Distribution theory

Distribution theory extends functions to continuous linear functionals on test functions, allowing generalized derivatives and point sources.

A distribution on an open set $U$ is a continuous linear functional on the space of test functions $\mathcal{D}(U)=C_c^{\infty }(U)$.

For distributions $u$ on ${\mathbb{R}}^m$ and $v$ on ${\mathbb{R}}^n$, their tensor product acts on a joint test function by $⟨u\otimes v,\Phi ⟩=⟨u_x,⟨v_y,\Phi (x,y)⟩⟩$. The inner function is smooth with compact support in the projection of $\mathrm{supp}\Phi$, and finite order of a distribution estimates prove continuity. Reversing the pairings gives the same value: finite sums of product kernels are dense in the required smooth seminorms, and the two orders agree on product kernels. Its support is contained in the Cartesian product of the two supports. This provides a rigorous replacement for formal integration in separate distribution variables.

A locally integrable function $f$ defines a distribution by $⟨u_f,\phi ⟩=\int f\phi$. The pairing is bilinear if distributions are taken complex-linear. This identifies ordinary functions with a subset of distributions without claiming that every distribution is a function.

The order of a distribution on a compact set is the least nonnegative integer $m$ for which its action is bounded by a constant times the sup norms of test-function derivatives through order $m$.

A distribution can have finite order of a distribution on every compact set without having one order bound valid on the whole domain. For example, the locally finite sum ${\sum }_{j\ge 1}{\partial }^j{\delta }_j$ on the real line has orders growing without bound near the integers.

An order-zero distribution is locally bounded by the sup norm of its test function. Such distributions are precisely the distributions represented locally by signed or complex Radon measures.

A Hadamard finite-part integral subtracts the divergent terms of a truncated integral and retains its finite limit. It defines distributions such as the extension of $1/x$ obtained by differentiating $\log x_{+}$.

For $m\ge 2$, the symmetric Hadamard finite-part integral defining ${\Lambda }_m$ subtracts the Taylor polynomial through degree $m-2$ from a test function before division by $x^m$. It has finite order of a distribution and satisfiesThe lower-order subtraction terms are integrable at infinity, while the remaining odd $1/x$ singularity cancels under symmetric truncation.

A sequence of distributions $u_j$ converges weakly to $u$ when $⟨u_j,\phi ⟩\to ⟨u,\phi ⟩$ for every test function $\phi$.

A sequence can converge as distributions off one point while retaining a nonconvergent multiple of a Dirac delta function at that point. If $u_j=v+c_j{\delta }_p+o_{{\mathcal{D}}^{\prime }}(1)$ with $c_j$ nonconvergent, restriction to the punctured domain tends to $v$, but a test nonzero at $p$ detects the obstruction to global convergence.

For $a>0$, polar-coordinate reduction and integration by parts give $m\sin (m||x{|}^2-a|)=2\delta (|x{|}^2-a)-\pi \cos (ma){\delta }_0+o_{{\mathcal{D}}^{\prime }}(1)$ in two dimensions. The circle term comes from the cusp in the folded phase, while the origin term is a radial endpoint contribution. A punctured-domain limit therefore need not extend across the quadratic critical point.

If ${\tau }_{h}f(x)=f(x-h)$ for functions, its extension to distributions is defined by

With translation of a distribution defined by $⟨{\tau }_{h}u,\phi ⟩=⟨u,\phi (\cdot +h)⟩$, differentiation of the test function gives $\frac{d}{dh}{\tau }_{h}u=-{\tau }_h{u}^{\prime }$ in weak convergence of distributions. The Taylor theorem shows convergence in the space of test functions, including a common compact support. Thus $({\tau }_{-h}u-u)/h\to u^{\prime }$; fixing the translation convention prevents a sign error.

A distribution is invariant under every translation parallel to the $i$th coordinate axis exactly when its $i$th distributional derivative vanishes. Differentiate the translated pairing for one direction; integrate that derivative in the translation parameter for the converse.

