x0∈Rd, we thus have(13)DkPL∞x0+rB≤Cn2rPL∞x0+rB,for any r>0 and any P∈Pn, where C is a constant (and does not depend on P, n, or r).Lemma 6.
Iff∈Λσ,Mx0 with M<s_σ-1, the sequence of polynomials Pjj occurring in (11) satisfies(14)DβPk-DβPjL∞x0+Bk≤C2jβσj,for any multi-index β such that |β|≤M and k≥j≥J.
In particular,DβPjx0j is a Cauchy sequence for any multi-index β such that β≤M.Proof.
Using the Markov inequality, we get(15)DβPj-DβPj+1L∞x0+Bj+1≤C2βjPj-Pj+1L∞x0+Bj+1≤C2βjPj-fL∞x0+Bj+1+f-Pj+1L∞x0+Bj+1≤C2βjσj+σj+1≤C′2βjσj,for any β such that β≤M. Therefore, if k satisfies k≥j≥J, one gets (16)DβPj-PkL∞x0+Bk≤∑l=jk-1DβPl-Pl+1L∞x0+Bk≤∑l=jk-1DβPl-Pl+1L∞x0+Bl+1≤C∑l=jk-12βlσl≤C2βjσj,which is the desired result.Lemma 7.
Iff∈Λσ,Mx0 with M<s_σ-1 and Pjj is a sequence of polynomials satisfying inequality (11), then, for any multi-index β such that β≤M, the limit(17)fβx0=limj→∞⁡DβPjx0is independent of the chosen sequence Pjj.Proof.
IfQjj is another sequence of polynomials satisfying inequality (11), one gets(18)DβQjx0-fβx0≤DβQjx0-DβPjx0+DβPjx0-fβx0.One has, using the Markov inequality, (19)DβPj-QjL∞x0+Bj≤C2βjPj-QjL∞x0+Bj≤C2βjPj-fL∞x0+Bj+f-QjL∞x0+Bj≤C2βjσj⟶0,as j→∞, which ends the proof.For such functions, we can introduce the notion of Peano derivative (see, e.g., [13, 19] for more information).Definition 8.
Under the hypothesis of Lemma7, the βth Peano derivative of f at x0 is fβx0=limj⁡DβPjx0.We can now obtain the result concerning the independence of the polynomials.Theorem 9.
IfM<s_σ-1, then f∈Λσ,Mx0 if and only if there exist C>0 and a unique polynomial P∈PM such that(20)f-PL∞x0+Bj≤Cσj,for any j sufficiently large.Proof.
LetPjj be a sequence of polynomials for which inequality (11) is satisfied and set(21)Px=∑β≤Mfβx0x-x0ββ!.One has(22)P-PjL∞x0+Bj=∑β≤Mfβx0-DβPjx0x-x0ββ!L∞x0+Bj≤∑β≤Mfβx0-DβPjx02-jβ.
Since Lemma6 implies(23)fβx0-DβPjx0≤C2jβσj,for any j sufficiently large, we have(24)P-PjL∞x0+Bj≤Cσj.
This inequality can be used to obtain(25)f-PL∞x0+Bj≤f-PjL∞x0+Bj+Pj-PL∞x0+Bj≤Cσj,which shows the existence of P.
If two polynomialsP,Q∈P[M] satisfy inequality (20), then(26)P-QL∞x0+Bj≤P-fL∞x0+Bj+f-QL∞x0+Bj≤Cσj,but if P≠Q,(27)P-QL∞x0+Bj≥C2-jM,for any j sufficiently large, so that 2jMσj does not tend to zero.The polynomialP in inequality (20) is the Taylor expansion of f, where the derivative is replaced with the Peano derivative.The spacesΛσ,Mx0 are a generalization of the usual Hölder spaces, defined by (2).Remark 10.
Letα∈0,∞; the sequence σ=(2-jα)j∈N is an admissible sequence with s_σ=s¯σ=-α, s_σ-1=s¯σ-1=α. Therefore, if α is not a natural number, we have(28)Λαx0=Λσ,αx0=Λσ,s_σ-1x0.
It is easy to check that the polynomial satisfying (2) is unique if and only if α∉N. If α∈N, one rather imposes P∈P[α-1] in order to obtain the uniqueness of the polynomial (one easily verifies that both definitions lead to the same spaces), so that Λαx0=Λσ,s_σ-1-1x0, with σj=2-jα. We will use the modified version in the sequel, ensuring the uniqueness of the polynomial.The following proposition rigorously expresses the idea that the spaceΛσ,Mx0 associated with a sequence σjj that decreases faster than 2-jM is included in the usual Hölder space ΛM(x0).Corollary 11.
Ifs_σ-1>M, one has Λσ,Mx0⊂ΛMx0.Proof.
Letf∈Λσ,Mx0, P be defined as in Theorem 9, that is,(29)Px=∑β≤Mfβx0x-x0ββ!,and let us set(30)Qx=∑β≤M-1fβx0x-x0ββ!.
One gets(31)f-QL∞x0+Bj≤f-PL∞x0+Bj+P-QL∞x0+Bj≤Cσj+C2-jM≤C′2-jM,since 2jMσj tends to zero.


## 2.1. Definition

The generalization of the Hölder spaces we propose here is based on the notion of admissible sequence [10].Definition 1.
A sequenceσ=(σj)j∈N of real positive numbers is called admissible if there exists a positive constant C such that(4)C-1σj≤σj+1≤Cσj,for any j∈N.Ifσ is such a sequence, we set(5)Θ_j=infk∈N⁡σj+kσk,Θ¯j=supk∈N⁡σj+kσk,and we define the lower and upper Boyd indices as follows:(6)s_σ=limj⁡log2⁡Θ_jj,s¯σ=limj⁡log2⁡Θ¯jj.Since (log⁡Θ_j)j∈N is a subadditive sequence, such limits always exist [17]. The following relations about such sequences are well known (see, e.g., [1]). If σ is an admissible sequence, let ε>0; there exists a positive constant C such that (7)C-12js_σ-ε≤Θ_j≤σj+kσk≤Θ¯j≤C2js¯σ+ε,for any j, k∈N. Let σ be an admissible sequence:(i)
ifs_σ>0, then there exists a positive constant C such that, for any J∈N,(8)∑j=J∞σj-1≤CσJ-1,(ii)
ifn∈N satisfies s¯σ<n, then there exists a positive constant C such that, for any J∈N,(9)∑j=1J2jnσj-1≤C2JnσJ-1.In this paper, σ will always stand for an admissible sequence and M for a natural number, possibly zero.Starting from the definitions of the pointwise Hölder spacesΛα(x0) (with α>0) and the generalized uniform Hölder spaces Λσ(Rd) introduced in [1], we are naturally led to the following definition.Definition 2.
Letx0∈Rd; a continuous function f belongs to Λσ,M(x0) if there exist C,J>0 such that(10)infP∈PMf-PL∞x0+Bj≤Cσj,for any j≥J.We trivially have the following alternative definition forΛσ,M(x0).Definition 3.
A continuous functionf belongs to Λσ,M(x0) if there exist C,J>0 such that, for any j≥J, there exists a polynomial Pj∈P[M] for which(11)suph∈Bj⁡fx0+h-Pjx0+h≤Cσj.Remark 4.
In [13], a generalization of the pointwise Hölder spaces is introduced by replacing the admissible sequence appearing in Definition 2 by a modulus of smoothness (more precisely, the definition is based on (34)). As shown in [1], given a modulus of smoothness ω, σj=ω(2-j) defines an admissible sequence. The converse is not necessarily true: only the decreasing admissible sequences converging to 0 give rise to a modulus of continuity [1].Sometimes, we will also need to impose a slightly stronger condition than continuity to the function.Definition 5.
A functionf is uniformly Hölder if there exists ε>0 such that f∈Λε(Rd).


