Giacomo Nardi

Franco

Maïtine Bergounioux

Antonio Leaci

Franco Tomarelli

FRACTIONAL SOBOLEV SPACES AND FUNCTIONS OF BOUNDED VARIATION OF ONE VARIABLE IN "FCAA" JOURNAL

Keywords: MSC 2010 Primary 26A30; Secondary 26A33, 26A45 Fractional calculus, Bounded variation functions, Riemann-Liouville derivative, Marchaud derivative, Sobolev spaces

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Introduction

The aim of this work is to investigate the fractional derivative concepts and make the connection between the related (so called fractional) Sobolev spaces and the spaces of functions of bounded variation whose derivatives are not functions but measures. Here, we only deal with the 1D case and we hope to extend results to higher dimensions via slicing theorems. Our main concern to investigate this connection is to consider variational models in the context of image processing. Indeed, variational models including many kinds of first order derivative have been studied from the seminal work of Rudin et al. [START_REF] Rudin | Nonlinear total variation based noise removal algorithms[END_REF]. We have used generalized second order derivatives in different segmentation and image analysis context [START_REF] Bergounioux | Second order variational models for image texture analysis[END_REF][START_REF] Bergounioux | Mathematical Analysis of a Inf-Convolution Model for Image Processing[END_REF][START_REF] Carriero | A Survey on Blake and Zisserman Functional[END_REF][START_REF] Carriero | Euler equations for Blake & Zisserman functional[END_REF]. The use of derivatives of order in (0,1) is not standard in image analysis though it could be a quite useful tool for texture analysis to our opinion. However, pioneer works have been done (see [START_REF] Pu | Fractional partial differential equation denoising models for texture image[END_REF] for example) using a finite dimensional setting. From a mathematical point of view, the natural underlying space is the space of functions of bounded variation. However, beyond the use of the total variation, it appears that more accurate penalization tools are needed. This is the case, for example, for texture analysis where the structure involves fractal dimension. To our opinion, the fractional derivative could be the suitable tool to describe images textures by providing quantitative information via the differentiation order. c Year Diogenes Co., Sofia pp. xxx-xxx, DOI: .................. There are two main classes of definitions for fractional differentiation whose connections are not fully explored to our knowledge. The fractional derivative in the sense of Gagliardo is not explicitely defined (not even almost everywhere) but it is implicitly assumed by the setting of fractional Sobolev Spaces and the underlying norm (see [START_REF] Caffarelli | An extension problem related to the fractional laplacian[END_REF] for example). It is, in some sense, a global definition which can be easily handled via the Fourier transform in the Hilbertian case. It is a priori well suited for variational analysis, especially in view of the Riesz distributional gradient. The second approach is based on the Riemann-Liouville fractional derivative (in short RL) and may be pointwise defined. We choose to focus on the RL derivative : there are many variants of the fractional derivatives/integrals definition as the Grunwald-Letnikov, Caputo, Weyl ones [START_REF] Butzer | An introduction to fractional calculus[END_REF][START_REF] Das | Functional Fractional Calculus[END_REF][START_REF] Mathai | The H-function[END_REF] but the RL derivative can be considered as a generic one. One can refer to [START_REF] De Oliveira | A review of definitions for fractional derivatives and integral[END_REF] for a review of the different definitions. For a complete study of these derivatives one can refer to the book by Samko and al. [START_REF] Samko | Fractional Integrals and Derivatives-Theory and Applications[END_REF] that contains an extensive bibliography in particular with respect to the pioneer work of Hardy-Littlewood. Moreover, in [START_REF] Tarasov | Fractional dynamics: Applications of fractional calculus to dynamics of particles, fields and media[END_REF], the connection is made with metric and measure spaces, in particular the Hausdorff measure. We decided to use this derivative concept because it seems more adapted to applications and numerical computations than the Gagliardo one. The RL derivative is widely used by physicists [START_REF] Uchaikin | Fractional derivatives for physicists and engineers: Background and theory[END_REF][START_REF] Uchaikin | Fractional derivatives for physicists and engineers: Applications[END_REF], in automatics, control theory and image processing as well, especially to deal with image enhancement and texture analysis [START_REF] Pu | Fractional partial differential equation denoising models for texture image[END_REF]; in [START_REF] Malinowska | Fractional variational calculus for non differentiable functions[END_REF] calculus of variations problems where the cost functional involves fractional derivative are investigated. Nevertheless, the context is often a discrete one and there is not much analysis (to our knowledge) in the infinite dimensional setting. In particular, the link between the classical spaces of bounded variation functions and the fractional Sobolev spaces is not clear. To our knowledge, there is no paper that compare the BV space and the fractional Sobolev spaces in the RL sense. Indeed, the concept of fractional Sobolev spaces is not much developed for the RL derivative, though this fractional derivative concept is commonly used in engineering. One can refer to [START_REF] Bourdin | A fractional fundamental lemma and a fractional integration by parts formula-Applications to critical points of Bolza functionals and to linear boundary value problems[END_REF][START_REF] Idczak | A fractional imbedding theorem[END_REF][START_REF] Idczak | Fractional Sobolev spaces via Riemann-Liouville derivatives[END_REF] however. The limitation of the present paper is twofold. Here we consider only the 1D case: indeed we extensively exploit the notion of good representative of BV functions of one variable, which is not available for several variables. However, 1D results are necessary to consider higher dimension cases via slicing theorems for example ([3] section 3.11). In addition, since we are interested in image processing applications, we consider only bounded intervals. Of course, it is still possible to extend the present analysis to unbounded intervals. However, in that case the different notions of fractional derivatives induce significant differences [START_REF] Cossar | A theorem on Cesaro summability[END_REF][START_REF] Butzer | An introduction to fractional calculus[END_REF][START_REF] Trebels | Multipliers for (C, alpha)-bounded Fourier expansions in Banach spaces and approximation theory[END_REF]. This implies more investigation to compare the effects of the different approaches and this does not fit our prior concern.

The paper is organized as follows. Section 2 is devoted to the presentation of the two main approaches with a special focus on the Riemann-Liouville fractional derivatives: the main tools are recalled. In Section 3, we define RL-fractional Sobolev spaces W s,1 RL,a+ and give basic properties. In the last section, we perform a comparison between these fractional Sobolev spaces, the classical BV space and the space SBV of functions whose distributional derivative is a special measure in the sense of De Giorgi (see [START_REF] Ambrosio | Functions of bounded variation and free discontinuity problems[END_REF]). In particular we prove that SBV ⊂ s∈(0,1)

W s,1 and s∈(0,1)

W s,1 \ BV = ∅ .

Fractional Calculus and Fractional Sobolev Spaces

In this section we present the two main (different) definitions of fractional Sobolev spaces that we can find in the literature. We are in particular interested to the case where the differentiation order is s ∈ [0, 1) in order to study the fractional spaces between L 1 and W 1,1 and their relationship with BV . In the sequel, we consider the 1D framework. We recall that the space AC(a, b) of absolutely continuous functions coincides with the Sobolev space W 1,1 (a, b) defined by

W 1,1 (a, b) := u ∈ L 1 (a, b) | u ∈ L 1 (a, b) , endowed with the norm u W 1,1 = u L 1 + u L 1 .
Here and in the sequel d dx u denotes the classical derivative of u, u denotes the distributional derivative of u, and u stands for the absolutely continuous part of u

([3]). Recall that W 1,1 (a, b) ⊂ C 0 ([a, b]), where C 0 ([a, b]
) is the space of continuous functions on [a, b] (see [1,[START_REF] Attouch | Variational analysis in Sobolev and BV spaces[END_REF][START_REF] Maz'ya | Sobolev spaces[END_REF] for example) and

∀u ∈ W 1,1 (a, b), ∀y ∈ [a, b] u L ∞ ≤ |u(y)| + u L 1 .
2.1. Gagliardo's fractional Sobolev Spaces. This section is devoted to recalling the classical definition of fractional Sobolev spaces in the sense of Gagliardo:

Definition 2.1 (Gagliardo's spaces). Let s ∈ (0, 1). For any p ∈ [1, +∞) we define the following space:

W s,p G (a, b) = u ∈ L p (a, b) : |u(x) -u(y)| |x -y| 1 p +s ∈ L p ([a, b] × [a, b]) . (2.1)
This is a Banach space endowed with the norm

u W s,p G (a,b) = [a,b] |u(x)| p dx + [a,b] [a,b] |u(x) -u(y)| p |x -y| 1+sp dx dy 1 p . W s,p G (a, b) is the interpolated space between L p (a, b) and W 1,p (a, b) and the term [u] W s,p G (a,b) = [a,b] [a,b] |u(x) -u(y)| p |x -y| 1+sp dx dy 1 p (2.2)
is the so-called Gagliardo semi-norm of u. We have in particular [START_REF] Di Nezza | Hitchhiker's guide to the fractional Sobolev spaces[END_REF] that

W s,p G (a, b) ⊆ W r,p G (a, b) ∀ 0 < r < s ≤ 1 . If s = m + τ > 1 with m ∈ N, τ ∈ [0, 1[
, such a definition can be generalized to higher orders by setting

W s,p G (a, b) = {u ∈ W m,p (a, b) : D m u ∈ W τ,p (a, b)}.
This point of view is related to interpolation theory (see [1,[START_REF] Aronszajn | Boundary values of functions with finite Dirichlet integral[END_REF][START_REF] Demengel | Functional spaces for the theory of elliptic partial differential equations[END_REF][START_REF] Gagliardo | Proprietà di alcune classi di funzioni in più variabili[END_REF][START_REF] Slobodeckij | Generalized Sobolev spaces and their applications to boundary value problems of partial differential equations[END_REF][START_REF] Tartar | An introduction to Sobolev spaces and interpolation spaces[END_REF] for example).

There is a huge literature concerning these fractional differentiation methods that we cannot mention here. However, let us quote a recent work by Shieh and Spector [START_REF] Shieh | On a new class of fractional partial differential equations[END_REF] who define the distributional Riesz fractional gradient for every C ∞ c (R) function (with compact support) as

D s u(x) = Γ(s/2) 2 s √ π Γ((1 -s)/2) R u (t) (x -t) s dt
and define the fractional Sobolev spaces X s,p (s ∈ (0, 1), 1 < p < ∞) as the closure of C ∞ c (R) with respect to the norm u := u L p (R) + D s u L p (R) 1/p . Here Γ stands for the classical Gamma function [START_REF] Oldham | An Atlas of Functions[END_REF]. Nevertheless, this definition is to be considered in a reflexive framework and will not be useful for our purpose since we need to choose p = 1 (see Section 3.) Moreover, we deal with bounded intervals and this definition should be applied only to functions with prescribed compact support (for example).

