Lebesgue Spaces

Set-Valued Anal (2007) 15:125–138 DOI 10.1007/s11228-006-0031-7

Lipschitzian Properties of Integral Functionals on Lebesgue Spaces L p, 1  p < ∞ Emmanuel Giner

Received: 20 June 2003 / Accepted: 25 September 2006 / Published online: 17 February 2007 © Springer Science + Business Media B.V. 2007

Abstract We give complete characterizations of integral functionals which are Lipschitzian on a Lebesgue space L p with p  = ∞. When the measure is atomless, we characterize the integral functionals which are locally Lipschitzian on such Lebesgue spaces. In every cases, the Lipchitzian properties of the integral functional can be described by growth conditions on the subdifferentials of the integrand which are equivalent to Lipschitzian properties of the integrand. Key words integral functional · calmness · stability · Lipschitzian functions · locally Lipschitzian functions · nonsmooth analysis. Mathematics Subject Classifications (2000) 26A16 · 26A24 · 26E15 · 28B20 · 49J52 · 54C35

1 Introduction With the study of the Clarke generalized gradient [6], locally Lipschitzian functions have played a key role and have been studied intensively. Using the mean value theorem, a Lipchitzian function can be characterized by the uniform boundedness of its subdifferential values, this property being valid for various subdifferentials [8, 9, 13], for example. Except in the convex case, the exact computation of the subgradients of an integral functional is not easy. Clarke gives in [6] Theorem 2.7.5, sufficient growth conditions on the subdifferential of the integrand ensuring the local Lipschitzian property of an integral functional on L p . Without nice formulae as in the Clarke and limiting generalized gradients cases, [8] chapter 3, Theorem 5.18, we can have a useful inclusion between the set of selections of the subdifferential of the

E. Giner (B) Laboratoire MIP, Université Paul Sabatier, UFR MIG, 118 Route de Narbonne, 31062 Toulouse, Cedex 04, France e-mail: [email protected]

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integrand, and the subdifferential of the functional integral, [11] Theorem 5.6. This fact permits to obtain a complete characterization of the Lipschitzian character of an integral functional by the conditions used by Clarke in [6] Theorem 2.7.5. In [11] the author, with the notion of calmness, makes a study of the contingent subdifferential of an integral functional defined on L p , 1  p < ∞. In the present paper, we use this work in order to give some characterizations of the Lipschitzian properties of integral functionals defined on such Lebesgue spaces. The main results of this article are the following: an integral functional I f is Lipschitzian on a L p , 1  p < ∞, if and only if the integrand f is Lipschitzian with a Lipschitz rate in the conjugate space Lq , 1/ p + 1/q = 1, Theorem 3. Moreover, if the measure is atomless, we obtain some characterizations of locally Lipschitzian integral functionals considered on the Lebesgue spaces, by polynomial growth conditions on the subdifferentials of the integrand f , these properties are equivalent to the following: an integral functional I f is locally Lipschitzian on L p , 1  p < ∞, if and only if, there exists a positive constant k an a non-negative element a of Lq , 1/ p + 1/q = 1 such for almost every ω the function fω is Lipschitzian with rate kr p−1 + a(ω) in the ball of center 0 and radius r (see Theorem 3.2). The tools we use are the mean value Theorems for the Clarke [6] and contingent subdifferentials [13], and also a refinement of a Lagrange duality result of the type of Aubin and Ekeland [1], obtained jointly with Bourass [4], which allow to obtain growth conditions on the contingent subdifferential of the integrand. Theorem 4 also gives a surprising result: when the measure is atomless a functional integral calm (see [11, 17]) and directionally Lipschitzian (in the sense of [15]) at some point of L p is locally Lipschitzian on the whole space L p , 1  p < ∞, and moreover, when p = 1, it is Lipschitzian.

2 Some Preliminaries We adopt the following notation: IR = IR ∪ {±∞} and IR+ is the set of non-negative real numbers. Let (X, |.|) be a normed space. For a numerical function defined on X, we consider its domain, domf = {x ∈ X: f (x) ∈ IR} and its epigraph, epi f = {(x, r) ∈ X × IR: f (x)  r}. For a real number r, the sublevel set associated to r is the set f r defined by: f r = {x ∈ X: f (x)  r}. An IR-valued function f is said to be proper if its domain is nonempty and if f does not take the value −∞. The function f is said to be calm at a point x0 of its domain if there exist some constant c in IR+ and some neighborhood V of x0 such that for every x in V: f (x0 ) − c|x − x0 |  f (x). The function f is said to be quiet at a point x0 if − f is calm at x0 . When f is both calm and quiet at a point x0 , f is said to be stable at x0 . We say that f is Lipschitzian on a subset Y of X if there exists a positive constant c, such that for all y, y in Y: | f (y) − f (y )|  c|y − y | . The function f is said to be Lipschitzian around a point x of X if f is Lipschitz on a neighborhood of x. It is said to be locally Lipschitzian or strictly continuous if f