For an open set $X\subseteq {\mathbb{R}}^n$, the space $\mathcal{E}(X)=C^{\infty }(X)$ carries the Fréchet topology in which $f_j\to f$ exactly when every derivative converges uniformly on every compact set contained in $X$.

The continuous dual ${\mathcal{E}}^{\prime }(X)$ of $\mathcal{E}(X)$ is the space of compactly supported distributions on $X$. Its weak topology is pointwise convergence on $\mathcal{E}(X)$.

A test function is a smooth function with compact support. Weak and distributional equations are identities obtained by integrating against every test function.

A cutoff function is a smooth function that equals one on a chosen inner set and vanishes outside a slightly larger set. On concentric balls $B_r\subset B_R$, it can be chosen with $|D\eta |\le C/(R-r)$.

The space of test functions is $\mathcal{D}(U)=C_c^{\infty }(U)$. A sequence converges in $\mathcal{D}(U)$ when all its supports eventually lie in one compact subset of $U$ and every derivative converges uniformly there.

The space of test functions has the locally convex inductive-limit topology of the spaces of smooth functions supported in a fixed compact set $K$. Each ${\mathcal{D}}_K$ uses seminorms $p_m(\phi )={\max }_{|\alpha |\le m}\Vert {\partial }^{\alpha }\phi {\Vert }_{\infty }$. Sequential convergence means a common compact support and uniform convergence of every derivative. A linear functional on this space is a distribution exactly when its restriction to each ${\mathcal{D}}_K$ satisfies a finite order of a distribution estimate; the order and constant may depend on $K$.

Every distribution is a limit of test functions in weak convergence of distributions. With a mollifier ${\rho }_{\epsilon }$ and an expanding smooth cutoff function $\chi (x/j)$, the functions $\chi (x/j)(u\ast {\rho }_{1/j})(x)$ have compact support and converge to $u$. For each fixed test, the cutoff eventually equals one on its support, while the reflected mollifier approximates that test with all derivatives in a common compact set.

A function is locally integrable when its absolute value has finite integral over every compact subset of its domain. It defines a regular distribution by integration against test functions.

A distributional identity is an equality that holds after both sides act on every smooth compactly supported test function.

The Dirac delta is the distribution defined by $⟨\delta ,\phi ⟩=\phi (0)$.

As distributions on the line, $m\sin (m|t|)\to 2{\delta }_0$. Pairing with a test function $A$ folds the integral to ${\int }_0^{\infty }m\sin (ms)[A(s)+A(-s)]ds$. One integration by parts yields $2A(0)$ plus a cosine integral tending to zero by the Riemann-Lebesgue lemma. Pointwise convergence is unnecessary.

For a smooth hypersurface $S$, its surface delta is the distribution $⟨{\delta }_S,\phi ⟩={\int }_S\phi dS$. Its normalization uses geometric surface measure rather than a defining function. On the unit circle, ${\delta }_{S^1}$ pairs by integrating $\phi (\cos t,\sin t)$ over $0\le t\le 2\pi$.

For the geometric surface delta distribution on the radius-$a$ sphere in ${\mathbb{R}}^3$, $⟨u_a,\varphi ⟩=a^2{\int }_{S^2}\varphi (a\omega )d\sigma (\omega )$. Angular integration gives the Fourier transform ${\hat{u}}_a(\xi )=4\pi a\sin (a|\xi |)/|\xi |$, with removable value $4\pi a^2$ at zero. The convolution of distributions with a compactly supported factor gives the regular distributionThis density has total mass $16{\pi }^2{a}^2{b}^2$. Values on endpoint spheres do not affect the distribution. When $a=b$, the inverse-distance singularity is locally integrable in three dimensions and is not an additional point mass. The annulus expresses the triangle inequality for the sum of two vectors of fixed lengths.

If $g$ is smooth and $\nabla g\ne 0$ on $S=\{g=0\}$, transverse local coordinates give $⟨\delta (g),\phi ⟩={\int }_S\phi /|\nabla g|dS$. Thus ${\delta }_S=|\nabla g|\delta (g)$ as distributions. In particular ${\delta }_{S^1}=2\delta (|x{|}^2-1)$. Omitting this Jacobian confuses the delta of a defining function with a geometric surface delta distribution.

