Abstract. A John function is a continuously differentiable function whose gradient is bounded by the reciprocal of the Euclidean distance to the boundary of the ...
JOHN FUNCTIONS FOR o-MINIMAL DOMAINS ANDREAS FISCHER Abstract. A John function is a continuously differentiable function whose gradient is bounded by the reciprocal of the Euclidean distance to the boundary of the domain. Here we construct John functions whose gradient norms have positive lower bounds for o-minimal domains. We prove their definability and give explicit estimates for number of John functions required for a given set to obtain uniform positive lower bounds.
1. Introduction Let n be a positive integer. A John function is a continuously differentiable function from some open subset U of Rn to R whose gradient is bounded by the reciprocal of the Euclidean distance to the boundary ∂U of U . These functions satisfy the bounded-mean-oscillation in the sense of John-Nirenberg, cf. [11, 12]. Conversely, every harmonic function satisfying bounded-meanoscillation is a John function, cf. [14, 17]. John functions on the unit disc in R2 are also related to the Bloch space on the unit disc in C, cf. [18, Proposition 5.4]. But Bloch functions are required to be holomorphic. For any subset X of Rn and x ∈ Rn let dX (x) denote the Euclidean distance between x and X. For two functions f and g we write f < g on X if f (x) < g(x) for all x ∈ X. The gradient-norms of John functions with domain U are bounded from above by C/d∂U for some constant C > 0. Naturally, the question arises whether some uniform positive lower bound may be found; that is, does there exist a John function f which additionally satisfies 1 k∇f k > d∂U on U ? With the exception of subsets of R, the answer to this question is unknown. However, for applications it often suffices to construct a finite sequence of John functions such that the sum of gradient-norms is bounded Date: 06.01.2008. 2000 Mathematics Subject Classification. Primary 26B99; Secondary 14P10, 14P15, 03C64. Key words and phrases. John functions, o-minimal structures. Research partially supported by NSERC Discovery Grant of Dr. Salma Kuhlmann and by EC-IHP-Network RAAG (Contract-No: HPRN-CT-2001-00271). 1
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from below by 1/d∂U . Such a sequence is called an exhaustive sequence of John functions for U . For general open domains of Rn the answer for this weaker problem is also unknown. Thus, we will restrict our considerations to sets which are definable in some o-minimal expansion of the real numbers. We fix the expansion M. Here, definable always means definable in M (with parameters in R). In order to investigate Green potential domains for the Laplacian operator, Kurdyka and Xiao constructed in [16] exhaustive sequences of John functions with definable domain. The key-result is the following Theorem. Theorem 1 (Kurdyka, Xiao). Let U ⊂ Rn be a proper definable open set. Then there exist finitely many C 1 functions f1 , . . . , fr : U → R and a constant C > 0 such that r X 1 C ≤ k∇fi k ≤ d∂U d ∂U i=1 on U . The construction of John functions in [16] makes essential use of integration, so that the functions fi in Theorem 1 are far away from being definable in any o-minimal expansion. The number r depends on the geometry of the set, and it can become arbitrarily large even in R2 . However, for the construction of exhaustive sequences of John functions, at least the logarithm is needed which is only definable in o-minimal structures with exponential function. The present paper is motivated by a question of K. Kurdyka addressed to the author: Can one assume that the functions fi are definable in the structure M expanded by the exponential function? By [22], this expanded structure is also o-minimal. We answer this question affirmatively by showing that one can choose logarithms of definable functions for the fi in Theorem 1. In addition, we compute explicit bounds for the number r of required John functions. We shall show the following Theorem. Theorem 2. Let U be a proper definable open subset of Rn . Then there are n + 1 definable C 1 functions g1 , . . . , gn+1 : Rn → R which vanish outside of U , and a constant C > 0 such that n+1 X 1 C ≤ k∇ ln ◦gi k ≤ d∂U d ∂U i=1 on U . In view of o-minimal geometry, Theorem 2 states a strong version of the zero-set property of definable C 1 functions. That is, every closed definable subset A of Rn is the zero-set of a definable C 1 function g : Rn → [0, ∞) whose logarithmic derivative in the complement of A is bounded by 1/dA . Schmid and Vilonen developed in [20, Chapter 6] a formula for calculating
JOHN FUNCTIONS FOR o-MINIMAL DOMAINS