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is Lipschitzian around any point of X; in this case, the Lipschitz rate at x ∈ X, [17] | f (x ) − f (x )| . As Rockafellar does [15, 16], we 9.1, is defined by: lipf (x) = lim sup |x − x | x ,x →x x =x

introduce the following notions: Definition 2.1 A numerical function f is said to be directionally Lipschitzian at x with respect to a vector y if f is finite at x and: lim sup r−1 ( f (x + ry ) − α  ) < +∞. f (x ,α  )− →(x, f (x)) 

r→0+ y →y

Here we use the notation: f

(x , α  ) − → (x, f (x)) ⇔ (x , α  ) ∈ epi( f ), and (x , α  ) → (x, f (x)). When f is lower semi-continuous, this last condition can be simplified to: lim sup r−1 ( f (x + ry ) − f (x )) < +∞. x → f x r→0+ y →y

with the notation: x → f x ⇔ (x , f (x )) → (x, f (x)). Definition 2.2 The function f is directionally Lipschitzian at x if there is at least one y such that f is directionally Lipschitzian at x with respect to y. Recall that when f is a strictly continuous function the generalized Clarke directional derivative at x0 in the direction y takes the following form: f ◦ (x0 ; y) =

lim sup r−1 ( f (x + ry) − f (x)).

(x,r)→(x0 ,0+ )

And the generalized Clarke subdifferential ∂ C f (x0 ) of f at x0 , [6] Chapter 2, is the subset of the topological dual X ∗ of X, given by: ∂ C f (x0 ) = {x∗ ∈ X ∗: ∀x ∈ X, x∗ , x  f ◦ (x0 ; x)}. The contingent (or the Hadamard) classical lower generalized derivative of the function f at a point x0 of its domain, is defined by: f  (x0 ; x) =

lim inf r−1 ( f (x0 + rx ) − f (x0 )).

(r,x )→(0+ ,x)

and the contingent subdifferential of f at the point x0 is the subset of X ∗ ∂ f (x0 ) = {x∗ ∈ X ∗: ∀x ∈ X, x∗ , x  f  (x0 ; x)}.

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When, for every x ∈ X, the following limit exists: f˙(x0 ; x) =

lim

(r,x )→(0+ ,x)



r−1 ( f (x0 + rx ) − f (x0 ),

the function f is said to be Hadamard directionally derivable at x0 . Observe that, in this last case we have necessarily: f˙(x0 ; 0) = 0, hence the function f is stable at x0 (see [11] Proposition 2.2 and Corollary 2.4). Let (, T , μ) be a measured space with a σ -finite positive measure μ, the tribe T being supposed to be μ-complete. For a measurable subset A ∈ T , we set Ac = {ω ∈  : ω ∈ / A}, and 1 A stands for the characteristic function of A, 1 A (w) = 1 if w ∈ A, 0 if ω ∈ / A. Let E be a finite dimensional linear space with Borel tribe B(E). We consider the space L0 (, E) of classes of measurable functions (for μ-almost everywhere equality) defined on  and with values in E. Let L p (, E), 1  p < ∞, be the Lebesgue space of classes of p-integrable functions defined on  with values in E and endowed with its strong natural topology. We denote by ||x|| the usual norm of an element x of L p (, E). Given v in L0 (, IR), we consider the upper integral of ∗ v, Iv or  vdμ defined by:    ∗ vdμ = inf udμ, u ∈ L1 (, IR), v  u, μ − a.e , Iv = 



