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A BV space function is approximately differentiable at almost every point with respect to Lebesgue measure. Its approximate gradient is the density of the absolutely continuous part of its distributional derivative. This does not imply membership in a Sobolev space.

For a sufficiently regular function on a domain $\Omega$, its boundary trace is the restriction to $\partial \Omega$, using the limit from within the domain when necessary. In a half-line evolution problem, the traces $f_j(t)={\partial }_x^{j}u(0,t)$ are boundary values of the spatial derivatives. An initial value does not generally determine every such trace without the partial differential equation and suitable boundary conditions. The Sobolev trace operator extends classical restriction continuously to appropriate Sobolev spaces, where pointwise values need not be defined.

The distributional derivative of this indicator function is $e_1{\mathcal{H}}^1$ on the vertical segment $x_1=1/2$, so its total variation seminorm on a domain is one. This is singular with respect to planar Lebesgue measure, whereas Sobolev space weak derivatives in $W^{1,1}$ have integrable densities. Its bounded-variation space norm is $1/2+1=3/2$.

This first-order operator vanishes exactly on J-holomorphic curves. Linearization at a map $u$ gives a Fredholm operator between suitable Sobolev spaces of sections. At a constant map it is the Dolbeault operator on the trivial pulled-back tangent bundle.

A compact embedding of Banach spaces $X,Y$ is an injective continuous linear map $i:X\to Y$ that sends every bounded set to a relatively compact subset. Equivalently, every bounded sequence in $X$ has a subsequence converging in $Y$. One identifies $X$ with its image when $i$ is the natural inclusion. The Rellich-Kondrachov theorem supplies many compact embeddings between Sobolev spaces and Lp spaces on bounded domains.

For arbitrary real $s,t$, the convolution of $u\in H^s(\mathbb{R})$ and $v\in H^t(\mathbb{R})$ has a canonical definition as a tempered distribution. With $⟨\xi ⟩=(1+{\xi }^2{)}^{1/2}$, write $A=⟨\xi {⟩}^s\hat{u}$ and $B=⟨\xi {⟩}^t\hat{v}$, both in the L2 space. Then $AB\in L^1$ by Cauchy-Schwarz inequality, andThus the product defines a tempered distribution, and the displayed inverse Fourier transform defines $u\ast v$. This extends ordinary convolution of Schwartz functions by the convolution theorem and density in the Sobolev norm.

The result need not belong to any Sobolev space, even when both inputs belong to every Sobolev space. For $\hat{u}=\hat{v}={\xi }^{-1/3}{1}_{(0,1)}(\xi )$, each weighted squared input is integrable, but the squared product is ${\xi }^{-4/3}{1}_{(0,1)}(\xi )$, which is not locally integrable at zero. Every weight $(1+{\xi }^2{)}^r$ is bounded above and below there, proving the failure for all real $r$.

A discrete Sobolev norm combines weighted norms of nodal values and their finite differences. For a real sequence $W_0,\dots ,W_N$ with zero endpoints, the displayed expression is a norm: it is the Euclidean norm of one vector containing both weighted lists, and zero norm forces all nodal values to vanish. It approximates the Sobolev space $H^1$ norm. Endpoint weights or prescribed zero endpoints must be specified when calling a discrete expression a norm.

An elliptic boundary value problem couples an elliptic differential equation in a domain to boundary conditions. Its weak formulation is often an equation for a bounded bilinear form on a Sobolev space.

On a two-dimensional domain containing the origin, a logarithmic singularity such as $u(r)=(\log (1/r){)}^{\alpha }$ with $0<\alpha <1/2$ belongs locally to $H^1$ but is unbounded. Thus the critical first-order Sobolev space does not embed into $L^{\infty }$.

For the unit ball and energy $W(F)=|F{|}^2+(\det F-1{)}^2$, boundary dilation $y=tx$ admits a singular incompressible radial competitor $r(R)=(R^3+t^3-1{)}^{1/3}$. Its radial stretch is $R^2/r^2$ and its double tangential stretch is $r/R$. Although the latter diverges at zero, the three-dimensional volume element makes its squared gradient integrable. Its energy is $4\pi {\int }_0^1[R^6/r^4+2r^2],dR\sim 8\pi t^2$.

The smooth homogeneous equilibrium $y=tx$ has constant stress and energy $(4\pi /3)[3t^2+(t^3-1{)}^2]$, of sixth-order growth. It is therefore beaten for large $t$ in the finite-energy Sobolev space class admitting the cavity. This conclusion depends on that class: the singular map is not continuous at the center and cannot simply be substituted as a smooth cavity-free competitor.

