Whole-space H² regularity from an L² distributional Laplacian

Given: The hypotheses and conventions of the Statement.

Assume the Axiom of Choice The Axiom of Choice; its countable instances supply the choice hypotheses inherited from the Fourier, Sobolev and regular-distribution interfaces below. No additional choice is used in the local computation.

Distributional derivatives are polynomial Fourier multipliers gives $\mathcal{F}(D^{\alpha }u)=(2\pi i\xi {)}^{\alpha }\mathcal{F}u$ for $u\in {\mathcal{S}}^{\mathrm{\prime }}({\mathbb{R}}^n)$. Summing the pure second derivatives gives $\mathcal{F}(\mathrm{\Delta }u)=-4{\pi }^2\mathrm{\mid }\xi {\mathrm{\mid }}^2\mathcal{F}u$. This use does not assume that the individual second derivatives belong to $L^2$.

Plancherel theorem gives a unitary transform ${\mathcal{F}}_2:L^2\to L^2$, with $\mathrm{\parallel }{\mathcal{F}}_{2}f{\mathrm{\parallel }}_2=\mathrm{\parallel }f{\mathrm{\parallel }}_2$.

Every $L^2$ class $f$ defines the regular tempered distribution $u_f(\phi )=\int f\phi$ by Polynomial growth functions define tempered distributions, and $\mathcal{F}{u}_f=u_{{\mathcal{F}}_{2}f}$ in ${\mathcal{S}}^{\mathrm{\prime }}$ by Fourier transform agrees with l one and plancherel transforms.

Smooth polynomially bounded multipliers on schwartz space defines multiplication by a smooth symbol with polynomially bounded derivatives through $⟨au,\phi ⟩=⟨u,a\phi ⟩$. In particular, for a polynomial $a$ and an $L^2$ class $f$, this pairing is $\int af\phi$, so $a{u}_f=u_{af}$; the product $af$ is locally integrable because $a$ is bounded on each compact set.

Locally integrable functions embed in distributions makes the regular-distribution map injective modulo almost-everywhere equality on $L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1$. Thus equality of regular distributions implies equality of their locally integrable representatives almost everywhere.

Weighted tempered-distribution characterization of Hˢ states that $u\in {\mathcal{S}}^{\mathrm{\prime }}$ belongs to the canonical image of $H^s$ if and only if $⟨\xi {⟩}^s\mathcal{F}u=u_g$ for some $g\in L^2$, with $\mathrm{\parallel }{E}_s^{-1}u{\mathrm{\parallel }}_{H^s}=\mathrm{\parallel }g{\mathrm{\parallel }}_2$. Here $⟨\xi ⟩=(1+\mathrm{\mid }\xi {\mathrm{\mid }}^2{)}^{1\mathrm{/}2}$; the application below is at $s=2$.

Integer-order W^k,2 and Hᵏ agree with equivalent norms identifies the canonical image of $H^2$ with the regular distributions of $W^{2,2}$ classes, and compares the weak-derivative norm with $\mathrm{\parallel }⟨\xi {⟩}^2{\mathcal{F}}_{2}f{\mathrm{\parallel }}_2$ and $\mathrm{\parallel }(1+4{\pi }^2\mathrm{\mid }\xi {\mathrm{\mid }}^2){\mathcal{F}}_{2}f{\mathrm{\parallel }}_2$, with constants depending only on $n$.

Tempered distributions embed continuously in distributions states that restriction from ${\mathcal{S}}^{\mathrm{\prime }}$ to compactly supported tests is injective. Thus equality of two tempered distributions on compactly supported tests is equality in ${\mathcal{S}}^{\mathrm{\prime }}$.