2.2.

Fractional integration and differentiation theory. Another point of view to deal with fractional derivatives is the one we describe in the sequel: the archetypal definition is the one known as Riemann-Liouville, though there are many variants [START_REF] De Oliveira | A review of definitions for fractional derivatives and integral[END_REF] that we do not consider. The point of view is different from the Gagliardo one. This viewpoint aims to a pointwise definition of derivatives by using fractional integrals while the Gagliardo's fractional Sobolev Spaces are defined by interpolation and global approach.

As we already mentioned it, we decided to focus on this second type which seems more suitable with respect to applications.

2.2.1. Fractional integrals. From now on [a, b] is a nonempty bounded interval of R. We start by defining the fractional integral for L 1 -functions:

Definition 2.2. Let u ∈ L 1 ([a, b]
). For every s ∈ (0, 1] we define the left-side and rightside Riemann-Liouville fractional integrals, by setting respectively

∀x ∈ [a, b] I s a+ [u](x) = 1 Γ(s) x a u(t) (x -t) 1-s dt, ∀x ∈ [a, b] I s b-[u](x) = 1 Γ(s) b x u(t) (t -x) 1-s dt.
The properties of left-side and right-side integrals are similar. In the sequel we list only the main results for the left-side integral I s a+ . The fractional integration theory has been extensively studied in [START_REF] Samko | Fractional Integrals and Derivatives-Theory and Applications[END_REF]. Next proposition recall the main properties of the fractional integral Proposition 2.1. For any s ∈ (0, 1), the fractional integral I s a+ is a continuous operator from

(i) L p (a, b) into L p (a, b) for every p ≥ 1, (ii) L p (a, b) into L r (a, b) for every p ∈ [1, 1/s) and r ∈ [1, p/(1 -sp)), (iii) L p (a, b) into C 0,s-1 p (a, b), for every p > 1/s (iv) L p (a, b) into L r (a, b) with r ∈ [1, ∞), for p = 1/s, (v) L ∞ (a, b) into C 0,s (a, b).
Moreover, with p ≥ 1, we have

∀u ∈ L p (a, b) lim s→0 + I s a+ u -u L p (a,b) = 0 . (2.3)
Here C 0,s (a, b) denotes the space of Hölder (continuous) functions of order s.

Remark 2.1 (Fractional integral of BV-functions). We point out that to ensure the Hölder-regularity of the fractional integral we need to work with L p -functions with p > 1.

The case p = 1 is not covered from the previous proposition. However, the point (v) guarantees such a regularity for bounded functions, which helps to study an important subset of L 1 (a, b), namely BV ([a, b]) (see section 4). Indeed, in dimension one, every function of bounded variation is bounded ([5] chapter 10), so we get

I s a+ (BV ([a, b])) ⊂ C 0,s (a, b) ∀ s ∈ (0, 1) . (2.4)

Fractional derivatives and representability.

There are several different definitions of fractional derivatives. We recall next the definition of Riemann-Liouville and Marchaud derivatives and refer to [START_REF] Ross | Functions that have no first order derivative might have fractional derivatives of all orders less that one[END_REF][START_REF] Samko | Fractional Integrals and Derivatives-Theory and Applications[END_REF] for a deeper analysis of the fractional differentiation theory.

Definition 2.3 (Riemann-Liouville fractional derivative). Let u ∈ L 1 (a, b) and n -1 ≤ s < n (n integer). The left Riemann-Liouville derivative of u at x ∈ [a, b] is defined by D s a+ u(x) = d n dx n I n-s a+ [u](x) = 1 Γ(n -s) d n dx n x a u(t) (x -t) s-n+1 dt (2.5)
at points where the classical derivative d n /dx n exists. If such a derivative exists at x for s = 0, n = 1, then it coincides with the function u at x. Similarly, we may define the right Riemann-Liouville derivative of u at x ∈ [a, b] as

D s b-u(x) = d n dx n I n-s b-[u](x) = 1 Γ(n -s) d n dx n b x u(t) (t -x) s-n+1 dt (2.6)
if the last term exists.

Example 2.1 (Power function). We consider the function u

(x) = x k (k ≥ 0) on [0, 1],
say a = 0 and b = 1. Then for every s ∈ [n -1, n) and any x ∈ (0, 1) the fractional derivative at x is defined as

D s 0+ x k = 1 Γ(n -s) d n dx n x 0 t k (x -t) n-s-1 dt = 1 Γ(n -s) d n dx n x n+k-s 1 0 (1 -v) n-s-1 v k dv = 1 Γ(n -s) d n dx n x n+k-s B(n -s, k + 1) = 1 Γ(n -s) Γ(k + 1)Γ(n -s) Γ(k + 1 + n -s) d n dx n x n+k-s = Γ(k + 1) Γ(k -s + 1) x k-s
where we exploited the Beta Euler function B(ν, µ) =

1 0 t ν-1 (1 -t) µ-1 = Γ(ν)Γ(µ) Γ(ν+µ) and d n dx n x τ = Γ(τ + 1) Γ(τ -n + 1) x τ -n ∀ τ ≥ 0 .
If s is a positive integer number, then the fractional derivative D s 0+ x k coincides with the classical one. If k = 0, then the left fractional derivative of the constant function (say D s 0+ ) is null if and only if s is a strictly positive integer number. Remark also that

D s 0+ x s-k = 0 ∀ s > 0 , k = 1, ..., 1 + [s]
where [s] denotes the integer part of the real number s.

Now we focus on the case

n = 1. Definition 2.4. (Representability) A function f ∈ L 1 (a, b
) is said to be representable, and it is represented by a fractional integral if f ∈ I s a+ (L 1 (a, b)) for some s ∈ (0, 1). Next Theorem [START_REF] Samko | Fractional Integrals and Derivatives-Theory and Applications[END_REF] gives a representability criterion:

Theorem 2.1. (L 1 -representability) Assume f ∈ L 1 (a, b). Then f ∈ I s a+ (L 1 (a, b)) for a given s ∈ (0, 1] if and only if I 1-s a+ [f ] ∈ W 1,1 (a, b) and I 1-s a+ [f ](a) = 0 . Moreover, if u ∈ L 1 (a, b) is such that f = I s a+ [u] then u = D s a+ f a.e.

on (a, b) .

As an immediate consequence, we have the following result:

Corollary 2.1. Assume s ∈ (0, 1); ∀u ∈ L 1 (a, b) D s a+ I s a+ [u] = u, (2.7
)

and ∀u ∈ I s a+ (L 1 (a, b)) I s a+ D s a+ u = u. (2.8)
In the previous Corollary, (2.7) proves that fractional differentiation can be seen as the inverse operator of the fractional integration. The converse is not true in general: a counterexample is given by the power function x s-k (k = 1, ..., 1 + [s]) whose s-fractional derivative is null if and only if k = 0 and s > 0 (Example 2.1). This is similar to the classical integration and differentiation theories where the integral of u differs from u for a constant. However, according to (2.8), for every function that can be represented as a fractional integral, the fractional integration acts as the reciprocal operator of the fractional differentiation.

Using Theorem 2.1, it is easy to verify that the power function x s-1 is not represented by a fractional integral. In fact

I 1-s a+ [x s-1 ] ≡ Γ(s -1) so that it belongs to W 1,1 (a, b) but it does not verify I 1-s a+ [x s-1 ](0) = 0.
Let us give a comment about the relationship between D s a+ and I s a+ . There are two kind of results on the fractional integral that can be very useful to study the properties of the fractional derivative :

• The first-one is a representability result (for instance Theorem 2.1) that gives conditions for a function f to be represented as the fractional integral of an other function u. This is quite important because it allows to easily prove that f admits a fractional derivative. However, the representability of a function u (i.e., u = I s a+ ϕ with ϕ ∈ L 1 ) is only a sufficient condition to get the existence of the derivative. The power function and the Heaviside function give two examples of functions that are not representable and whose fractional derivative exists (see previous discussion and Example 4.1).

• The second kind of result are embedding results, as (2.4), that give some informations on the regularity of the fractional integral to get the Riemann-Liouville fractional derivative existence.

u = I s a+ [f ] f ∈ L p (a, b) s ∈ (0, 1
) . In order to address this issue we need to introduce a slightly different definition/notion of fractional derivative. According to [START_REF] Samko | Fractional Integrals and Derivatives-Theory and Applications[END_REF] p. 110, we note that, for C 1 -functions and every s ∈ (0, 1), the use of integration by parts gives

D s a+ u(x) = u(x) Γ(1 -s)(x -a) s + s Γ(1 -s) x a u(x) -u(t) (x -t) 1+s dt ∀ x ∈ (a, b] .
(2.9)

The Marchaud fractional derivative is defined as the second summand in the right-hand side term of (2.9). To extend this setting to non-smooth functions we need to define the integral by a limit, which leads to the following definition:

D s a+,ε u(x) = u(x) Γ(1 -s)(x -a) s + s Γ(1 -s) ψ ε (x)
and

ψ ε (x) =        x-ε a u(x) -u(t) (x -t) 1+s dt if x ≥ a + ε , x-ε a u(x) (x -t) 1+s dt if a ≤ x ≤ a + ε . (2.10)
Note that the definition of ψ ε for a ≤ x ≤ a + ε is obtained by continuing the function u by zero beyond the interval [a, b]. The passage to the limit depends on the functional space we are working with. We remark that such a derivative is not defined at x = a and that a necessary condition for the derivative to exist is u(a) = 0

The right-side derivative can be defined similarly by using the integral between x and b. In the following we state the main results about Marchaud differentiation for the left-side derivative, but similar results can be obtained for the right-side one.

