Collared Sobolev conductivities have regular Liouville potentials and H1 recovery

Weighted Fourier Cauchy–Schwarz gives $\mathrm{\parallel }a{\mathrm{\parallel }}_{\mathrm{\infty }}+\mathrm{\parallel }\mathrm{\nabla }a{\mathrm{\parallel }}_{\mathrm{\infty }}\le C_s\mathrm{\parallel }a{\mathrm{\parallel }}_{H^s}$ because $\int (1+\mathrm{\mid }\xi {\mathrm{\mid }}^2{)}^{-s}(1+\mathrm{\mid }\xi \mathrm{\mid }{)}^{2}d\xi <\mathrm{\infty }$ for $s>5\mathrm{/}2$. The Fourier integrals for $a$ and its first derivatives converge absolutely and are continuous by dominated convergence, so these bounds hold for a $C^1$ representative. We shall also use $\mathrm{\parallel }fg{\mathrm{\parallel }}_{H^b}\le C\mathrm{\parallel }f{\mathrm{\parallel }}_{H^r}\mathrm{\parallel }g{\mathrm{\parallel }}_{H^b}$ when $r>3\mathrm{/}2$ and $0\le b\le r$. Here is a direct justification: split Fourier space into dyadic annuli, use $\mathrm{\parallel }{f}_j{\mathrm{\parallel }}_{\mathrm{\infty }}\le C{2}^{3j\mathrm{/}2}\mathrm{\parallel }{f}_j{\mathrm{\parallel }}_2$ for low-high products, and $\mathrm{\parallel }{Q}_j(f_k{g}_k){\mathrm{\parallel }}_2\le C{2}^{3j\mathrm{/}2}\mathrm{\parallel }{f}_k{\mathrm{\parallel }}_2\mathrm{\parallel }{g}_k{\mathrm{\parallel }}_2$ when comparable high shells $k$ produce a shell $j\le k+C$. The low-high sum is bounded by $\mathrm{\parallel }f{\mathrm{\parallel }}_{H^r}\mathrm{\parallel }g{\mathrm{\parallel }}_{H^b}$ since $r>3\mathrm{/}2$. For high-low products, sum the low $g$ blocks with bound $C{2}^{j(3\mathrm{/}2-b)}\mathrm{\parallel }g{\mathrm{\parallel }}_{H^b}$ if $b<3\mathrm{/}2$, a square-root logarithmic factor if $b=3\mathrm{/}2$, or $C\mathrm{\parallel }g{\mathrm{\parallel }}_{H^b}$ if $b>3\mathrm{/}2$; the excess $r-3\mathrm{/}2>0$ makes the resulting geometric sum converge. For high-high products, weighted summation over $k\ge j-C$ has geometric kernel $2^{-(k-j)(r+b)}$ and excess factor $2^{j(3\mathrm{/}2-r)}$; Cauchy–Schwarz for the input shell sequences completes the bound. The estimates are first proved for Fourier truncations and then passed to their norm limits.