## 2.2. Independence of the Polynomial from the Scale

It is important to remark that the polynomial occurring in inequality (11) is a function of the scale j. However, for the classical Hölder spaces, such polynomial is independent of j. Here, we look under which conditions the independence still holds in the generalized case, that is, under which conditions Pj=P∈P[M] for any j≥J. In this section, f will designate a continuous function of Lloc∞Rd (although the continuity hypothesis is very often dropped when dealing with L∞ norms, we keep it here to ensure the equivalence with Definition 3).We will need the following Markov inequality (see, e.g., [18]): Let p∈0,∞, k∈1,…,d, and S⊂Rd be a bounded convex set with nonempty interior; one has(12)DkPLpS≤Cn2PLpS,for any P∈P[n], where DkP denotes the partial derivative of P following the kth variable and where C depends only on S and p. If x0∈Rd, we thus have(13)DkPL∞x0+rB≤Cn2rPL∞x0+rB,for any r>0 and any P∈Pn, where C is a constant (and does not depend on P, n, or r).Lemma 6.
Iff∈Λσ,Mx0 with M<s_σ-1, the sequence of polynomials Pjj occurring in (11) satisfies(14)DβPk-DβPjL∞x0+Bk≤C2jβσj,for any multi-index β such that |β|≤M and k≥j≥J.
In particular,DβPjx0j is a Cauchy sequence for any multi-index β such that β≤M.Proof.
Using the Markov inequality, we get(15)DβPj-DβPj+1L∞x0+Bj+1≤C2βjPj-Pj+1L∞x0+Bj+1≤C2βjPj-fL∞x0+Bj+1+f-Pj+1L∞x0+Bj+1≤C2βjσj+σj+1≤C′2βjσj,for any β such that β≤M. Therefore, if k satisfies k≥j≥J, one gets (16)DβPj-PkL∞x0+Bk≤∑l=jk-1DβPl-Pl+1L∞x0+Bk≤∑l=jk-1DβPl-Pl+1L∞x0+Bl+1≤C∑l=jk-12βlσl≤C2βjσj,which is the desired result.Lemma 7.
Iff∈Λσ,Mx0 with M<s_σ-1 and Pjj is a sequence of polynomials satisfying inequality (11), then, for any multi-index β such that β≤M, the limit(17)fβx0=limj→∞⁡DβPjx0is independent of the chosen sequence Pjj.Proof.
IfQjj is another sequence of polynomials satisfying inequality (11), one gets(18)DβQjx0-fβx0≤DβQjx0-DβPjx0+DβPjx0-fβx0.One has, using the Markov inequality, (19)DβPj-QjL∞x0+Bj≤C2βjPj-QjL∞x0+Bj≤C2βjPj-fL∞x0+Bj+f-QjL∞x0+Bj≤C2βjσj⟶0,as j→∞, which ends the proof.For such functions, we can introduce the notion of Peano derivative (see, e.g., [13, 19] for more information).Definition 8.
Under the hypothesis of Lemma7, the βth Peano derivative of f at x0 is fβx0=limj⁡DβPjx0.We can now obtain the result concerning the independence of the polynomials.Theorem 9.
IfM<s_σ-1, then f∈Λσ,Mx0 if and only if there exist C>0 and a unique polynomial P∈PM such that(20)f-PL∞x0+Bj≤Cσj,for any j sufficiently large.Proof.
LetPjj be a sequence of polynomials for which inequality (11) is satisfied and set(21)Px=∑β≤Mfβx0x-x0ββ!.One has(22)P-PjL∞x0+Bj=∑β≤Mfβx0-DβPjx0x-x0ββ!L∞x0+Bj≤∑β≤Mfβx0-DβPjx02-jβ.
Since Lemma6 implies(23)fβx0-DβPjx0≤C2jβσj,for any j sufficiently large, we have(24)P-PjL∞x0+Bj≤Cσj.
This inequality can be used to obtain(25)f-PL∞x0+Bj≤f-PjL∞x0+Bj+Pj-PL∞x0+Bj≤Cσj,which shows the existence of P.
If two polynomialsP,Q∈P[M] satisfy inequality (20), then(26)P-QL∞x0+Bj≤P-fL∞x0+Bj+f-QL∞x0+Bj≤Cσj,but if P≠Q,(27)P-QL∞x0+Bj≥C2-jM,for any j sufficiently large, so that 2jMσj does not tend to zero.The polynomialP in inequality (20) is the Taylor expansion of f, where the derivative is replaced with the Peano derivative.The spacesΛσ,Mx0 are a generalization of the usual Hölder spaces, defined by (2).Remark 10.
Letα∈0,∞; the sequence σ=(2-jα)j∈N is an admissible sequence with s_σ=s¯σ=-α, s_σ-1=s¯σ-1=α. Therefore, if α is not a natural number, we have(28)Λαx0=Λσ,αx0=Λσ,s_σ-1x0.
It is easy to check that the polynomial satisfying (2) is unique if and only if α∉N. If α∈N, one rather imposes P∈P[α-1] in order to obtain the uniqueness of the polynomial (one easily verifies that both definitions lead to the same spaces), so that Λαx0=Λσ,s_σ-1-1x0, with σj=2-jα. We will use the modified version in the sequel, ensuring the uniqueness of the polynomial.The following proposition rigorously expresses the idea that the spaceΛσ,Mx0 associated with a sequence σjj that decreases faster than 2-jM is included in the usual Hölder space ΛM(x0).Corollary 11.
Ifs_σ-1>M, one has Λσ,Mx0⊂ΛMx0.Proof.
Letf∈Λσ,Mx0, P be defined as in Theorem 9, that is,(29)Px=∑β≤Mfβx0x-x0ββ!,and let us set(30)Qx=∑β≤M-1fβx0x-x0ββ!.
One gets(31)f-QL∞x0+Bj≤f-PL∞x0+Bj+P-QL∞x0+Bj≤Cσj+C2-jM≤C′2-jM,since 2jMσj tends to zero.


## 3. Alternative Definitions of Generalized Hölder Spaces

Since the uniform spacesΛσ(Rd) can be defined via finite differences or convolutions, one can wonder if such characterizations also hold for the pointwise version of these spaces.


