DEFINABLE SMOOTHING OF LIPSCHITZ CONTINUOUS ...

DEFINABLE SMOOTHING OF LIPSCHITZ CONTINUOUS FUNCTIONS ANDREAS FISCHER Abstract. Let M be an o-minimal structure over the real closed field R. We prove the definable smoothing of definable Lipschitz continuous functions. In the case of Lipschitz functions of one variable we are even able to preserve the Lipschitz constant.

1. Introduction Smoothing of Lipschitz continuous functions defined on separable Riemannian manifolds has recently been studied in [1]. In the present paper we discuss smoothing of Lipschitz continuous functions which are definable in an o-minimal structure expanding a real closed field. In the following, R is used to denote a real closed field; i.e., R is an ordered field which has no non-trivial ordered algebraic extension. The description of o-minimal structures requires the notion of semi-algebraic sets. These sets are by definition the Boolean combinations of sets of the form {x ∈ Rn : p(x) > 0} where p is a polynomial in n variables with coefficients in R. Definition 1.1. An o-minimal structure M on R is a sequence of sets (Sn )n∈N such that (1) each Sn is a Boolean algebra of subsets of Rn , (2) each Sn contains every semi-algebraic subset of Rn , (3) if A ∈ Sn , B ∈ Sm , then A × B ∈ Sn+m , (4) if A ∈ Sn+1 , then π(A) ∈ Sn , where π denotes the projection onto the first n coordinates, (5) the elements of S1 are precisely finite unions of points and open intervals. A set belonging to some Sn is called definable, and a function is called definable if its graph is definable. For a detailed introduction to o-minimal structures see [12] or [7]. The collection of semi-algebraic sets forms the smallest o-minimal structure on R, see for example [2]. On the field of real numbers R a very well studied o-minimal structure is the collection of globally subanalytic sets. In the last 20 years more examples have been constructed, see e.g. [9], [13], [14] and [15]. In the following we fix an o-minimal structure M on the fixed real closed field R. We endow Rn with the Euclidean R-norm k·k (note that an R-norm has the same definition as norm just taking its values in R) and the corresponding topology. The aim of this paper is to prove the following theorem. Date: 18.10.2006. 2000 Mathematics Subject Classification. Primary 03C64 Secondary 26B05. 1

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Theorem 1.2. Let f : U → R be a definable Lipschitz continuous function defined on the open set U ⊂ Rn , and let ε : U → (0, ∞) be a definable continuous function. Then there is a definable Lipschitz continuous and continuously differentiable function g : U → R such that (1.1)

|g(u) − f (u)| < ε(u), u ∈ U.

In classical Analysis, smoothing of Lipschitz continuous functions is an elegant application of integration. But this technique cannot be applied to o-minimal structures, even if we consider an expansion of the real numbers, since integration does not preserve definability. In o-minimal situation we avoid integration using Λm -regular stratification of definable sets. This concept was developed in [4]. Unfortunately, our method does not allow us to control the Lipschitz constant while smoothing the function. To be more precise, if the definable function depends on at least two variables, the Lipschitz constant of the approximating function may be much bigger than that of the original function. Some smoothing techniques of classical Analysis almost preserve the Lipschitz constant but they use convolution of functions. As indicated above, we obtain a stronger result for definable functions of one variable. Lemma 1.3. Let f : I → R be a definable Lipschitz continuous function with constant L defined on an open interval I, and let ε : I → (0, ∞) be a definable continuous function. Then there is a definable Lipschitz continuous and continuously differentiable function g : I → R such that |g(t) − f (t)| < ε(t) and |g 0 (t)| ≤ L, t ∈ I. Remark 1.4. The method of definable smoothing has some further property which may be of interest for some applications. As a definable function, f is continuously differentiable outside a definable set A ⊂ U of lower dimension. If V is an open definable neighbourhood of cl(A) ∩ U , we may assume that g coincides with f outside V . 2. Proof of Lemma 1.3 First we look at the one-dimensional case. The next lemma prepares the proof of Lemma 1.3. Lemma 2.1. Let f : (−1, 1) → R be definable, continuous and C 1 outside 0, such that limt%0 f 0 (t) = c and limt&0 f 0 (t) = d. Then for ε > 0 and δ > 0 there is a definable continuously differentiable function g : (−1, 1) → R satisfying (1) g(t) = f (t) for |t| > δ, (2) |f (t) − g(t)| ≤ ε, for t ∈ (−1, 1), and (3) inf t6=0 f 0 (t) ≤ g 0 (t) ≤ supt6=0 f 0 (t). Proof. We consider for 0 < σ < 1 the family of functions hσ : (−1, 1) → R which is defined by  c−d 2  − 4σ (σ + t) , −σ < t ≤ 0, c−d (2.1) hσ (t) = − 4σ (σ − t)2 , 0 < t < σ,   0, otherwise.