For $f\in L^2$ and $0<\theta <1$, Plancherel and [F6] give ${\int }_{{\mathbb{R}}^3}{\int }_{{\mathbb{R}}^3}\mathrm{\mid }f(x+h)-f(x){\mathrm{\mid }}^2\mathrm{\mid }h{\mathrm{\mid }}^{-3-2\theta }dxdh={\int }_{{\mathbb{R}}^3}\mathrm{\mid }\widehat{f}(\xi ){\mathrm{\mid }}^2{\int }_{{\mathbb{R}}^3}\mathrm{\mid }{e}^{i\xi \cdot h}-1{\mathrm{\mid }}^2\mathrm{\mid }h{\mathrm{\mid }}^{-3-2\theta }dhd\xi$. The translation formula follows by substitution on Schwartz functions and then $L^2$ density. Rotation and dilation make the inner integral $c_{\theta }\mathrm{\mid }\xi {\mathrm{\mid }}^{2\theta }$, where $0<c_{\theta }<\mathrm{\infty }$ because $\mathrm{\mid }{e}^{iu}-1\mathrm{\mid }\le \min \{2,\mathrm{\mid }u\mathrm{\mid }\}$. Adding $\mathrm{\parallel }f{\mathrm{\parallel }}_2^2$ proves equivalence with the Fourier $H^{\theta }$ norm. Applying this identity to the weak derivatives of order $k$ and using [F1] proves the analogous whole-space intrinsic/Fourier comparison at order $k+\theta$; at integer orders use the derivative rule alone. For $0<\theta <1$, multiplication by a bounded $C^1$ function $h$ is bounded on $H^{\theta }$. Indeed $\mathrm{\mid }h(x)f(x)-h(y)f(y){\mathrm{\mid }}^2\le 2\mathrm{\parallel }h{\mathrm{\parallel }}_{\mathrm{\infty }}^2\mathrm{\mid }f(x)-f(y){\mathrm{\mid }}^2+2\mathrm{\mid }f(y){\mathrm{\mid }}^2\mathrm{\mid }h(x)-h(y){\mathrm{\mid }}^2$. In the last term use $\mathrm{\mid }h(x)-h(y)\mathrm{\mid }\le \min \{2\mathrm{\parallel }h{\mathrm{\parallel }}_{\mathrm{\infty }},\mathrm{\parallel }\mathrm{\nabla }h{\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{\mid }x-y\mathrm{\mid }\}$; integrating the kernel $\mathrm{\mid }x-y{\mathrm{\mid }}^{-3-2\theta }$ near and far from the diagonal gives a finite constant times $\mathrm{\parallel }f{\mathrm{\parallel }}_2^2$. Apply the identity just proved to get the Fourier Sobolev bound. For $\theta =0$ this is just the $L^2$ multiplier bound.

If $F$ is any smooth scalar function on a neighborhood of $[m,M]$, then $F(1+a)-F(1)$ belongs to $H^s$ with a uniform bound over this class. To verify the fractional highest derivative directly, write $s=k+\theta$ with $k=2$ or $3$ and $0\le \theta <1$. For $k=2$, the chain rule gives terms $F^{\mathrm{\prime }}(\sigma )D^{2}a$ and $F^{\mathrm{\prime }\mathrm{\prime }}(\sigma )DaDa$. The first term is in $H^{\theta }$ by step 2.1, since $F^{\mathrm{\prime }}(\sigma )$ is bounded $C^1$ uniformly; for the second, $Da\in H^{s-1}$ with $s-1>3\mathrm{/}2$, so the algebra bound in step 1.1 and then step 2.1 apply. For $k=3$, the terms are $F^{\mathrm{\prime }}(\sigma )D^{3}a$, $F^{\mathrm{\prime }\mathrm{\prime }}(\sigma )D^{2}aDa$, and $F^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}(\sigma )(Da{)}^3$. Here $Da\in H^{s-1}$ is a multiplier on $H^{s-2}$ by step 1.1, so $D^{2}aDa\in H^{s-2}\subset H^{\theta }$; the remaining products follow from the same algebra and step 2.1. Lower derivatives have the corresponding easier estimates; the function itself has an $L^2$ bound by the mean value theorem and vanishes wherever $a=0$. These calculations may first be made for smooth approximations: convolve $a$ with a nonnegative mollifier, which preserves $m\le 1+a\le M$ and uniform $H^s$ and $C^1$ bounds, and then use convergence in lower Sobolev orders and weak compactness in the Hilbert $H^s$ space to identify the asserted weak derivatives and bound. A bounded sequence in this separable Fourier Hilbert space has a weakly convergent subsequence by diagonal convergence of its countable Fourier coordinates and the norm bound; hence only Countable Choice is used.