### 3.1. Characterization in Terms of Finite Differences

As usual,Δhnf will stand for the finite difference of order n: given a function f defined on Rd and x,h∈Rd,(32)Δh1fx=fx+h-fx,Δhn+1fx=Δh1Δhnfx,for any n∈N. We also set(33)BhMx0,j=x:x,x+M+1h⊂x0+Bj,where x,x+M+1h is the line segment with end points x and x+(M+1)h. In order to obtain a more general result, we drop the continuity condition of Definition 2 in this section.Proposition 12.
Letf∈Lloc∞Rd; one has f∈Λσ,M(x0) if and only if there exist C,J>0 such that(34)suph∈Bj⁡ΔhM+1fL∞BhMx0,j≤Cσj,for any j≥J.Proof.
The theorem of Whitney (see, e.g., [20]) directly implies that if f satisfies inequality (34), then f∈Λσ,Mx0: one has(35)infP∈PM⁡f-PL∞x0+Bj≤Csuph∈Bj⁡ΔhM+1fL∞BhMx0,j.
Let us now suppose thatf∈Λσ,M(x0) and let x∈BhM(x0,j), P∈P[M]. One has, using the Fréchet functional equation, (36)ΔhM+1fL∞BhMx0,j=ΔhM+1f-PL∞BhMx0,j≤M+1!M+2f-PL∞x0+Bj.
Now, since there exists a polynomialPj∈P[M] such that(37)f-PjL∞x0+Bj≤Cσj,for any j sufficiently large, one gets(38)suph∈Bj⁡ΔhM+1fL∞BhMx0,j≤Cσj,for any j sufficiently large.


### 3.2. Characterization in Terms of Convolutions

Let us denote the space of the infinitely differentiable functions with compact support included in a subsetE of Rd by Cc∞(E). In this section, ρ will denote a radial function of Cc∞(B) such that ρ(x)∈[0,1] for any x∈Rd and ρ1=1. Moreover, one sets ρj=2jdρ(2j·), for any j∈N.In [1], the following result has been obtained.Lemma 13.
LetN∈N0; if f∈Lloc1Rd satisfies(39)supk≥j⁡f∗ρk-fL∞x0+Bj≤Cσj,for j≥J, then, for any multi-index β such that β≤N, one has(40)Dβf∗ρj-f∗ρj-1L∞x0+Bj≤C2jNσj,for j≥J+1.Using the same ideas as in [1], one gets a similar characterization.Theorem 14.
Iff∈Λσ,M(x0), then there exists a function Φ∈Cc∞(Rd) such that(41)supk≥j⁡f-f∗ΦkL∞x0+Bj≤Cσj,for any j sufficiently large.
Conversely, ifσ→0, if f is uniformly Hölder and satisfies inequality (41) for a function Φ∈Cc∞(Rd), then f∈Λσ,M(x0) for any M∈N0 such that M+1>s¯(σ-1).Proof.
Assumef∈Λσ,M(x0). As in [1] (see also [3]), let us set(42)Ψx=∑j=0m/2-1-1jm!j!m-j!12j-mdρx2j-m,where m is large enough (larger than M+1) and Φ=Ψ/∫Ψdx. Using the same arguments as in [1], one gets(43)f∗Φkx-fx=C∫Δ2-ktmfxρtdt,which, due to Proposition 12, leads to inequality (41).
Let us show the converse. Letα∈(0,1) be such that f∈Λα(Rd) and set, as in [1],(44)f1=f∗Φ1,fj=f∗Φj-Φj-1,for j>1. Since f is uniformly Hölder, f is uniformly equal to ∑j≥1fj on Rd and(45)ΔhM+1f=∑j≥1ΔhM+1fj,uniformly on Rd, for any h∈Rd. For j∈N, let n0∈N0, h∈Rd, and j0∈N0 be such that M+1<2n0, |h|≤2-(j+n0), and 2-(j0+1)α≤σj≤2-j0α. One has (46)ΔhM+1fL∞x0+Bj≤∑k=1j-1ΔhM+1fkL∞x0+Bj+∑k=jj0ΔhM+1fkL∞x0+Bj+∑k≥j0+1ΔhM+1fkL∞x0+Bj,where the second term in the right-hand side only appears if j≤j0.
Using Lemma13 and the fact that M+1>s¯(σ-1), the mean value theorem allows writing (47)∑k=1j-1ΔhM+1fkL∞x0+Bj≤∑k=1j-1ChM+1supβ=M+1⁡DβfkL∞x0+Bj-1≤C2-jM+1∑k=1j-12kM+1σk≤Cσj.Moreover,(48)∑k=jj0ΔhM+1fkL∞x0+Bj=ΔhM+1f∗Φj0-f∗Φj-1L∞x0+Bj≤Cf∗Φj0-f∗Φj-1L∞x0+Bj-1≤Cf∗Φj0-fL∞x0+Bj-1+f-f∗Φj-1L∞x0+Bj-1≤Cσj-1+Cσj-1≤Cσj.Finally, (49)∑k≥j0+1ΔhM+1fkL∞x0+Bj≤C∑k≥j0+1fkL∞Rd≤C∑k≥j0+12-kα≤2-j0α≤Cσj.One then has (50)suph∈Bj+n0⁡ΔhM+1fL∞x0+Bj+n0≤suph∈Bj+n0⁡ΔhM+1fL∞x0+Bj≤Cσj≤Cσj+n0,as wanted, in view of Proposition 12.


## 3.1. Characterization in Terms of Finite Differences

As usual,Δhnf will stand for the finite difference of order n: given a function f defined on Rd and x,h∈Rd,(32)Δh1fx=fx+h-fx,Δhn+1fx=Δh1Δhnfx,for any n∈N. We also set(33)BhMx0,j=x:x,x+M+1h⊂x0+Bj,where x,x+M+1h is the line segment with end points x and x+(M+1)h. In order to obtain a more general result, we drop the continuity condition of Definition 2 in this section.Proposition 12.
Letf∈Lloc∞Rd; one has f∈Λσ,M(x0) if and only if there exist C,J>0 such that(34)suph∈Bj⁡ΔhM+1fL∞BhMx0,j≤Cσj,for any j≥J.Proof.
The theorem of Whitney (see, e.g., [20]) directly implies that if f satisfies inequality (34), then f∈Λσ,Mx0: one has(35)infP∈PM⁡f-PL∞x0+Bj≤Csuph∈Bj⁡ΔhM+1fL∞BhMx0,j.
Let us now suppose thatf∈Λσ,M(x0) and let x∈BhM(x0,j), P∈P[M]. One has, using the Fréchet functional equation, (36)ΔhM+1fL∞BhMx0,j=ΔhM+1f-PL∞BhMx0,j≤M+1!M+2f-PL∞x0+Bj.
Now, since there exists a polynomialPj∈P[M] such that(37)f-PjL∞x0+Bj≤Cσj,for any j sufficiently large, one gets(38)suph∈Bj⁡ΔhM+1fL∞BhMx0,j≤Cσj,for any j sufficiently large.


