View source for Lipschitz condition

{{MSC|54E40}}
[[Category:Analysis]]
{{TEX|done}}

====Definition====
The term is used for a bound on the [[Continuity, modulus of|modulus of continuity]] a function. In particular, a function $f:[a,b]\to \mathbb R$ is said to satisfy the Lipschitz condition if there is a constant $M$ such that
\begin{equation}\label{eq:1}
|f(x)-f(x')| \leq M|x-x'|\qquad \forall x,x'\in [a,b]\, .
\end{equation}
The smallest constant $M$ satisfying \eqref{eq:1} is called [[Lipschitz constant]]. The condition has an obvious generalization to vector-valued maps defined on any subset of the euclidean space $\mathbb R^n$: indeed it can be easily extended to maps between metric spaces (see [[Lipschitz function]]). 

====Historical remarks====
The condition was first considered by Lipschitz in {{Cite|Li}} in his study of the convergence of the Fourier series of a periodic function $f$. More precisely, it is shown in {{Cite|Li}} that, if a periodic function $f:\mathbb R \to \mathbb R$ satisfies the inequality
\begin{equation}\label{eq:2}
|f(x)-f(x')|\leq M |x-x'|^\alpha \qquad \forall x,x'\in \mathbb R
\end{equation}
(where $0<\alpha\leq 1$ and $M$ are fixed constants), then the Fourier series of $f$ converges everywhere to the value of $f$. This conclusion can be derived, for instance, from the [[Dini-Lipschitz criterion]] and the convergence is indeed uniform. For this reason some authors (especially in the past) use the term Lipschitz condition for the weaker inequality \eqref{eq:2}. However, the most common terminology for such condition is [[Hölder condition]] with Hölder exponent $\alpha$.

====Properties====
Every function that satisfies \eqref{eq:2} is [[Uniform continuity|uniformly continuous]]. Lipschitz functions of one real variable are, in addition, [[Absolute continuity|absolutely continuous]]; however such property is in general false for Hölder functions with exponent $\alpha<1$. Lipschitz functions on Euclidean sets are almost everywhere differentiable (cf. [[Rademacher theorem]]; again this property does not hold for general Hölder functions). By the mean value theorem, any function $f:[a,b]\to \mathbb R$ which is everywhere differentiable and has bounded derivative is a Lipschitz function. In fact it can be easily seen that in this case the Lipschitz constant of $f$ equals 
\[
\sup_x |f'(x)|\, .
\] 
The statement can generalized to differentiable functions on convex subsets of $\mathbb R^n$.

If we denote by $\omega(\delta,f)$ the modulus of continuity of a function $f$, namely the quantity
\[
\omega (\delta, f) = \sup_{|x-y|\leq \delta} |f(x)-f(y)|\, ,
\]
then \eqref{eq:2} can be restated as the inequality $\omega (\delta, f) \leq M \delta^\alpha$.

====Function spaces====
Consider $\Omega\subset \mathbb R^n$. It is common to endow the space of Lipschitz functions on $\Omega$, often denoted by ${\rm Lip}\, (\Omega)$ with the seminorm
\[
[f]_1 := \sup_{x\neq y} \frac{|f(x)-f(y)|}{|x-y|}\, ,
\]
which is just the Lipschitz constant of $f$. Similarly, for functions as in \eqref{eq:2} it is customary to define the Hölder seminorm
\[
[f]_\alpha := \sup_{x\neq y} \frac{|f(x)-f(y)|}{|x-y|^\alpha}\, .
\]
If the functions in question are also bounded, one can define the norm $\|f\|_{0, \alpha} = \sup_x |f(x)| + [f]_\alpha$. The corresponding normed vector spaces are [[Banach space|Banach spaces]], usually denoted by $C^{0,\alpha} (\Omega)$, which are just particular examples of [[Hölder space|Hölder spaces]]. For $C^{0,1}$ some authors also use the notation ${\rm Lip}_b$. Under appropriate assumptions on the domain $\Omega$, $C^{0,\alpha} (\Omega)$ coincides with the [[Sobolev space|Sobolev spaces]] $W^{\alpha, \infty} (\Omega)$.

====References====
{|
|-
|valign="top"|{{Ref|Ad}}|| R. A. Adams, J. J. F. Fournier, "Sobolev Spaces", Academic Press, 2nd edition, 2003
|-
|valign="top"|{{Ref|EG}}|| L.C. Evans, R.F. Gariepy, "Measure theory  and fine properties of     functions" Studies in Advanced Mathematics. CRC  Press, Boca Raton, FL,      1992. {{MR|1158660}} {{ZBL|0804.2800}} 
|-
|valign="top"|{{Ref|GT}}|| D. Gilbarg,   N.S.  Trudinger,   "Elliptic partial differential equations of second order" ,  Springer  (1983)
|-
|valign="top"|{{Ref|Li}}|| R. Lipschitz,   "De explicatione per series trigonometricas insttuenda functionum unius variablis arbitrariarum, et praecipue earum, quae per variablis spatium finitum valorum maximorum et minimorum numerum habent infintum disquisitio"  ''J. Reine Angew. Math.'' , '''63'''  (1864)  pp. 296–308  {{MR|}} {{ZBL|}} 
|-
|valign="top"|{{Ref|Na}}|| I.P. Natanson,   "Constructive function theory" , '''1–3''' , F. Ungar   (1964–1965)  (Translated from Russian)  {{MR|1868029}} {{MR|0196342}}  {{MR|0196341}} {{MR|0196340}} {{ZBL|1034.01022}} {{ZBL|0178.39701}}  {{ZBL|0136.36302}} {{ZBL|0133.31101}} 
|-
|valign="top"|{{Ref|Zy }}||A. Zygmund,   "Trigonometric series" , '''1–2''' , Cambridge Univ. Press  (1988)  {{MR|0933759}} {{ZBL|0628.42001}} 
|-
|}