The identityis interpreted distributionally: symmetric frequency cutoffs give $\sin (nx)/(\pi x)$, which converge to the Dirac delta function.

With inverse Fourier transform normalization $(2\pi {)}^{-1}$, $\int e^{is\theta }{\theta }^{j}d\theta =2\pi i^{-j}{\delta }^{(j)}(s)$ as an oscillatory integral. The factor $i^{-j}$ and the sign convention $⟨{\delta }^{\prime },g⟩=-g^{\prime }(0)$ are essential. For instance, amplitude $-i{x}_2\theta /(2\pi )$ with phase $x_1\theta$ gives $-x_2{\delta }^{\prime }(x_1)$.

For every test function $\phi$,so $H^{\prime }=\delta$ as distributions.

For a smooth function $u$, the distributional derivative of the product of $u$ with the Heaviside function is $D(Hu)=H{u}^{\prime }+u(0){\delta }_0$. Induction gives the displayed formula. It records all boundary sources, including derivatives of the Dirac delta distribution, and determines which initial derivatives must vanish in a retarded fundamental solution.

For nonzero $a$, $\delta (ax)=\delta (x)/|a|$ in the distributional sense.

The derivative of the Dirac delta is the distributional derivative defined byfor every test function $\phi$.

A multi-index is a tuple of nonnegative integers used to abbreviate products and partial derivatives: $|\alpha |={\sum }_j{\alpha }_j$, $x^{\alpha }={\prod }_j{x}_j^{{\alpha }_j}$, and $D^{\alpha }={\prod }_j{D}_j^{{\alpha }_j}$.

For a distribution $u$ and a multi-index $\alpha$, its distributional derivative is defined by

The locally integrable function $\log |x|$ has distributional derivative equal to the principal-value reciprocal distribution. Symmetric integration by parts makes the boundary term $\log \epsilon [\phi (\epsilon )-\phi (-\epsilon )]$ tend to zero.

If $v^{\prime }=0$ on $\mathbb{R}$, then $v$ is a constant distribution. Every test function of integral zero is the derivative of its compactly supported primitive, so $v$ vanishes on those test functions. Choosing one test function $\eta$ of integral one gives $⟨v,\phi ⟩=⟨v,\eta ⟩\int \phi$. Iteration shows that a distribution with $m$th derivative zero is a polynomial of degree at most $m-1$.

A locally integrable function $g$ is the $i$th weak derivative of $u$ whenfor every test function $\phi \in C_c^{\infty }(U)$. Thus a weak derivative is a distributional derivative that is represented by a locally integrable function.

The weak gradient is the vector of first-order weak derivatives of a function. Its $i$th component is specified by $\int u{\partial }_i\phi =-\int ({\partial }_{i}u)\phi$ for compactly supported test functions. For a differentiable function with appropriate local integrability it agrees with the classical gradient.

If $u\in W_{\mathrm{loc}}^{1,2}$ and $\Phi$ is continuously differentiable with bounded derivative on the range in use, then $D(\Phi (u))={\Phi }^{\prime }(u)Du$ almost everywhere. One proves this first for smooth functions and then uses density of smooth functions in a Sobolev space and boundedness of ${\Phi }^{\prime }$. On compact sets one may truncate $\Phi$ away from the range of a bounded $u$.

If $u\in W_{\mathrm{loc}}^{1,2}$, then $Du=0$ almost everywhere on every level set $\{u=t\}$. Choose smooth truncations ${\Phi }_{\epsilon }(s)$ with $|{\Phi }_{\epsilon }|\le 2\epsilon$, derivative one near zero, bounded derivative, and derivative zero outside $(-2\epsilon ,2\epsilon )$. The Sobolev chain rule gives $D{\Phi }_{\epsilon }(u-t)\to 1_{\{u=t\}}Du$ in local $L^2$ by dominated convergence theorem, while ${\Phi }_{\epsilon }(u-t)\to 0$. The limiting weak derivative is therefore zero.

