Lipschitz condition

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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   in his study of the convergence of the Fourier series of a periodic function $f$. More precisely, it is shown in   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

	R. A. Adams, J. J. F. Fournier, "Sobolev Spaces", Academic Press, 2nd edition, 2003
	L.C. Evans, R.F. Gariepy, "Measure theory and fine properties of functions" Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992. MR1158660 Template:ZBL
	D. Gilbarg, N.S. Trudinger, "Elliptic partial differential equations of second order" , Springer (1983)
	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 Template:ZBL
	I.P. Natanson, "Constructive function theory" , 1–3 , F. Ungar (1964–1965) (Translated from Russian) MR1868029 MR0196342 MR0196341 MR0196340 Template:ZBL Template:ZBL Template:ZBL Template:ZBL
	A. Zygmund, "Trigonometric series" , 1–2 , Cambridge Univ. Press (1988) MR0933759 Template:ZBL