On the Gradient Inequality
by
S. L
à ojasiewicz and M.-A. Zurro
Summary. Some proof of the Gradient Inequality is given.
K. Kurdyka and A. Parusiński in [1] have done a very elegant proof of the
Gradient Inequeliy. The aim of this note is to give another some other
version of their proof. It is rather conceptual. Let us recall this inequality :
Let f be an analytic function in a neighborhood of 0 ∈ Rn such that f (0) = 0,
then we have the inequality:
|f (x)|ϑ 6 |grad f (x)|
,
with
0 < ϑ < 1,
in a neighborhood of 0.
As a consequence of this proof, we obtain a qualitative statement which is
easily equivalent to the above inequality (see the remark 1).
Proof. We can assume that grad f (0) = 0 (the remaining case being trivial).
Let us consider the mapping
g : B 37−→ ( |f (x)| , |grad f (x)| ) ∈ R2uv ,
with B a compact ball sufficientely small and centered at 0. Consider the set
E = g(B) ⊂ [0, ∞) × [0, ∞) ; it is compact and subanalytic. It is enough
to prove that for every L > 0 the point 0 does not belong to the closure
of the set F = E ∩ {0 6 v < Lu} . In fact, the function ϕ(u) = inf Eu
for small u > 0 is bouded semianalytic, hence then we must have ϕ(u) =
0
0
a + cuϑ + o (uϑ ) with 0 < ϑ0 < 1 . It follows that some neighborhood of
0 in E is contained in {v > uϑ } with some 0 < ϑ < 1 and this implies
the Gradient Inequality.
Let us suppose the contrary i.e. that with some L > 0 we have 0 ∈ F . Then
the set g −1 (F ) ∩ g −1 (0) is non empty and so it contains a point b. By the
curve sellecting lemma, there is a semianalytic arc {x = γ(t) : 0 < t 6 ε}
contained in g −1 (F ) with γ analytic in some neighborhood of 0 and such
that γ(0) = b . Then we have g ◦ γ ⊂ {0 6 v < u} , and therefore for the
function h = f ◦ γ we have the inequalities
(1)
|h0 (t)| 6 M |grad f (γ(t))| 6 M L|h(t)| ,
with a constant M . But h(0) = 0 and h 6≡ 0, hence h(t) = ak tk + . . .
with ak 6= 0 , k > 1 , and so h0 (t) = kak tk−1 + . . . , which contradicts the
inequality (1).
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The argument of the above proof implies the following :
Remark 1. The gradient inequality (for any f 6≡ 0) is equivalent to each
of the following qualitative statements:
(A) Let F be an analytic function in a neighborhood of 0 ∈ Rn , such that
F 6≡ 0 and F (0) = 0. Then
|F (x)|
−→ 0
| grad F (x) |
for x → 0 , grad F (x) 6= 0 .
(B) Let F be an analytic function in a neighborhood of 0 ∈ Rn , such
that F 6≡ 0 and F (0) = 0. For any analytic function x(t) ∈ Rn in a
neighborhood of 0 ∈ R , such that x(0) = 0 and grad F (x(t)) 6≡ 0 , we
have
|F (x(t))|
−→ 0
for t → 0 .
| grad F (x(t)) |
In fact, the gradient inequality implies of course the statement (A) which implies the statement (B).
Assume now the statement (B) with
grad f (x) = 0 (for any function F). We have the semianalytic function
0
0
ϕ(u) = inf Eu = cuϑ + o(uϑ ) for small u > 0 , with c > 0 and some
ϑ0 > 0 . Then the set g −1 (ϕ) ∩ g −1 (0) is non empty and so it contains a point b. By the curve sellecting lemma, there is a semianalytic arc
{x = ξ(t) : 0 < t 6 ε} contained in g −1 (ϕ) with ξ(t) analytic in some
neighborhood
of¢ 0 and such that ξ(0) = b . As g(ξ(t)) ∈ ϕ for small t > 0 ,
¡
(i.e. ϕ |f (ξ(t))| = |grad f (ξ(t))|) the statement for F (x) = f (x + b) and
x(t) = z(t) − b implies that we must have ϑ0 < 1 . This implies the gradient
inequality with some ϑ < ϑ0 .
Obsere that (A) can be easily shown by proving by courve sellecting lemma
that for every ε > 0 zero can not be an adherent point of the set {x :
ε|grad f (x)| < |f (x)|} .
Observe that the equivalence of both statements is a particular case of the
following fact :
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Proposition. Let g : M → N be a subanalytic mapping of real analytic
manifolds such the image g(M ) is relatively compact. Let a ∈ M b ∈ N .
Then g(x) → b for x → a+ iff for any analytic function x(t) in a neighborhood of 0 ∈ R and such that x(0) = a we have g(x(t)) −→ b for t → 0 + .
One shows the sufficiency of the condition in the following way.
It is
enaugh to consider the case where M is a neighborhood of 0 in Rn ,
N = R , a = 0 , b = 0 and g is bounded and 6≡ 0 . One considers the
function γ(r) = sup |x|=r |g(x)| and the set {x : |g(x)| > γ(|x|) − |x|} . It
is subanalytic, and so by curve sellecting lemma it contains a semianalytic
arc {x(t) : t > 0} where x(t) is analytic
¡ in¢a neighborhood of 0 in R and
such that x(0) = 0 . It implies that γ x(t) → 0 for t → 0+ .
Observe that if we do not assume that the image is relatively compact, the
proposition does not hold. One checks it on the following exemple : one
takes the caracteristic function ϕn of the set {0 <P
x < 1/n , 0 < y < xn }
2
in R , n = 1, 2, . . . , and then one puts g(x) =
ϕn .
It is easy to modify the ϕn ’s in order to have g continuous except the
origine.
Remark 2. The given proof works for the Kurdyka - Parusiński generalization of the Gradient Inequality (see [1]) : one suppose that f : G −→ R
(with an open G) is analytic, subanalytic and such that 0 ∈ f ; then the
inequality holds in a set {|x| < δ, |f (x)| < δ} with some δ > 0 .
StanisÃlaw L
à ojasiewicz, Instytut Matematyki Uniwersytetu
Jagiellońskiego, Reymonta 4, 30059 Kraków, Poland.
Marı́a Angeles Zurro Moro, Departamento de Matemáticas,
Universidad Autónoma de Madrid, Canto Blanco, 28106 Madrid,
Spain.
References
[1]
Kurdyka K., Parusiński A. , wf -stratification of subanalytic functions
and the L
à ojasiewicz inequality , C.R.Acad.Sci.Paris, I, 318 (1994) .
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[2]
L
à ojasiewicz S. , Ensembles semi-analytiques , Preprint IHES (1965) .
[3]
L
à ojasiewicz S., Zurro M. A. , Una introducción a la geometrı́a Semiy Sub- analı́tica , Publicaciones de la Universidad de Valladolid
(Spain) (1993) .
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