For a distribution $u$ and a smooth function $a$, the product $au$ is defined by $⟨au,\phi ⟩=⟨u,a\phi ⟩$. It obeys the Leibniz rule; in one dimension, $(au{)}^{\prime }=a^{\prime }u+a{u}^{\prime }$.

If a smooth function $a$ on $\mathbb{R}$ has finitely many zeros $r_j$, all simple, then $aw=0$ exactly when $w={\sum }_j{c}_j{\delta }_{r_j}$. Away from the zeros, divide a test function by the nonvanishing $a$ to show $w$ vanishes. Near $r_j$, write $a(x)=(x-r_j)b_j(x)$ with smooth nonzero $b_j$. The kernel of multiplication by a coordinate, after translation and multiplication by $b_j$, says that the local distribution is a multiple of ${\delta }_{r_j}$. A partition of unity gives the global sum. Simplicity of the zeros excludes derivatives of deltas.

In one dimension, $xw=0$ if and only if $w=c{\delta }_0$. Choose a cutoff function $\eta =1$ near zero and write every test function as $\phi =\phi (0)\eta +x\psi$. The pairing with $x\psi$ vanishes, leaving $⟨w,\phi ⟩=\phi (0)⟨w,\eta ⟩$.

For smooth $a$ and a point $r$,These follow by applying both sides to a test function and using the Leibniz rule. In particular $(x-r){\delta }_r^{\prime }=-{\delta }_r$.

The Cauchy principal valuedefines a distribution. Near zero the odd constant contribution cancels, and the remaining numerator is $O(x)$. It satisfies $x\mathrm{pv}(1/x)=1$. Hence all solutions of $xv=1$ are $v=\mathrm{pv}(1/x)+c{\delta }_0$ by the kernel of multiplication by a coordinate.

For $t>0$, the dilation ${\delta }_{t}u$ of a distribution on ${\mathbb{R}}^n$ is defined byFor a regular distribution this agrees with the function $x\mapsto u(tx)$.

For $t>0$, the support of a distribution satisfiesThe duality formula $⟨{\delta }_{t}u,\phi ⟩=t^{-n}⟨u,\phi (\cdot /t)⟩$ proves one inclusion by transporting test functions; applying the inverse dilation proves the other. For a compact support, the Paley–Wiener–Schwartz theorem gives another proof: $\widehat{{\delta }_{t}u}(\zeta )=t^{-n}\hat{u}(\zeta /t)$ rescales the exponential growth indicator by $1/t$.

A distribution $u$ is homogeneous of degree $\sigma$ when ${\delta }_{t}u=t^{\sigma }u$ for every $t>0$. With the unnormalized angular-frequency convention, its Fourier transform is homogeneous of degree $-n-\sigma$.

The scaling property of the Fourier transform extends by duality toConsequently a homogeneous distribution of degree $\sigma$ transforms into one of degree $-n-\sigma$.

For a real symmetric positive-definite three-dimensional matrix, the reciprocal quadratic form is locally integrable and defines a regular tempered distribution. With the Fourier kernel $e^{-ix\cdot \xi }$, radial Abel regularization gives $\mathcal{F}(|x{|}^{-2})=2{\pi }^2/|\xi |$. The linear change of variables $y=G^{1/2}x$ gives the displayed formula. No principal value or origin-supported correction is required. The result is homogeneous of degree $-1$ on the frequency side.

For a complex symmetric three-dimensional $A=G+iB$ with real symmetric $B$ and positive-definite $G$, the reciprocal is a regular tempered distribution since $|x^{T}Ax|\ge x^{T}Gx$. Holomorphic continuation of the real quadratic transform proves the displayed expression. The determinant branch is the analytic determinant square root for accretive symmetric matrices. The other root has positive real part because $\mathrm{Re}({\xi }^T{A}^{-1}\xi )={\xi }^T{G}^{-1/2}(1+C^2{)}^{-1}{G}^{-1/2}\xi >0$ for nonzero real $\xi$. Both the original and transformed singularities are locally integrable, so the continued identity holds as distributions at the origin too.

For $0<\alpha <n$, the Riesz potential is convolution with a constant multiple of $|x{|}^{\alpha -n}$. Its Fourier multiplier is a constant multiple of $|\xi {|}^{-\alpha }$.

