Sobolev space

For real $s$, the $L^2$-based Sobolev space $H^s({\mathbb{R}}^n)$ consists of tempered distributions satisfying

This is the closure of the compactly supported test functions $C_c^{\infty }(U)$ in the $H^2(U)$ norm. On a smooth bounded domain it consists precisely of the $H^2$ functions with zero value and zero normal derivative traces. Tangential first derivatives then vanish as well. The clamped Hessian identity makes the Laplacian norm equivalent to the full $H^2$ norm on this space, enabling a Lax-Milgram theorem treatment of the clamped biharmonic problem.

For $u\in H_0^2(U)$, ${\sum }_{i,j}\Vert {\partial }_{ij}u{\Vert }_2^2=\Vert \Delta u{\Vert }_2^2$. Two integrations by parts prove this for compactly supported test functions, and $H^2$ density extends it to the clamped second-order Sobolev space. Together with the Poincare-Wirtinger inequality applied to the mean-zero first derivatives and the zero-boundary Poincare inequality, it controls the full $H^2$ norm by $\Vert \Delta u{\Vert }_2$. The boundary conditions are essential to this exact identity on bounded domains.

A homogeneous Sobolev space controls derivatives without an independent low-frequency L2 norm: formally $\Vert f{\Vert }_{{\dot{H}}^s}^2=\int |\xi {|}^{2s}|\hat{f}(\xi ){|}^{2}d\xi$, with the normalization chosen according to the Fourier transform. Completions or quotient conventions must be specified when polynomial ambiguities occur. In three dimensions ${\dot{H}}^1$ is represented by $L^6$ functions with gradient in $L^2$, and $\Vert f{\Vert }_6\le C\Vert \nabla f{\Vert }_2$.

The first Gaussian Sobolev space consists of functions and their weak first derivatives square-integrable for standard Gaussian measure. If $f=\sum c_n{h}_n$ in the normalized Probabilists' Hermite polynomial basis, Gaussian integration by parts gives $⟨f^{\prime },h_{n-1}⟩=\sqrt{n}{c}_n$. Thus Parseval identity identifies it with $\sum (1+n)|c_n{|}^2<\infty$, and polynomial truncations converge in its norm. It is the form domain of the Gaussian number operator, with Gaussian Dirichlet energy $\sum n|c_n{|}^2$.

For $1\le p<\infty$, an inhomogeneous Sobolev space norm is $\Vert u{\Vert }_{W^{k,p}(D)}=({\sum }_{|\alpha |\le k}\Vert D^{\alpha }u{\Vert }_{L^p(D)}^p{)}^{1/p}$, using weak derivatives. In particular, $\Vert u{\Vert }_{H^1(D)}^2={\int }_D(|u{|}^2+|\nabla u{|}^2)$. The gradient seminorm alone defines a different completion, the Dirichlet energy space, unless a suitable Poincare inequality makes the two norms equivalent on the chosen zero-boundary space.

Multiplication by a test function is bounded on every real-index Sobolev space. Its Fourier transform acts by convolution with a Schwartz function; frequency-weight estimates and Young's convolution inequality give the norm bound.

The continuous dual space of $H^s({\mathbb{R}}^d)$ is $H^{-s}({\mathbb{R}}^d)$ under the extension of the L2 inner product. The Fourier transform definition $\Vert f{\Vert }_{H^s}^2=\int (1+|\xi {|}^2{)}^s|\hat{f}(\xi ){|}^{2}d\xi$ proves boundedness of the pairing by the Cauchy-Schwarz inequality, and the Riesz representation theorem proves that every continuous functional has this form. Consequently a bounded linear operator $K:H^r\to H^{r+a}$ has an adjoint operator, relative to this pairing, from $H^{-r-a}$ to $H^{-r}$. For an operator that smooths by two orders for every $r$, this yields $K^{\ast }:H^2\to H^4$ by taking $r=-4$.

If $u\in W^{1,p}(U)$, then after modification on a null set its restriction to almost every line parallel to a coordinate axis is absolutely continuous and satisfieswhenever the line segment lies in $U$.

For $u\in W^{1,q}((0,1{)}^2)$ with $q>1$, almost every coordinate-line slice has an absolutely continuous representative and is Hölder continuous with exponent $1-1/q$. Slice bounds need not be uniform. Global Hölder continuity follows from Morrey's inequality only when $q>2$, with exponent $1-2/q$; unbounded examples exist at and below $q=2$.