Apply step 3.1 to $F(t)=\sqrt{t}$, $\frac{1}{2}\log t$, and $t^{-1\mathrm{/}2}$ to obtain the stated regularity and support of $g_j-1$, $\log g_j$, and $g_j^{-1}-1$. To bound $q_j=g_j^{-1}\mathrm{\Delta }{g}_j$ in $H^{s-2}$, for $s<3$ use step 2.1 with $\theta =s-2\in (1\mathrm{/}2,1)$ and the uniform $C^1$ norm of $g_j^{-1}$; for $s\ge 3$ use the multiplier bound in step 1.1 with $r=s$ and $b=s-2$. Split $g_j^{-1}=1+(g_j^{-1}-1)$ for that whole-space bound. The support statement follows from $g_j=1$ outside the collar ball and from the local definition of weak derivatives.

Put $w=\log g_1-\log g_2$, $H=g_1{g}_2$ and $Q=q_1-q_2$. Since $H\mathrm{\nabla }w=g_2\mathrm{\nabla }{g}_1-g_1\mathrm{\nabla }{g}_2$, distributional differentiation gives $\mathrm{div}(H\mathrm{\nabla }w)=g_2\mathrm{\Delta }{g}_1-g_1\mathrm{\Delta }{g}_2=HQ$. The equality follows first for smooth positive approximations and then in distributions using step 4.1 and the Sobolev products from steps 1.1–2.1; all terms also have their ordinary weak-product interpretation because $g_j$ are $C^1$ and $\mathrm{\Delta }{g}_j\in L^2$. The functions $w$ and its gradient vanish outside $\overline{B_{\rho }}$. By [F3], extend $w$ by zero to ${\mathbb{R}}^3$. Choose a nonnegative unit-mass smooth mollifier from [F4]. For small $\epsilon$, its convolution with the zero extension is supported in $B_2$; by [F5], the mollifications converge to $w$ in $H^1$. Thus the distributional identity may be tested against $w$: $Q\in H^{-1}$ and multiplication by the bounded $C^1$ function $H$ is bounded on $H^1$. It gives $\int H\mathrm{\mid }\mathrm{\nabla }w{\mathrm{\mid }}^2=-⟨Q,Hw⟩$.

A compactly supported $H^1$ function in $B_{\rho }\subset (-2,2{)}^3$ obeys $\mathrm{\parallel }w{\mathrm{\parallel }}_2\le 4\mathrm{\parallel }{\mathrm{\partial }}_{1}w{\mathrm{\parallel }}_2$: for a smooth approximation supported in the same larger cube, write $w(x_1,x^{\mathrm{\prime }})={\int }_{-2}^{x_1}{\mathrm{\partial }}_{1}w(t,x^{\mathrm{\prime }})dt$, apply Cauchy–Schwarz, and integrate in $x_1,x^{\mathrm{\prime }}$; passing to $H^1$ limits gives the inequality. Since $H\ge m$, step 5.1 and the Fourier dual bound yield $m\mathrm{\parallel }\mathrm{\nabla }w{\mathrm{\parallel }}_2^2\le \mathrm{\parallel }Q{\mathrm{\parallel }}_{H^{-1}}\mathrm{\parallel }Hw{\mathrm{\parallel }}_{H^1}\le C\mathrm{\parallel }Q{\mathrm{\parallel }}_{H^{-1}}\mathrm{\parallel }w{\mathrm{\parallel }}_{H^1}$. Poincaré in the first sentence then implies $\mathrm{\parallel }w{\mathrm{\parallel }}_{H^1}\le C\mathrm{\parallel }Q{\mathrm{\parallel }}_{H^{-1}}$, dividing by its norm if nonzero. Finally ${\sigma }_1-{\sigma }_2={\sigma }_2(e^{2w}-1)$; the uniform bounds for $w$, ${\sigma }_2$ and $\mathrm{\nabla }{\sigma }_2$, the mean value theorem, and the ordinary weak product rule give $\mathrm{\parallel }{\sigma }_1-{\sigma }_2{\mathrm{\parallel }}_{H^1}\le C\mathrm{\parallel }w{\mathrm{\parallel }}_{H^1}$. Also $\mathrm{\parallel }Q{\mathrm{\parallel }}_{H^{-1}}\le \mathrm{\parallel }Q{\mathrm{\parallel }}_{H^{-1\mathrm{/}2}}$ by the Fourier weights. This proves all recovery claims.