The $p=n$ endpoint: no $L^{\mathrm{\infty }}$ or Holder embedding

Let $n\ge 2$. On nonempty bounded $C^1$ domains in ${\mathbb{R}}^n$ (or domains satisfying the all-exponents extension hypothesis of the critical embedding theorem), $W^{1,n}$ embeds into $L^q$ for every finite $q$ but not into $L^{\mathrm{\infty }}$, and some $W^{1,n}$ classes have no representative in $C^{0,\alpha }(\bar{\mathrm{\Omega }})$ for any $\alpha >0$. The exponential improvement of Trudinger-Moser and the BMO/John-Nirenberg route to the same integrability are outside this pair's scope and are recorded in the planning scope-denial ledger; the companion page carries the witnesses.

The dimension restriction is essential: on a bounded interval $I$, every $W^{1,1}(I)$ class has an absolutely continuous representative and satisfies $\mathrm{\parallel }u{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(I)}\le \mathrm{\mid }I{\mathrm{\mid }}^{-1}\mathrm{\parallel }u{\mathrm{\parallel }}_{L^1(I)}+\mathrm{\parallel }{u}^{\mathrm{\prime }}{\mathrm{\parallel }}_{L^1(I)}$.

The endpoint limitation is Kinnunen's Remark 3.16 and the surrounding discussion of the critical exponent, with Hunter's Example 3.30 as the explicit logarithmic witness with Laugesen's Theorem 3.23 supplying the supercritical comparison. This remark records the limitation only: the positive embedding $W^{1,n}\hookrightarrow L^q$ for finite $q$ and the failure of the $L^{\mathrm{\infty }}$ and Holder bounds are proved elsewhere on this page and on its companion, and the Trudinger-Moser and BMO routes are deliberately not built in this pair.