DEFINABLE SMOOTHING OF LIPSCHITZ CONTINUOUS FUNCTIONS

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We note that hσ is continuously differentiable outside 0 and that limt%0 h0σ (t) = −(c − d)/2 and limt&0 h0σ (t) = (c − d)/2. Let gσ := f + hσ . Then gσ is continuously differentiable even at 0. For σ chosen small enough, gσ (t) = f (t), |t| > σ. Since hσ is bounded by |c − d|σ we may also assume that |gσ (t) − f (t)| ≤ ε if σ is sufficiently small. The derivative of hσ (outside 0) is bounded by |c − d|/2. So if we choose σ sufficiently small we obtain the estimates |f 0 (t) − c| < |c − d|/4 for −σ < t < 0 and |f 0 (t) − d| < |c − d|/4 for 0 < t < σ. So item (3) is satisfied. ¤ Proof of Lemma 1.3. As a definable function of one variable, f is continuously differentiable outside of a finite set {a1 , ..., ak }, cf. [12] Chapter 7 Theorem 3.2. Moreover, the limits limt&0 f (ai + t)/t and limt%0 f (ai + t)/t exist in R ∪ {±∞}, cf. [12] Chapter 3 Corollary 1.6. Since f is Lipschitz continuous, both limits are bounded, and so they exist in R. By definability of f , there is a pointed definable neighbourhood Ui of ai such that f is continuously differentiable in Ui . We apply Lemma 2.1 to each ai and obtain a g with the desired properties. ¤ 3. Preliminaries We prepare the proof of Theorem 1.2 by several lemmas. For the next lemma we recall the well known fact that a definable Lipschitz continuous function f : U → R can always be extended to a definable Lipschitz continuous function f defined on cl(U ) (closure of U ). This extended function is unique. In general, a continuously differentiable function of several variables with bounded derivative is not Lipschitz continuous. Lemma 3.1. Let U ⊂ Rn be definable and f : U → R be definable and Lipschitz continuous. Let V ⊂ U be open and g : V → R be definable and continuously differentiable with bounded gradient such that ( g(ξ), ξ ∈ V, (3.1) F (ξ) := f (ξ), ξ ∈ cl(U ) \ V is continuous. Then F is Lipschitz continuous. Proof. We select L > 0 large enough such that f is Lipschitz continuous with constant L and k∇gk is bounded by L. For x, y ∈ cl(U ) we put [x, y] := {x+t(y−x) : 0 ≤ t ≤ 1}. The set [x, y] is not necessarily contained in cl(U ). But, according to o-minimality, there exist 0 = t1 ≤ ... ≤ t2N = 1 such that [x, y] ∩ cl(U ) = ∪N i=1 [ξ2i−1 , ξ2i ] where we have set ξi = x + ti (y − x), i = 1, ..., 2N . We may further assume that for 1 ≤ i ≤ N either [ξ2i−1 , ξ2i ] ⊂ cl(U )\V or [ξ2i−1 , ξ2i ]\{ξ2i−1 , ξ2i } ⊂ V applies. F restricted to [ξ2i−1 , ξ2i ] is Lipschitz continuous with constant L. By the assumption, for j = 1, ..., 2N − 1 we have |F (ξj ) − F (ξj+1 )| ≤ Lkξj+1 − ξj k = Lky − xk(tj+1 − tj ). So (3.2)

2N −1 X

|F (y) − F (x)| = |

F (ξj+1 ) − F (ξj )|

j=1

≤

2N −1 X

L(tj+1 − tj )ky − xk = Lky − xk.