As expected, if u is a C 1 -function, the Marchaud and Riemann-Liouville derivative coincide for every s ∈ (0, 1), and their expression is given by (2.9): (1) For every

∀ u ∈ C 1 ([a, b]) D s a+ u(x) = D s a+ u(x) ∀ x ∈ (a
f = I s a+ [u] where u ∈ L p (a, b) with p ≥ 1, we get D s a+ f = u . (2) Let be p ∈ (1, ∞). For any f ∈ L p (a, b), we get f = I s a+ [u] with u ∈ L p (a, b
) if and only if the limit of the family {ψ ε } as ε → 0, where ψ ε is defined in (2.10) exists (for the L p norm topology). D s a+ u(x) = D s a+ u(x), if s + α < 1. This is a useful result in order to study the fractional derivative because the Marchaud derivative it is easier to handle.

Riemann-Liouville Fractional Sobolev space (p = 1)

In this section, we define the Sobolev spaces associated to the Riemann-Liouville fractional derivative for p = 1. The case p = 1 is of particular interest since we aim to study the relationship between these spaces and the spaces of functions of bounded variation.

A first possible definition could be given by the following set for s ∈ (0, 1):

{u ∈ L 1 (a, b) | D s a+ u ∈ L 1 (a, b
)} , which contains all the L 1 -functions such that the Riemann-Liouville fractional derivative or order s for a given s ∈ (0, 1) belongs to L 1 . We noticed that if the Riemann-Liouville fractional derivative of u exists for some s, then, referring to the same s, I s a+ [u] is differentiable almost everywhere. However, we have no information on the differential properties of the fractional integral. These differential properties are not completely described by the pointwise derivative though it exists a.e. This shows that the previous definition is not suitable to obtain a generalized integration by parts formula.

Therefore, to develop a satisfactory theory of fractional Sobolev spaces we use a more suitable definition in the next section.

3.1. Riemann-Liouville Fractional Sobolev spaces. Following [START_REF] Bourdin | A fractional fundamental lemma and a fractional integration by parts formula-Applications to critical points of Bolza functionals and to linear boundary value problems[END_REF][START_REF] Idczak | Fractional Sobolev spaces via Riemann-Liouville derivatives[END_REF] where these spaces are denoted AC s,1 a + , we may define the Riemann-Liouville Fractional Sobolev spaces as follows:

Definition 3.1. Let s ∈ [0, 1). We denote by W s,1 RL,a+ (a, b) := {u ∈ L 1 (a, b) | I 1-s a+ [u] ∈ W 1,1 (a, b) }.
A similar space W s,1 RL,b-(a, b) can be defined by using the right-side fractional integral :

W s,1 RL,b-(a, b) = {u ∈ L 1 (a, b) | I 1-s b-[u] ∈ W 1,1 (a, b) }. Short notations W s,1 RL,a+ , W s,1
RL,b-will be used whenever there is no risk of confusion. For any s ∈ (0, 1), the space W s,1 RL,a+ is a Banach space endowed with the norm

u W s,1 RL,a+ := u L 1 (a,b) + I 1-s a+ u W 1,1 (a,b) .
Note that this definition does not mean that u is representable but its fractional integral f = I 1-s a+ u is representable. We shall describe the representable functions of W s,1 RL,a+ next (it is related to their trace). a+ [u] is differentiable at every point with a derivative as an L 1 -function, then u ∈ W s,1 RL,a+ . Indeed, in such case I s-1 a+ [u] ∈ W 1,1 (a, b) (see [START_REF] Royden | Real analysis[END_REF]). 3.2. Embedding properties. Before performing comparisons between these fractional Sobolev spaces and the spaces of bounded variation functions, we investigate some embedding relations.

Theorem 3.1. Assume 0 < s < r < 1 and consider u ∈ L 1 (a, b) ∩ I r a+ (L 1 (a, b)). Then u ∈ W s,1 RL,a+ (a, b) and u W s,1 RL,a+ (a,b) ≤ C s,r u W r,1 RL,a+ (a,b) . Proof. Let be u ∈ L 1 (a, b) ∩ I r a+ (L 1 (a, b)).
Then u is representable and Theorem 2.1 yields that u ∈ W r,1 RL,a+ . As a direct consequence of Theorem 31 in [START_REF] Idczak | Fractional Sobolev spaces via Riemann-Liouville derivatives[END_REF], we get that for every 0 < s < r ≤ 1 the embedding W r,1 RL,a+ ⊂ W s,1 RL,a+ is compact. Therefore u ∈ W s,1 RL,a+ and (by continuity of the embedding) we get

u W s,1 RL,a+ ≤ C s,r u W r,1 RL,a+
.

Next theorem gives a relationship between Riemann-Liouville Sobolev spaces and Gagliardo Sobolev spaces: Theorem 3.2. Let be s, r ∈ (0, 1) such that r > s. Then

W r,1 G (a, b) ∩ I s a+ (L 1 (a, b)) ⊂ W s,1 RL,a+ (a, b) with continuous injection. More precisely, ∀u ∈ W r,1 G (a, b) ∩ I s a+ (L 1 (a, b)) u W s,1 RL,a+ (a,b) ≤ C u W r,1 G (a,b) .
Proof. Let us choose s ∈ (0, 1) and r > s (in (0, 1)). Let be u ∈ I s a+ (L 1 (a, b)). It is represented by a fractional integral of a L 1 -function, so its Riemann-Liouville and Marchaud derivative coincide. Due to Remark 2.2, the RL fractional derivative coincide with the Marchaud derivative for functions in the RL fractional Sobolev space of the same order. Therefore to achieve the claim, it will be enough showing that the Gagliardo norm of a fractional Sobolev space keeps under control the the L 1 norm of the Marchaud derivative of the same order. Recall that

D s a+,ε u(x) = u(x) ε s Γ(1 -s) if a ≤ x ≤ a + ε and D s a+,ε u(x) = 1 Γ(1 -s) u(x) (x -a) s + s Γ(1 -s) x-ε a u(x) -u(t) (x -t) 1+s dt if x ≥ a + ε. D s a+,ε u L 1 (a+ε,b) ≤ 1 Γ(1 -s) b a+ε |u(x)| (x -a) s dx + s Γ(1 -s) b a+ε x-ε a |u(x) -u(t)| |x -t| 1+s dt dx ≤ 1 Γ(1 -s) b a+ε |u(x)| (x -a) s dx + s Γ(1 -s) [u] W s,1 G (a,b) .
where the Gagliardo semi-norm [u] W s,1 G (a,b) is given by (2.2). Moreover

D s a+,ε u L 1 (a,a+ε) ≤ 1 ε s Γ(1 -s) a+ε a |u(x)|dx; finally D s a+,ε u L 1 (a,b) ≤ 1 Γ(1 -s) b a+ε |u(x)| (x -a) s dx + ε -s a+ε a |u(x)|ds + s[u] W s,1 G (a,b) . (3.1)
Now, we know ([18] -section 6. for example) that

W r,1 G (a, b) ⊂ L p (a, b) with continuous embedding for p ∈ [1, 1 1 -r ] and compact embedding if p ∈ [1, 1 1 -r ).As u ∈ W r,1 G (a, b), then u ∈ L p (a, b
) where p can be chosen such as 1 1-s < p < 1 1-r since that r > s. Let us call p = p/(p -1) the conjugate exponent and apply Hölder inequalities to relation 3.1. Note that p satisfies 1 -sp > 0. We get

D s a+,ε u L 1 (a,b) ≤ u L p (a+ε,b) Γ(1 -s) (b -a) 1-sp -ε 1-sp 1 -sp 1/p + u L p (a,a+ε) Γ(1 -s) ε (1-sp )/p + s Γ(1 -s) [u] W s,1 G (a,b) ≤ u L p Γ(1 -s)   (b -a) 1-sp -ε 1-sp 1 -sp 1/p + ε (1-sp )/p   + s Γ(1 -s)
[u] W s,1 G (a,b) . Passing to the limit as ε → 0 gives

D s a+ u L 1 (a,b) ≤ u L p Γ(1 -s) (b -a) (1-sp ) 1 -sp 1/p + s Γ(1 -s) [u] W s,1 G (a,b) . As u L p ≤ C u W r,1 G (a,b) and [u] W s,1 G (a,b) ≤ [u] W r,1 G (a,b) ≤ u W r,1 G (a,b)
we finally get

D s a+ u L 1 (a,b) ≤ C(s, r, a, b) u W r,1 G (a,b
) . This ends the proof. Remark 3.2.

(1) Note that for every s, r ∈ (0, 1) with s < r, the function u

: x → (x-a) s+1 Γ(s+2) belongs to W r,1 G (a, b)∩I s a+ (L 1 (a, b)) so that this space is not reduced to {0}. Indeed u is a W 1,1 (a, b) function and lso a W r,1 G (a, b) function since W 1,1 (a, b) ⊂ W r,1
G (a, b). Moreover a short computation similar to the one of example 2.1 gives that u = I s a+ (x -a) which proves that u ∈ I s a+ (L 1 (a, b)).

(2) Note that s = r is the critical case in the above theorem. We cannot handle this case with the same arguments. Next, we investigate the relationship between W 1,1 (a, b) and the fractional Sobolev spaces W s,1 RL,a+ . A crucial tool is the following integration by parts formula.