similarly, we can define the lower integral of v: Jv = −I−v . If v− = inf(0, v) and v+ = sup(0, v), then the only case where Iv is not identical to Jv is when both v− and v+ are not integrable, and in this case we have: −∞ = Jv < Iv = ∞. Let f :  × E −→ IR be a T ⊗ B(E)-measurable integrand. When the multifunction ω ⇒ epi( f (ω, .)) is with closed values and is measurable in the sense of [5] and [14], we say that f is a normal integrand [14, 17]. Given a subset X of L0 (, E), if M :  ⇒ E is a measurable multifunction, we denote by X(M) the set of measurable selections of M which are in X. For an element x of L p (, E), we denote by f (x) the function: ω −→ f (ω, x(ω)). The integral functional I f associated to f is the functional defined at a point x of L p (, E) by: ∗ I f (x) = I f (x) =  f (x)dμ. In the sequel, x0 is an element of the domain of I f , that is, an element such that f (x0 ) is integrable. We will say that the integrand f is calm at x0 , (respectively, quiet, stable, locally lipschitzian, at x0 ) if for almost every ω in , the function f (ω, .) is calm, (respectively, quiet, stable, locally lipschitzian) at x0 (w). Moreover, f  (x0 ; .), (respectively, f ◦ (x0 ; .) ) stands for the integrand (ω, e) −→ f  (ω, x0 (ω); e), (respectively, (ω, e) −→ f ◦ (ω, x0 (ω); e)). Similarly ∂ f (x0 ) stands for the multifunction: ω ⇒ ∂ f (ω, x0 (ω)), (respectively, ∂ C f (x0 ) stands for ω ⇒ ∂ C f (ω, x0 (ω)). If f is a normal integrand, [17] Theorem 14.56 asserts that f  (x0 ; .) is a normal integrand, and ∂ f (x0 ) is a measurable multifunction. An integrand f:  × E −→ IR is said to be proper if for almost every ω in  the function f (ω, .) is proper. An integrand f will be called strictly continuous, if for almost every ω in  the function f (ω, .) is strictly continuous. In this last case, the measurability of ∂ C f (x0 ) is classic, see for example, [6] Lemma of Theorem 2.7.2 or [18] Corollary I.21. An increasing sequence (n )n of measurable subsets of finite measure is said to be   n = 0. Given a multifunction M defined on  a σ -finite covering of  if μ  \ n

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and with values in E and with measurable graph, we denote the set of measurable selections of M by L0 (M). Let us recall the following definitions: Definition 2.3 (See [12] Section 3). A subset X of L0 (, E) is said to be decomposable if, for every x, y in X and every measurable set A, the function x1A + y1Ac is an element of X. Definition 2.4 (See [4] Section 3). Let X and Y be two decomposable subsets of L0 (, E). The set X is said to be rich in Y, if X is a subset of Y, and if, for any element y of Y, there exist a σ -finite covering (n )n of  and a sequence (xn )n of elements of X satisfying for all n in IN: y1n = xn 1n . In the sequel we will use the following results: Theorem 2.5 ([4] Corollary 3.9): Suppose the tribe T is μ-complete. Let M be a multifunction with nonempty values and T ⊗ B(E) measurable graph and let f :  × E −→ IR be a T ⊗ B(E)-measurable integrand. If X is a decomposable subset which is rich in L0 (M) then: inf I f (x) = I inf f (., e) ,

x∈X

e∈M(.)

provided the left hand side is distinct from ∞. Theorem 2.6 ([4] Corollary 5.7): Let us suppose the measure μ is atomless, and let M be an E-valued multifunction defined on  and with measurable graph, and let X be a decomposable set rich in L0 (M). Let f0 be a measurable integrand and ( fi )1in be a finite family of measurable proper integrands. The functional I fi is considered as a functional defined on X. Besides, let us suppose that there exist a family c = (ci )0in of real numbers and an element x of X such that the Slater condition is satisfied: for each i = 1, ..., n, −∞ < I fi (x) < ci and I− f0 (x) < +∞. Then, the following assertions are equivalent: (a)



I fi ci ⊆ I f0 c0 .

1in

(b) There exists y∗ in IRn+ , such that the function u y∗ given by: ⎧ ⎨

u y∗ (ω) = inf − f0 (ω, e) + ⎩

n

yi∗ fi (ω, e), e ∈ M(ω) ∩

1in

i=1

is integrable and satisfies: −c0 + y∗ , c 



 

u y∗ dμ.

domfi (ω, .)