j=1

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We now use some aspects of the concept of Λm -regular cells, cf. [4] Definition 4.1. (We do not need the full strength of this concept.) For definable open U ⊂ Rn we denote by Cbm (U, Rk ) the m times continuously differentiable functions from U to Rk with bounded (first) derivative. By definition, each Λm -regular function is Cbm , and actually, Λ1 -regular means the same as Cb1 . We define Cbm cells as follows. A Cbm cell of R is either an open interval or a singleton. Suppose that we know all Cbm cells of R` , 1 ≤ ` ≤ n. Then a Cbm cell of Rn+1 is either a set of the form {(x, h(x)) ∈ X × Rn+1−d } where X ⊂ Rd is an open Cbm cell in Rd and h : X → Rn+1−d is a definable Cbm function; or M is of the form {(x, y) : x ∈ X, f (x) < y < g(x)} where X ⊂ Rn is an open Cbm cell and f, g ∈ Cbm (X, R) ∪ {±∞} such that for all x ∈ X, f (x) < g(x) applies; or M is a singleton. Note that by construction, Cbm cells are definable sets. An interesting property of Λ1 -regular cells is that a Λ1 -regular function defined on a Λ1 -regular cell is Lipschitz continuous, cf. [4] Corollary 9.9. This implies that Cbm functions on Cbm cells are Lipschitz continuous. We extend the notion of C m functions to functions with not necessarily open domain as follows. A definable function f : A → Rd is m times continuously differentiable if there exists an open definable set B containing A and a definable C m function g : B → Rd which coincides with f on A. The dimension of a definable set is the maximal integer d such that A contains a definable set which is definably homeomorphic to Rd . This definition is well-defined, c.f. [12], and we refer the reader to [12] Chapter 4 for a detailed description of dimension. Moreover, it is easy to check that a Cbm cell is definably homeomorphic to some Rd . Lemma 3.2. Let M ⊂ Rn be a Cbm cell of dimension d < n and M ⊂ V ⊂ U definable open neighbourhoods of M . Let f : U → R be definable and Lipschitz continuous such that both f |U \M and f |M are C m . Then for every definable continuous ε : U → (0, ∞) there is a Lipschitz continuous C m function g : U → R such that (3.3)

|g(u) − f (u)| < ε(u), u ∈ U

and f = g outside V . Proof. The dimension of M is less than n. So M is the graph of a definable Cbm function h : X → Rn−d where X ⊂ Rd is an open Cbm cell. Let U 0 := U ∩ X × Rn−d . For each ξ ∈ M , ε(ξ) > 0. So the continuity of f implies that there is an open definable neighbourhood V 0 of M such that |f (ξ) − f (ξ + η)| < ε(ξ + η) whenever ξ ∈ M and η ∈ {0} × Rn−d with ξ + η ∈ V 0 . We may further choose V 0 in such a way that M ⊂ V 0 ⊂ (V ∩ (X × Rn−d )). We define a function ψ : X × Rn−d → X × Rn−d by ψ(x, y) = (x, y − h(x)). ψ is obviously Cbm and so, since M × Rn−d is a Cb1 cell, ψ is Lipschitz continuous. Hence, we can extend ψ to a Lipschitz continuous function ψ defined on cl(U 0 ). In addition, ψ is bijective with Lipschitz continuous inverse. The function F = f ◦ ψ −1 is, as composition of Lipschitz continuous functions, also Lipschitz continuous. In addition, F is m times continuously differentiable in ψ(U 0 ) \ X × {0} and X × {0}. Step 1: We construct a Cbm function σ : X → R which tends to 0 as x tends to the boundary of X or infinity such that ψ(V 0 ) contains the set W := {(x, y) ∈ X × Rn−d : kyk < σ(x)}. p p Let φ : Rd → (−1, 1)d be given by φ(x1 , ..., xd ) = (x1 / 1 + x21 , ..., xd / 1 + x2d ).