Proposition 3.1 ([8]). If 0 ≤ 1 p < s < 1 and 0 ≤ 1 r < s < 1, then for every u ∈ W s,1 RL,a+ and v ∈ W r,1 RL,b-we get b a (D s a+ u)(t)v(t) dt = b a (D s b-v)(t)u(t) dt + u(b) I 1-s b-[v](b) -I 1-s a+ [u](a) v(a) . (3.2) Example 3.1 (Smooth functions). Set [a, b] = [0, 1]. For every u ∈ C ∞ ([0, 1], R
) and for every s ∈ [0, 1) we get for any x = 0, using integration by parts twice:

D s 0+ u(x) = 1 Γ(1 -s) d dx x 0 u(t)(x -t) -s dt = 1 (1 -s)Γ(1 -s) d dx u(0)x 1-s + x 0 u (t)(x -t) 1-s dt = 1 Γ(1 -s) u(0)x -s + x 0 u (t)(x -t) -s dt = 1 Γ(1 -s) u(0)x -s + 1 Γ(2 -s) u (0)x 1-s + 1 Γ(2 -s) x 0 u (t)(x -t) 1-s dt
since Γ(z + 1) = zΓ(z) for every z > 0. As x → x -s belongs to L 1 (0, 1) and u is bounded and s ∈ [0, 1), we get that D s 0+ u belongs to L 1 (0, 1). Moreover, we have

u -D s 0+ u-u(0) L 1 ([0,1],R) = 1 0 u (x) - u(0)x -s Γ(1 -s) - u (0)x 1-s Γ(2 -s) - x 0 u (t)(x -t) 1-s Γ(2 -s) dt-u(0) dx ≤ |u(0)| 1 0 1 - x -s Γ(1 -s) dx + 1 0 u (x) - u (0)x 1-s Γ(2 -s) - x 0 u (t)(x -t) 1-s Γ(2 -s) dt dx ≤ |u(0)| 1 - 1 Γ(2 -s) + 1 0 u (x) - u (0)x 1-s Γ(2 -s) - x 0 u (t)(x -t) 1-s Γ(2 -s) dt dx.
Indeed, we observe that x → 1 -x -s Γ(1-s) does not vanish on (0, 1) so that

1 0 1 - x -s Γ(1 -s) dx = 1 0 (1 - x -s Γ(1 -s) dx = 1 - 1 Γ(2 -s) . Since lim s→1 Γ(2 -s) = Γ(1) = 1, by the Lebesgue convergence theorem, we get u -D s 0+ u -u(0) L 1 ([0,1],R) s→1 -→ 1 0 u (x) -u (0) - x 0 u (t)dt dx = 0. Moreover, if u(0) = 0 then D s 0+ u is well defined at x = 0 and ∀x ∈ [0, 1] lim s→1 D s 0+ (u)(x) = u (x),
and not only almost everywhere. Now, we compare the usual Sobolev space W 1,1 (a, b) and the Riemann-Liouville fractional Sobolev spaces. We already know that W 1,1 (a, b) is compactly embedded in W s,1 RL,a+ for any s ∈ [0, 1) thanks to Theorem 31 of [START_REF] Idczak | Fractional Sobolev spaces via Riemann-Liouville derivatives[END_REF] again. Now, we investigate the behavior of D s a+ u with respect to s as s → 1. In the sequel M a,b is the space of (finite) Radon measures on R with support contained

µ ε µ) as ε → 0 if ∀ϕ ∈ C([a, b]) lim ε→0 µ ε (ϕ) = µ(ϕ) .
Next L 1 denotes the 1D Lebesgue measure and δ a the Dirac measure at a say

∀ϕ ∈ C(R) δ a , ϕ = ϕ(a) .
We recall that any L p (a, b) function f may be considered as a measure by the identification

f → f L 1 with ∀ϕ ∈ C([a, b]) f L 1 , ϕ = R ϕ(x)f (x) dx . Theorem 3.3. Assume that u ∈ W 1,1 (a, b). Then, u = u and D s a u M a,b u L 1 + u(a)δ a as s → 1.
In addition, if u(a) = 0, then we have

D s a+ u L 1 (a,b) -→ u as s → 1. Proof. Let be u ∈ W 1,1 (a, b). Then u ∈ W s,1
RL,a+ for any s ∈ [0, 1). Moreover u is defined everywhere since W 1,1 (a, b) is continuously embedded in C([a, b])(see [1]). Integration by parts provide the next representation for fractional integral and derivative:

I 1-s a+ [u](x) = 1 Γ(2 -s) u(a) (x -a) 1-s + x a u (t)(x -t) 1-s dt (3.3)
and

D s a+ u(x) = 1 Γ(1 -s) u(a) (x -a) -s + x a u (t)(x -t) -s dt = u(a)(x -a) -s Γ(1 -s) + I 1-s a+ [u ](x) . (3.4)
As s ∈ (0, 1) we get that the fractional derivative is defined almost everywhere on (a, b] and belongs to L 1 (a, b) with

D s a+ u L 1 (a,b) ≤ |u(a)|(b -a) 1-s Γ(2 -s) + I 1-s a+ [u ] L 1 (a, b) ≤ C |u(a)| + u L 1 (a, b) . (3.5)
Now, as u ∈ L 1 (a, b) then by relation (2.3) we get the strong convergence of I 1-s a u to u in L 1 (a, b) as s → 1. Therefore we get the result in the case where u(a) = 0 using (3.4).

Moreover, for every

ϕ ∈ C 1 (a, b) u(a) Γ(1 -s) b a (x -a) -s ϕ(x)dx = u(a) Γ(2 -s) ϕ(b)(b -a) 1-s - b a (x -a) 1-s ϕ (x)dx s→1 → u(a)ϕ(a). So D s a+ u, ϕ = u(a) Γ(1 -s) b a (x -a) -s ϕ(x)dx + b a I 1-s a [u ](x)ϕ(x) dx → u(a)ϕ(a) + b a u (x)ϕ(x) dx = u(a)δ a + u L 1 , ϕ .
We conclude with the density of the C 1 functions in the space of continuous functions: let be ϕ ∈ C(a, b) and ϕ n ∈ C 1 such that ϕ n -ϕ ∞ → 0. With (3.5) we get for every s ∈ (0, 1)

| D s a+ u, ϕ -ϕ n | ≤ D s a+ u L 1 ϕ n -ϕ ∞ ≤ C |u(a)| + u L 1 (a, b) ϕ n -ϕ ∞ n→+∞ → 0. Similarly | u(a)δ a + u L 1 , ϕ -ϕ n | ≤ |u(a)| + u L 1 (a, b) ϕ n -ϕ ∞ n→+∞ → 0 .
Let be ε > 0. There exists n 0 ∈ N such that

∀n ≥ n 0 , ∀s ∈ (0, 1) | D s a+ u, ϕ -ϕ n | + | u(a)δ a + u L 1 , ϕ -ϕ n | ≤ ε 2 . Fix n * ≥ n 0 . As ϕ n * ∈ C 1 (a, b), there exists η * > 0 such that ∀s such that 1 -s ≤ η * | D s a+ u -u(a)δ a + u L 1 , ϕ n * | ≤ ε 2 .
Finally, as

| D s a+ u -u(a)δ a -u L 1 , ϕ | ≤ | D s a+ u, ϕ -ϕ n * | + | u(a)δ a + u L 1 , ϕ -ϕ n * | + D s a+ u -u(a)δ a -u L 1 , ϕ n * .
we get for every s such that 1

-s ≤ η * | D s a+ u -u(a)δ a + u L 1 , ϕ | ≤ ε . This proves that ∀ϕ ∈ C(a, b) D s a+ u, ϕ → u(a)δ a + u L 1 , ϕ .
Remark 3.3. Previous Theorem shows that, even if u is a smooth function,its fractional derivative cannot converge in L 1 to u whenever u(a) = 0, since some of the its mass must concentrates at x = a. The simplest example is provided by the constant function u ≡ 1 in (0, 1). Then

u ≡ 0 , D s a+ u(x) = (x -a) -s Γ(1 -s)
and

D s a+ u L 1 (a,b) = (b -a) 1-s Γ(2 -s) s→1 -→ 1 = u L 1 (a,b) .

Comparison with BV and SBV

Let us recall the definition and the main properties of the space of functions of bounded variation (see [START_REF] Ambrosio | Functions of bounded variation and free discontinuity problems[END_REF][START_REF] Attouch | Variational analysis in Sobolev and BV spaces[END_REF] for example), defined by

BV (a, b) = {u ∈ L 1 (a, b) | T V (u) < +∞}, where (a, b) is a bounded, open subset of R, T V (u) := sup b a u(x) ξ (x) dx | ξ ∈ C 1 c (a, b), ξ ∞ ≤ 1 . ( 4
= u L 1 + T V (u), is a Banach space.
The derivative in the sense of distributions of every u ∈ BV (a, b) is a bounded Radon measure, denoted u , and T V (u) is the total variation of u. For more details, one can refer to [START_REF] Ambrosio | Functions of bounded variation and free discontinuity problems[END_REF].

Proposition 4.1 ([3]). Let (a, b) be a bounded open subset of R (1)
For any u ∈ BV (a, b), the total variation coincides with the essential pointwise variation. Precisely,

T V (u) = inf v=u a.e. sup T i |v(t i+1 ) -v(t i )|
where T stands for any subdivision

T = {t 0 = a < t 1 < • • • < t n = b} of (a, b) . (2)
For every u ∈ BV (a, b), the Radon measure u can be decomposed into u = u dx + ( u) ⊥ , where u dx is the absolutely continuous part of u with respect of the Lebesgue measure and ( u) ⊥ is the singular part.

(3) The mapping u → T V (u) is lower semi-continuous from BV (a, b) to R + for the L 1 (a, b) topology. (4) BV (a, b) ⊂ L σ (a, b) with continuous embedding, for σ ∈ [1, ∞]. (5) BV (a, b) ⊂ L σ (a, b) with compact embedding, for σ ∈ [1, ∞) .
The singular part ( u) ⊥ of the derivative has a jump part and a Cantor component. The SBV (a, b) space (see [START_REF] Ambrosio | Functions of bounded variation and free discontinuity problems[END_REF] for example) is the space of functions in BV (a, b) whose derivative has no singular Cantor component. The functions of SBV (a, b) have two components: one is regular and belongs to W 1,1 (a, b) and the other one is a countable summation of characteristic functions. More precisely, any increasing function in SBV (a, b) can be written as

u(x) = u(a) + x a u(t) dt + x k ∈Ju p k χ [x k ,1] (x) x ∈ [a, b]
where J u denotes the (at most countable) set of jump points of u and

p k = u + (x k )-u -(x k )
denotes the positive jump of u at x k . This describes all the functions of SBV (a, b) since any SBV -function can be written as the difference of two SBV increasing functions. Indeed, any SBV -function can be written as the difference of two BV increasing functions. These functions are obviously in SBV thanks to the unique decomposition (up to additive constant) of the derivative ([3]-Corollary 3.33).

Next example shows that there exists a SBV -function that belongs to W s,1 RL,a+ for any s ∈ (0, 1). This confirms the regularizing behavior of the fractional integral operator and represents a preliminary result in order to prove the relationship between SBV function and fractional Sobolev space.