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characteristic cycles of constructible sheaves using the logarithmic derivatives of definable C 1 functions f : U → [0, ∞) which vanish outside of U . By Theorem 2 we may assume that the derivative of this function is bounded by 1/d∂U on U . In Section 2 we briefly recall some concepts about o-minimality to which we refer throughout the paper. The proof of Theorem 2 is performed in Section 3. In Section 4 we discuss higher regularity of the functions gi in Theorem 2. 2. Preliminaries and Notation For a set U the symbols U and ∂U denote the closure and the boundary of U , respectively. We briefly recall the definition of o-minimal structures. For an introduction addressed to geometers we refer the reader to [4]. A subset of Rn that is described by a finite set of polynomial inequalities is called semialgebraic. An o-minimal structure M over R is a sequence (Mn )n∈N of sets, where each Mn is the collection of definable subsets of Rn , such that (a) each Mn is a Boolean algebra of sets containing all semialgebraic subsets of Rn , (b) finite Cartesian products of definable sets are definable (c) linear projections of definable sets are definable, (d) the definable subsets of R have finite boundary. A function f is called definable if its graph Γ(f ) is definable. The structure consisting of all semialgebraic sets is an example of an ominimal structure, cf. [1]. Further examples are known, see for example [3, 5, 6, 19, 23]. In the following, the structure M is fixed. Our construction of John functions is based on a special kind of partition of definable sets into Λ1 -regular cells. A Λ1 -regular function, cf. [8, Def. 1.3] or [13], is a definable C 1 function with bounded first derivative. The set of Λ1 -regular functions from U to V is denoted by Λ1 (U, V ). The symbols ±∞ are regarded as constant functions defined on arbitrary sets. Definition 1. A Λ1 -regular standard cell in R is either a single point or an open interval. Suppose we know the standard Λ1 -regular cells in Rn−1 , then a Λ1 -regular standard cell S of Rn is either a single point, or a definable set S of either the form S = Γ(h) where h ∈ Λ1 (X, Rk ) is defined on some open Λ1 -regular standard cell X ⊂ Rn−k ; or of the form S = {(x, y) : x ∈ X, f (x) < y < g(x)} for some open Λ1 -regular standard cell X ⊂ Rn−1 and f, g ∈ Λ1 (X, R) ∪ {±∞} such that f < g on X. A set S ⊂ Rn is called Λm -regular cell if there is an orthogonal linear automorphism of Rn mapping S to a standard Λm -regular cell. Let δ > 0. A Λ1 -regular cell is called δ-flat if it is either an open set, or the function h of one of its corresponding standard cells S = Γ(h) is additionally Lipschitz-continuous with constant δ.
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Let L > 1. A set X is called L-quasi-convex if any two points x and y can be connected within X by a rectifiable curve whose length is bounded by L kx − yk. Our interest in Λ1 -regular cells is based on the following kind of stratification, cf. [8, Theorem 1.4 and proof of Proposition 4.1]. Theorem 3. Let L > 1 and δ > 0. For every finite family of definable subsets A1 , . . . , Ak of Rn , there exists a δ-flat L-quasi-convex Λ1 -stratification of Rn compatible with the sets A1 , . . . , Ak . That is, there is a finite partition of Rn into subsets S1 , . . . , Sr , called strata, such that (a) each stratum is a δ-flat L-quasi-convex Λ1 -regular cell, (b) for each stratum Si , the frontier Si \ Si is the union of some of the strata, (c) each Aj is the union of some of the strata. We will extend Lipschitz-continuous functions. The situation is the following. Let f : A → R be a definable Lipschitz continuous function with constant L, and let K ≥ L. We apply the standard extension operator to f with constant K to obtain the function F : Rn → R which is given by F (x) := inf {K kx − yk + f (y) : y ∈ A} . This function is definable and Lipschitz-continuous with constant K such that F = f on A. Note that if the function f is non-negative and K > 0, then F > 0 on Rn \ A. In general, every definable function f : X → R is piecewise C 1 -smooth. That is, there exists a finite partition of A into definable sets A1 , . . . , Ar such that, for every i = 1, . . . , r, there exists a definable open neighborhood Vi of Ai and a definable C 1 function fi : Vi → R with fi |Ai = f |Ai . This is actually equivalent to the concept of C 1 cell decomposition, cf. [2, Chapter 7.3]. In particular, this implies that every definable function is C 1 -smooth outside of a definable nowhere dense subset. We denote by D1 (f ) the set of points at which the definable function f is not C 1 -smooth. This set is definable. We will also smooth definable Lipschitz-continuous functions. This is made possible by the following theorem, cf. [9]. Theorem 4. Let f : U → R be a definable Lipschitz-continuous function, and let e : U → R+ be a definable continuous function. Then there is a definable Lipschitz-continuous C 1 function g : U → R, such that for all u ∈ U , (1)
|g(u) − f (u)| < e(u).
