Lipschitz condition

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

Definition[edit]

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[edit]

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[edit]

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[edit]

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[edit]