Example 4.1 (Step functions). Let u : [0, 1] → R (a = 0), u(x) = χ [α,1] (x) with α ∈ (0, 1). We consider s ∈ [0, 1). For every x ∈ [α, 1] we get

I 1-s 0+ [χ [α,1] ](x) =    0 if x ∈ [0, α) (x -α) 1-s Γ(2 -s) if x ∈ [α, 1] (4.2)
which proves that

I 1-s 0+ [χ [α,1] ] ∈ W 1,1 (0, 1) so that χ [α,1] ∈ W s,1
RL,0+ (0, 1). The fractional derivative is given by

D s 0+ χ [α,1] (x) =    0 if x ∈ [0, α] (x -α) -s Γ(1 -s) if x ∈ (α, 1] (4.3)
and, for every s ∈ (0, 1), we have that

1 α (x -α) -s Γ(1 -s) = (x -α) 1-s Γ(2 -s) 1 α = (1 -α) 1-s Γ(2 -s) which implies that D s 0+ u L 1 ([0,1])

s→1

-→ 1 = T V (u) (4.4) where T V (u) denotes the total variation of u on (0, 1).

Next result is useful to prove Theorem 4.1: Lemma 4.1. Let {f k } ⊂ W 1,1 (a, b) a sequence of non-negative functions with f k (a) = 0 and non-negative derivative. We suppose also that

k f k , k f k ∈ L 1 (a, b) . (4.5) Then k f k (x) = k f k (x) a.e. x ∈ [a, b] . (4.6)
Proof. The result follows from the monotone convergence theorem and the hypothesis on

f k . In fact, for every x ∈ [a, b] we get k f k (x) = k x a f k (t) dt = x a k f k (t) dt ,
which proves that k f k ∈ W 1,1 (a, b) and (4.6) follows.

Theorem 4.1. For every s ∈ (0, 1), it holds that SBV (a, b) ⊂ W s,1 RL,a+ (a, b) and

D s a+ u M a,b u L 1 + u(a + )δ a + x k ∈Ju p k δ x k as s → 1 .
Moreover, if u(a + ) = 0 then

D s a+ u L 1 (a,b) → T V (u) as s → 1 .
Here u(a + ) denotes the right limit of u at zero and J u ⊂ (a, b) is the jump set of u.

Proof. With a simple change of variables, we can assume that [a, b] = [0, 1]. This will make the proof easier to read. As every SBV -function can be written as the difference of two increasing SBV functions, we prove the result for a SBV -increasing function. Such a function u can be written as

u(x) = u(0 + ) + x 0 u(t) dt + x k ∈Ju p k χ [x k ,1] (x) a.e. x ∈ [0, 1]
where u(0 + ) denotes the right limit of u at zero, J u denotes the (at most countable) set of jump points of u and p k = u + (x k ) -u -(x k ) denotes the positive jump of u at x k . In particular

T V (u) = 1 0 | u(t)| dt + x k ∈Ju p k .
The function u can be written as the sum of two functions u = u a + u j where u a is the absolutely continuous part (u a ∈ W 1,1 (0, 1)) and u j is the jump part. They are given by:

u a (x) = u(0 + ) + x 0 u(t) dt a.e. x ∈ [0, 1] , u j = x k ∈Ju p k χ [x k ,1] .
• We first consider u a . Since u a belongs to W 1,1 (a, b), for every s ∈ (0, 1) it belongs to W s,1 RL,a+ and its fractional derivative is given by (3.4):

D s 0+ (x) = u(0 + )x -s Γ(1 -s) + I 1-s 0+ [ u](x) a.e. x ∈ [0, 1].
Moreover with Theorem 3.3 we have

if u(0 + ) = 0 D s 0+ u L 1 (0,1) -→ u as s → 1 , otherwise D s 0+ u M 0,1 u L 1 + u(0 + )δ 0 as s → 1 . (4.7)
• We consider now u j with s ∈ (0, 1). We use Example 4.1 and set

f k = I 1-s 0+ [p k χ [x k ,1] ] =    0 if x ∈ [0, x k ) p k (x -x k ) 1-s Γ(2 -s) if x ∈ [x k , 1]
and

f k (x) =    0 if x ∈ [0, x k ) p k (x -x k ) -s Γ(1 -s) if x ∈ [x k , 1].
Functions f k and f k are L 1 for every k and f k (0) = 0. Moreover, we get

sup n   1 0 k≤n f k (t) dt   ≤ sup n k≤n p k Γ(3 -s) ≤ T V (u) Γ(3 -s) , and
sup n   1 0 k≤n f k (t) dt   ≤ sup n k≤n p k Γ(2 -s) ≤ T V (u) Γ(2 -s) .
So, with the monotone convergence theorem, we have that the series

I 1-s 0+ u j = x k ∈Ju f k and x k ∈Ju f k are normally convergent in L 1 (0, 1). Using Lemma 4.1 we get (I 1-s 0+ u j ) (x) = x k ∈Ju p k (x -x k ) -s Γ(1 -s) χ [x k ,1] a.e. x ∈ (0, 1).
This means that I 1-s 0+ u j ∈ W 1,1 (0, 1) and consequently, u j ∈ W s,1 RL,a+ . So we may conclude that SBV (a, b) ⊂ W s,1 RL,a+ . It remains to prove the weak * convergence of D s 0+ u j towards

x k ∈Ju p k δ x k . Using again the monotone convergence theorem one gets 1] a.e. x ∈ (0, 1).

D s 0+ u j (x) = x k ∈Ju p k (x -x k ) -s Γ(1 -s) χ [x k ,
Let ϕ ∈ C 1 (0, 1) with compact support.

D s 0+ u j , ϕ = 1 0 D s 0+ u j (x)ϕ(x) dx = x k ∈Ju p k Γ(1 -s) 1 0 (x -x k ) -s χ [x k ,1] (x)ϕ(x) dx .
Here, we used again that the series is normally convergent. An integration by parts gives

1 Γ(1 -s) 1 0 (x -x k ) -s χ [x k ,1] (x)ϕ(x) dx = - 1 Γ(2 -s) 1 x k (x -x k ) 1-s ϕ (x) dx.

So the Lebesgue dominating theorem gives that lim

s→1 1 Γ(1 -s) 1 0 (x -x k ) -s χ [x k ,1] (x)ϕ(x) dx = - 1 x k ϕ (x) dx = ϕ(x k ) .
We finally obtain, always with the same tools, that

lim s→1 D s 0+ u j , ϕ = x k ∈Ju p k ϕ(x k ) = x k ∈Ju p k δ k , ϕ .
We conclude by density as in the proof of Theorem 3.3. The subsequent remarks point out that the fractional Sobolev spaces are larger than SBV and give some relationship between BV and W s,1 . Remark 4.2 (Cantor-Vitali function). The Cantor-Vitali function is an example of increasing continuous function on [0, 1] whose classical derivative is defined and null at a.e. point. It is well known that such a function is of bounded variation but is not a SBVfunction. Precisely such a function is Hölder-continuous with exponent α = ln 2/ ln 3 (i.e., the Hausdorff dimension of the Cantor set). Then, the Riemann-Liouville derivative is well defined at every point for every s ∈ (0, α). In fact, it is surely possible to prove the result for any s ∈ (0, 1) using an adapted Cantor-like function whose Hölder exponent would be s. Moreover the Riemann-Liouville derivative of order s belongs to C 0,α-s ([0, 1]) , and we get that the Cantor-Vitali function belongs to W s,1 RL,0+ ([0, 1]) for every s ∈ (0, α). This proves in particular that in general

(BV \ SBV ) ∩ W s,1
RL,0+ = ∅ , since the function we exhibit belongs to BV and W s,1 RL,0+ and not to SBV . Remark 4.3 (A continuous but non Hölder-continuous function in RL spaces). Set u(x) = ln(x/2) -1 if x ∈ (0, 1) and u(0) = 0 . This u provides an example of monotone, continuous function which is not α Hölder-continuous for any α ∈ (0, 1), but u belongs to s∈(0,1)

W s,1 .

Remark 4.4 (Relationship between BV and W s,1 RL,0+ ). In [START_REF] Ross | Functions that have no first order derivative might have fractional derivatives of all orders less that one[END_REF] the authors investigate the relationship between usual a.e. differentiation and the fractional Riemann-Liouville derivative definition. Several interesting examples are given. One of them is given by the Weierstrass function defined as

W (x) = ∞ n=0
q -n (e iq n x -e iq n a )

x ∈ [a, b]

where q > 1. It is proved that W has continuous and bounded fractional Riemann-Liouville derivatives of all orders s < 1. However, since W is nowhere differentiable it cannot be of bounded variation. This implies that Riemann-Liouville fractional Sobolev spaces are not contained in BV ([a, b]). Then we can state SBV ⊂ s∈(0,1) W s,1 , s∈(0,1) W s,1 \ BV = ∅ .

The question to be addressed now is the relation between BV and W s,1 RL,0+ . Indeed, we have either BV ⊂ W s,1 RL,0+ or they are completely different spaces whose intersection contains SBV (the case W s,1 RL,0+ ⊂ BV is excluded).

Conclusion

In this paper we try to make connections between the two main definitions of fractional derivatives : the pointwise one whose typical representation is the RL derivative and the global one which is typically the Gagliardo one. In view of a more precise description of the derivative of order s ∈ (0, 1) with respect to BV functions we have also proved preliminary results to compare SBV (a, b) and W s,1 RL,a+ . Open problems are numerous. In particular, it remains to strongly connect the Riemann-Liouville theory with the Gagliardo one. This would allow to perform comparison between W s,1 RL,a+ and Besov-spaces for example. In addition, we have to understand precisely how W s,1 RL,a+ behaves with respect to BV (a, b) to get some information about the BV \W 1,1 functions. In particular, we proved that SBV (a, b) ⊂ W s,1 RL,a+ and exhibit a function in W s,1 RL,a+ , that does not belong to BV (a, b); however we still don't know if BV (a, b)W s,1 RL,a+ . In addition, it remains to prove density results and continuity/compactness results in view of variational models involving the RL derivative.