For any definable open neighborhood V of D1 (f ) ∩ U , we may assume that g = f outside of V .
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3. Constructing John functions First we need to investigate some useful neighborhoods of Λ1 -regular cells. Lemma 5. Let c ∈ (0, 1). Then there is a δ > 0 such that for every 2-quasiconvex Λ1 -regular cell S ⊂ Rn of the form S = Γ(h), where h : U → Rn−d is a Lipschitz-continuous function with constant δ, V := {dS (x) < cd∂S (x) : x ∈ Rn } ⊂ U × Rn−d . Proof. The set V is connected, and ∂S ∩ V = ∅. Hence, if there is an x ∈ V outside of U × Rn−d , then there exists also an x ∈ ∂U × Rn−d contained in V . If x = (x1 , . . . , xn ), then, by shifting, we may assume that (0, . . . , 0, xn−d+1 , . . . , xn ) ∈ ∂S. Let y ∈ ∂S such that d∂S (x) = kx − yk. Note that the set S is a subset of © ¡ ¢ ª A := y + tν : kνk = 1 ∧ ^ Rν, Rd × {0} ≤ arctan(2δ), t ∈ R . By assumption, there exists a z ∈ S with kx − zk < c kx − yk. Let B denote the closed ball with center x and radius c kx − yk. Hence, z ∈ B. The ball B is contained in the closed cone generated by the lines through y that intersect with B. This cone has an apex angle α = arctan (c) which is strictly less than π/2. The tangent of the angle between the vectors x and x − y is bounded by 2δ, as U is 2-quasi-convex. So the angle between x and the axis of the cone is bounded by arctan(2δ). By choosing δ very small, say δ satisfies π 2 arctan(2δ) < − α, 2 the intersection of the cone and A is the set {y}. As y 6∈ B, the intersection B ∩ S = ∅ which contradicts z ∈ B. Thus, the lemma is proved. ¤ In the next Lemma we construct definable functions whose domain are complements of the closure of non-open Λ1 -regular cells A and which approximate the distance function dA on a sufficiently large subset of Rn . Lemma 6. Let ² > 0. Let U ⊂ Rd be an open (1 + ²/3)-quasi-convex Λ1 regular cell. Then there exists a δ > 0 such that for every definable Lipschitzcontinuous C 1 function h : U → Rn−d with constant δ, there is a Lipschitzcontinuous definable function f : Rn → [0, ∞) with the following properties: (a) (b) (c) (d) (e)
f is C 1 -smooth in U × Rn−d \ Γ(h), f = 0 on Γ(h) 1 ≤ k∇f k ≤ 1 + ² on U × Rn−d \ Γ(h), dΓ(h) ≤ f on Rn , f ≤ (1 + ²)dΓ(h) on U × Rn−d .
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Proof. We may assume that ² < 1/3 and that δ < ²/3. Define the function g : U × Rn−d → [0, ∞) by g(x, y) := ky − h(x)k ,
(x, y) ∈ U × Rn−d .
This function is definable and C 1 -smooth outside of Γ(h). Moreover, for any y 6= h(x), ° ° ° ¡ ¢° 1 °, (y − h(x))D h(x), y − h(x) k∇g(x, y)k = ° x ° ° ky − h(x)k where Dx h denotes the derivative of h with respect to x. Note that Dx h is bounded by ²/3, since h is Lipschitz-continuous with constant δ. Hence k∇gk is bounded from above by 1+²/3 so that g is locally Lipschitz-continuous with constant (1 + ²/3). As U is quasi-convex with constant 1 + ²/3, the function g is Lipschitz-continuous with constant ³ ² ´2 1+ < 1 + ². 3 Evidently, k∇gk is bounded from below by 1. As dΓ(h) (x, y) = inf{k(x, y) − (x0 , h(x0 ))k : x0 ∈ U }, the function g is bounded from below by dΓ(h) . Therefore, the definable Lipschitz-continuous function f : Rn → [0, ∞), obtained by applying the standard extension operator to g with constant K = 1 + ², is also bounded from below by dΓ(h) . Thus f satisfies property (d). The functions f and g coincide on U × Rn−d , so that the properties (a), (b) and (c) are evident by the properties of g. Finally, (e) is a consequence of (c). ¤ The previous lemma in connection with Theorem 4 is a strengthening of the Lemma 2.2 in [16] which is the key point in the proof of Theorem 1. However we are interested in an explicit bound of the required John functions which is prepared by the following lemma. Lemma 7. Let U be a definable open subset of Rn , and let ² > 0. Then there are n + 1 definable Lipschitz-continuous functions g1 , . . . , gn+1 : U → [0, ∞), and a constant M > 1, such that for each i = 1, . . . , n + 1, (a) (b) (c) (d) (e)
gi |∂U ≡ 0, gi |U is C 1 -smooth, d∂U ≤ gi ≤ (1 + ²)d∂U , Pn+1 1 ≤ j=1 k∇gj k, k∇gi k ≤ M .
