Smooth Approximation of Lipschitz Functions on Finsler Manifolds

and Applied Analysis Hindawi Publishing Corporation http://www.hindawi.com Volume 2013 ISRN Applied Mathematics Hindawi Publishing Corporation http://www.hindawi.com Volume 2013 Hindawi Publishing Corporation http://www.hindawi.com Volume 2013 International Journal of Combinatorics Hindawi Publishing Corporation http://www.hindawi.com Volume 2013 Journal of Function Spaces and Applications International Journal of Mathematics and Mathematical Sciences Hindawi Publishing Corporation http://www.hindawi.com Volume 2013


Introduction
There are many geometrically significant functions on a Riemannian manifold which are typically Lipschitz but not smooth, as it is the case, for example, of distance functions.Thus it is interesting to study the regularization and smooth approximation of Lipschitz functions on Riemannian manifolds.This has been done in the classical work of Greene and Wu [1], where in particular it is proved that every Lipschitz real function on a (connected, second countable, and finite dimensional) Riemannian manifold can be approximated, in the  0 -fine topology, by smooth Lipschitz functions whose Lipschitz constants can be made arbitrarily close to the Lipschitz constant of the original function.This result has been extended in [2] to the case of infinite-dimensional Riemannian manifolds, where some interesting applications are also given.Recently, related approximation results in the setting of the so-called Banach-Finsler manifolds have been obtained in [3].
Our purpose here is to study the analogous approximation problem in the context of (finite-dimensional) Finsler manifolds, where the Finsler structure is supposed to be positively (but in general not absolutely) homogeneous.The contents of the paper are as follows.In Section 2 we collect some basic preliminary facts about Finsler manifolds.Section 3 is devoted to give a mean value inequality in this context.Next, in Section 4, we obtain our main result.Namely, we prove in Theorem 8 that every Lipschitz real function on a connected, second countable Finsler manifold can be approximated, in the  0 -fine topology, by  1 -smooth Lipschitz functions with Lipschitz constants arbitrarily close to the Lipschitz constant of the original function.This approximation result has been used in [4] in order to obtain a version of the Myers-Nakai Theorem for reversible Finsler manifolds (that is, in the case that the Finsler structure is absolutely homogeneous).In Section 5 we introduce the class of quasi-reversible Finsler manifolds, which can be described as those Finsler manifolds where distance functions are in fact Lipschitz.As a consequence of our main result, we obtain a completeness criterium for quasi-reversible Finsler manifolds, in terms of the existence of a proper  1 -smooth function with uniformly bounded derivative.In this way we extend the completeness criterium for Riemannian manifolds given by Gordon in [5].Finally, in Section 6 we consider the normed algebra  1  () of all  1 functions with bounded derivative on a quasi-reversible Finsler manifold , and we obtain a characterization of normed algebra isomorphisms  :  1  () →  1  () as composition operators.From this we obtain a variant of Myers-Nakai Theorem in the context of complete reversible Finsler manifolds.

Preliminaries
We start with the basic notion of Minskowski norm.