Another important issue is also to address engineering and/or imaging problems in a rigorous mathematical framework. From that point of view, the fractional derivative concept widely used in engineering is the Caputo one : The main advantage of Caputo derivatives, which makes them the preferred ones in many engineering applications, is the fact that the initial conditions for fractional differential equations with Caputo derivatives are expressed by integer derivatives at time 0, say quantities with a straightforward physical interpretation. This relies on the Laplace transform of Caputo fractional derivative, formally identical to the classical formula for integer derivatives (in contrast to the formula for RL fractional derivatives):

L C D α a+ u (s) = s α L{u} - n-1 j=0 s α-j-1 u j (0) n -1 < α ≤ n .
The connection between Riemann-Liouville and Caputo derivatives, when they both exist, is given by the relationship (see [START_REF] Almeida | Necessary and sufficient conditions for the fractional calculus of variations with Caputo derivatives[END_REF] for example):

2. 2 . 3 .

 23 Marchaud derivative and representability for p ∈ (1, ∞). The representability result given by Theorem 2.1 can be improved by characterizing the set of the functions u ∈ L p (a, b) (p > 1) represented by another L p -function

Definition 2 . 5 (

 25 Marchaud fractional derivative). Let u ∈ L p (a, b) and s ∈ (0, 1). The left-side Marchaud derivative of u at x ∈ (a, b] is defined by D s a+ u = lim ε→0 D s a+,ε u in L p with respect to the strong topology with

  , b]. Next result generalizes Theorem 2.1 and Corollary 2.1 : Theorem 2.2 ([30]-Theorems 13.1 -13.2 ). Let be s ∈ (0, 1).

Remark 2 . 2 (

 22 Marchaud vs Riemann-Liouville derivative). We point out that, for every s ∈ (0, 1), we have ∀ u ∈ I s a+ (L 1 (a, b)), for a.e. x ∈ (a, b] D s a+ u(x) = D s a+ u(x), because of Theorems 2.1 and 2.2.This implies in particular that ∀ u ∈ C 0,s+α (a, b), for a.e. x ∈ (a, b]

Remark 3 . 1 .

 31 If u ∈ L 1 (a, b) and I s-1

( 3 )

 3 The equality issue is an open question. In other terms, can we find a function in W s,1 RL,a+ and not in W s,1 G (a, b) ∩ I s a (L 1 (a, b)) ?

  in [a, b], that can be identified to the dual of C([a, b]) the space of continuous functions with support in [a, b] The notation •, • stands for the duality product between M a,b and C([a, b]). We recall that a family of Radon measures µ ε weakly converges to µ in M a,b (notation

. 1 )

 1 and C 1 c (a, b) denotes the space of functions of class C 1 with compact support in (a, b). The space BV (a, b), endowed with the norm u BV (a,b)

Remark 4 . 1 .

 41 We may note that if we extend the functions u with support in [a, b] by 0 below a and denote them similarly, we may consider (-∞, b] instead of [a, b]. Then the appearance of the Dirac measure at a is consistent with the distributional derivative of u on (-∞, b) since there is a jump at a.

Definition 5 . 1 (

 51 Caputo fractional derivative).[START_REF] Almeida | Necessary and sufficient conditions for the fractional calculus of variations with Caputo derivatives[END_REF][START_REF] Caputo | Linear models of dissipation whose Q is almost frequency independent[END_REF] Let u ∈ L 1 (a, b) and n -1 ≤ s < n (where n integer). The left Caputo fractional derivatives of u at x ∈ [a, b] is defined by C D s a+ u(x) = I n-s a+ d n dx n u (x) = 1 Γ(n -s) x a u (n) (t) (x -t) s-n+1 dt (5.1)when the right-hand side is defined. The right Caputo fractional derivatives of u at x ∈ [a, b] is defined in a similar way.

  -α + 1) (x -a) j-α .In particular:C D α a+ u(x) = D α a+ u(x) if u(a) = u (a) = • • • = u n-1 (a) = 0 .

Università del Salento, Dipartimento di Matematica e Fisica "Ennio De Giorgi", I 73100 Lecce, Italy e-mail: antonio.leaci@unisalento.it

Institut Pasteur, Laboratoire d'Analyse d'Images Biologiques, CNRS, UMR 3691, Paris, France, e-mail:giacomo.nardi@pasteur.fr

Politecnico di Milano, Dipartimento di Matematica, Piazza "Leonardo da Vinci", 32, I 20133 Milano, Italy e-mail:franco.tomarelli@polimi.it

Therefore all the results and comments stated in the present paper concerning Riemann-Liouville derivatives can be easily transferred to Caputo derivatives. This will be precisely addressed in a forthcoming work.


J-M Huriot

T E Smith

J-F Thisse

MINIMUM-COST DISTANCES IN SPATIAL ANALYSIS

An axiomatic approach to distance is developed which focuses on those behavioral concepts of distance related to movement in space. In particular, spatial movement by behaving units is postulated to involve a choice from among some set of abstract trips in space, and implicitly, to involve the minimization of some relevant notion of trip costs. In this context, the relevant behavioral notion of distance in space is taken to be the minimum-cost distance generated by this choice process. These trip-cost concepts extend the classical notions of paths, path lengths, and shortest paths in metric spaces. Hence many of the analytical results of the paper involve extensions of classical shortest-path distance properties to minimum-cost distances. In addition to these extensions, a characterization theorem is given which specifies the possible functional relationships between trip costs and their associated path lengths. These relationships include most functional forms which are commonly employed in the literature.

 [START_REF] Busemann | G eomet r y of G e o d e s i c s[END_REF], [START_REF] Blumenthal | Distance Geometry[END_REF], [START_REF] Beguin | An Axiomatic Approach to Geographical Space[END_REF], and Smith (1987aSmith ( ,1987b))], and closely parallels that literature. In particular, we begin by assuming that trips can be meaningfully added together to form trip chains (in a manner analogous to concatenations of paths). It is then assumed that such trip chains can always be treated as sequences of individual trips, so that (from a cost-minimizing viewpoint) the relevant cost of a trip chain never exceeds the total cost of its component trips. However, unlike simple path length, there may be significant economies of trip chaining.

Finally, given these structural assumptions, the minimum-cost distance between any two locations is defined (in a manner completely paralleling shortest-path distance) to be the greatest lower bound on all trip costs between these locations.

Our first result (Theorem 3. Finally, in the concluding section of the paper, we briefly discuss some of the limitations and possible extensions of our approach. In particular, we consider behavioral situations in which the assumption of economies of trip chaining may fail to hold. This is especially relevant when the psychological costs of longer trips are signicantly greater than those of shorter trips. We also describe situations in which trip costs may not be monotone increasing in their associated path lengths.

To develop our results, we begin in the next section with a review of the basic concepts of paths and path lengths in general spaces. This review serves to clarify the formal parallel between these concepts and the more general concepts of trips and trip costs in the present paper. In addition, the results summarized here provide the basis for the comparative analysis of trip costs and path lengths in sections 4 and 5 below.

SHORTEST-PATH DISTANCES

The following development of relevant distance concepts is given in much greater detail in Smith (1987aSmith ( ,1987b)). Let R, R , and R denote the real denote the positive integers. Throughout the analysis, we take X to be a given nonempty set of abstract locations. Observe also that if for each x e X we define the null path, p , to be the unique path in P with p (a) -x for all a € [0,1], then it follows at once A X from N1 and N3 that the set of null paths, P° -{p : x € X } , on X is contained A X in every path network on X.

Distances

In order to construct path networks, it is convenient to introduce the following concept of a path base. In particular, if for any path set, P C P^, the n ^-concatenation power, Pn , is defined recursively by P^ -P and pn+^ -PoPn for all n e Z , then we now say that: In these terms, our objective is to establish conditions on trip structures, T, under which the associated minimum-cost distance function, d^, on X is always realizable. To do so, we begin by observing that if any trip set, T(x,y), is finite, then the existence of minimum-cost trips in T(x,y) is trivial.

However, if T(x,y) is infinite, then the existence of minimum-cost trips can only be insured by imposing additional structural conditions on T(x,y).

With this in mind, our approach is to assume that it is meaningful to make similarity comparisons between abstract trips in T, and that such comparisons are representable in terms of a measure of "trip dissimilarity", which serves as an appropriate notion of "distance" in T To clarify the nature of this relationship, our approach will be to show that each standard path assignment, p, generates a natural s u b s t r u c t u r e , -(X,N^,dp), of the path structure, P, for which there always exists a precise functional relationship between dc and the shortest N^-path distance, d^. To do so, we first show that:

LEMMA 5. then implies that each path in must be the reparameterization of some path in p(T) . Finally, since p(T) C implies by definition that each path in p(T) must be its own d-standard parameterization, and since it may verified [Proposition 11.5 in Smith (1987a)] that the d-standard parameterization of a path is identical with that of each of its reparameterizations (i.e. that q €

[p] ■ * <1^ " P d ) » we may thus conclude that p(T) -* Given this result, observe next if for any given standard path assignment, (ii) To establish the equality in (5.5), observe simply that if -N, then Np(x,y) -N(x,y) =* d^(x,y) -d(x,y) for all x,y € X, so that for this case,

(5.2) now implies (5.5). ■

CONCLUDING REMARKS

In this final section, we consider briefly a number of possible extensions of the present analysis. To begin with, it is important to emphasize once again that while the above framework was formulated in terms of travel behavior, the basic concepts are meaningful in essentially any behavioral context which involves movement between locations (or transitions between states) in some space. Additional economic examples include production processes, in which efficient transformation sequences from given inputs to desired outputs can be treated as "minimum-cost chains". Similar interpretations can be given to a wide range of social transition processes, such as educational or training programs in which efficient training sequences can also be viewed as minimum-cost chains. Thus, in all those processes where our basic hypothesis of economies o f trip c h a i n i n g is appropriate, minimizing behavior induces a well behaved notion of distance on the relevant state space: such as "minimum-transformation-cost" distances on production possibility spaces, or "minimum-transition-cost" distances on social state spaces.