## 3.2. Characterization in Terms of Convolutions

Let us denote the space of the infinitely differentiable functions with compact support included in a subsetE of Rd by Cc∞(E). In this section, ρ will denote a radial function of Cc∞(B) such that ρ(x)∈[0,1] for any x∈Rd and ρ1=1. Moreover, one sets ρj=2jdρ(2j·), for any j∈N.In [1], the following result has been obtained.Lemma 13.
LetN∈N0; if f∈Lloc1Rd satisfies(39)supk≥j⁡f∗ρk-fL∞x0+Bj≤Cσj,for j≥J, then, for any multi-index β such that β≤N, one has(40)Dβf∗ρj-f∗ρj-1L∞x0+Bj≤C2jNσj,for j≥J+1.Using the same ideas as in [1], one gets a similar characterization.Theorem 14.
Iff∈Λσ,M(x0), then there exists a function Φ∈Cc∞(Rd) such that(41)supk≥j⁡f-f∗ΦkL∞x0+Bj≤Cσj,for any j sufficiently large.
Conversely, ifσ→0, if f is uniformly Hölder and satisfies inequality (41) for a function Φ∈Cc∞(Rd), then f∈Λσ,M(x0) for any M∈N0 such that M+1>s¯(σ-1).Proof.
Assumef∈Λσ,M(x0). As in [1] (see also [3]), let us set(42)Ψx=∑j=0m/2-1-1jm!j!m-j!12j-mdρx2j-m,where m is large enough (larger than M+1) and Φ=Ψ/∫Ψdx. Using the same arguments as in [1], one gets(43)f∗Φkx-fx=C∫Δ2-ktmfxρtdt,which, due to Proposition 12, leads to inequality (41).
Let us show the converse. Letα∈(0,1) be such that f∈Λα(Rd) and set, as in [1],(44)f1=f∗Φ1,fj=f∗Φj-Φj-1,for j>1. Since f is uniformly Hölder, f is uniformly equal to ∑j≥1fj on Rd and(45)ΔhM+1f=∑j≥1ΔhM+1fj,uniformly on Rd, for any h∈Rd. For j∈N, let n0∈N0, h∈Rd, and j0∈N0 be such that M+1<2n0, |h|≤2-(j+n0), and 2-(j0+1)α≤σj≤2-j0α. One has (46)ΔhM+1fL∞x0+Bj≤∑k=1j-1ΔhM+1fkL∞x0+Bj+∑k=jj0ΔhM+1fkL∞x0+Bj+∑k≥j0+1ΔhM+1fkL∞x0+Bj,where the second term in the right-hand side only appears if j≤j0.
Using Lemma13 and the fact that M+1>s¯(σ-1), the mean value theorem allows writing (47)∑k=1j-1ΔhM+1fkL∞x0+Bj≤∑k=1j-1ChM+1supβ=M+1⁡DβfkL∞x0+Bj-1≤C2-jM+1∑k=1j-12kM+1σk≤Cσj.Moreover,(48)∑k=jj0ΔhM+1fkL∞x0+Bj=ΔhM+1f∗Φj0-f∗Φj-1L∞x0+Bj≤Cf∗Φj0-f∗Φj-1L∞x0+Bj-1≤Cf∗Φj0-fL∞x0+Bj-1+f-f∗Φj-1L∞x0+Bj-1≤Cσj-1+Cσj-1≤Cσj.Finally, (49)∑k≥j0+1ΔhM+1fkL∞x0+Bj≤C∑k≥j0+1fkL∞Rd≤C∑k≥j0+12-kα≤2-j0α≤Cσj.One then has (50)suph∈Bj+n0⁡ΔhM+1fL∞x0+Bj+n0≤suph∈Bj+n0⁡ΔhM+1fL∞x0+Bj≤Cσj≤Cσj+n0,as wanted, in view of Proposition 12.


## 4. Generalized Pointwise Hölder Spaces and Wavelets

The usual Hölder spaces can “nearly” be characterized in terms of wavelets [12]: for the sufficiency of the condition, the function has to be uniformly Hölder and a logarithmic correction appears. We show here that such a result still holds in the generalized case.


### 4.1. Definitions

Let us briefly recall some definitions and notations (for more precisions, see, e.g., [21–23]). Under some general assumptions, there exist a real-valued function ϕ and 2d-1 real-valued functions ψi1≤i<2d defined on Rd, called wavelets, such that(51)ϕ·-k:k∈Zd∪ψi2j·-k:1≤i<2d,k∈Zd,j∈N0,form an orthogonal basis of L2(Rd). Any function f∈L2(Rd) can be decomposed as follows:(52)fx=∑k∈ZdCkϕx-k+∑j=0+∞∑k∈Zd∑1≤i<2dcj,kiψi2jx-k,where(53)cj,ki=2dj∫Rdfxψi2jx-kdx,Ck=∫Rdfxϕx-kdx.Let us remark that we do not choose the L2Rd normalization for the wavelets, but rather an L∞ normalization, which is better fitted to the study of the Hölderian regularity. Hereafter, the wavelets are always supposed to belong to CnRd with n>M, and the functions Dsϕs≤γ, Dsψis≤γ are assumed to have fast decay (where γ is a sufficiently large number, i.e., strictly greater than M).A dyadic cube of scalej is a cube of the form(54)λ=k12j,k1+12j×⋯×kd2j,kd+12j,where k=k1,…,kd∈Zd. From now on, wavelets and wavelet coefficients will be indexed with dyadic cubes λ. Since i takes 2d-1 values, we can assume that it takes values in 0,1d-0,…,0; we will use the following notations: (i)
λ=λi,j,k=k/2j+i/2j+1+0,1/2j+1d,(ii)
cλ=cj,ki,(iii)
ψλ=ψj,ki=ψi2j·-k.The pointwise Hölderian regularity of a function is closely related to the decay rate of its wavelet leaders.Definition 15.
The wavelet leaders are defined by(55)dλ=supλ′⊂λ⁡cλ′.Two dyadic cubes λ and λ′ are adjacent if they are at the same scale and if dist(λ,λ′)=0. We denote by 3λ the set of the 3d dyadic cubes adjacent to λ and by λj(x0) the dyadic cube of side length 2-j containing x0; we then set(56)djx0=supλ⊂3λjx0⁡dλ.