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This map is obviously Cbm and definable. Moreover, the set φ(X) is bounded and open. We select a definable C m function θ : Rd → R which vanishes outside φ(X) and is positive on φ(X). The support of θ is bounded, so θ is Cbm . Note that the zero-set of D : Rd → R, x 7→ dist((x, 0), Rd \ ψ(V 0 )), is contained in the zero-set of θ. This allows us to apply the generalised L Ã ojasiewicz inequality, [7] Theorem C14, to θ and D. So we obtain a bijective definable continuous map ρ : R → R with ρ(0) = 0 such that ρ ◦ θ(x) ≤ D(x) for x ∈ Rd . By definability, ρ is C m in (0, δ) for some 0 < δ < 1. We define ρ˜ : R → R by µ ¶ t2 δt2 (3.4) t 7→ ρ . 1 + t2 1 + t2 Hence, ρ˜ is differentiable at 0 and by [4] Proposition 7.2, ρ˜ is even continuously differentiable at 0. So σ = ρ˜ ◦ θ ◦ φ is the desired function. We may further assume that the derivative of σ is bounded by 1. Step 2: Let ϕ : [0, ∞) → [0, 1] be a definable C m function with ϕ|[0,1/2] = 1 and ϕ|[1,∞) = 0. Then ϕ0 is bounded by some constant K > 0. Note that x 7→ F (x, 0) is an m times continuously differentiable Lipschitz continuous function on X. We define G : ψ(U 0 ) → R by µ ¶ µ µ ¶¶ kyk kyk (3.5) G(x, y) := F (x, 0)ϕ + F (x, y) 1 − ϕ . σ(x) σ(x) G is definable and C m in ψ(U 0 ). Since for (x, y) ∈ W the value G(x, y) lies between F (x, 0) and F (x, y), we obtain the inequality |G(x, y) − F (x, y)| < ε ◦ ψ −1 (x, y). We now prove that Lipschitz continuity of G. By the assumption, |F (ξ) − F (η)| ≤ Lkξ − ηk, and k∇F (x, y)k is bounded by L outside X × {0} as well as k∇(F (x, 0))k on X × Rn−d . We first show that k∇ϕ(kyk/σ(x))k is bounded by 2K/σ(x). Indeed, ° ¶° ¯ µ ¶¯ ° µ ¶° µ ° ¯ 0 kyk ¯ ° kyk ° kyk ° ° ° ¯ ° ¯ ° (3.6) °∇ϕ σ(x) ° ≤ ¯ϕ σ(x) ¯ · °∇ σ(x) ° °µ ¶° ° kyk ° y ° ° ≤K° 2 ∇x σ(x), σ (x) kykσ(x) ° °µ ¶° ∇x σ(x)kyk y ° K ° ° ° , = σ(x) ° σ(x) kyk ° 2K ≤ . σ(x) So for 0 < kyk ≤ σ(x) (3.7)

k∇G(x, y)k =k(∇((F (x, 0) − F (x, y))ϕ(kyk/σ(x)) + (F (x, 0) − F (x, y))∇ϕ(kyk/σ(x)) + ∇F (x, y)k ≤L + Lkyk2K/σ(x) + L ≤2L(1 + K).

As a consequence we see that G is Cbm on W , and that G = F outside W . For the Lipschitz continuity of G we use Lemma 3.1; i.e., we have to show that G extends continuously to cl(ψ(U 0 )) and G = F on ∂ψ(U 0 ). This is evident for the points outside ∂X ×{0} since there G = F , and F is Lipschitz continuous by construction.

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We further note that |G(x, y) − F (x, y)| ≤ |(F (x, 0) − F (x, y))| ≤ Lσ(x). So for ξ ∈ ∂X and (x, y) ∈ ψ(U 0 ) (3.8)

|G(x, y) − F (ξ, 0)| ≤ |G(x, y) − G(x, 0)| + |G(x, 0) − F (ξ, 0)| ≤ Lσ(x) + Lkx − ξk.

Therefore, G(u) − F (u) tends to 0 as u ∈ ψ(U 0 ) tends to ∂ψ(U 0 ). Step 3: Now we define g : U → R by ( G ◦ ψ(ξ), if ξ ∈ V 0 (3.9) g(ξ) := f (ξ), otherwise. Using Lemma 3.1 we easily obtain the desired properties for g.

