Higher-order Sobolev embedding

Whole-space iteration. Let $B$ be the fixed bounded support ball from [F6]. Suppose $1\le q_0<n$ and $w\in W^{\mathrm{\ell },q_0}({\mathbb{R}}^n)$ is supported in $B$, with $\mathrm{\ell }\ge 1$, and put $1\mathrm{/}{q}_1=1\mathrm{/}{q}_0-1\mathrm{/}n$, so $q_1=q_0^{\ast }$. For every $\mathrm{\mid }\beta \mathrm{\mid }\le \mathrm{\ell }-1$, [F1] gives $D^{\beta }w\in W^{1,q_0}({\mathbb{R}}^n)$ with norm bounded by $C\mathrm{\parallel }w{\mathrm{\parallel }}_{W^{\mathrm{\ell },q_0}}$. Applying the whole-space inequality [F7] to each derivative gives $D^{\beta }w\in L^{q_1}$; summing the finitely many norms shows $w\in W^{\mathrm{\ell }-1,q_1}({\mathbb{R}}^n)$ with controlled norm. Compact support is retained, and [F2] gives the invariant $\mathrm{\ell }-n\mathrm{/}{q}_0=(\mathrm{\ell }-1)-n\mathrm{/}{q}_1$. The $q_0=1$ case uses the p=1 inequality and its density passage in [F7].

Assertion (3): global Holder representatives. Assume $kp>n$ and choose $m\ge 0$, $0<\alpha <1$ with $m+\alpha \le k-\frac{n}{p}$; for the endpoint equality, require $k-\frac{n}{p}\mathrm{\notin }\mathbb{Z}$ and $m=\lfloor k-\frac{n}{p}\rfloor$. Fix $\beta$ with $\mathrm{\mid }\beta \mathrm{\mid }\le m$ and put $\mathrm{\ell }:=k-\mathrm{\mid }\beta \mathrm{\mid }$ and $t:=n\mathrm{/}p$, so $\mathrm{\ell }-t\ge \alpha$. By [F1] and [F6], $w:=D^{\beta }v$ is compactly supported in $W^{\mathrm{\ell },p}({\mathbb{R}}^n)$ with norm at most $C\mathrm{\parallel }u{\mathrm{\parallel }}_{W^{k,p}(\mathrm{\Omega })}$. Starting from $(\mathrm{\ell },p)$, whenever the current exponent $q<n$ and current order $r\ge 2$, apply [F7] to every derivative of $w$ of order at most $r-1$; this gives $w\in W^{r-1,q^{\ast }}$, where $1\mathrm{/}{q}^{\ast }=1\mathrm{/}q-1\mathrm{/}n$, with controlled norm. The invariant $r-n\mathrm{/}q=\mathrm{\ell }-n\mathrm{/}p\ge \alpha >0$ shows the process cannot stop with order $1$ and exponent below $n$. If it reaches $q>n$, write the remaining order as $r$. When $r=1$, take $q^{\mathrm{\prime }}=q$; Morrey gives exponent $1-n\mathrm{/}q=\mathrm{\ell }-n\mathrm{/}p\ge \alpha$. When $r\ge 2$, the first derivatives of $w$ lie in $W^{1,q}$; [F8] bounds them and $w$ on the fixed compact support, so $w\in W^{1,q^{\mathrm{\prime }}}({\mathbb{R}}^n)$ for every finite $q^{\mathrm{\prime }}$, and choose $q^{\mathrm{\prime }}>n$ with $1-n\mathrm{/}{q}^{\mathrm{\prime }}>\alpha$. If the iteration reaches $q=n$, then the invariant gives $r-n\mathrm{/}q=r-1\ge \alpha >0$, hence $r\ge 2$. Thus $w$ and its first derivatives are compactly supported in $W^{1,n}$; on their common bounded support, Holder puts them in $W^{1,s}$ for any $1<s<n$ sufficiently close to $n$, and [F7] then puts them in $L^{s^{\ast }}$ for $s^{\ast }=ns\mathrm{/}(n-s)$, which can be chosen arbitrarily large. Hence again $w\in W^{1,q^{\mathrm{\prime }}}$ for some $q^{\mathrm{\prime }}>n$ with $1-n\mathrm{/}{q}^{\mathrm{\prime }}>\alpha$. In every case $w\in W^{1,q^{\mathrm{\prime }}}$ for an exponent $q^{\mathrm{\prime }}>n$ with $1-n\mathrm{/}{q}^{\mathrm{\prime }}\ge \alpha$, and its norm is bounded by $C\mathrm{\parallel }u{\mathrm{\parallel }}_{W^{k,p}}$.