However, it is also important to emphasize the limitations of this framework. In particular, our basic hypothesis of economies of trip chaining may fail to hold in many important behavioral contexts where trip chaining is meaningful. This is even true in economic settings such as the airline example mentioned in the introduction. Aside from physical costs such as connection costs, there may also exist many important psychological costs of movement which influence behavior. In the simple case of travel, for example, even if all physical costs exhibit such perceived costs are signifant, the relevant notion of "minimum-cost distance" may not only fail to be a quasimetric, but may in fact be very difficult to measure. In particular, when such costs are identifiable only in an ordinal sense, the resulting distance representations will be highly nonunique. Hence, as a second extension of the present analysis, it is of interest to consider the representational properties of ordinal cost structures with respect to minimum-cost distance. Results of this type have been established by [START_REF] Beals | Metrics and Geodesics Induced by Order Relations[END_REF] for metric representations of perceived distance orderings. Possible extensions of these results to the present framework will be considered in a subsequent paper.

  human spatial behavior are based on some underlying notion of distance. Our most familiar concept of distance is of course euclidean (or straight-line) distance between points, which represents the shortest-path distance in euclidean space. But the notion of shortest-path distance is itself much broader, and indeed is meaningful for essentially any kind of spatial movement [as studied for example in Smith (1987a,1987b)]. However, from a behavioral viewpoint, not even shortest-path distances are sufficiently general to cover the full range of distance concepts which are relevant for movement in space. In airline travel, for example, it is clear that while shortest-path concepts (such as great-circle routes on the globe) are useful for determining flight times, they are often poorly correlated with actual ticket prices. Hence from the viewpoint of potential air travelers, the economic distance between points is often shaped more by forces of supply and demand than by physical separation alone. With this in mind, the central objective of the present paper is to extend the classical notion of shortest-path distances to a more general class of minimum-cost distances, which is sufficiently broad to include most types of economic distances observed in practice. This extension is primarily motivated by the desire to incorporate explicit economic concepts of space within the more general axiomatic approach to geographic space proposed by Beguin and Thisse (1979). However, this framework is potentially meaningful for many types of non-economic distance as well, including both social and psychological concepts of distance. To model such behavioral notions of distance, it is useful to begin by considering the act of spatial movement itself. For in any given behavioral situation, the relevant notion of distance can depend critically on the type of movement involved (work trip by a commuter, vacation trip by a household, or commodity shipment by a firm). Moreover, each movement decision generally involves a choice from among some set of options (such as route and mode choice by a commuter, or carrier choice by a commodity shipper). Hence our present approach is to model all such options in terms of abstract trips, and to treat each movement decision as a choice from some given set of feasible trips. In this context, it is also postulated that there exists some relevant measure of travel cost associated with each potential trip. Hence, to the 1 extent that such trip choices are based cost-minimizing behavior, one may take the minimum trip costs between locations to define the relevant behavioral notion of distance in space. Within this framework, the specific goals of the present paper are to formalize this notion of minimum-cost distance, and to study a number of its more important properties from a behavioral viewpoint. This study is largely inspired by the classical development of shortest-path distances in metric spaces [as summarized in

  1 below) is to show that the well-known triangularity property of shortest-path distances extends to all minimum-cost distances, i.e. that such distances must always be quasimetrics. This general formulatipn has the additional analytical advantage of ensuring that minimum-c&st distances are well defined whenever there exist finite-cost trips between all locations. But from a behavioral viewpoint this formulation leaves open the question of whether such distances can actually be realized by any trips, i.e. whether there exist minimum-cost trips. Hence, as an extension of the well-known existence results for shortest paths (geodesics) between locations, we next consider the existence question for minimum-cost trips between locations. But unlike the classical notion of paths (which are generally assumed to be continuous functions with respect to the neighborhood topology generated by shortest-path distance), there is no prior structure on the set of abstract trips. Hence to establish meaningful results, we first postulate the existence of some appropriate measure of trip similarity by which trips can be compared. In this context, our main result (Theorem 3.2 below) is to establish regularity conditions on the similarity structure of trips (both with respect to their cost attributes and origin-distination attributes) which ensure the existence of minimum-cost trips between all locations. Next, we consider the possible relationships between trip costs and path lengths associated with these trips. In certain cases, these concepts may of course be identical. For example, if the relevant set of trips for a given car owner correspond to the set of alterative route choices leading to his desired destination, and if the relevant cost variable is travel time, then minimum-cost trips in this case may correspond precisely to shortest routes in the given transport network (assuming that traffic congestion and other factors are not significant). More generally, the relevant notion of trip cost may be some monotone increasing function of some underlying measure of path length. In this context, our central result (Theorem 4.1 below) is to identify the precise nature of the functional relationships which are possible between trip costs and path lengths. In addition, the corresponding functional relations between minimum-cost distances and shortest-path distances are also established (Theorems 5.1 and 5.2 below).

  x,y,z G X. Each distance satisfying both D2 and D3 is designated as a metric on X. For any x € X and e > 0, the set, S£(x) -{y e X: d(x,y) < e), is designated as the d-neighborhood of size c about x in X. A subset, BCX, is then said to be d-bounded iff B C S£(x) for some x in uniform-distance spaces is that they share many of the convergence properties of familiar euclidean spaces. To state these properties, we require the following additional concepts for general distance spaces, (X,d). First, a sequence (x^) in X said to be d -convergent iff there exists some x e X such that d(x,x ) 0. Similarly, (x ) is said to be a n n d -Cauchy sequence iff for each e > 0 there is some n e all n,m ^ n^. Next, a set, B C X, and is said to be totally d-bounded iff for each e > 0 there exists a finite subset {x.,..,x } C B such that B e, of some finite subset of B. Hence this condition essentially requires that each d-neighborhood in X be "uniformly approxi mated" by its finite subsets.To apply these concepts to uniform-distance spaces, we require the following additional concepts. First, a subset, B C X, is said to be dpaths p and q are said to be parametrically equivalent, and written as p ~ q. The set of all reparameterizations of p is designated as the reparameterization class, [p], for p in P^. Next we define an appropriate notion of "path addition". In particular, if for any paths, P^.the concatenation of p^ and p£. Tftis definition is readily extended recursively to all finite concatenations, p,°p0o •••o p ,,

  notion of path length on networks. We begin by observing that paths may in general have infinite length, and hence may have values ranging over the extended nonnegative real numbers, R+ («) -R+ u {«>). class of all lengths on N is denoted by L(N). Condition LI simply requires that each null path should have zero length. Condition L2 requires that the length of any path concatenation should be equal to the sum of the lengths of its component paths. Finally, condition L3 requires that the length of any path should be the same as the length , is a shortest-path distance on X iff d is a quasimetric. From a behavioral viewpoint, this result shows that the classical triangularity condition (D3) is synomous with shortest-path distance. In other words, whenever a given distance, d: X -+ R+ , satisfies triangularity, there must always be some implicit path network, N, on X and path length, £, on N, with respect to which d is the shortest-path distance. In general, both the path network, N, and path length, SL, may be highly nonunique. However, there is always a unique representative path length for d in the following sense. If L^(N) denotes the set of path lengths in L(N) for which d is a shortest N-path distance, and if path length, i € L^(N), is said to be minimal iff i(p) ^ i'(p) for all i' € L^iN) and p € N, then there always exists a unique minimal path length in I*d (N). This minimal path length may be defined as follows. Let the class of finite ordered subsets of [0structure, P -(X,N,d), is designated as a continuous path structure on X if and only if N C C^(X). (ii) If in addition, (X,d) is a regular uniform-quasimetric space, then P is designated as a d-regular path structure. One important feature of d-regular path structures, P -(X,N,d), is that the path network, N, is itself a uniform-quasimetric space with respect to an appropriate notion of distance between paths. In particular, if the function, d+ : N -+ R+ («°), defined for all paths, p,q e N, by (2.11) d+ (p,q) -sup(d[p(o),q(a)]: a € [0,1]} is designated as the supremum distance on N generated by d, then it can be shown that d+ is in fact a uniform-quasimetric on N [Proposition 4.5 in Smith (1987a)], and hence that (N,d+ ) constitutes a uniform-quasimetric space. Two additional properties of uniform-quasimetric spaces will prove useful in the analysis below. In particular, if a path, p € C^(X), is now said to be uniformly d -continuous iff for each e > 0 there is some S > 0 such that for all a,/3 € [0,1], |a -/i| < S =* d[p(a) ,p(/i) ] < «. then it can be shown [Proposition 4.4(ii) and Proposition 4.6(ii) in Smith (1987a)] that: PROPOSITION 2.4. (i) If (X,d) is a uniform-quasimetric space, then for all x,y e X and all sequences (x ) in X, In addition, each path, p e C^(X), is uniformly d-continuous. Several additional properties of d-regular path structures are important for our purposes. First of all, it can be shown [Proposition 11.3 in Smith (1987a)] that the d-length function, i^, is itself continuous along paths, i.e. that PROPOSITION 2.5. For any d-regular path structure, P -(X,N,d), and any infinite path, p 6 N, the function, ¿d (p(0,-)), is continuous. Moreover, this continuity property in turn implies that it is possible to parameterize all (¿^-finite) paths in a manner directly proportional to their d-length. This notion can be made more precise as follows: DEFINITION 2.8. (i) For any path structure, P -(X,N,d), and finite path, p 6 N, a reparameterization, pd e [p], is designated as a d-standard parameterization of p iff for all intervals, [a,j8] £ [0subset, Nd -{pd : p e N ) , of all d-standard parameterizations of paths in N is designated as the d -standard representation of N. With this definition, we have the following additional property of all d-regular path structures [see Proposition 11.4 in Smith (1987a)]: PROPOSITION 2.6. For any d-regular path structure, P -(X,N,d), and any J ?d -finite path, p e N, there exists a unique d -standard parameterization of p in N. 2.6. Shortest Paths Finally, for purposes of comparison with minimum-cost trips below, we record the following results on the existence of shortest paths in d-regular path structures. To begin with, let us now say that: DEFINITION 2.9. (i) For any path structure, P -(X,N,d), the shortest N-path distance function, d, is said to be realizable iff for each pair of locations, x,y € X, there exists a path, p e N(x,ypath, p 6 N(x,y), satisfying (2.11) is designated as a shortest path from x to y in P. To ensure the existence of shortest paths, we require the following additional regularity condition. If for any path structure, P -(X,N,d), we now designate a path sequence (p^) in N as an ¿^-bounded sequence whenever there exists some finite upper bound, b > 0, with i.(p ) ^ b for all n 10. (i) For any d-regular path structure, P -(X,N,d), a family of paths, P C N, is said to be finitely closed in P iff for each p e Cd (X) and i^-bounded sequence (p^) in Pthe full path network, N, is finitely closed in P, then P is said to be a finitely closed path structure. With this additional condition, it can be shown [Proposition 12.5 in Smith (1987a)] that: PROPOSITION 2.7. (Existence of Shortest Paths) (i) For any substructure, P' -(X',N',d'), of a d-regular path structure, P -(X,N,d), if N' is finitely closed in P, then the shortest N '-path distance, d', is realizable. (ii) In particular, if P is itself a finitely-closed path structure, then the shortest W-path distance, d, is realizable. 3. MINIMUM-COST DISTANCE Given this overview of shortest-path distance concepts, we turn now to the major focus of the present paper, namely, to the more general notion of minimum-cost distance. To do so, we begin in section 3.1 below by developing a model of abstract trips in terms of the notion of a trip structure, which parallels the above concept of a path structure. In addition it is shown that the associated minimum-cost distance between possible origins and destinations of trips exhibits the quasimetric property of shortest>path distances. As in section 2 above, we begin with an arbitrary set, X, of locations (or addresses), x, and consider an abstract collection, T, of possible trips, t, between locations in X. Such abstract trips may involve physical travel by individuals, or shipments of economic commodities. More generally, such abstract trips may involve pairwise interactions of any type between behaving units associated with attributes, x, in some relevant space of attributes, X. For our present purposes, the relevant spatial attributes of each trip, t e T, are taken to be its origin location, o(t), and its terminal location, r(t), in X. All other relevant aspects of trip, t, are taken to be summarized by some given nonnegative measure of travel cost, c(t). For physical trips, such costs must implicitly involve some choice of path (or route) in X from o(t) to r(t) [as analyzed in section 4 below], as well as a choice of travel mode (such as car, bus, or plane). For more general types of interactions, such costs may also involve implicit routing choices (through some communication network or organizational decision structure), and choices of interaction modes (such as personal contacts, phone calls, or letters). Our basic structural hypotheses about trips focus on the possibility of trip chaining. In particular, for any two trips, t^,t2 e resulting trip-chain operator, ©, on T is here taken to be associative, i.e. to satisfy the identity, (tj©t2)®tj ■ t^©(t2®tj) [so that recursively defined n -trip chains, t^©t2© ' • *®tn , are independent of the order in which chains are formed]. Hence, it suffices to consider only simple pairwise trip chains. For each such trip chain, t^©t2 , it follows by definition that both o(t^®t2) -o(t^) and T(tj©t2) -r(t2) hold identically. However, the relevant travel costs associated with trip chains are somewhat more problematic. For unlike the simple physical addition of path lengths, there may in general exist significant economies of trip chaining. This is seen most clearly, for example, in the discounting of many types of round-trip fares. More generally, if each possible trip chain is regarded as an economic commodity, then various types of economic discounts may be available in the market place. On the other hand, it may also be argued on economic grounds that such costs should never exceed the combined costs of individual trips. Indeed if this were the case, then it would be more economical to treat y) -{t € T: o(t) -x and r(t) -y) denotes the set of all trips in T from x to y, then it is assumed that T(x,y) is nonempty for all x,y € X. Finally, in a manner analogous to the notion of null paths in section 2 above, it is convenient to assume the existence of null trips, 0(x) e T(x,x), at each location, x e X, which may be interpreted as representing the "no trip" option for behaving units located at x. Each null trip is characterized by a zero travel cost, while all other travel costs are positive. In addition, it is assumed that for any pair of distinct locations, x,y e X, there do not exist trips in T(x,y) with travel costs arbitrarily close to zero. To formalize these assumptions, it is convenient to represent the assignment of origin locations, terminal locations, and travel costs to each trip in T by functions, o: T •+ X, r: T -* • X, and c: T -* R+ , respectively. Similarly, the assignment of each location in X to a unique null trip in T in terms of the following notion of a trip structure: DEFINITION 3.1. For any nonempty sets, X, T, and collection of functions, 2 o: T -* X, r: T -* • X, c: T -* R+ , $: X -♦ T, and ©: A(T ) -» T, the ordered array, T -(X,T,o,r,c,6 ,©), is designated as a trip structure on X iff the following four conditions are satisfied for all t € T, x,y,z € X, tx e T(x,y) and t2 e T(y,ztrip, t e T(x,y), satisfying (3.7) is designated as a minimumcost trip from x to y in T.

