Endpoint and rough-domain limitations of the trace theorems

Assume the Axiom of Choice. The positive results of this page are the bounded trace operator $T:W^{1,p}(\mathrm{\Omega })\to L^p(\mathrm{\partial }\mathrm{\Omega })$ of The $L^p$ trace operator on a bounded $C^1$ domain, its sharp range $W^{1-1\mathrm{/}p,p}(\mathrm{\partial }\mathrm{\Omega })$ for $1<p<\mathrm{\infty }$ with the bounded right inverse of A bounded right inverse of the trace, supported in a prescribed collar, and the kernel identification of The kernel of the trace is the closure of the test functions. Four limitations belong to the statement of the theory.

(i) The sharp range statement is proved here for $1<p<\mathrm{\infty }$. At $p=1$ the trace operator $T:W^{1,1}(\mathrm{\Omega })\to L^1(\mathrm{\partial }\mathrm{\Omega })$ is still bounded and onto, but the range must not be renamed $W^{0,1}(\mathrm{\partial }\mathrm{\Omega })$: that notation describes no space constructed on this page. Moreover, at $p=1$ there is no bounded linear right inverse $L^1(\mathrm{\partial }\mathrm{\Omega })\to W^{1,1}(\mathrm{\Omega })$. The $p=1$ surjectivity of Gagliardo and the nonexistence of a bounded linear extension are classical facts attributed below; they are not proved on this page, and the constructions of A bounded right inverse of the trace, supported in a prescribed collar are used only in the range $1<p<\mathrm{\infty }$.

(ii) The trace theorems of this page use bounded $C^1$ domains with the local one-sided graph property of Bounded C^k domains and boundary charts. Derivatives of the flattening maps and their inverses are bounded on compact patches; shrinking the charts and taking a finite cover of the compact boundary gives bounds depending on the chosen domain and cover. The definition imposes no uniform constants across all charts or domains. Individual $C^1$ boundary arcs do not suffice: at an outward-cusp tip the local one-sided graph property fails. For the planar model ${\mathrm{\Omega }}_{\alpha }=\{(x,y):0<y<1,\mathrm{\mid }x\mathrm{\mid }<y^{\alpha }\}$, the companion page's concentrating sequence disproves the unweighted $W^{1,p}$ to boundary $L^p$ trace bound when $\alpha >p$. Weighted trace results require their own hypotheses; Zuppa supplies context for cusp models and weighted estimates, without a claim here that weighting is necessary or that every cusp has the same threshold.

(iii) Gagliardo's original hypotheses are Lipschitz, not $C^1$; the same statements hold for bounded Lipschitz domains with a Lipschitz boundary atlas, and the $C^1$ statements proved here are special cases. No Lipschitz-domain strengthening beyond the finite-dimensional Euclidean domains treated on this page is claimed, and no sharpness of the Lipschitz class is asserted.

(iv) The zero-boundary identification $\ker T=W_0^{1,p}(\mathrm{\Omega })$ is a statement about the $W^{1,p}$-closure of $C_c^{\mathrm{\infty }}(\mathrm{\Omega })$; no pointwise boundary parametrisation of an arbitrary Sobolev class is claimed, and no claim is made that a Sobolev class has boundary values at individual points.

The $p=1$ surjectivity is Gagliardo's Teorema [1.II], as proved again by Mironescu; the absence of a bounded linear right inverse at $p=1$ is Peetre's theorem as quoted by Hajlasz and Martio; Hunter's Section 3.9 records both the $p=1$ onto statement and the Besov description for $1<p<\mathrm{\infty }$. The outward-cusp limitation is documented by Zuppa and the weighted-space literature he cites. None of these attributed facts is used as a proof obligation elsewhere on this page.

Gagliardo, Teorema [1.II] and no. 4 (printed pp. 290 and 300-305), treats the summable case; Mironescu (printed pp. 99-101) gives a complete proof that every $L^1$ function on the boundary is a trace; Hajlasz and Martio (Remarks after Theorem 10, part (4), printed p. 243) record the failure of a bounded linear right inverse at $p=1$; Hunter (printed p. 73) states both endpoint descriptions; Zuppa's Section 1 (Condition A1, Theorems 2 and 4) is the cusp source. This remark records scope, not new mathematics.