3.1 Lipschitz functions and their properties

Lipschitz functions are key players in geometric measure theory. They're special because they don't stretch distances too much, making them useful for studying shapes and measures in complex spaces.

These functions have cool properties that link them to other important concepts. They're always continuous, often differentiable, and help us understand things like rectifiable sets and Hausdorff dimension.

Lipschitz Functions: Definition and Properties

Definition and Lipschitz Constant

• A function $f: X \to Y$ between metric spaces $(X, d_X)$ and $(Y, d_Y)$ is Lipschitz continuous if there exists a real constant $K \geq 0$ such that for all $x_1, x_2 \in X$, $d_Y(f(x_1), f(x_2)) \leq K d_X(x_1, x_2)$
  • The smallest such $K$ is the Lipschitz constant of $f$, denoted as $\text{Lip}(f)$
  • Examples: $f(x) = 2x$ is Lipschitz continuous with $\text{Lip}(f) = 2$, and $f(x) = \sin(x)$ is Lipschitz continuous with $\text{Lip}(f) = 1$

Continuity and Differentiability Properties

• Lipschitz functions are uniformly continuous
  • For every $\varepsilon > 0$, there exists a $\delta > 0$ such that for all $x, y \in X$ with $d_X(x, y) < \delta$, we have $d_Y(f(x), f(y)) < \varepsilon$
  • This is a stronger form of continuity than regular continuity
• Lipschitz functions are absolutely continuous
  • They are differentiable almost everywhere and can be represented as an integral of their derivative
  • Example: $f(x) = |x|$ is Lipschitz continuous and absolutely continuous, but not everywhere differentiable

Algebraic Properties

• The composition of Lipschitz functions is also Lipschitz
  • If $f: X \to Y$ and $g: Y \to Z$ are Lipschitz continuous with constants $K_1$ and $K_2$, then $g \circ f: X \to Z$ is Lipschitz continuous with constant $K_1K_2$
• Lipschitz functions form a vector space
  • They are closed under addition, subtraction, and scalar multiplication
  • If $f, g: X \to Y$ are Lipschitz continuous with constants $K_1$ and $K_2$, then $af + bg$ is Lipschitz continuous with constant $|a|K_1 + |b|K_2$ for any $a, b \in \mathbb{R}$

Fundamental Theorems of Lipschitz Functions

Rademacher's Theorem

• A Lipschitz function $f: \mathbb{R}^n \to \mathbb{R}^m$ is differentiable almost everywhere with respect to the Lebesgue measure
  • This means the set of points where $f$ is not differentiable has Lebesgue measure zero
  • Example: $f(x) = |x|$ is Lipschitz continuous but not differentiable at $x = 0$, which is a set of Lebesgue measure zero

Kirszbraun Theorem (Lipschitz Extension Theorem)

• If $U \subset \mathbb{R}^n$ and $f: U \to \mathbb{R}^m$ is Lipschitz continuous, then there exists a Lipschitz continuous function $F: \mathbb{R}^n \to \mathbb{R}^m$ such that $F|_U = f$ and $\text{Lip}(F) = \text{Lip}(f)$
  • This theorem allows extending a Lipschitz function defined on a subset to the entire space while preserving the Lipschitz constant
  • Example: If $f: [0, 1] \to \mathbb{R}$ is Lipschitz continuous, it can be extended to a Lipschitz continuous function on $\mathbb{R}$

Stepanov's Theorem

• If $f: [a, b] \to \mathbb{R}$ is Lipschitz continuous, then $f$ is absolutely continuous, and $f'(x)$ exists for almost every $x \in [a, b]$
  • This theorem connects Lipschitz continuity with absolute continuity and differentiability
  • Example: $f(x) = |x|$ is Lipschitz continuous and absolutely continuous on $[a, b]$, and $f'(x)$ exists for all $x \neq 0$

Whitney Extension Theorem

• Given a closed set $E \subset \mathbb{R}^n$ and a function $f: E \to \mathbb{R}$, there exists a function $F \in C^\infty(\mathbb{R}^n)$ such that $F|_E = f$ and $F$ is Lipschitz continuous if and only if $f$ satisfies a certain compatibility condition involving its jets (Taylor polynomials)
  • This theorem characterizes the extendability of a function to a smooth Lipschitz function on the entire space
  • Example: If $E = \{0, 1\}$ and $f(0) = 0, f(1) = 1$, then $f$ can be extended to a smooth Lipschitz function on $\mathbb{R}$, such as $F(x) = x$