Definition 1.
Let  be a finite-dimensional real vector space.One says that a functional  :  → [0, ∞) is a Minkowski norm on  if the following conditions are satisfied (i) Positivity: (V) = 0 if and only if V = 0.
(v) Strong convexity: for every V ∈  \ {0}, the quadratic form  V associated to the second derivative of the function  2 at V, that is, is positive definite on .
We note that conditions (i) and (ii) in the above definition are, in fact, consequence of conditions (iii)-(v) (see Theorem 1.2.2 of [6]).It is clear that every norm associated to an inner product is a Minkowski norm.Recall that, in general, a Minkowski norm needs not to be symmetric, and there are indeed very interesting examples of nonsymmetric Minkowski norms, such as, for example, Randers spaces (see [6]).We say  is symmetric or absolutely homogeneous if  (V) = ||  (V) , for every V ∈  and every  ∈ R. ( In this case,  is a norm in the usual sense.Now the definition of Finsler manifold is as follows. Definition 2. A Finsler manifold is a pair (, ), where  is a finite-dimensional  ∞ -smooth manifold and  :  → [0, ∞) is a continuous function defined on the tangent bundle , satisfying (ii) For every  ∈ , (, ⋅) :    → [0,∞) is a Minkowski norm on the tangent space   .
In particular, a Riemannian manifold is a special case of Finsler manifold, where the Minkowski norm on each tangent space    is given by an inner product.The Finsler structure  is said to be reversible if, for every  ∈ , (, ⋅) is symmetric.This is of course the case of Riemannian manifolds.Now suppose that (, ) is a connected Finsler manifold.The Finsler distance   on  is defined by where the Finsler length of a piecewise  1 path  : [, ] →  is defined as: In this way we have (see Section 6.2 of [6]) that the Finsler distance   is the so-called an asymmetric distance on , in the sense that it verifies (i)   (, ) ≥ 0.
In general,   needs not to be symmetric.Nevertheless, when  is reversible the Finsler distance   is symmetric, and therefore (,   ) is a metric space in the usual sense.In general, for each  ∈  and  > 0, the forward ball of center  and radius  is defined as In the same way, the backward ball of center  and radius  is defined as Note that, as can be seen in [6], the family of forward balls and also the family of backward balls are both neighborhood basis for the topology of the manifold .
If (, ) is a Minkowski space, that is, a vector space endowed with a Minkowski norm, then the associated asymmetric distance   is given by In this case, we will denote by  0 () the forward ball of center 0 ∈  and radius , and we call it the Minkowski ball of center 0 and radius .That is, We next recall the following result by Deng and Hou (see Theorem 1.2 in [7]) concerning the exponential mapping in a Finsler manifold, which will be useful in what follows.
Theorem 3 (Deng and Hou [7]).Let (, ) be a connected Finsler manifold, let  ∈ , and consider  > 0 such that the exponential mapping exp  :  0 () → B +  () is a  1diffeomorphism.Then, for , V ∈  0 () with  ̸ = V, one has that lim A terminological remark is now in order.Suppose that  and  are two nonempty sets, endowed with asymmetric distances   and   , respectively.We say that a mapping  : (,   ) → (,   ) is Lipschitz, with constant  > 0 (or, briefly, -Lipschitz) if, for every  1 ,  2 ∈ , As usual, we will say that  : (,   ) → (,  ) is bi-Lipschitz when  is bijective and both  and  −1 are -Lipschitz mappings.With this terminology at hand and as a direct consequence of Theorem 3, we obtain the following result, describing the bi-Lipschitz behavior of the exponential mapping associated to a Finsler manifold in small balls.
Corollary 4. Let (, ) be a connected Finsler manifold.For each  ∈  and each  > 0, there exists  > 0 such that the exponential mapping

Mean Value Inequality
In this section we obtain a kind of mean value inequality in the context of Finsler manifolds.If (, ) is a connected Finsler manifold, we define the Lipschitz constant of a function  :  → R as Of course  is Lipschitz if and only if Lip() < ∞.We denote by Lip() the space of all real Lipschitz functions defined on .If  :  → R is now a  1 -smooth function, we define as usual the norm of its differential () at a point  ∈  by Next, we give the following result providing the desired mean value inequality.
Theorem 5. Let (, ) be a connected Finsler manifold and  :  → R a  1 -smooth function.Then, Thus, for every ,  ∈ , one has that Proof.For the proof, fix a number  ≥ 0, and we are going to see that the following conditions are equivalent: (1)  is -Lipschitz.
We finish this section with the following simple result giving a local characterization of Lipschitz mappings, which will be useful later.
In this way we obtain that  is -Lipschitz.The converse is clear.