Proof. Step 1: Let 1 < L < 1 + ²/(60n), and let 0 < δ < ²/20 be so small that Lemma 6
JOHN FUNCTIONS FOR o-MINIMAL DOMAINS
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holds with ²/(20n) in place of ². By Theorem 3, there is a δ-flat L-quasiconvex Λ1 -regular stratification of Rn that is compatible with ∂U . Denote by S1 , . . . , Sr the strata contained in ∂U . We may assume that, for each i = 1, . . . , r, there is a linear orthogonal automorphism of Rn mapping the stratum Si to the graph of a definable C 1 function hi : Ui → Rn−dim(Si ) which is Lipschitz-continuous with constant δ. Thus, there is a function fi satisfying the conclusion of Lemma 6 with fi , hi and Ui in place of f , h and U . We define the function Gn as follows. For any η ∈ Rn let ´ o n³ ² dim(Si ) fi (η) : i = 1, . . . , r . Gn (η) := inf 1+ 3n The functions fi are Lipschitz-continuous with constant 1 + ²/(20n), so that Gn is Lipschitz-continuous with constant ³ ² ´³ ² ´ 1+ 1+ 20n 3n which is bounded by 1 + ²/2. Step 2: Select a Λ1 -regular stratification of Rn that is compatible with the sets U, D1 (Gn ) and {fi = fj } for 1 ≤ j < i ≤ r. Denote by T1 , . . . , Ts the strata contained in U ∩ D1 (Gn ). Then, for every j, there is an i such that Gn = fi on Tj . Claim: If Gn |Tj = fi |Tj , then n ³ o ² ´ Tj ∩ U ⊂ x ∈ U : dSi (x) < 1 − d∂Si (x) . 20n The function fi is then called a matching function to Tj . Let x ∈ Tj , and assume that ³ ² ´ (2) dSi (x) ≥ 1 − d∂Si (x). 20n By Theorem 3 (b), the frontier of the stratum Si itself partitions into some of the strata. So there is a k such that Sk ⊂ ∂Si , and dSk (x) = d∂Si (x). Note that the function fk is Lipschitz-continuous with constant ³ ² ´ 1+ , 20n and that fk vanishes in Sk .
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On the one hand we obtain an upper estimate for Gn (x), ³ ´ ² (3) Gn (x) ≤ 1 + dim(Sk ) fk (x) 3n ³ ´³ ² ² ´ ≤ 1+ dim(Sk ) 1 + dS (x) 3n 20n ´ k ³ ´ ³ ² ² ≤ 1+ dim(∂Si ) 1 + d∂Si (x). 3n 20n On the other hand the value Gn (x) is bounded from below by ´ ³ ² (4) dim(Si ) fi (x) Gn (x) ≥ 1 + 3n ³ ´ ² ≥ 1+ dim(Si ) dSi (x) 3n ³ ´³ ² ´ ² ≥ 1+ (dim(∂Si ) + 1) 1 − d∂Si (x). 3n 20n As dim(∂Si ) ≤ n − 1, the inequalities (3) and (4) imply that Gn (x) < Gn (x) which is a contradiction. Thus, the claim is proved. Step 3: Construction of the functions G0 , . . . , Gn−1 . For each i select a definable open neighborhood Vi of Ti contained in n ³ o ² ´ x ∈ U : dSi (x) < 1 − d∂Si (x) 20n such that if Tj and Ti have the same dimension, then Vj ∩ Vi = ∅ for i 6= j. This is possible because of Theorem 3 (b) and Step 2. Furthermore, for each i we select a definable C 1 function ∆i : Rn → [0, ∞) which is positive in Vi and vanishes elsewhere. This function exists by [4, Theorem C.11]. Moreover, by [4, Corollary C.11], we may assume that both ∆i and k∇∆i k are bounded by ²/(20n2 ). The functions G0 , . . . , Gn−1 are constructed similar to Gn , but the functions fi must be slightly modified. Therefore, for every j, we assign to Tj a matching function. For k = 0, . . . , n − 1, we define the modified function fik by X fik := fi − ∆j where the sum is taken over all j for which dim(Tj ) = k and fi is assigned to Tj . Define Gk by n³ ´ o ² Gk (x) = inf 1+ dim(Si ) fik (x) : i = 1, . . . , r . 3n Then, whenever x 6∈ D1 (Gk ), ² 1 ≤ k∇Gk (x)k ≤ 1 + . 2