On a sufficiently regular domain in ${\mathbb{R}}^n$, $H^s$ is a Banach algebra when $s>n/2$:

For an integer $k\ge 0$ and sufficiently regular bounded functions, $\Vert fg{\Vert }_{H^k}\le C_k(\Vert f{\Vert }_{\infty }\Vert g{\Vert }_{H^k}+\Vert g{\Vert }_{\infty }\Vert f{\Vert }_{H^k})$. The Leibniz rule and derivative interpolation give the estimate. In a differentiated semilinear wave equation, this separates a controlled low-order coefficient from an arbitrarily high Sobolev norm, allowing propagation of smoothness after a fixed base norm has been controlled.

The periodic Sobolev space $H^s({\mathbb{T}}^d)$ consists of periodic distributions whose Fourier coefficients obeyIn one dimension, $H^1(\mathbb{T})$ embeds continuously into $L^{\infty }(\mathbb{T})$ and is a Banach algebra.

A distribution $u$ on an open set $X$ belongs to $H_{\mathrm{loc}}^s(X)$ when $\chi u\in H^s({\mathbb{R}}^n)$ for every test function $\chi \in C_c^{\infty }(X)$.

For an open set $U\subseteq {\mathbb{R}}^n$ and $1\le p\le \infty$, the first-order Sobolev space iswhere each $D_{i}u$ is a weak derivative. Its standard norm is $\Vert u{\Vert }_{W^{1,p}}=\Vert u{\Vert }_p+{\sum }_i\Vert D_{i}u{\Vert }_p$.

For a Sobolev space function, almost every line parallel to a coordinate axis gives an absolutely continuous restriction. An absolutely continuous function has derivative zero almost everywhere on each of its level sets. Slicing and Fubini's theorem give the assertion in higher dimension. If $w\in W^{2,p}$, apply it to $w$ and then to each component of its gradient to get $D^{2}w=0$ almost everywhere on $\{w=0\}$. This supplies the contact differentiation used in noncontact of ROF level boundaries.

On a connected open set, a Sobolev space function with zero weak gradient is constant almost everywhere. Local mollification gives constant smooth representatives on interior balls; overlaps and connectedness identify their constants. Without connectedness, the constant may differ between components. This explains the constant ambiguity in a Neumann Poisson problem.

The H1 space is the first-order Sobolev space whose elements and first weak derivatives belong to L2 space. It is a Hilbert space with inner product given by the sum of the L2 inner products of the functions and of their first weak derivatives.

For $f\in H^1(\mathbb{R})$, its difference quotient equals $D_{h}f={\int }_0^1{f}^{\prime }(\cdot +\theta h)d\theta$ in L2 space. Thus $\Vert D_{h}f{\Vert }_2\le \Vert f^{\prime }{\Vert }_2$, and translation of a function gives $D_{h}f\to f^{\prime }$ in L2 space. Conversely, bounded difference quotients have a weakly convergent subsequence as $h\to 0$. Against a test function, $\int D_{h}fv\to -\int f{v}^{\prime }$, so that weak limit is the weak derivative of $f$. It lies in L2 space, which is precisely the first-order Sobolev space condition.

The zero-boundary Sobolev space $H_0^1(U)$ is the closure of the space of test functions $C_c^{\infty }(U)$ in $H^1(U)$. Under standard regularity assumptions its elements have zero boundary trace.

On a nonempty bounded domain where the Poincare inequality holds, no $a\in H_0^1(\Omega )$ can satisfy $a\ge a_{-}>0$ almost everywhere. For $g(s)=\min (\max (s,0),a_{-})$, Lipschitz truncation preserves zero-boundary Sobolev spaces, so $g(a)\in H_0^1(\Omega )$. But $g(a)$ would be the nonzero constant $a_{-}$, whose zero gradient contradicts the Poincare inequality. In a uniformly elliptic divergence-form elliptic operator, the homogeneous boundary condition belongs to the unknown and test functions, not to a coefficient bounded below by a positive constant.

If $v\in H_0^1(\Omega )$ and $g:\mathbb{R}\to \mathbb{R}$ is Lipschitz continuous with $g(0)=0$, then $g(v)\in H_0^1(\Omega )$. The Lipschitz version of the Sobolev chain rule controls its weak derivatives, and its boundary trace is $g(0)=0$. Equivalently, approximate $v$ by compactly supported smooth functions, apply the derivative bound to their compositions, and pass weakly in $H^1$ and strongly in $L^2$. The closed linear space $H_0^1$ is weakly closed. This allows positive-part and bounded truncations in weak formulations without losing the Dirichlet boundary condition.