The locally integrable homogeneous function $x\mapsto |x{|}^{-\alpha }$, $0<\alpha <n$, is a Riesz kernel. For $\hat{f}(\xi )=\int e^{-ix\cdot \xi }f(x)dx$,

If $1<p<n/\alpha$, the pointwise Hedberg inequality isSplit the integral at radius $r$. Dyadic annuli bound the near part by $C{r}^{\alpha }Mf(x)$, while Hölder's inequality bounds the far part by $C\Vert f{\Vert }_p{r}^{\alpha -n/p}$. Choosing $r^{n/p}=\Vert f{\Vert }_p/Mf(x)$ balances the terms.

A standard mollifier is a nonnegative test function $\rho$ with integral one. The rescaling ${\rho }_{\epsilon }(x)={\epsilon }^{-n}\rho (x/\epsilon )$ forms an approximation to the identity, and $u\ast {\rho }_{\epsilon }$ is smooth wherever the convolution stays inside the domain.

Mollification is the regularization of a function or distribution by convolution with a rescaled mollifier. It creates smooth functions on interior regions, preserves constant-coefficient distributional equations there, and converges back to the original object in the appropriate local topology as the radius tends to zero. The process is distinct from the kernel used to perform it.

The distributional equation $\Delta u=\mathrm{div}(bu)+cu+f$ becomes the smooth interior equation $\Delta u_{\sigma }=\mathrm{div}(bu{)}_{\sigma }+(cu{)}_{\sigma }+f_{\sigma }$. The products are convolved as wholes. Only coefficient bounds on compact interior sets are needed, and the divergence-forcing interior H1 estimate provides a uniform first derivative bound.

The support of a distribution is the complement of the largest open set on which its pairing vanishes for every supported test function. A partition of unity shows that the union of such open sets still has this vanishing property. For example, the Dirac delta distribution and all its derivatives have support $\{0\}$.

A distribution has compact support when it vanishes on every test function supported outside some compact set. Every compactly supported distribution has finite order.

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 $s<-M-n/2$.

For $u\in {\mathcal{E}}^{\prime }({\mathbb{R}}^n)$, its Fourier transform is the smooth functionIt has at most polynomial growth, and its extension to complex frequency is controlled more precisely by the Paley–Wiener–Schwartz theorem.

Let a distribution have support in a compact convex set $K$. Fixed neighborhood cutoffs initially give an exponential bound larger than the support function $H_K$. Cutoffs of width $\epsilon$ cost powers of ${\epsilon }^{-1}$ in derivative estimates but enlarge the exponential by only $e^{C\epsilon |\mathrm{Im}z|}$. Choosing $\epsilon =(1+|z|{)}^{-1}$ absorbs the cutoff cost into a polynomial and retains the exact exponential $e^{H_K(\mathrm{Im}z)}$.

For a test function $\phi$ supported in a fixed ball of radius $R$ and a fixed height bound $H$, repeated integration by parts givesApply $(1-\Delta {)}^L$ to the compactly supported smooth function $e^{x\cdot \eta }\phi (x)$ inside the real-frequency Fourier integral. The bounded $\eta$ absorbs its derivative and exponential factors into the constant. This estimate makes the numerator of a Hörmander staircase integral absolutely integrable and makes truncated contour endpoints vanish.

Every compactly supported distribution is a finite sum of distributional derivatives of bounded continuous functions. One proof convolves it with a sufficiently high-order Bessel potential and then applies a power of $1-\Delta$.

Every compactly supported distribution on an open set $X$ is a finite sum $u={\sum }_{\alpha }{\partial }^{\alpha }{f}_{\alpha }$ with continuous $f_{\alpha }$ compactly supported inside $X$. One can first write $u={\partial }^{\gamma }F$ for a continuous global convolution primitive, then multiply by a cutoff function equal to one near the support. The identity $\chi {\partial }^{\gamma }F={\sum }_{\beta \le \gamma }(-1{)}^{|\beta |}(\genfrac{}{}{0ex}{}{\gamma }{\beta }){\partial }^{\gamma -\beta }[({\partial }^{\beta }\chi )F]$ localizes every coefficient. Finite global representation can fail for a distribution of unbounded order.

