Higher elliptic regularity cannot gain more than two derivatives

Given: Countable Choice; a fixed integer $k\ge 0$; the interval $\mathrm{\Omega }=(-1,1)$; the datum $f(x)=\mathrm{\mid }x{\mathrm{\mid }}^{k+1\mathrm{/}2}$; and the function $u(x)=-A\mathrm{\mid }x{\mathrm{\mid }}^{k+5\mathrm{/}2}$ with $A=((k+\frac{5}{2})(k+\frac{3}{2}){)}^{-1}>0$.

On each half-interval the classical derivative of order $j$ of $f(x)=\mathrm{\mid }x{\mathrm{\mid }}^{k+1\mathrm{/}2}$ is a nonzero constant times $\mathrm{\mid }x{\mathrm{\mid }}^{k+1\mathrm{/}2-j}$, with a possible sign change across $0$. For $j\le k$ these derivatives tend to $0$ at $0$ and are in $L^2$; the order $k+1$ derivative is locally integrable with magnitude a positive constant times $\mathrm{\mid }x{\mathrm{\mid }}^{-1\mathrm{/}2}$, but is not in $L^2$ near $0$. Since all derivatives through order $k$ extend continuously across $0$, the piecewise classical derivatives are the weak derivatives through order $k+1$, with no point-mass terms. Thus $f\in H^k((-1,1))$ but $f\mathrm{\notin }{H}_{\mathrm{l}\mathrm{o}\mathrm{c}}^{k+1}((-1,1))$ near $0$. (Integer-order Sobolev spaces and their norms, Weak derivative of a locally integrable function)

A class $u\in H^1((-1,1))$ is a local weak solution of $-u^{\mathrm{\prime }\mathrm{\prime }}=f$ if ${\int }_{-1}^1{u}^{\mathrm{\prime }}\overline{{\phi }^{\mathrm{\prime }}}dx={\int }_{-1}^{1}f\bar{\phi }dx$ for every $\phi \in C_c^{\mathrm{\infty }}((-1,1))$. (Local weak solutions of a divergence-form operator)

Put $p=k+\frac{5}{2}$ and $A=(p(p-1){)}^{-1}$. For $u=-A\mathrm{\mid }x{\mathrm{\mid }}^p$, its derivatives through order $k+2$ are piecewise constant multiples of $\mathrm{\mid }x{\mathrm{\mid }}^{p-j}$, hence lie in $L^2((-1,1))$; the $(k+3)$rd derivative has magnitude a positive constant times $\mathrm{\mid }x{\mathrm{\mid }}^{-1\mathrm{/}2}$ and is not locally in $L^2$ at $0$. The derivatives through order $k+2$ extend continuously across $0$, so these piecewise formulas are the weak derivatives and no delta mass occurs. (Integer-order Sobolev spaces and their norms, Weak derivative of a locally integrable function)

Assume Countable Choice. For the operator $L=-u^{\mathrm{\prime }\mathrm{\prime }}$ in dimension one, $a^{11}=1$ and $b=c=0$, so it is uniformly elliptic with $\theta =M_a=1$, $M_b=M_c=0$, and its constant coefficients lie in $W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{k+1,\mathrm{\infty }}$ for every $k$. If $f\in H_{\mathrm{l}\mathrm{o}\mathrm{c}}^k(\mathrm{\Omega })$ and $u\in H^1(\mathrm{\Omega })$ is a local weak solution of $Lu=f$, the interior theorem gives $u\in H_{\mathrm{l}\mathrm{o}\mathrm{c}}^{k+2}(\mathrm{\Omega })$. (Interior $H^{k+2}$ elliptic regularity, Uniformly elliptic divergence-form operators and their sesquilinear forms, The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$))