On a domain $U\subseteq {\mathbb{R}}^n$, a common normalization of the Dirichlet inner product isThe Poincare inequality makes its induced seminorm a norm on $H_0^1(U)$.

For a conformal bijection $f:D\to \tilde{D}$ and smooth $u,v$ of compact support on $\tilde{D}$, $\boxed{(u\circ f,v\circ f{)}_{\nabla ,D}=(u,v{)}_{\nabla ,\tilde{D}}}$. The Jacobian matrix of $f$ is a rotation times $|f^{\prime }|$. Consequently the chain rule introduces $|f^{\prime }{|}^2$ into the gradient pairing, while the change of variables formula introduces precisely the same factor into area. They cancel. The identity extends by completion to the Dirichlet energy spaces; it does not preserve the $L^2$ term of an inhomogeneous Sobolev norm.

For planar Gaussian free field theory, this is the completion of real test functions $C_c^{\infty }(D)$ in the norm induced by the Dirichlet inner product, $(f,g{)}_{\nabla }=(2\pi {)}^{-1}{\int }_D\nabla f\cdot \nabla g$. On a general unbounded domain this convention must be distinguished from completion in the inhomogeneous $H^1$ norm. A conformal map identifies its energy norm with that on the unit disc by conformal invariance.

Let $U\subseteq D$ be open. Regard $H_{\mathrm{supp}}=H_0^1(U)$ as functions on $D$ by zero extension of H01, and let $H_{\mathrm{harm}}$ consist of the weakly harmonic Sobolev functions on $U$ belonging to $H_0^1(D)$. For $h\in H_{\mathrm{harm}}$, the Dirichlet inner product $(h,\varphi {)}_{\nabla }$ vanishes for each test function supported in $U$. Approximate any $u\in H_{\mathrm{supp}}$ by those test functions in the gradient norm and use the Cauchy-Schwarz inequality. This gives $\boxed{(h,u{)}_{\nabla }=0}$. The proof also works with the inhomogeneous zero-boundary Sobolev space convention, and with a homogeneous completion realized as weak functions.

Every element of $W^{1,p}(a,b)$ for $1\le p<\infty$ has an absolutely continuous representative. Its ordinary derivative exists almost everywhere, equals its weak derivative, and belongs to $L^p(a,b)$.

On $I=(-a,a)$, a first-order Sobolev space function has a continuous representative and obeysBound the average by the Cauchy-Schwarz inequality and average the identity $u(x)-u(y)={\int }_y^x{u}^{\prime }$. This also gives $[u{]}_{C^{0,1/2}}\le \Vert u^{\prime }{\Vert }_2$. A unit constant with the standard Sobolev norm cannot be asserted on arbitrary interval lengths: $u=1$ already disproves it when $2a<1$.

For every open set $U\subseteq {\mathbb{R}}^n$, smooth functions in $U$ are dense in $W^{1,p}(U)$ for $1\le p<\infty$. Near a flat boundary, one may first translate the function into the domain and then apply a mollifier.

By the definition of $W_0^{1,p}(U)$ as the closure of $C_c^{\infty }(U)$, extending a function by zero outside $U$ gives a bounded map into $W^{1,p}({\mathbb{R}}^n)$ and does not create a boundary distribution.

A Sobolev extension operator is a bounded linear map $E:W^{1,p}(U)\to W^{1,p}({\mathbb{R}}^n)$ whose restriction to $U$ is the original function. For a half-space, reflection across the boundary gives such an operator.

For $U=(0,\infty )\times {\mathbb{R}}^{n-1}$, the formula $Eu(x_1,x^{\prime })=u(|x_1|,x^{\prime })$ defines a bounded Sobolev extension operator. Its weak normal derivative changes sign across the boundary, while its tangential weak derivatives are reflected unchanged.

The odd reflection of a function on a half-space is $Eu(x^{\prime },x_n)=\mathrm{sgn}(x_n)u(x^{\prime },|x_n|)$. When $u$ has zero trace on the boundary, this extension preserves the relevant weak regularity and often converts a homogeneous boundary problem into an interior problem.

A function $f:[a,b]\to \mathbb{R}$ is absolutely continuous exactly when there is a function $g\in L^1(a,b)$ such thatThen $f$ is differentiable almost everywhere and $f^{\prime }=g$ almost everywhere.