⎫ ⎬ ⎭

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3 Characterizations of Lipschitzian Properties of Integral Functionals on Lp , 1  p < ∞ Let us give first the statements of the main results of this article. Hereafter 1  p < ∞ and 1/ p + 1/q = 1. Theorem 3.1 Let f be a normal integrand such for an element x of L p (, E), f(x) is integrable. The following assertions are equivalent: (a) The functional I f is Lipschitzian on L p (, E). (b) The integrand f is strictly continuous and there exists a non-negative function a of Lq (, IR), satisfying for every (ω, e) ∈  × E and e∗ ∈ ∂ C f (ω, e), |e∗ |  a(ω). (c) The integrand f has finite values and there exists a non-negative function a of Lq (, IR) verifying for every (ω, e) ∈  × E and e∗ ∈ ∂ C f (ω, e), |e∗ |  a(ω). (d) There exists a non-negative function a of Lq (, IR), such that for almost every ω ∈ , the function f (ω, .) is Lipschitzian with rate a(ω). Theorem 3.2 Let f be a normal integrand such for an element x of L p (, E), f (x) is integrable. Let us consider the following assertions: (a) The functional I f is finite, calm and directionally Lipschitzian at some point of the space L p (, E). (b) The functional I f is finite, continuous and calm at each point of a nonempty open subset of L p (, E). (c) The functional I f is finite, continuous and quiet at each point of a nonempty open subset of L p (, E). (d) The functional I f is Lipschitzian around some point of L p (, E). (e) The functional I f is strictly continuous on L p (, E). (f) The functional I f is Lipschitzian on every ball of L p (, E). (g) The integrand f is strictly continuous and there exist a positive constant k a non-negative function a of Lq (, IR) such that for every (ω, e) ∈  × E and e∗ ∈ ∂ C f (ω, e), |e∗ |  k|e| p−1 + a(ω). (h) The integrand f has finite values and there exist a positive constant k a nonnegative function a of Lq (, IR) such that for every (ω, e) ∈  × E and e∗ ∈ ∂ f (ω, e), |e∗ |  k|e| p−1 + a(ω). (i) There exist a positive constant k and a non-negative function a of Lq (, IR), such that for almost every ω, for every r > 0, the function f (ω, .) is Lipschitzian with rate kr p−1 + a(ω) on the ball of center 0 and radius r of E. Then, one has: (i) ⇔ (h) ⇔ (g) ⇒ (f) ⇒ (e) ⇒ (d) ⇔ (c) ⇔ (b) ⇒ (a). If the measure μ is atomless, all the assertions are equivalent. Let us give an immediate consequence of Theorems 3.1 (d) and 3.2 (i), when p = 1: Corollary 3.3 Suppose the measure μ is atomless, and let f be a normal integrand. The integral functional I f is Lipschitzian on L1 (, E) if an only if it is finite, calm and directionally Lipschitzian at some point of L1 (, E). In [3] Proposition 2.2, Bismut proves, when the measure μ is atomless, that a convex integral functional continuous at a point of L p (, E) is everywhere continuous on

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L p (, E). A convex function on L p (, E) is continuous at some point if and only if it is Lipschitzian around this point; thus, Theorem 3.2 (d)⇔(e), extends the result of Bismut in the nonconvex case. In [6] Theorem 2.7.5, the assertion (g)⇒(f) is already proved. Since any functional which is contingent subdifferentiable at some point is calm at this point, Corollary 3.4 is a consequence of Theorem 3.2 (b)⇒ (f). Corollary 3.4 Suppose the measure is atomless. Let f be a normal integrand. If there exists a nonempty open subset V of L p (, E) such that I f is finite, continuous and with a nonempty contingent subdifferential at each point of V, then I f is Lipschitzian on every ball of L p (, E). Since any functional which is directionnally Hadamard derivable at some point is stable at this point, Corollary 3.5 is a consequence of Theorem 3.2 (b)⇒(f) or (c)⇒ (f). Corollary 3.5 Suppose the measure is atomless. Let f be a normal integrand. If the integral functional I f is Hadamard directionally derivable at each point of an open subset of L p (, E), 1  p < ∞, then it is Lipschitzian on every ball of L p (, E). Proof of Theorem 3.1 From [13] Corollary 2.5 and [6] Theorem 2.3.7 we have the equivalences: (d)⇔(c)⇔(b). The assertion (d)⇒(a) is a consequence of Hölder’s inequality. Let us prove (a)⇒(c). For this purpose, we need Lemma 3.8, but before let us prove the following lemmas:   Lemma 3.6 Let f be a normal integrand. Then the multifunction  : (ω, e) ⇒ ∂ f (ω, e) is measurable. Proof of Lemma 3.6  × E is endowed with the tribe T ⊗ B(E). Let  = {(ω, e) ∈  × E : f (ω, e) ∈ IR}. Since f is normal, by Corollary 1 K of [14] the multifunction F(ω, e) = epi f (ω, .) − (e, f (ω, e)) is measurable with closed nonempty values as a multifunction defined on the measurable set . For every n ∈ IN, nF is also measurable and from Theorem 14.20 [17] the pointwise limit G = lim sup nF is n

measurable. From Lemma 3.5 of [11] and Theorem 8.2 [17], for every (ω, e) ∈ , G(ω, e) = T(e, f (ω,e)) epi f (ω, .) = epi f  (ω, e; .); thus, the integrand defined on  × E by ((ω, e), e )  → f  (ω, e; e ) is normal. Let S be the set {(ω, e) ∈  : f  (ω, e; 0) = 0}. From [17] Corollary 14.34, S is T ⊗ B(E)-measurable. The integrand defined on S × E by ((ω, e), e )  → f  (ω, e; e ) is normal and proper. From Theorem 14.56 [17] the multifunction : (ω, e) ⇒ ∂( f  (ω, e; .))(0) = ∂ f (ω, e) is measurable as a multifunction defined on S. Since the domain of the multifunction  is contained in S, we deduce the measurability of .   Lemma 3.7 Let f be a finite valued normal integrand, and let ∂ f be the multifunction ω ⇒ {(e, e∗ ) : e∗ ∈ ∂ f (ω, e)}. If the functional I f is stable at every point of an open subset of L p (, E), then the set (L p × Lq )(∂ f ) of measurable selections of ∂ f which are in L p (, E) × Lq (, E∗ ), is nonempty. Proof of Lemma 3.7 Without loss of generality we may suppose that the functional integral I f is stable at each point of the open ball of L p (, E) with center 0 and