### 4.2. Result

From now on, we will suppose that the wavelets are compactly supported (such wavelets are constructed in [24]) and j0 will stand for a natural number such that the support of ψi is included in 2j0B, for any i∈1,…,2d-1.Theorem 16.
Iff∈Λσ,Mx0, then there exist C>0 and J∈N0 such that(57)djx0≤Cσj,for any j≥J.
Conversely, letf be a uniformly Hölder function; if inequality (57) is satisfied for an admissible sequence σ that tends to zero, then f∈Λτ,M(x0), where τ is the admissible sequence defined by τj=σj|log2σj| and M∈N0 is any number satisfying M+1>s¯(σ-1).Proof.
In what follows,Pj-k0∈P[M] is according to (11). If f∈Λσ,Mx0, let k0∈N0 be such that 2j0+1+4d≤2k0. For j≥k0+1 and λ=λi,j′,k′⊂3λj(x0), one has (58)cλ=2dj′∫fxψλxdx=2dj′∫fx-Pj-k0xψλxdx=2dj′∫k′/2j′+Bj′-j0fx-Pj-k0xψλxdx≤2dj′∫x0+Bj-k0fx-Pj-k0xψλxdx≤C2dj′σj-k0∫ψλxdx≤Cσj,which is the desired result.
Now, let us suppose that inequality (57) is satisfied for a function f∈Λε(Rd) and an admissible sequence σ tending to 0. Let us set(59)f-1=∑kCkϕ·-k,fj=∑i,kcλψλ,for j∈N0. In [2], it has been shown that these functions have the same regularity as the wavelets and that f is uniformly equal to ∑j≥-1fj. Let us define(60)PJx=∑β≤Mx-x0ββ!∑j=-1JDβfjx0,and let us choose nd∈N such that R>2-j and k/2j∈x+RB (x∈Rd) implies(61)k2j+i2j+1+0,12jd⊂x+2ndRB.Let us also choose md∈N such that any ball x+Bj (x∈Rd, j∈N0) is included in a dyadic cube of length 2md-j. If J′ is such that σj<1 for any j≥J′, we finally choose J such that J≥sup⁡J′,j0+nd+md+1. One has(62)f-PJL∞x0+BJ≤∑j=-1Jfjx-∑β≤Mx-x0ββ!Dβfjx0L∞x0+BJ+∑j≥J+1fjL∞x0+BJ.
Let us look at the first term of the right-hand side. Letj≤J; using the Taylor expansion, one gets (63)fjx-∑β≤Mx-x0ββ!Dβfjx0L∞x0+BJ≤C2-JM+1supβ=M+1⁡DβfjL∞x0+BJ.If β satisfies |β|=M+1, we have, for any x∈x0+BJ, (64)Dβfjx≤∑i,k2jM+1cλDβψλx=∑i∑k2-j∈x+Bj-j02jM+1cλDβψλx.Each coefficient cλ in the last sum is such that λ⊂x+Bj-j0-nd. Therefore, if j≥j0+nd+md+1, then(65)cλ≤Cσj-j0-nd-md-1.Otherwise, since f is uniformly Hölder, cλ≤C≤Cσj. Therefore,(66)DβfjL∞x0+BJ≤C2jM+1σj,for any j≤J, which implies (67)∑j=-1Jfjx-∑β≤Mx-x0ββ!Dβfjx0L∞x0+BJ≤C2-JM+1∑j=-1J2jM+1σj≤CσJ.
For the second term in the right-hand side of (62), let us define J1∈N as the number such that 2-εJ1≤σJ<2-εJ1-1 and decompose the sum as follows:(68)∑j≥J+1fjL∞x0+BJ=∑j≥J1+1fjL∞x0+BJ+∑j=J+1J1fjL∞x0+BJ.We have (69)∑j≥J1+1fjL∞x0+BJ≤∑j≥J1+1fjL∞Rd≤C∑j≥J1+12-εj≤C2-εJ1≤CσJ.Now, for j∈J+1,…,J1 and x∈x0+BJ, one has(70)fjx≤∑i∑k2-j∈x+Bj-j0cλψλx.If j≥J+j0+nd, the wavelet coefficients cλ in the last sum are such that(71)λ⊂x+Bj-j0-nd⊂x0+BJ-1,and therefore(72)cλ≤CσJ-md-1≤CσJ.In the other case,(73)λ⊂x+Bj-j0-nd⊂x0+Bj-j0-nd-1,and thus(74)cλ≤Cσj-j0-nd-md-1≤Cσj≤CσJ.These inequalities lead to(75)∑j=J+1J1fjL∞x0+BJ≤CJ1σJ≤Clog2⁡σJσJ.Putting all these inequalities together, one gets(76)f-PJL∞x0+BJ≤Clog2⁡σJσJ,as desired.The converse part of the previous theorem requires a uniform regularity condition. As shown in [25], a stronger condition than continuity is necessary in the usual case (see also [26], where similar results are obtained (in the usual case) with a Besov regularity assumption). Similarly, the logarithmic correction is best possible in the usual case [25].


## 4.1. Definitions

Let us briefly recall some definitions and notations (for more precisions, see, e.g., [21–23]). Under some general assumptions, there exist a real-valued function ϕ and 2d-1 real-valued functions ψi1≤i<2d defined on Rd, called wavelets, such that(51)ϕ·-k:k∈Zd∪ψi2j·-k:1≤i<2d,k∈Zd,j∈N0,form an orthogonal basis of L2(Rd). Any function f∈L2(Rd) can be decomposed as follows:(52)fx=∑k∈ZdCkϕx-k+∑j=0+∞∑k∈Zd∑1≤i<2dcj,kiψi2jx-k,where(53)cj,ki=2dj∫Rdfxψi2jx-kdx,Ck=∫Rdfxϕx-kdx.Let us remark that we do not choose the L2Rd normalization for the wavelets, but rather an L∞ normalization, which is better fitted to the study of the Hölderian regularity. Hereafter, the wavelets are always supposed to belong to CnRd with n>M, and the functions Dsϕs≤γ, Dsψis≤γ are assumed to have fast decay (where γ is a sufficiently large number, i.e., strictly greater than M).A dyadic cube of scalej is a cube of the form(54)λ=k12j,k1+12j×⋯×kd2j,kd+12j,where k=k1,…,kd∈Zd. From now on, wavelets and wavelet coefficients will be indexed with dyadic cubes λ. Since i takes 2d-1 values, we can assume that it takes values in 0,1d-0,…,0; we will use the following notations: (i)
λ=λi,j,k=k/2j+i/2j+1+0,1/2j+1d,(ii)
cλ=cj,ki,(iii)
ψλ=ψj,ki=ψi2j·-k.The pointwise Hölderian regularity of a function is closely related to the decay rate of its wavelet leaders.Definition 15.
The wavelet leaders are defined by(55)dλ=supλ′⊂λ⁡cλ′.Two dyadic cubes λ and λ′ are adjacent if they are at the same scale and if dist(λ,λ′)=0. We denote by 3λ the set of the 3d dyadic cubes adjacent to λ and by λj(x0) the dyadic cube of side length 2-j containing x0; we then set(56)djx0=supλ⊂3λjx0⁡dλ.


