Traces of Sobolev functions and higher integrability

We give a sharp characterization of how additional integrability in the interior improves the integrability of boundary traces of $\mathrm{W}^{1,p}$-Sobolev functions. The optimality of our results relies on a novel nonlinear extension or lifting operator.

💡 Research Summary

The paper investigates how additional interior integrability of Sobolev functions influences the integrability of their boundary traces. Classical trace theory tells us that for a Lipschitz domain Ω and 1 ≤ p ≤ ∞ the trace operator γ_{∂Ω}: W^{1,p}(Ω) → Y(∂Ω) is bounded and surjective, where Y(∂Ω) equals L¹(∂Ω) for p=1, the fractional Sobolev space W^{1‑1/p,p}(∂Ω) for 1<p<∞, and Lip(∂Ω) for p=∞. However, if a function u∈W^{1,p}(Ω) also belongs to a higher Lebesgue space L^{q}(Ω) with q>p^{*}=np/(n‑p), the classical description does not capture any possible improvement of the trace’s integrability.

The main result, Theorem 1.1, provides a sharp description for 1<p<n and p^{}<q≤∞:  γ_{∂Ω}(W^{1,p}(Ω)∩L^{q}(Ω)) = W^{1‑1/p,p}(∂Ω) ∩ L^{r}(∂Ω), where the exponent r is determined by the scaling relation  r = 1 + q(1‑1/p). Thus the trace belongs simultaneously to the usual fractional Sobolev space and to a Lebesgue space whose exponent depends non‑linearly on q. When q↘p^{}, r→p, and the result collapses to the classical trace theorem; when q=∞, r=∞, reflecting the maximal possible gain.

The proof splits into two inclusions. The “⊂” direction is relatively straightforward: using the fundamental theorem of calculus along the normal direction and Hölder’s inequality, one obtains a multiplicative estimate  ‖u(·,0)‖{L^{r}(∂Ω)} ≤ C‖∇u‖{L^{p}(Ω)}‖u‖_{L^{q}(Ω)}^{r‑1}, which shows that any u∈W^{1,p}∩L^{q} has a trace in the claimed space.

The challenging “⊃” direction requires constructing an extension operator that maps a given boundary datum f∈W^{1‑1/p,p}(∂Ω)∩L^{r}(∂Ω) back into the interior while preserving both the W^{1,p}‑norm and the L^{q}‑norm. The authors first apply the Poisson kernel K_{P} to obtain a linear harmonic extension v of f to the half‑space ℝ^{n}{+}. This extension satisfies  ‖∇v‖{L^{p}(ℝ^{n}_{+})} ≤ C