For a continuous function $f$ on $[0,1]$, let $G_n$ be its dyadic slope martingale. Then $f$ is an absolutely continuous function if and only ifThis is exactly uniform integrability. The uniformly integrable martingale convergence theorem gives convergence in L1 $G_n\to g$, while their integrated linear interpolations converge uniformly to $f$. Hence $f(x)=f(0)+{\int }_0^{x}g$. Conversely, if $f$ has density $g\in L^1$, its slopes are $\mathbb{E}[g\mid {\mathcal{F}}_n]$, and the uniform integrability of conditional expectations proves the criterion. The dyadic tail integral is also the sum of the absolute endpoint increments in cells whose slope is at least $\lambda$ in magnitude.

For a boundary portion ${\Gamma }_D\subseteq \partial U$, the spaceis the kernel of the restricted trace map, hence a closed vector subspace of $H^1(U)$.

For prescribed boundary data $\phi$, the affine Sobolev spaceconsists of the $W^{1,p}$ functions whose trace equals the trace of $\phi$. It is the natural admissible set for a variational Dirichlet problem.

To solve a linear Dirichlet problem with boundary value $g$, choose a sufficiently regular extension $G$ whose trace is $g$ and write $v=w+G$. Then $w$ has zero trace, while moving the known term $LG$ to the right-hand side turns the equation $Lv=f$ into the homogeneous-boundary problem $Lw=f-LG$.

The Campanato seminorm measures mean oscillation byOn regular bounded domains, ${\mathcal{L}}^{p,n+p\alpha }=C^{0,\alpha }$ for $0<\alpha \le 1$, with equivalent norms.

If a nondecreasing oscillation quantity satisfiesfor $0<\rho \le R$, with $\gamma >\beta$, then iteration gives $\Phi (\rho )\le C(\rho /R{)}^{\beta }\Phi (R)+CB{\rho }^{\beta }$.

The Hardy operator is the averaging operatoron functions over the positive half-line or an interval beginning at zero.

For $1<p<\infty$, the Hardy operator is bounded on $L^p(0,1)$ with operator norm at most $p/(p-1)$.

If $u\in W^{1,p}(0,1)$ has zero trace at zero, thenIndeed, the one-dimensional Sobolev representative satisfies $u(x)={\int }_0^x{u}^{\prime }(t)dt$, so the claim is the Hardy averaging inequality applied to $u^{\prime }$.

A Poincare inequality controls a function by its derivatives after the constant ambiguity has been removed. For example, on a bounded connected Lipschitz domain,

For $n\ge 2$, a round annulus $A_R=B_{2R}\setminus B_R$ is connected and obeys ${\int }_{A_R}|v-v_{A_R}{|}^2\le C(n)R^2{\int }_{A_R}|\nabla v{|}^2$. Scaling reduces the claim to a fixed annulus. In dimension one the annulus has two components, so the inequality fails with a single average; one needs a separate average on each component.

On a bounded connected Lipschitz domain $U$, every $u\in H^1(U)$ satisfiesEquivalently, the gradient norm is equivalent to the $H^1$ norm on the mean-zero Sobolev space.

For the cube $Q=(0,L{)}^n$ and $u\in H^1(Q)$,This follows from the pairwise-difference identityand a coordinate-by-coordinate path from $x$ to $y$. The Cauchy-Schwarz inequality bounds the squared sum of its $n$ increments by $n$ times the sum of their squares, while the one-dimensional fundamental theorem of calculus bounds each integrated increment by $L^{n+2}\Vert D_{i}u{\Vert }_2^2$. Division by $2|Q|=2L^n$ gives the stated constant.

The mean-zero Sobolev space is the closed subspaceOn a bounded connected Lipschitz domain, the Poincare-Wirtinger inequality makes $\Vert \nabla u{\Vert }_2$ an equivalent Hilbert norm on this space.

If $U$ is bounded and connected and a boundary portion ${\Gamma }_D$ has positive surface measure, thenfor every $v\in H^1(U)$ whose trace vanishes on ${\Gamma }_D$. Otherwise a normalized counterexample sequence, the Rellich-Kondrashov compactness theorem, and continuity of the trace would produce a nonzero constant with zero trace on ${\Gamma }_D$.

On a bounded connected Lipschitz domain $U$,for every $u\in H^1(U)$. If this failed, the Rellich-Kondrachov compactness theorem would produce a normalized strong $L^2$ limit with zero gradient, hence a constant function; continuity of the Sobolev trace theorem would force that constant to vanish, contradicting its unit norm.

For a probability density function $w$ on a domain, a weighted Poincare inequality has the formThe optimal $\lambda$ is the spectral gap of the associated weighted diffusion operator.

For $w=e^{-\Phi }$ and $g=h{e}^{-\Phi /2}$, integration by parts givesThis conjugates the weighted Dirichlet energy to a Schrodinger operator with effective potential $|{\Phi }^{\prime }{|}^2/4-{\Phi }^{\prime \prime }/2$.