## 4.2. Result

From now on, we will suppose that the wavelets are compactly supported (such wavelets are constructed in [24]) and j0 will stand for a natural number such that the support of ψi is included in 2j0B, for any i∈1,…,2d-1.Theorem 16.
Iff∈Λσ,Mx0, then there exist C>0 and J∈N0 such that(57)djx0≤Cσj,for any j≥J.
Conversely, letf be a uniformly Hölder function; if inequality (57) is satisfied for an admissible sequence σ that tends to zero, then f∈Λτ,M(x0), where τ is the admissible sequence defined by τj=σj|log2σj| and M∈N0 is any number satisfying M+1>s¯(σ-1).Proof.
In what follows,Pj-k0∈P[M] is according to (11). If f∈Λσ,Mx0, let k0∈N0 be such that 2j0+1+4d≤2k0. For j≥k0+1 and λ=λi,j′,k′⊂3λj(x0), one has (58)cλ=2dj′∫fxψλxdx=2dj′∫fx-Pj-k0xψλxdx=2dj′∫k′/2j′+Bj′-j0fx-Pj-k0xψλxdx≤2dj′∫x0+Bj-k0fx-Pj-k0xψλxdx≤C2dj′σj-k0∫ψλxdx≤Cσj,which is the desired result.
Now, let us suppose that inequality (57) is satisfied for a function f∈Λε(Rd) and an admissible sequence σ tending to 0. Let us set(59)f-1=∑kCkϕ·-k,fj=∑i,kcλψλ,for j∈N0. In [2], it has been shown that these functions have the same regularity as the wavelets and that f is uniformly equal to ∑j≥-1fj. Let us define(60)PJx=∑β≤Mx-x0ββ!∑j=-1JDβfjx0,and let us choose nd∈N such that R>2-j and k/2j∈x+RB (x∈Rd) implies(61)k2j+i2j+1+0,12jd⊂x+2ndRB.Let us also choose md∈N such that any ball x+Bj (x∈Rd, j∈N0) is included in a dyadic cube of length 2md-j. If J′ is such that σj<1 for any j≥J′, we finally choose J such that J≥sup⁡J′,j0+nd+md+1. One has(62)f-PJL∞x0+BJ≤∑j=-1Jfjx-∑β≤Mx-x0ββ!Dβfjx0L∞x0+BJ+∑j≥J+1fjL∞x0+BJ.
Let us look at the first term of the right-hand side. Letj≤J; using the Taylor expansion, one gets (63)fjx-∑β≤Mx-x0ββ!Dβfjx0L∞x0+BJ≤C2-JM+1supβ=M+1⁡DβfjL∞x0+BJ.If β satisfies |β|=M+1, we have, for any x∈x0+BJ, (64)Dβfjx≤∑i,k2jM+1cλDβψλx=∑i∑k2-j∈x+Bj-j02jM+1cλDβψλx.Each coefficient cλ in the last sum is such that λ⊂x+Bj-j0-nd. Therefore, if j≥j0+nd+md+1, then(65)cλ≤Cσj-j0-nd-md-1.Otherwise, since f is uniformly Hölder, cλ≤C≤Cσj. Therefore,(66)DβfjL∞x0+BJ≤C2jM+1σj,for any j≤J, which implies (67)∑j=-1Jfjx-∑β≤Mx-x0ββ!Dβfjx0L∞x0+BJ≤C2-JM+1∑j=-1J2jM+1σj≤CσJ.
For the second term in the right-hand side of (62), let us define J1∈N as the number such that 2-εJ1≤σJ<2-εJ1-1 and decompose the sum as follows:(68)∑j≥J+1fjL∞x0+BJ=∑j≥J1+1fjL∞x0+BJ+∑j=J+1J1fjL∞x0+BJ.We have (69)∑j≥J1+1fjL∞x0+BJ≤∑j≥J1+1fjL∞Rd≤C∑j≥J1+12-εj≤C2-εJ1≤CσJ.Now, for j∈J+1,…,J1 and x∈x0+BJ, one has(70)fjx≤∑i∑k2-j∈x+Bj-j0cλψλx.If j≥J+j0+nd, the wavelet coefficients cλ in the last sum are such that(71)λ⊂x+Bj-j0-nd⊂x0+BJ-1,and therefore(72)cλ≤CσJ-md-1≤CσJ.In the other case,(73)λ⊂x+Bj-j0-nd⊂x0+Bj-j0-nd-1,and thus(74)cλ≤Cσj-j0-nd-md-1≤Cσj≤CσJ.These inequalities lead to(75)∑j=J+1J1fjL∞x0+BJ≤CJ1σJ≤Clog2⁡σJσJ.Putting all these inequalities together, one gets(76)f-PJL∞x0+BJ≤Clog2⁡σJσJ,as desired.The converse part of the previous theorem requires a uniform regularity condition. As shown in [25], a stronger condition than continuity is necessary in the usual case (see also [26], where similar results are obtained (in the usual case) with a Besov regularity assumption). Similarly, the logarithmic correction is best possible in the usual case [25].


## 5. A Generalized Definition of the Hölder Exponent

The usual Hölder spaces are embedded:α<β implies Λβx0⊂Λαx0. A notion of regularity for a function f∈Lloc∞Rd at x0 can thus be given by the so-called Hölder exponent:(77)hfx0=sup⁡α>0:f∈Λαx0.To do so in the generalized case, one needs some conditions under which Λσ,Mx0⊂Λσ′,M′x0.This generalized exponent naturally leads to the definition of an alternative multifractal formalism, yet similar to the one developed in [12], where, for example, logarithmic corrections can appear.


### 5.1. A Trivial Illustration

The classical version of Theorem16 theoretically allows estimating the Hölder regularity at a given point x0 by looking at the behavior of djx0 versus the scales j [12]. This notion of regularity is given by the Hölder exponent hfx0, defined by equality (77). Following the standard wavelet characterization [12], one should have(78)log2⁡djx0~hfx0j+C,and a log-log plot can be used to estimate the slope h0. This method is simpler than directly fitting the curve(79)γC,ht=Cωh2-j,with ω(h)(r)=rh to the function j↦dj(x0), since in this latter case one has to estimate two parameters (C and h) to retain only one (namely, h, which is the estimation of hfx0). However, this approach allows fine-tuning the computation of h0 using Theorem 16. In the case of a Brownian motion, for example, having the law of the iterated logarithm in mind [14, 15], one should rather choose(80)ωhr=rlog⁡log⁡rh,in the definition of γC,h. Fitting dj(x0) with different definitions of ω(h) should help to discern between specific models. In the case proposed here, it could support the detection of the presence of a logarithmic correction, which could be the signature of a Brownian motion.As an illustration, the wavelet leader of a Brownian motionj↦dj(x0) for some x0 (the middle was arbitrarily chosen) is represented in Figure 1. When trying to fit the curve C2-jh to dj(x0) using the Levenberg-Marquardt algorithm [27], one gets hf(x0)=0.554 (see Figure 1). The same computation with the logarithmic correction (using Definition (80)) gives hfx0=0.495, which is closer to the expected value 1/2. Computing the distance, for each point x0 of the signal (220 points), between the estimated Hölder exponent hf(x0) and the expected value 1/2 gives rise to the boxplot represented in Figure 2.Figure 1
The functionj↦djx0 in black, with the standard curve C2-jh (blue) and the logarithmically corrected curve (using equality (80)) (red) obtained with the Levenberg-Marquardt algorithm.Figure 2
Boxplot of the distance between1/2 and the estimated Hölder exponent. The left boxplot corresponds to the usual method and the other corresponds to a method involving the logarithmic correction.Of course, this application is only a simple illustrative example; more sophisticated work is required to show the effectiveness of this approach and is out of the scope of this work (see [28]). From a numerical point of view, the estimation of hf(x0) is unstable (in particular, the detection of fast and slow points in the Brownian motion seems to be a difficult task, if not insufferable); this is why one rather tries to estimate the size of the set of points sharing the same Hölder exponent. This is the purpose of the multifractal formalism for functions (see, e.g., [12]), which will not be recalled here.