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radius . Let α be an element of L p (, IR) such that for all ω ∈ , α(ω) > 0, and ||α|| < . Let  be the multifunction of 3.6. We observe that the graphs of ∂ f and  coincide, therefore from [17] Theorem 14.8, ∂ f is with measurable graph. From [5] III 40,41 and [5] IV 10, the multifunction F(ω) = ∂ f (ω) ∩ (α(ω)B × E∗ ) is also with measurable graph (here B is the closed unit ball of E). Since f is normal with finite values, Lemma 2.4 of [13], (see also Corollary 1 of [7] or VIII Proposition 1.4 of [10]), ensures the nonemptiness of the values of F and, using [5] III 22, we obtain the existence of a measurable selection (x0 , x∗0 ) of this multifunction. As, for every ω ∈ , we have: |x0 (ω)|  α(ω), we deduce that x0 is in the open ball of L p (, E) of center 0 and radius . Let us show that x∗0 is in Lq (, E). For every (ω, e) ∈  × E, we have x∗0 (ω), e  f  (ω, x0 (ω); e), and, by integration, we deduce for every x in L p (, E): I x∗0 ,x  I f  (x0 ;x) . The functional I f is stable at x0 . Then by [11] Corollary 2.4, and [11] Theorem 5.6 there exists a positive constant c such that for all x in L p (, E): I x∗0 ,x  I f  (x0 ;x)  I f  (x0 ; x)  c||x||. We deduce that for all x in L p (, E), the function x∗0 , x + is integrable, and hence x∗0 , x is integrable: thus, the functional x → I x∗0 ,x is finite valued, linear and continuous. This proves that x∗0 is in the dual   space of L p (, E), that is to say Lq (, E∗ ). Lemma 3.8 Let M be a multifunction with measurable graph and with values in E × E∗ ; if (L p × Lq )(M) is nonempty, it is a decomposable set rich in L0 (M). When f is a finite valued normal integrand, and the functional I f is stable at every point of an open subset of L p (, E), (L p × Lq )(∂ f ) is a decomposable set rich in L0 (∂ f ). Proof of Lemma 3.8 Let x be an element of (L p × Lq )(M). Given an element y of L0 (M) and a σ -finite covering (n )n of , let n = {ω ∈ n : |y(ω)|  n}, and xn = y1n + x1n c . Then, for every n ∈ IN, xn is in (L p × Lq )(M) and, moreover: y1n = xn 1n . This proves the richness of (L p × Lq )(M) in L0 (M). In case M = ∂ f , Lemma 3.6 ensures that ∂ f is with measurable graph and the use of Lemma 3.7 give the second part of Lemma 3.8.   Lemma 3.9 If the functional I f is finite valued on L p (, E), then the integrand f is almost everywhere with finite values. Proof of Lemma 3.9 Let ∞ = {ω ∈  : ∃e ∈ E, f (ω, e) = ∞}. Then ∞ is the projection on  of f −1 (∞), this last set being T ⊗ B(E)-measurable (Theorem 2.A [14]), using [5] Theorem III 23 we obtain the measurability of ∞ . Suppose that ∞ is of positive measure. Let M be the multifunction with nonempty values defined on ∞ by: M(ω) = {e ∈ E : f (ω, e) = ∞}. Since the graph of M is equal to f −1 (∞) ∩ (∞ × E), it is T ⊗ B(E)-measurable, and from [5] Theorem III 22, M admits a measurable selection y defined on ∞ . Let us define z by: z(ω) = y(ω) if ω ∈ ∞ and z(ω) = 0 if not. Let (n )n be a σ -finite covering of . Define n = {ω ∈ n: |z(ω)|  n}, then (n )n is also a σ -finite covering of , and thus there exists an integer m such Am = ∞ ∩ m is of positive measure. Given an element x of L p (, E), the function w = z1Am + x1Acm is an element of L p (, E) such that f (w) is not integrable because this function is infinite valued on Am : this contradicts the

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assumption. Thus ∞ is a set of null measure. Applying this result to − f , we obtain that f is almost everywhere with finite values.   End of the proof of Theorem 3.1. Let us assume that I f is Lipschitzian on L p (, E), hence I f is finite valued on L p (, E). Moreover, since I f is Lipschitzian on L p (, E), there exists a positive constant k such for every x ∈ L p (, E), the set ∂ I f (x) is included in kBq , the ball of Lq (, E∗ ) of center the origin and radius k. For every x in L p (, E), we have, from [11] Theorem 5.6 (d): Lq (∂ f (x)) ⊆ ∂ I f (x) ⊆ kBq .