The Bessel potential of order $s$ is the Fourier multiplier $(1+|\xi {|}^2{)}^{-s/2}$. Sufficiently high order turns a compactly supported finite-order distribution into a bounded continuous function.

A fundamental solution of a linear differential operator $P(D)$ is a distribution $E$ satisfying $P(D)E={\delta }_0$.

Every nonzero constant-coefficient linear partial differential operator has a distributional fundamental solution.

Let $P({\xi }^{\prime },z)$ be a polynomial that is monic in the complex variable $z$. A Hörmander staircase partitions the real ${\xi }^{\prime }$-space into measurable sets ${\Delta }_j$ and assigns bounded heights $c_j$ so that $P({\xi }^{\prime },s+i{c}_j)$ stays uniformly away from zero for ${\xi }^{\prime }\in {\Delta }_j$ and $s\in \mathbb{R}$. Integrating $1/P$ over the resulting union of horizontal contours constructs a fundamental solution of a linear differential operator.

Suppose $P({\xi }^{\prime },z)$ has degree $m$ in $z$ and a nonzero constant leading coefficient $a$. Among the $m+1$ heights $h_{\ell }=3\ell$, at least one satisfiesat each real ${\xi }^{\prime }$. Factor into $m$ linear factors with multiplicity. Each root's imaginary part can be within distance less than one of at most one candidate height. The pigeonhole principle leaves a height whose distance from every root is at least one, proving the product bound. The height can be chosen measurably: the set where a candidate succeeds is the closed intersection of $|P({\xi }^{\prime },q+i{h}_{\ell })|\ge |a|$ over rational $q$, and selecting the first successful candidate gives a Borel set partition. This avoids any need for continuous global root labels.

A retarded fundamental solution is supported in a chosen forward region, such as $[0,\infty )$ for an ordinary differential operator.

For $P(z)=a_N{z}^N+\cdots +a_0$ with $N\ge 1$ and $a_N\ne 0$, the unique retarded fundamental solution is $E=Hu$, where $P(D)u=0$, $u^{(j)}(0)=0$ for $j<N-1$, and $u^{(N-1)}(0)=1/a_N$. The distributional jump formula for a Heaviside product gives $P(D)E={\delta }_0$. Its explicit expression isThe residue theorem gives the initial derivatives from the coefficients at infinity. Equivalently, the inverse Fourier transform is integrated below all its poles, producing support of a distribution in $[0,\infty )$. Uniqueness follows because the difference of two retarded solutions is an analytic homogeneous solution vanishing on the negative half-line.

A constant-coefficient operator of order $N$ is elliptic when its homogeneous principal symbol $P_N(\xi )$ is nonzero for every real $\xi \ne 0$.

A square system whose entries have common maximal order $N$ is elliptic when its matrix principal symbol is invertible at every nonzero real frequency. Unlike scalar operators, a system can be first-order elliptic in three dimensions: the Pauli matrices give $A(\xi )={\sum }_j{\sigma }_j{\xi }_j$ with $A(\xi {)}^2=|\xi {|}^{2}1$.

The principal symbol of an odd-order scalar differential operator is an odd map from the frequency sphere to the complex plane. The Borsuk-Ulam theorem forces a zero on a two-dimensional sphere, contradicting ellipticity. Restriction to three variables proves the same obstruction in every greater dimension.

A scalar elliptic differential operator of order $N$ has a principal symbol nonzero on the unit sphere. Compactness gives a positive minimum there, and homogeneity makes that term dominate all lower-degree terms at sufficiently large real frequencies.

Elliptic regularity says that solutions of an elliptic equation gain derivatives wherever the coefficients and forcing are regular. For the Laplacian, local estimates typically control two derivatives of $u$ by $\Delta u$ together with a lower-order norm of $u$.