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Moreover, there exists an open definable neighborhood Wk of the union of the strata Tj of dimension k, such that Gk is C 1 -smooth in Wk . Step 4: Construction of g0 , . . . , gn−1 . Each function gk is obtained by applying Theorem 4 to Gk . We only have to make sure that there is a suitable definable neighborhood of D1 (Gk ) in which the smoothing process may change the values of Gk . Let Rk denote the union of all Tj of dimension k. We define the required open definable neighborhoods Vk of D1 (Gk ) as follows: ½ Vk :=
x ∈ U : dU \Wk (x)
dRi (x) . 3 3 i=0
For every k = 0, . . . , n − 1 we apply Theorem 4 to Gk to obtain the function gk which coincides with Gk outside of Vk . The functions g0 , . . . , gn−1 are definable Lipschitz-continuous C 1 functions on U , with a constant M > 1. Thus, their gradient-norms are all bounded from above by M . Claim: There is a definable open neighborhood W of U ∩D1 (Gn ) such that on W , 1≤
n−1 X
k∇gi k .
i=0
This neighborhood W is given by the union of the sets U \Vk , k = 0, . . . , n−1. Step 5: Construction of gn . By Theorem 4, we smooth Gn , and obtain a definable Lipschitz-continuous C 1 function gn : U → (0, ∞) which coincides with Gn outside of the set W . Verifying the properties (a)-(e) is straight forward. ¤ Proof of Theorem 2. We let ² > 0 be a small rational number, and select definable C 1 functions g1 , . . . , gn+1 : U → R that satisfy the conclusions of Lemma 7 with some M > 1. Note that the gradients of the functions gi restricted to U are bounded. Thus, the functions ( g˜i :=
(gi ) 0,
1+²
,
on U, otherwise,
are definable C 1 functions for i = 1, . . . , n + 1. On U , ∇ ln ◦g˜i = (1 + ²)∇ ln ◦gi = (1 + ²)
∇gi . gi
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Thus, (5)
n+1 X
k∇ ln ◦g˜i k =
i=1
n+1 X
(1 + ²)
i=1
≤ M (1 + ²)
k∇gi k gi
n+1 X i=1
1 gi
≤ M (1 + ²)(n + 1)
1 d∂U
.
Moreover, n+1 X
k∇ ln ◦g˜i k =
i=1
n+1 X
(1 + ²)
i=1
≥ (1 + ²)
n+1 X i=1
≥
1 d∂U
k∇gi k gi k∇gi k (1 + ²)d∂U
. ¤
Remark 1. The inequality (5) shows exactly where we loose the control for the constant C of Theorem 2. This is due to the smoothing process used in Step 4 and Step 5 of the proof of Proposition 7. Assume that, for every constant ∆ > 0, every definable Lipschitz-continuous map with constant L can be approximated in the sense of Theorem 4 by a definable Lipschitz-continuous C 1 function with constant L+∆. Then the upper constant C in Theorem 2 can be bounded by n + 2 by taking the constants δ and ² sufficiently small in Lemma 5, Lemma 6 and Lemma 7. So far, such approximation is not known to the author. 4. Regularity of the functions gi in Theorem 2 We may always assume that the functions gi restricted to U are C m -smooth for every m ∈ N. This is provided by Escribano’s Approximation Theorem, cf. [7]. For the semialgebraic structure we may even assume analycity, cf. [21]. For the class of o-minimal expansions of the real exponential field which admit C ∞ cell decomposition, see for example [3, 6, 23], we may assume C ∞ smoothness, cf. [10]. The question whether the functions gi in Theorem 2 can be chosen to be of class C m for 1 < m ≤ ∞ is more subtle. First we consider polynomially bounded o-minimal structures, that is, every definable unary function is ultimately bounded by a polynomial. For polynomially bounded o-minimal structures we obtain the following corollary of Theorem 2.