(1)

I f being finite-valued and Lipschitzian on L p (, E), Lemmas 3.9, and 3.8, ensure that (L p × Lq )(∂ f ) is a decomposable set rich in L0 (∂ f ). If 1 < p < ∞, define the integrand g on  × E × E∗ by g(ω, e, e∗ ) = |e∗ |q . Then due to Eq. 1, and Theorem 2.5, the function a(ω) = sup |e∗ | verifies the assumption (c) of e∈E, e∗ ∈∂ f (ω,e)

Theorem 3.1, since:  a(ω)q μ(ω) = 

sup (x,x∗ )∈(L p ×Lq )(∂ f )

Ig (x, x∗ ) 

sup

x∈L p (,E), x∗ ∈∂ If (x)

x∗ q  kq .

Now if p = 1, define the integrand g on  × E × E∗ by: g(ω, e, e∗ ) = 0 if |e∗ |  k, +∞ if |e∗ | > k. We have from Eq. 1: sup

x∈L1 (,E), x∗ ∈ ∂ I f (x)

hence Theorem 2.5 gives, with b (ω) =  

b (ω)dμ(ω) =

sup (x,x∗ )∈(L1 ×L∞ )(∂ f )

x∗ ∞  k,

sup e∈E, e∗ ∈∂ f (ω,e)

Ig (x, x∗ ) 

g(ω, e, e∗ ): sup

x∈L1 (,E), x∗ ∈∂ I f (x)

Ig (x, x∗ ) = 0.

This proves that b is the null function and therefore the function a(ω) = |e∗ | is essentially bounded by k. This completes the proof of Theorem 3.1. sup e∈E, e∗ ∈∂ f (ω,e)

Proof of Theorem 3.2 Let us show (i)⇒(h). Let fω = f (ω, .). It suffices to prove that lipfω (e)  k|e| p−1 + a(ω). Setting r = |e|, assertion (i) ensures that f is Lipschitzian with rate k(r + ) p−1 + a(ω), on the ball of center e and radius , thus for every , lipfω (e)  k(r + ) p−1 + a(ω), and this give the desired inequality. Let us prove (h)⇒(g). It suffices to prove that lipfω (e)  k|e| p−1 + a(ω). If e  = e are in the ball of center e and radius , then from [13] Theorem 3.1, there exist c ∈ [e , e ] and sequences (cn )n , (c∗n )n such that c∗n ∈ ∂ fω (cn ) and (cn )n converges to c with: fω (e ) − fω (e )  lim inf c∗n , e − e . From assumption 4 (h) we have: n

c∗n , e − e  (k|cn | p−1 + a(ω))|e − e |, and thus: fω (e ) − fω (e )  k(|e| + ) p−1 + a(ω). |e − e |

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Since this last inequality is valid for arbitrary e  = e in the ball of center e and radius , we obtain lipfω (e)  k|e| p−1 + a(ω), and thus, assertion (g) is satisfied. With the mean value Theorem [6] Theorem 2.3.7 we show easily (g)⇒(i). We have proved the equivalences: (i)⇔(h)⇔(g) The proof of (g)⇒(f) can be found in [6] Theorem 2.7.5 when  is of finite measure and the function a is bounded. The proof in the general case is similar. It is clear that (f)⇒(e)⇒(d)⇒(a). The equivalences: (d)⇔(c)⇔(b) are consequences of the following Proposition which is adapted from Benyamini and Lindenstrauss [2] Proposition 6.45.   Proposition 3.10 Let X be a Banach space and f be a numerical function defined on X. The following assertions are equivalent: (a) (b) (c)

f is finite, continuous and calm at every point of a nonempty open set. f is finite, continuous and quiet at every point of a nonempty open set. f is locally Lipschitz at some point of X.

The proof of Proposition 3.10 is a slight modification of that given in [2] Proposition 6.45. For convenience let us give it. Suppose the second assertion is satisfied on the open set U. We will prove that f is locally Lipschitzian at some point of U. Let Bδ be the open ball of X of center 0 and radius δ. There exists an open subset V of U, and > 0 such that: V + B ⊂ U. Since f is continuous on U, for each 0 < t < 1, the function q(x, u, t) = t−1 ( f (x + tu) − f (x)) is continuous on V × B . For each integer n, let us define An as:   An = (x, u) ∈ V × B :

sup sup(q(x, u, t), q(x, −u, t))  nu .