### 5.2. Preliminary Results

We first need some technical easy results. From now on, iff∈Λσ,Mx0, Pjj will stand for the sequence of polynomials of P[M] corresponding to the definition. We will write(81)Pjx=∑β≤Majβxβ,Qjx=∑β≤M-1ajβxβ.Lemma 17.
Letf∈Λσ,Mx0; one has(82)supβ=M⁡ajβ≤C∑k=1j-12MΘ¯1k+1,(83)supβ=M⁡ajβ≤CσjΘ_1-j∑k=1j-12MΘ_1k+1.Proof.
Using the Markov inequality (13), we get (84)DβPj-Pj+1L∞x0+Bj+1≤C2jMPj-Pj+1L∞x0+Bj+1≤C2jMPj-fL∞x0+Bj+f-Pj+1L∞x0+Bj+1≤C2jMσj,for any β such that β≤M and j sufficiently large. Therefore, we have (85)DβP1-PjL∞x0+Bj≤∑k=1j-1DβPk-Pk+1L∞x0+Bj≤∑k=1j-1DβPk-Pk+1L∞x0+Bk+1≤C∑k=1j-12kMσk≤C∑k=1j-12MΘ¯1k,for any j.
Now, letβ be a multi-index such that β=M; inequality (82) follows from(86)DβP1-PjL∞x0+Bj≥Cajβ-a1β,while inequality (83) can be obtained in the same way, using(87)DβP1-PjL∞x0+Bj≤C∑k=1j-12kMσk≤CσjΘ_1-j∑k=1j-12MΘ_1k,valid for any j.Corollary 18.
Letf∈Λσ,Mx0; one has the following inequalities: (i)
if2MΘ¯1<1,(88)f-QjL∞x0+Bj≤Cσj+2-jM,(ii)
if2MΘ¯1>1,(89)f-QjL∞x0+Bj≤Cσj+Θ¯1j,(iii)
if2MΘ¯1=1,(90)f-QjL∞x0+Bj≤Cσj+2-jMj.Corollary 19.
Letf∈Λσ,M(x0); one has the following inequalities: (i)
if2MΘ_1<1,(91)f-QjL∞x0+Bj≤Cσj2MΘ_1-j+2-jM,(ii)
if2MΘ_1>1,(92)f-QjL∞x0+Bj≤Cσj+2-jM,(iii)
if2MΘ_1=1,(93)f-QjL∞x0+Bj≤Cσjj+2-jM.


### 5.3. Definitions

Before introducing the definition of a generalized Hölder exponent, we must first consider embedded spaces of typeΛσ,Mx0. Once the definitions are given, we provide sufficient conditions for generalized Hölder spaces to be embedded.Definition 20.
If, for anyα>0, σα is an admissible sequence, the application(94)σ·:α>0⟼σαis called a family of admissible sequences.Definition 21.
Letx0∈Rd; a family σ(·) of admissible sequences is decreasing for x0 if α>β implies Λσα,αx0⊂Λσβ,βx0.Definition 22.
Letσ· be a decreasing family of admissible sequences for x0; if f∈Lloc∞Rd, then the Hölder exponent of f at x0 for the family σ(·) is given by(95)hfσ·x0=sup⁡α>0:f∈Λσα,αx0.The following proposition is a simple corollary of the results obtained in the previous section; it helps to check if a family of admissible sequences is decreasing. Ifσ(·) is a family of admissible sequences, we set(96)Θ_jα=infk∈N⁡σj+kασkα,Θ¯jα=supk∈N⁡σj+kασkα.Proposition 23.
Letσ(·) be a family of admissible sequences and x0∈Rd; σ(·) is decreasing for x0 if it satisfies the following two conditions: (1)
ifm≤α<β<m+1, with m∈N0, then there exist C,J>0 such that(97)σjβ≤Cσjα,for anyj≥J,(2)
for anym∈N, at least one of the following two conditions is satisfied:
(a)
there existsε0>0 such that, for any ε∈(0,ε0), there exist C,J>0 for which σj(m)≤Cσj(m-ε) and
(i)
if1<2mΘ¯1(m), then (Θ¯1(m))j≤Cσj(m-ε),(ii)
if1>2mΘ¯1(m), then 2-jm≤Cσj(m-ε),(iii)
if1=2mΘ¯1(m), then j2-jm≤Cσj(m-ε),for anyj≥J,(b)
there existsε0>0 such that, for any ε∈(0,ε0), there exist C,J>0 for which 2-jm≤Cσj(m-ε) and
(i)
if1<2mΘ_1m, then σjm≤Cσjm-ε,(ii)
if1>2mΘ_1m, then σjm2mΘ_1m-j≤Cσjm-ε,(iii)
if1=2mΘ_1m, then jσjm≤Cσjm-ε,for anyj≥J.This result is similar to the one obtained in [1] (under the hypothesis of Proposition 23, one gets a decreasing family of admissible sequences for the uniform case), but the proof given for these generalized uniform Hölder spaces cannot be adapted for the pointwise case.Remark 24.
Since the family of admissible sequencesσ· defined by σjα=2-jα satisfies the conditions of Proposition 23, this result implies the classical inclusion Λβx0⊂Λαx0, for any α, β satisfying α<β.Let us consider the following example to grasp the practicality of Proposition23.Example 25.
Letg:α>0↦gα be a positive function and set(98)σjα=2-jαjgα.One directly checks that condition (2)(b) is satisfied, so that σ· is a decreasing family of admissible sequences.
Both conditions can be used to obtain decreasing families of admissible sequences; here is an example where the first condition is used.Example 26.
Let(99)gα=-1ifα∈N0otherwise.If σ· is defined using identity (98), condition (2)(a) is satisfied so that this sequence is also a decreasing family of admissible sequences.