Lipschitz Functions vs Other Function Classes

Lipschitz Functions and Uniform Continuity

• Every Lipschitz function is uniformly continuous, but the converse is not true
  • Example: $f(x) = \sqrt{x}$ is uniformly continuous on $[0, \infty)$ but not Lipschitz continuous
  • Lipschitz continuity is a stronger condition than uniform continuity

Lipschitz Functions and Absolute Continuity

• Every Lipschitz function is absolutely continuous, but the converse is not true
  • Example: $f(x) = x^2 \sin(1/x)$ for $x \neq 0$ and $f(0) = 0$ is absolutely continuous but not Lipschitz continuous
  • Lipschitz continuity implies absolute continuity, but not vice versa

Lipschitz Functions and Hölder Continuity

• Lipschitz functions are a proper subset of Hölder continuous functions
  • A function $f$ is Hölder continuous with exponent $\alpha \in (0, 1]$ if there exists a constant $C > 0$ such that $|f(x) - f(y)| \leq C|x - y|^\alpha$ for all $x, y$ in the domain
  • Lipschitz functions correspond to the case $\alpha = 1$
  • Example: $f(x) = \sqrt{x}$ is Hölder continuous with $\alpha = 1/2$ but not Lipschitz continuous

Lipschitz Functions and Functions of Bounded Variation

• Lipschitz functions are closely related to functions of bounded variation
  • A function $f: [a, b] \to \mathbb{R}$ is of bounded variation if and only if it can be expressed as the difference of two increasing Lipschitz functions
  • Example: $f(x) = x\sin(1/x)$ for $x \neq 0$ and $f(0) = 0$ is of bounded variation but not Lipschitz continuous

Applications of Lipschitz Functions in Geometric Measure Theory

Hausdorff Measure and Dimension

• Lipschitz functions are used to define the Hausdorff measure and dimension of sets in metric spaces
  • The Hausdorff dimension of a set $E$ is the infimum of all $s > 0$ such that the $s$-dimensional Hausdorff measure of $E$ is zero
  • Lipschitz functions preserve Hausdorff dimension, i.e., if $f: X \to Y$ is Lipschitz continuous and $E \subset X$, then $\dim_H(f(E)) \leq \dim_H(E)$

Rectifiable Sets

• Lipschitz functions play a crucial role in the theory of rectifiable sets
  • A set $E \subset \mathbb{R}^n$ is countably $k$-rectifiable if it can be covered, up to a set of $\mathcal{H}^k$-measure zero, by a countable union of Lipschitz images of subsets of $\mathbb{R}^k$
  • Example: A smooth curve in $\mathbb{R}^n$ is countably 1-rectifiable, as it can be covered by a countable union of Lipschitz images of intervals

Tangent Spaces and Approximate Tangent Spaces

• Lipschitz functions are used to prove the existence of tangent spaces and approximate tangent spaces for rectifiable sets
  • The tangent space of a rectifiable set $E$ at a point $x$ is the unique $k$-dimensional subspace $T_x E$ such that the Hausdorff distance between $E$ and $x + T_x E$ goes to zero as we zoom in around $x$
  • Approximate tangent spaces are a generalization of tangent spaces that exist almost everywhere for rectifiable sets

Area and Coarea Formulas

• The area and coarea formulas for Lipschitz functions relate the Hausdorff measures of sets and their images under Lipschitz mappings
  • The area formula states that for a Lipschitz function $f: \mathbb{R}^n \to \mathbb{R}^m$ and a measurable set $A \subset \mathbb{R}^n$, $\mathcal{H}^n(f(A)) = \int_A J_f(x) d\mathcal{H}^n(x)$, where $J_f(x)$ is the Jacobian of $f$ at $x$
  • The coarea formula is a generalization of the area formula that relates the integrals of a function over a set and its level sets

Currents

• Lipschitz functions are used in the study of currents, which are generalized surfaces in geometric measure theory
  • The boundary of a current is defined using the pushforward of Lipschitz functions
  • The mass of a current is defined using the Lipschitz constant of the defining function
  • Example: A smooth oriented submanifold with boundary can be represented as a current, and its boundary is the current defined by the boundary of the submanifold