For a fixed sufficiently regular image edge set, the Mumford–Shah functional is a strictly convex quadratic energy on the Sobolev space of its complement. A first variation gives $u-\alpha \Delta u=g$ within each region and separate one-sided Neumann boundary conditions on free edges. A jump need not vanish across an edge. The weak solution is unique by the Lax-Milgram theorem; this does not prove uniqueness when the edge set is also optimized.

For the positive Dirichlet Laplacian $A$ on a bounded smooth domain, with normalized eigenfunctions $e_j$ and positive eigenvalues ${\mu }_j$, define $A^{\alpha }u={\sum }_j{\mu }_j^{\alpha }(u,e_j)e_j$ on the vectors for which ${\sum }_j{\mu }_j^{2\alpha }|(u,e_j){|}^2<\infty$. The resulting Hilbert space embeds in the Sobolev space $H^{2\alpha }$, with the boundary conditions appropriate to the exponent. In particular $H_{\alpha }$ embeds in $C^{0,\beta }$ whenever $0<\beta <\min (1,2\alpha -n/2)$. High powers impose further boundary compatibility; ordinary smoothness alone does not imply membership in every fractional domain.

A function space is a collection of functions with common domain and codomain, often supplied with a vector space structure and a norm or topology. Examples include L2 spaces and Sobolev spaces.

A grey-value image signal is a scalar function on an image signal domain or an array of pixel values. Continuous image processing models use spaces such as $L^2$, Sobolev spaces or the BV space, while discrete models use finite-dimensional arrays. The chosen representation distinguishes smooth transitions, noise and image edges.

The Japanese bracket is a smooth positive frequency weight, comparable to $1+|\xi |$. It defines Sobolev space norms and the growth estimates of a symbol class without singularities at zero frequency.

A Lipschitz domain is an open connected subset of Euclidean space whose boundary is locally the graph of a Lipschitz continuous function after rotating and translating coordinates. Locally the domain lies on one side of that graph. This permits corners but excludes boundary cusps that cannot be represented with a finite Lipschitz constant. Bounded Lipschitz domains are a standard setting for Sobolev spaces and the Poincaré inequality for total variation.

For a cube $Q$ of side length $r$ in ${\mathbb{R}}^n$, $p>n$, and a continuously differentiable function $v$, its average $v_Q=|Q{|}^{-1}{\int }_{Q}v$ satisfiesAveraging the fundamental theorem of calculus along a line segment from $y$ to $z\in Q$ and changing variables gives $|v(y)-v_Q|\le C_n{\int }_Q|Dv(w)||w-y{|}^{1-n}dw$. The Holder inequality applies because $(n-1)p^{\prime }<n$, and the kernel norm is $O(r^{1-n/p})$. Approximation extends the estimate to the continuous representative of a Sobolev space function. On ${\mathbb{R}}^n$ it yields a Hölder continuous function of exponent $1-n/p$ representing every $W^{1,p}$ class.

For $d<p<\infty$, every Sobolev space element $u\in W^{1,p}({\mathbb{R}}^d)$ has a Hölder continuous function representative of exponent $1-d/p$, with the displayed bound. It also satisfies $\Vert u{\Vert }_{\infty }\le C_{d,p}(\Vert u{\Vert }_p+\Vert \nabla u{\Vert }_p)$. Averaging the fundamental theorem of calculus along a line segment over a ball givesThe Holder inequality bounds this by $C_{d,p}{r}^{1-d/p}\Vert \nabla u{\Vert }_p$, because $(d-1)p^{\prime }<d$. To compare two ball averages, translate a ball along the segment between their centres and use the same fundamental theorem of calculus along a line segment. These bounds give the displayed estimate. Density of smooth functions in a Sobolev space supplies the representative for nonsmooth $u$. For $p=\infty$, the corresponding conclusion is Lipschitz continuity.

A compactly supported distribution of order of a distribution at most $M$ has a smooth function as its Fourier transform, bounded by $C⟨\xi {⟩}^M$. It therefore belongs to every Sobolev space $H^s$ with $s<-M-n/2$.

Suppose an invertible real matrix $C$ intertwines symmetric orthogonal tile-reflection matrices: $C{M}_s=N_{s}C$. Then $C^{T}C$ commutes with every $M_s$, so its positive inverse square root does too. The polar decomposition of an invertible real matrix therefore gives an orthogonal matrix $O=C(C^{T}C{)}^{-1/2}$ satisfying $O{M}_s=N_{s}O$. On a disjoint union of congruent tiles, $O$ preserves both the $L^2$ inner product and Dirichlet energy. The intertwining identities preserve the face trace conditions defining the assembled Sobolev space, so $O$ gives a unitary equivalence of the assembled Laplacians.