Once an $L^2$ distributional solution of $\Delta u=\mathrm{div}(bu)+cu+f$ gains $H^1$ regularity, expand the divergence term. Smooth multiplication makes the right side belong to $H_{\mathrm{loc}}^{m-1}$ whenever $u\in H_{\mathrm{loc}}^m$. Interior Poisson regularity gives $H_{\mathrm{loc}}^{m+1}$. Iteration and the Sobolev embedding theorem produce a smooth representative.

For $\Delta v=\mathrm{div}F+G$ and $U\subset \subset V$, a cutoff function energy estimate gives $\Vert v{\Vert }_{H^1(U)}\le C(\Vert v{\Vert }_{L^2(V)}+\Vert F{\Vert }_{L^2(V)}+\Vert G{\Vert }_{L^2(V)})$. One derivative of forcing is permitted because integration by parts places it on the test function. This estimate allows a mollified distributional solution initially in $L^2$ to gain its first weak derivative.

For a smooth uniformly elliptic homogeneous equation away from other boundaries, an interior gradient estimate on a ball of radius comparable to the distance $s$ from a flat side gives $|\nabla u|\le C{s}^{-1}{\sup }_{B_{cs}}|u|$. If $u$ extends continuously with value zero on that side, $s|\nabla u|\to 0$ locally along it. This justifies vanishing boundary terms such as $u_x\sin (nx)$ in a distributional Fourier sine projection.

For a classical solution of $\Delta u=f$ on a half-ball with zero data on its flat face, a tangential Caccioppoli inequality controls every second partial derivative having at least one tangential index. The equation recovers the remaining second normal derivative. The resulting Sobolev norm estimate has a dimension-only constant for the fixed radii one and two. Nonzero data are handled by homogenization of Dirichlet boundary data, adding the $W^{2,2}$ norm of the supplied boundary extension. No condition is imposed on the curved face because the smooth cutoff function vanishes there.

For a constant-coefficient elliptic differential operator $P(D)$ of order $N\ge 1$, localizing the equation gives $P(D)(\chi u)=\chi P(D)u+[P(D),\chi ]u$. The commutator has order at most $N-1$, so a global Fourier transform estimate raises the known Local Sobolev space index by one until the forcing index plus $N$ is reached.

Let $(w_m)$ be the orthonormal basis of Dirichlet eigenfunctions for a strictly positive Dirichlet realization of an elliptic operator, with eigenvalues ${\lambda }_m$. ThenFor integers $k$, these domains are the Sobolev domains of powers of an elliptic Dirichlet operator. At $k=0$ the condition is just Parseval identity and imposes no boundary condition. At $k=1,2$ it describes $H_0^1$ and $H^2\cap H_0^1$, respectively. Higher $k$ impose traces of powers of $L$, not only the trace of $u$. For $L=-d^2/d{x}^2$ on $(0,\pi )$, the smooth Dirichlet function $u=x(\pi -x)$ has sine coefficients proportional to $m^{-3}$ for odd $m$, so the weighted sum diverges at $k=3$ despite its ordinary $H^3$ regularity.

For a strictly positive Dirichlet realization of an elliptic operator with smooth coefficients and smooth boundary, put $X_0=L^2(U)$. For each integer $k\ge 1$,These are the domains $D(A_D^{k/2})$, with norms equivalent to the indicated Sobolev space norms. The bilinear forms $((u,v){)}_{2l}=(L^{l}u,L^{l}v{)}_{L^2}$ and $((u,v){)}_{2l+1}=B[L^{l}u,L^{l}v]$ are inner products on $X_{2l}$ and $X_{2l+1}$. The higher boundary conditions are essential: for $L=-d^2/d{x}^2$ and $u=x(\pi -x)$, $u$ is smooth and zero at the boundary but $L^{2}u=0$, so $\Vert L^{2}u{\Vert }_2$ is not a norm on all of $H^4\cap H_0^1$. Repeated elliptic regularity proves $A_D:X_{k+2}\to X_k$ is an isomorphism, which gives the norm equivalence inductively from $X_0$ and $X_1=H_0^1$.

On a smooth bounded domain, a solution offor an invertible shift satisfiesInterior regularity and boundary flattening give the derivative gain, while invertibility controls the lower-order norm.