Assertion (1). Assume $kp<n$ and take the compactly supported extension $v$ from [F6]. Iterating step 1.1 for $j=1,\dots ,k-1$ gives $v\in W^{1,p_{k-1}}({\mathbb{R}}^n)$ with $1\mathrm{/}{p}_j=1\mathrm{/}p-j\mathrm{/}n$; all exponents are finite and $p_j<n$ because $(j+1)p<n$ for $j\le k-1$. One more application of [F7] gives $v\in L^{p_k^{\ast }}({\mathbb{R}}^n)$, where $1\mathrm{/}{p}_k^{\ast }=1\mathrm{/}p-k\mathrm{/}n$ by [F2]. For every $1\le q\le p_k^{\ast }$, Holder [F4] on the fixed support ball $B$ gives $\mathrm{\parallel }v{\mathrm{\parallel }}_{L^q({\mathbb{R}}^n)}\le C(B,p,q)\mathrm{\parallel }v{\mathrm{\parallel }}_{L^{p_k^{\ast }}({\mathbb{R}}^n)}$; the whole-space endpoint estimate and [F6] bound this by $C(n,k,p,q,\mathrm{\Omega })\mathrm{\parallel }u{\mathrm{\parallel }}_{W^{k,p}(\mathrm{\Omega })}$. Since $v=u$ on $\mathrm{\Omega }$, restriction proves (1).

Assertion (2). Assume $kp=n$, and take $v$ from [F6]. Iterating step 1.1 through $k-1$ reductions gives $v\in W^{1,n}({\mathbb{R}}^n)$ with support in $B$. If $1\le q\le n$, Holder [F4] on $B$ bounds $\mathrm{\parallel }v{\mathrm{\parallel }}_{L^q}$ by $C(B,q)\mathrm{\parallel }v{\mathrm{\parallel }}_{L^n}$. If $q>n$, set $s=nq\mathrm{/}(n+q)$, so $1<s<n$ and $s^{\ast }=q$; compact support and Holder give $v\in W^{1,s}({\mathbb{R}}^n)$ with $\mathrm{\parallel }v{\mathrm{\parallel }}_{W^{1,s}}\le C(B,s)\mathrm{\parallel }v{\mathrm{\parallel }}_{W^{1,n}}$, and [F7] gives $\mathrm{\parallel }v{\mathrm{\parallel }}_{L^q}\le C(n,s)\mathrm{\parallel }v{\mathrm{\parallel }}_{W^{1,s}}$. In both cases the norm is bounded by $C(n,k,q,\mathrm{\Omega })\mathrm{\parallel }u{\mathrm{\parallel }}_{W^{k,p}(\mathrm{\Omega })}$ using [F6]; restriction to $\mathrm{\Omega }$ proves (2), with no $L^{\mathrm{\infty }}$ endpoint asserted.

Apply [F3] and [F8] with this exponent $q^{\mathrm{\prime }}$ to $w$ on a ball containing $\bar{\mathrm{\Omega }}$. This gives a representative $g_{\beta }\in C^{0,\alpha }(\bar{\mathrm{\Omega }})$ with $\mathrm{\parallel }{g}_{\beta }{\mathrm{\parallel }}_{C^{0,\alpha }(\bar{\mathrm{\Omega }})}\le C\mathrm{\parallel }u{\mathrm{\parallel }}_{W^{k,p}(\mathrm{\Omega })}$, uniformly over the finitely many $\mathrm{\mid }\beta \mathrm{\mid }\le m$. By [F9], whenever $\mathrm{\mid }\beta \mathrm{\mid }<m$, the classical derivatives of $g_{\beta }$ are the continuous representatives $g_{\beta +e_i}$, since these represent the weak derivatives of $D^{\beta }v$ and weak derivatives are unique. Therefore $g_0\in C^{m,\alpha }(\bar{\mathrm{\Omega }})$, represents $u$, and its $C^{m,\alpha }$ norm is bounded by the sum of the finitely many bounds just obtained. This proves (3) for the strict range and also the stated fractional-endpoint case: there $\mathrm{\ell }-t=\alpha$ for $\mathrm{\mid }\beta \mathrm{\mid }=m$, and the final Morrey exponent is exactly $\alpha$, which is allowed by [F8].