Smooth Approximation of Lipschitz Functions
In this section we present our results about regularization of Lipschitz functions on Finsler manifolds.In particular, as consequence of Theorem 8 as follows, we can derive that if  :  → R is a Lipschitz function defined on a connected and second-countable Finsler manifold  and  > 0 is given, there exists a Suppose that  :   → R is Lipschitz and let  > 0. Then there exists a  ∞ -smooth function  :  → R such that |(V) − (V)| ≤ , for every V ∈ , and Lip(|  ) ≤ Lip(|  ).
Proof.By choosing a basis of , we may assume that  = R  .Note that, by local compactness, it follows that the Minkowski norm  is equivalent to the usual Euclidean norm ‖ ⋅ ‖ in R  , in the sense that there exists some  ≥ 1 such that for every V ∈ R  .Now, if  :   → R is a -Lipschitz function for the Minkowski norm , then  is a ( ⋅ )-Lipschitz function for the Euclidean norm.Hence, using, for example, the well-known MacShane extension result, we can obtain a Lipschitz extension f : R  → R. Now consider a sequence (  )  of usual  ∞ -smooth mollifiers on R  , where each   is nonnegative, supp(  ) is contained in the Euclidean ball B(0, 1/), and ∫ R    = 1.For each , define   : R  → R by Each   is  ∞ -smooth, and, since f is uniformly continuous, we have that the sequence (  ) converges to f uniformly on R  .Given  > 0, choose  > / and large enough so that ‖  − f‖ ∞ <  and define  =   .Then, if , V ∈ , Therefore, we have that Lip(|  ) ≤ Lip(|  ), as we wanted.
We next give the main result of the paper.Proof.Let us denote  = Lip().Without loss of generality we may assume that, for every  ∈ , () > 0 is small enough so that () ≤ /2 and Using Corollary 4, for each  ∈ , we can choose   > 0 such that the exponential mapping exp  is a  1 -diffeomorphism and (1 + ())-bi-Lipschitz from the Minkowski ball  0  (3  ) ⊂    onto the forward ball B +  (3  ) ⊂ , where 0  denotes the null vector of   .In addition, by the continuity of  and , we can also assume that () ≥ ()/2 and |() − ()| ≤ ()/2, for every  ∈ B +  (3  ).Since  is second countable, there is a sequence where we denote   =    .Now, for each  ∈ N, we define and we then have that   is  ⋅ (1 + (  ))-Lipschitz.
Next, we are going to construct a partition of unity subordinated to the covering {B +   (2  )} ∈N of , estimating the Lipschitz constant of the respective functions.Thus, for each  ∈ N, let   : R → [0, 1] be a  ∞ -smooth function such that and define   :  → [0, 1] by It is clear that each   is  1 -smooth and Lipschitz.Furthermore,   = 1 on the forward ball B +   (  ) and   = 0 on  \ B +   (2  ).Now we define the functions   :  → [0, 1] by setting  1 =  1 and, for  ≥ 2, Then, it is easy to check that, for every  ∈ N, (i)   is  1 -smooth and   -Lipschitz, where   = ∑ ≤ Lip(  ).
Remark 9.In general, the exponential mapping in a Finsler manifold is only  1 -smooth.According to a result of Akbar-Zadeh in [8] (see also [6], page 127), the exponential mapping is  2 -smooth if and only if it is  ∞ -smooth, and this property characterizes a special class of Finsler manifolds, called manifolds of Berwald type.Thus, if (, ) is a connected, second countable manifold of Berwald type, the same proof above gives that the approximating function  in Theorem 8 can be chosen to be  ∞ -smooth.

Quasi-Reversible Manifolds and a Completeness Criterium
In this section, as an application of the approximation result given in the above section, we obtain a completeness criterium for the class of manifolds that we call quasi-reversible.These are defined as follows.
Definition 10.A Finsler manifold (, ) is said to be quasireversible if there exists some  ≥ 1 such that It is clear that every reversible Finsler manifold is quasireversible.In fact, a Finsler manifold is reversible if and only if it is quasi-reversible for  = 1.On the other hand, a remarkable class of quasi-reversible (not necessarily reversible) manifolds are those manifolds of Berwald type.Indeed, we can deduce this, using a result due to Ichijyō [9] (see also [6], page 258) saying that if  is a manifold of Berwald type, then all its tangent spaces (  , (, ⋅)), for every  ∈ , are linearly isometric to each other.
We next give a useful characterization of connected quasireversible manifolds.
This implies that   (, ) ≤  ⋅   (, ).Interchanging the roles of  and , we obtain the reverse inequality.
Note that, by choosing  = 1 in the above result, we can deduce at once the following characterization of connected reversible manifolds.
As an application of Theorem 8, we are going to obtain a completeness criterium in the context of quasi-reversible manifolds.This will extend the corresponding result by Gordon [5] for Riemannian manifolds.First recall that a sequence (  ) in a Finsler manifold (, ) is said to be forward Cauchy (resp., backward Cauchy) if, for every  > 0, there exists some  0 ∈ N such that, if  0 ≤  ≤ , then   (  ,   ) < , (resp.,   (  ,   ) < ).We say then that (, ) is forward complete (resp., backward complete) if every forward Cauchy sequence is convergent (resp., every backward Cauchy sequence is convergent).It is clear that, for quasi-reversible manifolds, forward and backward completeness are equivalent.On the other hand, recall that a continuous function  :  → R is said to be proper if, for every compact set  ⊂ R, its preimage  −1 () is compact.Theorem 13.Let (, ) be a connected, second countable, and quasi-reversible Finsler manifold.The following conditions are equivalent: (1) (, ) is forward complete.
(3) There exists a proper  1 -smooth function  :  → R whose differential is uniformly bounded in norm.
Proof.(1) ⇒ (2) Fix  ∈  and consider the forward distance function   =   (, ⋅).The Hopf-Rinow Theorem (see Theorem 6.6.1 in [6]) gives that (, ) is forward complete if and only if every closed and (forward) bounded subset of  is compact.This implies that   is a proper function.Furthermore, by Theorem 11, we have that   is Lipschitz.