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

As a map into $L^2$, $T$ factors asThe first arrow is bounded by part 3 and the second is the compact embedding from part 2; hence $T:L^2\to L^2$ is a compact operator.

Writing $u=T(f)$ in the defining identity givesIn particular this holds for every test function $v$, so the definition of a distributional derivative yields

Choose a test function $\chi$ that equals one on the unit ball and vanishes outside the ball of radius two. Sinceintegration by parts givesThe first term on the right is supported where $1\le |x|\le 2$, where $u^2\le |x{|}^{\alpha }{u}^2$. The Cauchy-Schwarz inequality and Young inequality bound the second term byAfter absorbing the first integral,

This estimate also proves completeness. Indeed, a Cauchy sequence in $\Sigma$ is Cauchy in $H^1$ locally, while its gradients and the functions $|x{|}^{\alpha /2}{u}_n$ converge in $L^2$. The limits agree locally with a function $u$, so $u\in \Sigma$ and $u_n\to u$ in the energy norm. Thus $\Sigma$ is a Hilbert space; it is the confining-potential energy space for $V(x)=|x{|}^{\alpha }$.

WriteDifferentiating $\log J(u+th)$ at $t=0$ for a real test function $h$ givesAfter integration by parts,Rescaling the dependent and independent variables reduces this Euler-Lagrange equation to the ground-state equation for $Q$. The uniqueness of its positive radial solution and the equality cases in rearrangement show that all minimizers areThis is the classification of Weinstein-functional minimizers.

The space of test functions $C_0^{\infty }(D)$ consists of infinitely differentiable real-valued functions whose support of a function is a compact subset of $D$.

The space of test functions isA sequence ${\phi }_j$ converges to $\phi$ in $\mathcal{D}$ when all supports lie eventually in one compact set $K$ andfor every multi-index $\alpha$. A distribution is a linear functional $u:\mathcal{D}\to \mathbb{C}$ such that, for every compact $K$, there are $C_K>0$ and $m_K\in {\mathbb{N}}_0$ withwhenever $\mathrm{supp}\phi \subseteq K$. Convergence in ${\mathcal{D}}^{\prime }$ is pointwise convergence on test functions.

Continuity plainly implies sequential continuity. Conversely, suppose the linear form is sequentially continuous but the displayed estimate fails for some compact $K$. For every $j$, choose ${\phi }_j$ supported in $K$ such thatThen ${\phi }_j\to 0$ in $\mathcal{D}$, while $u({\phi }_j)\nrightarrow 0$, a contradiction. Hence the seminorm estimate holds on every compact set and $u\in {\mathcal{D}}^{\prime }$.

For a translation vector $h$ and a multi-index $\alpha$, defineThese definitions extend ordinary translation and differentiation to distributions.

If ${\tau }_{t{e}_i}u=u$ for every $t$, differentiating its pairing at $t=0$ gives ${\partial }_{i}u=0$. Conversely, if ${\partial }_{i}u=0$, then for every test functionThe pairing is constant in $t$, hence ${\tau }_{t{e}_i}u=u$. This proves both directions of translation invariance and vanishing distributional derivative.

Finally, the distributional differentiation obeys the linear chain rule under the linear coordinates $s=x-y$ and $t=x+y$. Thusand likewiseas distributions. Adding the two identities givesthe one-dimensional wave equation; this is the low-regularity form of the travelling waves in the D'Alembert formula.

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.