## 5.1. A Trivial Illustration

The classical version of Theorem16 theoretically allows estimating the Hölder regularity at a given point x0 by looking at the behavior of djx0 versus the scales j [12]. This notion of regularity is given by the Hölder exponent hfx0, defined by equality (77). Following the standard wavelet characterization [12], one should have(78)log2⁡djx0~hfx0j+C,and a log-log plot can be used to estimate the slope h0. This method is simpler than directly fitting the curve(79)γC,ht=Cωh2-j,with ω(h)(r)=rh to the function j↦dj(x0), since in this latter case one has to estimate two parameters (C and h) to retain only one (namely, h, which is the estimation of hfx0). However, this approach allows fine-tuning the computation of h0 using Theorem 16. In the case of a Brownian motion, for example, having the law of the iterated logarithm in mind [14, 15], one should rather choose(80)ωhr=rlog⁡log⁡rh,in the definition of γC,h. Fitting dj(x0) with different definitions of ω(h) should help to discern between specific models. In the case proposed here, it could support the detection of the presence of a logarithmic correction, which could be the signature of a Brownian motion.As an illustration, the wavelet leader of a Brownian motionj↦dj(x0) for some x0 (the middle was arbitrarily chosen) is represented in Figure 1. When trying to fit the curve C2-jh to dj(x0) using the Levenberg-Marquardt algorithm [27], one gets hf(x0)=0.554 (see Figure 1). The same computation with the logarithmic correction (using Definition (80)) gives hfx0=0.495, which is closer to the expected value 1/2. Computing the distance, for each point x0 of the signal (220 points), between the estimated Hölder exponent hf(x0) and the expected value 1/2 gives rise to the boxplot represented in Figure 2.Figure 1
The functionj↦djx0 in black, with the standard curve C2-jh (blue) and the logarithmically corrected curve (using equality (80)) (red) obtained with the Levenberg-Marquardt algorithm.Figure 2
Boxplot of the distance between1/2 and the estimated Hölder exponent. The left boxplot corresponds to the usual method and the other corresponds to a method involving the logarithmic correction.Of course, this application is only a simple illustrative example; more sophisticated work is required to show the effectiveness of this approach and is out of the scope of this work (see [28]). From a numerical point of view, the estimation of hf(x0) is unstable (in particular, the detection of fast and slow points in the Brownian motion seems to be a difficult task, if not insufferable); this is why one rather tries to estimate the size of the set of points sharing the same Hölder exponent. This is the purpose of the multifractal formalism for functions (see, e.g., [12]), which will not be recalled here.


## 5.2. Preliminary Results

We first need some technical easy results. From now on, iff∈Λσ,Mx0, Pjj will stand for the sequence of polynomials of P[M] corresponding to the definition. We will write(81)Pjx=∑β≤Majβxβ,Qjx=∑β≤M-1ajβxβ.Lemma 17.
Letf∈Λσ,Mx0; one has(82)supβ=M⁡ajβ≤C∑k=1j-12MΘ¯1k+1,(83)supβ=M⁡ajβ≤CσjΘ_1-j∑k=1j-12MΘ_1k+1.Proof.
Using the Markov inequality (13), we get (84)DβPj-Pj+1L∞x0+Bj+1≤C2jMPj-Pj+1L∞x0+Bj+1≤C2jMPj-fL∞x0+Bj+f-Pj+1L∞x0+Bj+1≤C2jMσj,for any β such that β≤M and j sufficiently large. Therefore, we have (85)DβP1-PjL∞x0+Bj≤∑k=1j-1DβPk-Pk+1L∞x0+Bj≤∑k=1j-1DβPk-Pk+1L∞x0+Bk+1≤C∑k=1j-12kMσk≤C∑k=1j-12MΘ¯1k,for any j.
Now, letβ be a multi-index such that β=M; inequality (82) follows from(86)DβP1-PjL∞x0+Bj≥Cajβ-a1β,while inequality (83) can be obtained in the same way, using(87)DβP1-PjL∞x0+Bj≤C∑k=1j-12kMσk≤CσjΘ_1-j∑k=1j-12MΘ_1k,valid for any j.Corollary 18.
Letf∈Λσ,Mx0; one has the following inequalities: (i)
if2MΘ¯1<1,(88)f-QjL∞x0+Bj≤Cσj+2-jM,(ii)
if2MΘ¯1>1,(89)f-QjL∞x0+Bj≤Cσj+Θ¯1j,(iii)
if2MΘ¯1=1,(90)f-QjL∞x0+Bj≤Cσj+2-jMj.Corollary 19.
Letf∈Λσ,M(x0); one has the following inequalities: (i)
if2MΘ_1<1,(91)f-QjL∞x0+Bj≤Cσj2MΘ_1-j+2-jM,(ii)
if2MΘ_1>1,(92)f-QjL∞x0+Bj≤Cσj+2-jM,(iii)
if2MΘ_1=1,(93)f-QjL∞x0+Bj≤Cσjj+2-jM.


## 5.3. Definitions

Before introducing the definition of a generalized Hölder exponent, we must first consider embedded spaces of typeΛσ,Mx0. Once the definitions are given, we provide sufficient conditions for generalized Hölder spaces to be embedded.Definition 20.
If, for anyα>0, σα is an admissible sequence, the application(94)σ·:α>0⟼σαis called a family of admissible sequences.Definition 21.
Letx0∈Rd; a family σ(·) of admissible sequences is decreasing for x0 if α>β implies Λσα,αx0⊂Λσβ,βx0.Definition 22.
Letσ· be a decreasing family of admissible sequences for x0; if f∈Lloc∞Rd, then the Hölder exponent of f at x0 for the family σ(·) is given by(95)hfσ·x0=sup⁡α>0:f∈Λσα,αx0.The following proposition is a simple corollary of the results obtained in the previous section; it helps to check if a family of admissible sequences is decreasing. Ifσ(·) is a family of admissible sequences, we set(96)Θ_jα=infk∈N⁡σj+kασkα,Θ¯jα=supk∈N⁡σj+kασkα.Proposition 23.
Letσ(·) be a family of admissible sequences and x0∈Rd; σ(·) is decreasing for x0 if it satisfies the following two conditions: (1)
ifm≤α<β<m+1, with m∈N0, then there exist C,J>0 such that(97)σjβ≤Cσjα,for anyj≥J,(2)
for anym∈N, at least one of the following two conditions is satisfied:
(a)
there existsε0>0 such that, for any ε∈(0,ε0), there exist C,J>0 for which σj(m)≤Cσj(m-ε) and
(i)
if1<2mΘ¯1(m), then (Θ¯1(m))j≤Cσj(m-ε),(ii)
if1>2mΘ¯1(m), then 2-jm≤Cσj(m-ε),(iii)
if1=2mΘ¯1(m), then j2-jm≤Cσj(m-ε),for anyj≥J,(b)
there existsε0>0 such that, for any ε∈(0,ε0), there exist C,J>0 for which 2-jm≤Cσj(m-ε) and
(i)
if1<2mΘ_1m, then σjm≤Cσjm-ε,(ii)
if1>2mΘ_1m, then σjm2mΘ_1m-j≤Cσjm-ε,(iii)
if1=2mΘ_1m, then jσjm≤Cσjm-ε,for anyj≥J.This result is similar to the one obtained in [1] (under the hypothesis of Proposition 23, one gets a decreasing family of admissible sequences for the uniform case), but the proof given for these generalized uniform Hölder spaces cannot be adapted for the pointwise case.Remark 24.
Since the family of admissible sequencesσ· defined by σjα=2-jα satisfies the conditions of Proposition 23, this result implies the classical inclusion Λβx0⊂Λαx0, for any α, β satisfying α<β.Let us consider the following example to grasp the practicality of Proposition23.Example 25.
Letg:α>0↦gα be a positive function and set(98)σjα=2-jαjgα.One directly checks that condition (2)(b) is satisfied, so that σ· is a decreasing family of admissible sequences.
Both conditions can be used to obtain decreasing families of admissible sequences; here is an example where the first condition is used.Example 26.
Let(99)gα=-1ifα∈N0otherwise.If σ· is defined using identity (98), condition (2)(a) is satisfied so that this sequence is also a decreasing family of admissible sequences.



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*Source: 8276258-2018-05-29.xml*