Figure 1

 1 Figure 1 below) includes most of the travel cost functions typically employed in the literature. In particular, travel costs typically involve nonzero start-up costs which are independent of trip length, and which thus create a discontinuity at i -0 (in view of the zero cost of all null trips). In addition, it has been observed by many writers [see for example, Hoover (1948), Locklin (1954), and Isard (1956)] that direct transport expenditures tend to increase at a decreasing rate, reflecting long-haul scale economies, as well as typical quantity-discount pricing policies.

  Figure1

  readily be verified [from the f i niteness of the path structure, P -(X,N,d), together with Proposition 9.3 inSmith (1987a)] that the triple, P-(X,Np,dp), yields a well defined substructure of P -(X,N,d). With these observations, we may now establish the following functional relationship between the minimum-cost distance, d , and the shortest N -distance, d : c ' p p THEOREM 5.1. F o r any standard p a t h a s s i g n m e n t , p: T -* • N^, f r o m a p -r e g u l a r trip structure, T -(X,T,o,t,c,0,®), to a f i n i t e d-regular path structure, P -(X,N,d), if c is g e n e r a t e d a n d if <f>: id [p(T+ )] -* • R ( | is a n y trip-ge n e r a t o r function satisfying (4.2), then f o r all distinct l o c a t i o n s , x,y e X, (5.2) dc (x,y) -*[d (x,y)]Proof: Simply observe that since there exist minimum-cost trips, say t"" xy e T(x,y) for each x,y e X by Theorem 4.1, it follows at once from the increasing monotonicity of <f> that for all distinct x,y e X and t e T(x,y)(5.3) dc (x,y) -c(t ) £ c (t) * > J < ¿d [p(t)]=> ¿d [p(t )] -min(jed [p(t)]: t e T(x,y)} * Ad Ip(tx y )] " dP (x-y) Finally, since x * y * t € 0(X), it follows from Al that p(t ) e T , and xy xy + hence from (4.2) that d (x,y) -c(t ) -¿(¿.[p(t )]) -^[d (x,y)]. Thus (5.2) distinct x,y € X, and the result is established. ■ Notice also from the proof of Theorem 5.1 that whenever trip costs, c, are ¿d *generated, each mimimum-cost trip, t , in T must always be assigned to a shortest path, , from x to y in P^. As a final consequence of the general result in Theorem 5.1, we obtain the following desired relationships between dc and and the shortest N-path distance, d: THEOREM 5.2. (i) F o r any standard p a t h a s s i g n m e n t , p: T -* ■ N^, f r o m a p -r e g u l a r trip s t r u c t u r e , T -(X,T,o,r,c,0,©), to a f i n i t e d -regular path structure, P -(X,N,d), if c is £ .-generated a n d if ij[p(T,)lt r i p -generator function o f the f o r m (4.2), then f o r all distinct l o c a t i o n s , x,y e X, (5.4) dc (x,y) S: rf[d(x,y)] (ii) I f in addition it is true that N -N, then f o r all distinct P locations x,y e X, (5.5) dc (x,y) -¿[d(x,y)]. Proof: (i) First, to verify that the values, ^[d(x,y)], are actually well defined, observe from the subpath-closure property (A6) of p(T) £ N, together with the continuity of the function, ¿d [p(0,*)], in Proposition 2.5, that for any value, ^( p ) e J0^[p(T) ], it must be true that (0, ^^(p) ] £ >?d [p(T+ )]. But for any distinct x,y € X, it then follows from the connectedness property of T, together with (2.6) and the locational-consistency property (A2) of p, that 0 < d(x,y) :S ^¿(p) for each p e p[T(x,y)] . Hence d(x,y) e id [p(T+ )], and it follows that ^[d(x,y)] is well defined for each distinct x,y € X. Finally, since Np(x,y) £ N(x,y) by definition, it follows from (2.6) and (5.1) that dp(x,y) £ d(x,y), and we may conclude from (5.2) together with the increasing monotonicty of <f> that (5.4) must hold.

  For while airline fares may exhibit economies with respect to trip chaining, such fares are often not the only relevant costs. In particular, if one considers the time costs of travel, then it is clear that flight connections can result in significant time costs which are independent of actual flight times. More generally, there may exist significant connection costs in trip chaining which may lead to violations of condition (3.1). Hence as one possible extension of the present framework, it is of interest to consider the minimum-cost distances generated by trip chains with connection costs. Even in simple examples, it is clear that such distances may fail to satisfy the triangularity condition (D3), and hence may even fail to be quasimetrics.

Finally

  , turning to the relationships between minimum-cost distance and underlying measures of physical distance, it should be clear that the notion of generated travel costs in Definition 4.2 is highly restrictive. Indeed, even the physical relations between travel time and path length are seldom that simple. In particular, while such relationships may exhibit certain overall monotone tendencies, their exact nature on, say, a given road network is complicated by a host of other factors including road capacity, stop lights, and traffic congestion. More generally, it should be clear that path length is only one among many variables which influence travel costs. Hence the present notion of trip-cost generators should be regarded as only the simplest possible types of structural relations which may be considered.

Figure 1 .

 1 Figure 1 . A Concave Trip-Cost: Generator

T XT' numbers, nonnegative reals, and positive reals, respectively, and let Z T T

T T DEFINITION 2.3. A set of paths, P C Px> is designated as a path base on X if and only if P satisfies the following two conditions for all x,y € X, pe Px , and [a,0] C [0,1]: Bl. (Weak Connectedness) Pn (x,y) * 0 for some n e Z T r B2. (Subpath Closure) p € P => p(a,/J) e P Each path base allows the construction of a path network in a simple way. More specifically, if for each path set, P £ P^, we denote that class of all reparameterizations of paths in P by [P] -{[p]: p € P}, then it can be shown [see Proposition 6.1 in Smith (1987a)] that: PROPOSITION 2.2. For any path base, P, on X, the associated family of paths, (2.5) N(P) -U [Pn ] C PY + + X is a path network on X. Moreover, N(P) is the smallest path network containing P [i.e. if P C N for any path network, N, then N(P) C N ] . Hence it is natural to designate N(P) as the path network generated by P.

[More explicit relationships between these two concepts will be considered in section 5 below].