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[[Category:Analysis]]
 
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A restriction on the behaviour of increase of a function. If for any points $x$ and $x'$ belonging to an interval $[a,b]$ the increase of a function $f$ satisfies the inequality
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====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}
 
\begin{equation}\label{eq:1}
|f(x)-f(x')| \leq M|x-x'|^{\alpha},
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|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]]).
 +
 
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====Historical remarks====
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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}
 
\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 $\alpha$.
  
where $0<\alpha\leq1$ and $M$ is a constant, then one says that $f$ satisfies a Lipschitz condition of order $\alpha$ on $[a,b]$ and writes $f\in\operatorname{Lip}\alpha$, $f\in\operatorname{Lip}_M\alpha$ or $f\in H^{\alpha}(M)$. Every function that satisfies a Lipschitz condition with some $\alpha>0$ on $[a,b]$ is uniformly continuous on $[a,b]$, and functions that satisfy a Lipschitz condition of order $\alpha=1$ are absolutely continuous (cf. [[Absolute continuity|Absolute continuity]]; [[Uniform continuity|Uniform continuity]]). A function that has a bounded derivative on $[a,b]$ satisfies a Lipschitz condition on $[a,b]$ with any $\alpha\leq 1$.
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====Properties====
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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
 +
\[
 +
\sup_x |f'(x)|\, .
 +
\]  
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The statement can generalized to differentiable functions on convex subsets of $\mathbb R^n$.
  
The Lipschitz condition \eqref{eq:1} is equivalent to the condition
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If we denote by $\omega(\delta,f)$ the modulus of continuity of a function $f$, namely the quantity
\begin{equation}
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\[
\omega(\delta,f)\leq M\delta^{\alpha},
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\omega (\delta, f) = \sup_{|x-y|\leq \delta} |f(x)-f(y)|\, ,
\end{equation}
+
\]
 +
then \eqref{eq:2} can be restated as the inequality $\omega (\delta, f) \leq M \delta^\alpha$.
  
where $\omega(\delta,f)$ is the modulus of continuity (cf. [[Continuity, modulus of|Continuity, modulus of]]) of $f$ on $[a,b]$. Lipschitz conditions were first considered by R. Lipschitz [[#References|[1]]] as a sufficient condition for the convergence of the [[Fourier series|Fourier series]] of $f$. In the case $0<\alpha<1$ the condition \eqref{eq:1} is also called a [[Hölder condition|Hölder condition]] of order $\alpha$.
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====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====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> 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|}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Zygmund,  "Trigonometric series" , '''1–2''' , Cambridge Univ. Press  (1988)  {{MR|0933759}} {{ZBL|0628.42001}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  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}} </TD></TR></table>
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{|
 +
|-
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|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}}
 +
|-
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|}

Latest revision as of 09:19, 17 June 2014

2020 Mathematics Subject Classification: Primary: 54E40 [MSN][ZBL]

Definition

The term is used for a bound on the 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 [Li] in his study of the convergence of the Fourier series of a periodic function $f$. More precisely, it is shown in [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 uniformly continuous. Lipschitz functions of one real variable are, in addition, 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 spaces, usually denoted by $C^{0,\alpha} (\Omega)$, which are just particular examples of 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 spaces $W^{\alpha, \infty} (\Omega)$.

References

[Ad] R. A. Adams, J. J. F. Fournier, "Sobolev Spaces", Academic Press, 2nd edition, 2003
[EG] L.C. Evans, R.F. Gariepy, "Measure theory and fine properties of functions" Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992. MR1158660 Zbl 0804.2800
[GT] D. Gilbarg, N.S. Trudinger, "Elliptic partial differential equations of second order" , Springer (1983)
[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
[Na] I.P. Natanson, "Constructive function theory" , 1–3 , F. Ungar (1964–1965) (Translated from Russian) MR1868029 MR0196342 MR0196341 MR0196340 Zbl 1034.01022 Zbl 0178.39701 Zbl 0136.36302 Zbl 0133.31101
[Zy ] A. Zygmund, "Trigonometric series" , 1–2 , Cambridge Univ. Press (1988) MR0933759 Zbl 0628.42001
How to Cite This Entry:
Lipschitz condition. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lipschitz_condition&oldid=28839
This article was adapted from an original article by A.V. Efimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article