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Corollary 8. Let U be a proper definable open subset of Rn . Let m ∈ N. Then there are n + 1 definable C m functions g1 , . . . , gn+1 : Rn → R which vanish outside of U , and a constant C > 0 such that n+1 X 1 C ≤ k∇ ln(gi (x))k ≤ d∂U (x) d (x) ∂U i=1
for all x ∈ U . Proof. We may assume that every function gi is C m -smooth outside of ∂U . By [4, Proposition C.9], there exists a bijective definable C m function φi : R → R which is strictly increasing and C m -flat at 0 such that φi ◦ gi is C m -smooth. The interesting point is that the zero-order at 0 of these functions φi imply the smoothness of φ ◦ gi . Choose a natural number p such that for t > 0 small enough tp < φi (t) for all i = 1, . . . , n+1. Then the pth powers of the functions g1 , . . . , gn+1 satisfy the conclusion of the corollary. ¤ The argument in the previous proof does not hold true, when the exponential function is definable in the o-minimal structure. We even do not know if in this case we may assume that the functions gi are C 2 -smooth on Rn . In general we can exclude that the gi are C ∞ -smooth. This is evident for polynomially bounded o-minimal structures, as there the ring of definable C ∞ functions from Rn to R is quasianalytic, cf. [15]. This property implies that not even the zero-set property of definable C ∞ functions holds. We prove that even with exponential function the gi are not C ∞ . To present an example, we need the next lemma. Lemma 9. Let f : (0, δ) → (0, ∞) be a definable C 1 function such that lim f (t) = 0.
t&0
Let c > 0 be a constant. Then tf 0 (t) ≤c t&0 f (t)
(6)
lim
implies that there is an ² > 0 and a k ∈ N such that f (t) ≥ tk for t < ². Proof. We assume that inequality (6) holds. The Monotonicity Theorem, cf. [2, Theorem 3.1.2] implies, that f 0 is positive in some neighborhood of 0. Therefore, there is some δ > 0 such that 0< Thus,
Z s
δ
f 0 (t) c+1 ≤ , f (t) t
f 0 (t) dt ≤ f (t)
Z s
δ
c+1 dt, t
t < δ.
0 < s < δ.
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So ln(f (s)) ≥ ln(f (δ)) − (c + 1) ln(δ) + (c + 1) ln(s). Hence, there is a constant C > 0 such that f (s) ≥ Csc+1 . Therefore, f (t) can be bounded by tk for t > 0 close to 0 and a sufficiently large natural number k. ¤ Here is the announced example. Example 1. Let U = (0, ∞). Then there is no definable C ∞ function g : R → R which vanishes outside of U and which is positive on U such that g 0 (t) C ≤ , t > 0, g(t) d∂U (t) for some constant C > 0. Proof. This follows from the fact that t = d∂U (t) for t > 0 and Lemma 9.
¤
5. Open Question Of course one is interested in the construction of exhaustive sequences of John functions with arbitrary open domain. The bound for the number of John functions in Theorem 2 does not depend on the special geometry of the open domain. We conjecture that for any open subset of Rn there exist exhaustive sequences of John functions. References [1] J. Bochnak, M. Coste, M.-F. Roy, Real Algebraic Geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), Springer Verlag Berlin - Heidelberg 1998 [2] L. van den Dries, Tame Topology and O-minimal Structures, LMS Lecture Notes 248, Cambridge University Press 1998 [3] L. van den Dries, A. Macintyre, D. Marker, The elementary theory of restricted analytic fields with exponentiation, Ann. Math. (2) 140, No.1, 183–205 (1994). [4] L. van den Dries; C. Miller, Geometric categories and o-minimal structures, Duke Math. J. 84, 497–540 (1996) [5] L. van den Dries; P. Speissegger, The real field with convergent generalized power series, Trans. Amer. Math. Soc. 350 (1998), no. 11, 4377–4421. [6] 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. [7] J. Escribano, Approximation Theorems in o-minimal structures, Illinois J. Math. 46(1), 2002, 111–128 [8] A. Fischer, O-minimal Λm -regular stratification, Ann. Pure Appl. Logic 147 (2007), no. 1-2, 101–112. [9] A. Fischer Definable Smoothing of Lipschitz-continuous Functions, Illinois J. Math., accepted, June 2007 [10] A. Fischer Smooth functions in o-minimal structures, preprint, October 2007 [11] F. John, Functions whose gradients are bounded by the reciprocal distance from the boundary of their domain, Russian Math. Surveys 29 (1974), 170–175.
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