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4. Proof of Theorem 1.2 For the proof of Theorem 1.2 we use a consequence of Λm -regular stratification. A Cbm (resp. Λm -regular) stratification is a finite partition of Rn into disjoint definable sets X1 , ..., Xr ; for each i = 1, ..., r there is a linear orthogonal isomorphism φi : Rn → Rn such that φi (Xi ) is a Cbm (resp. Λm -regular) cell; in addition, the frontier ∂Xi is the union of some of the Xj . We call a stratification compatible with the definable sets A1 , ..., As ⊂ Rn , if each Aj is the union of some Xi . In every o-minimal structure we can use Λm -regular stratification, cf. [4] Theorem 4.5. So we may apply Cbm stratifications, too. Proof of Theorem 1.2. By [12] chapter 7 §3, we can partition U into finitely many definable sets X1 , ..., Xr such that the restrictions of f to Xi are C m . We select a Cbm stratification of cl(U ) compatible with the X1 , ..., Xr . We use N to denote the number of Cbm cells Zi of dimension less than n which are contained in U . Moreover, we may assume that dim(Zi ) ≥ dim(Zi+1 ) for i = 1, ..., N − 1. We choose for each Zi a definable open neighbourhood Ui . Since we deal with a stratification, we may assume that Ui ∩ Uj = ∅ if j > i. For each i = 1, ..., N we choose a further neighbourhood Vi of Zi by (4.1)

Zi ⊂ Vi ⊂ {x : dist(x, Zi )
j Zi , fj is Lipschitz in U by Lemma 3.1, and |fj (u) − fj−1 (u)| < jε(u)/N . So fN is a Lipschitz continuous function which is continuously differentiable and |fN (u) − f (u)| < N ε(u)/N = ε(u). ¤


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In the proof of Theorem 1.2 we modified the values of the function f : U → R only in a special open neighbourhood of cl(D)∩U where D is the set of points at which f is not continuously differentiable. Lemma 3.1 requires no special neighbourhoods as long as they are definable and open. So we assume that there is an approximating g : U → R which coincides with f outside an arbitrarily small definable open neighbourhood of cl(D) ∩ U . References [1] D. Azagra, J. Ferrera, F. Lopez-Mesas, et al Smooth Approximation of Lipschitz functions on Riemannian manifolds to appear in J. Math. Anal. Appl [2] J. Bochnak, M. Coste, M.-F. Roy Real Algebraic Geometry Ergebnisse der Mathematik und ihrer Grenzgebiete (3) Springer Verlag Berlin - Heidelberg 1998 [3] J. Escribano Approximation Theorems in o-minimal structures Illinois Journal of Mathematics 46(1), 2002, 111-128 [4] A. Fischer Peano-Differentiable Functions in O-minimal Structures Dissertation 2006 available at http://www.opus-bayern.de/uni-passau/volltexte/2006/67/ [5] K. Kurdyka On a subanalytic sratification satisfying a Whitney-Property with exponent 1 Proceeding Conference Real Algebraic Geometry - Rennes 1991, Springer LNM 1524 (1992), 316-322. [6] D. Marker Model Theory: An Introduction Graduate Texts in Mathematics, 217, Springer New York, 2002 [7] C. Miller, L. van den Dries Geometric categories and o-minimal structures Duke Math. J. 84, 497-540 (1996) [8] A. Parusi´ nski Lipschitz stratification of subanalytic sets Ann. Sci. cole Norm. Sup. (4) 27 (1994), no. 6, 661–696 [9] J.-P. Rolin, P. Speissegger, A. J. Wilkie Quasianalytic Denjoy-Carleman classes and ominimality J. Amer. Math. Soc. 16 (2003), no. 4, 751-777 [10] M. Shiota Nash manifolds Lecture Notes in Mathematics, 1269 Springer-Verlag, Berlin, 1987 [11] L. van den Dries o-minimal structures Logic: from foundations to applications (Staffordshire, 1993), 137-185, Oxford Sci. Publ., Oxford Univ. Press, New York, 1996 [12] L. van den Dries Tame Topology and O-minimal Structures LMS Lecture Notes 248, Cambridge University Press 1998 [13] L. van den Dries, P. Speissegger The field of reals with multisummable series and the exponential function Proc. London Math. Soc. (3) 81 (2000), no. 3, 513-565. [14] L. van den Dries, P. Speissegger The real field with convergent generalized power series Trans. Amer. Math. Soc. 350 (1998), no. 11, 4377-4421. [15] A. J. Wilkie Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function J. Amer. Math. Soc. 9 (1996), no. 4, 1051-1094. University of Saskatchewan, Department of Mathematics & Statistics, 106 Wiggins Road, Saskatoon, SK, S7N 5E6, Canada, Tel.: (306) 966 6110 E-mail address: [email protected]