0
For each integer n, An is a relatively closed subset of V × B . Moreover, since f is quiet on V we have: V × B = ∪n An . From the Baire property, there exists (x0 , u0 ) ∈ V × B , a natural number m and a positive real number r such that: sup(q(x, u, t), q(x, −u, t))  mu

(2)  2 whenever 0 < t  1/m, x is in x0 + Br , and u is in u0 + Br . Let r0 = min mr , 2(ur0 +r) . Let us show that f is Lipschitz on x0 + B r20 . Let x, y ∈ x0 + B r20 , so that x − y  

y−x r0 . Define t = x−y , v = r x−y , then y = x + tv. Since x − y  r0 , we have 0 < r t  1/m. We claim that z = y + tu0 (= x + t(u0 + v)) is in x0 + Br because tu0   r0 u0   r/2 . Now, since v = r, the following equality r

f (x) − f (y) = f (x) − f (z) + f (z) − f (y) = f (z − t(u0 + v)) − f (z) + f (y + tu0 ) − f (y) gives, using Eq. 2: f (x) − f (y)  mt(u0 + v + u0 ) 

m(2u0  + r) x − y. r

This last inequality shows that f is Lipschitzian around x0 . We have proved: (b)⇒ (c). Taking − f we deduce (a)⇒ (c). Since a locally Lipschitz function around some point

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135

is calm and quiet on some neighbourhood of this point, the proof of Proposition 3.10 is complete. Now, let us suppose the measure μ is atomless. For the reverse implications, we prove first: (a)⇒(e). Let x be a point of L p (, E) where assertion (a) is satisfied. Since I f is calm at x, from [11] Definition 2.2 and [11] Lemma 4.4, there exist positive constants c and such that for every ||y || < : I f (x) − c||y ||  I f (x + y ) = −I− f (x + y ).

(3)

Let us prove first that I f is stable at x. Since I f is directionally Lipschitzian at x respect to some vector y, we obtain: lim sup r−1 (I f (x + ry ) − I f (x)) < +∞. r→0+ y →y

From the above inequality, we deduce the existence of positive constants k, δ, such that: sup{r−1 (I f (x + ry ) − I f (x)) : 0 < r < δ, ||y − y|| p  } < k.

(4)

We apply Theorem 2.6 with M(ω) = E, X = L p (, E) and: f0,r (ω, e) = r−1 ( f (ω, x(ω) + re) − f (ω, x(ω))),

f1 (ω, e) = |e − y(ω)| p .

Relation (4) can be written: for every 0 < r < δ, I f1  ⊆ I f0,r k . The assumptions of Theorem 2.6 are satisfied since I f1 (y) = 0, and, for δ  = inf (δ, (1 + ||y||)−1 ), using Eq. 3 we also obtain: for every 0 < r  δ  ,

I− f 0,r (y)  c||y|| < ∞.

We deduce the existence of a non-negative scalar yr∗ such that:  ur dμ − yr∗ , −k  

(5)

(6)

with: ur (ω) = inf {− f0,r (ω, e) + yr∗ f1 (ω, e)}  − f 0,r (ω, y(ω)). e∈E

(7)



From Eqs. 5, 6, and 7, the family (  ur dμ)0
(9)

with k = sup{k ||y − y|| + k, ||y || = 1} < ∞. Thus Eq. 9 gives the quietness of I f at x. Then I f is stable at x, hence continuous at x. By the use of [11], Theorem 4.1 (c) and Corollary 4.3 (c), we obtain the finiteness of I f on L p (, E), hence for every p

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element x of L p (, E), I− f (x) = −I f (x) . But I f is directionally Lipschitzian at x with respect the direction y, therefore the condition of Definition 2.1 becomes: lim sup r−1 (I f (x + ry ) − I f (x )) < +∞. r→0+ x →x y →y

We deduce the existence of positive constants γ , , l verifying: sup{Ir−1 ( f (x +ry )− f (x )): 0 < r < γ , ||x − x||  , ||y − y||  } < l. p

p

(10)

Once again we apply Theorem 2.6, with M(ω) = E2 , X = L p (, E)2 and f0,r (ω, e, e ) = r−1 ( f (ω, e + re ) − f (ω, e)); f1 (ω, e, e ) = |e − x(ω)| p ; f2 (ω, e, e ) = |e − y(ω)| p . Condition (8) can be written: for every 0 < r < γ , I f1  ∩ I f2  ⊆ I f0,r l . The assumptions of Theorem 2.6 are satisfied since I f1 (x, y) = I f2 (x, y) = 0, and for 0 < r < γ  with γ  = inf(δ  , γ ), using (5) we have: I− f 0,r (x, y)  c||y||.