For every multi-index $\alpha$ and real $s$,This follows from the Fourier multiplier $(i\xi {)}^{\alpha }$.

If a periodic function $u$ has zero mean, its nonzero Fourier frequencies giveThus inversion of the Laplacian on mean-zero periodic functions gains two Sobolev derivatives.

For every real $s$,is an isomorphism. Its inverse is the Fourier multiplier $(1+|\xi {|}^2{)}^{-1}$.

For real $m\ne 0$ and constant real $\alpha$, the Fourier symbol of $-\Delta -\alpha \cdot \nabla +m^2$ has modulus at least $|\xi {|}^2+m^2$. Its inverse therefore maps $L^2$ into $H^2$ with norm at most $\max (1,m^{-2})$, uniformly in the drift.

A sequence bounded in $H^2({\mathbb{R}}^n)$ whose members are supported in one bounded set has a subsequence converging strongly in $H^1({\mathbb{R}}^n)$. Apply Rellich compactness simultaneously to the functions and their first derivatives.

For $s>1/2$, restriction to the coordinate hyperplane extends uniquely to a bounded linear mapIn Fourier variables, Cauchy--Schwarz bounds the integral over the normal frequency because $(1+t^2{)}^{-s}$ is integrable exactly when $s>1/2$.

On a bounded smooth domain, the Sobolev trace theorem can be strengthened toFor smooth $u$, extend the outward unit normal to a smooth vector field $X$ near the closure of $U$ with $X\cdot \nu =1$ on its boundary. The divergence theorem gives ${\int }_{\partial U}{u}^2={\int }_U(\mathrm{div}X)u^2+2uX\cdot Du$, and the Cauchy-Schwarz inequality gives the estimate. Density extends it to $H^1(U)$. A bound containing only the product $\Vert u{\Vert }_2\Vert Du{\Vert }_2$ is false for nonzero constant functions. Equivalently, for every $\epsilon >0$, the Young inequality gives $\Vert Tu{\Vert }_2^2\le \epsilon \Vert Du{\Vert }_2^2+C_{\epsilon }\Vert u{\Vert }_2^2$.

For $1\le p<\infty$, boundary restriction on smooth functions extends uniquely to a bounded linear map $T:W^{1,p}({\mathbb{R}}_{+}^n)\to L^p({\mathbb{R}}^{n-1})$. The estimate follows by applying the fundamental theorem of calculus on normal line segments and then integrating over the boundary variables.

For $1\le p<\infty$, no bounded map from $L^p({\mathbb{R}}^n)$ to $L^p({\mathbb{R}}^{n-1})$ can agree with restriction on continuous functions. Indeed,has a fixed nonzero trace while $\Vert u_k{\Vert }_p=k^{-1/p}\Vert \varphi {\Vert }_p\Vert \psi {\Vert }_p\to 0$.

If $({\Delta }^m+1)u=f$ distributionally on ${\mathbb{R}}^n$, $m$ is even, and $f\in H^r$, thenand $u\in H^{r+2m}({\mathbb{R}}^n)$.

If $s>d/2$, then $H^s({\mathbb{R}}^d)$ embeds continuously into bounded continuous functions.

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.

For $p>n$, the Hölder continuous function representative $u^{\ast }$ of $u\in W^{1,p}({\mathbb{R}}^n)$ has a classical Frechet derivative at almost every point, equal to its weak derivative. At a point $x$ where the $p$-mean oscillation of $Du$ tends to zero, apply the Morrey inequality on a cube to $u^{\ast }(y)-u^{\ast }(x)-Du(x)\cdot (y-x)$. On a cube of side comparable to $|y-x|$, its oscillation is at mostThe required points have full measure by the Lebesgue differentiation theorem.

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.

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 $L^{\infty }$.

A Sobolev inequality bounds a function in one $L^q$ norm by its weak derivatives in another $L^p$ norm. In three dimensions, the first-order case $\Vert u{\Vert }_{L^6}\le C\Vert \nabla u{\Vert }_{L^2}$ is the endpoint estimate behind the continuous embedding $H^1\hookrightarrow L^6$.

For a continuously differentiable function $F$ on an interval $I=[t-\delta /2,t+\delta /2]$,To prove it, average $|F(t){|}^2-|F(u){|}^2$ over $u\in I$ using the fundamental theorem of calculus. On either half of $I$, the resulting derivative kernel has absolute value at most $1/2$. Since $|(|F{|}^2)\prime |\le 2|FF\prime |$, the displayed bound follows.