(11)

We obtain the existence of non-negative scalars y∗1,r , y∗2,r satisfying:  ur dμ − (y∗1,r + y∗2,r ) , −l  

with: ur (ω) =

inf

(e,e )∈E×E

(12)

 − f0,r (ω, e, e ) + y∗1,r f1 (ω, e, e )

 +y∗2,r f2 (ω, e, e )  − f 0,r (ω, x(ω), y(ω)) (13)  From Eqs. 11 and 13 we deduce that the family (  ur dμ)0
0
r−1 (I f (x + rz) − I f (x )) < m

sup

||x −x0 || ||z||=

(||x − x|| + ||z − y|| p ) + l < ∞. p

This proves the finiteness of the Lipschitz rate of I f at x0 . Therefore I f is strictly continuous on L p (, E), and the proof of (a)⇒(e) is complete. Let us prove the assertion (e)⇒(h). Let us observe first that, when assumption (e) holds, I f is finite at any point of L p (, E), hence, the use of Lemma 3.9 ensures the finiteness of the values of the integrand f . Using Lemma 3.7, let (x0 , x∗0 ) be in (L p × Lq )(∂ f ). The functional I f is locally Lipschitz around the point x0 of L p (, E). There exist positive constants , c so that I f has Lipschitz constant c in the ball of L p (, E) centered at x0 and with radius 1/ p . Let cBq be the ball of Lq (, E∗ ) centered at the origin and

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137

with radius c. For every x in the ball of center x0 and radius 1/ p , we have, from [11] Theorem 5.6 (d): Lq (∂ f (x)) ⊆ ∂ I f (x) ⊆ cBq .

(15)

Let us assume first 1 < p < ∞. Introduce the integrands: f0 (ω, e, e∗ ) = |e∗ |q ; f1 (ω, e, e∗ ) = |e − x0 (ω)| p and the set X = (L p × Lq )(∂ f ). From Lemma 3.8, X is a decomposable set rich in L0 (∂ f ), and the inclusions (15) can be written: I f1  ⊆ I f0 c . q

sup{I f0 (x, y) : I f1 (x, y)  , (x, y) ∈ X}  cq , or

Since I f1 (x0 , x∗0 ) = 0 < and I f0 (x0 , x∗0 ) is finite, we deduce again from Theorem 2.6, the existence of a non-negative y∗ such that the function u defined by: u(ω) = inf − f0 (ω, e, e∗ ) + y∗ f1 (ω, e, e∗ ) is integrable. Let v = (−u)+ . Thus, for every ∗ (e,e )∈∂ f (ω)

(ω, e) ∈  × E, and e∗ ∈ ∂ f (ω, e), we obtain: |e∗ |q  y∗ |e − x0 (ω)| p + v(ω). 

(16)

p

)|  2 p−1 (|t| p + |t | p ), taking k = (2 p−1 y∗ ) Since |t + t | p = |2( t+t 2 p−1 ∗

(2 y |x0 (ω)| + v(ω)) e∗ ∈ ∂ f (ω, e), p

1/q

1/q

and a(ω) =

, we deduce from Eq. 16: for every (ω, e) ∈  × E and

|e∗ |  k|e| p−1 + a(ω), with a ∈ Lq (, IR). This last inequality is just assertion (h) of Theorem 3.2. In order to complete the proof of assertion (e)⇒(h) in Theorem 3.2, now we consider the case p = 1. Introduce the integrands: f1 (ω, e, e∗ ) = |e − x0 (ω)|, f0 (ω, e, e∗ ) = 0 if |e∗ |  c, +∞ if |e∗ | > c, and X = (L1 × L∞ )(∂ f ). Assumption (15) can be written: sup{I f0 (x, y): I f1 (x, y)  , (x, y) ∈ X}  1, or

I f1  ⊆ I f0 1 .

Since I f1 (x0 , x∗0 ) = 0 < and I f0 (x0 , x∗0 ) = 0, we can apply Theorem 2.6. There exists a non-negative constant y∗ such that the function: u(ω) =

inf

(e,e∗ )∈∂ f (ω)

− f0 (ω, e, e∗ ) + y∗ f1 (ω, e, e∗ ).

is integrable. Thus, for every (ω, e) ∈  × E, and e∗ ∈ ∂ f (ω, e), we obtain: f0 (ω, e, e∗ )  f1 (ω, e, e∗ ) − u(ω). which is equivalent to: e∗ ∈ ∂ f (ω, e) ⇒ |e∗ |  c . We have proved assertion (h) of Theorem 3.2 in case p = 1. The proof of Theorem 3.2 is complete.  

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