Linear extension operators for restrictions of function spaces to irregular open sets

Let an open set ${\displaystyle \Omega \subset {ℝ}^n}$ satisfy for some 0≤d≤n and ε > 0 the condition: the ${\displaystyle d}$-Hausdorff content ${\displaystyle H_d(\Omega \cap B)\ge \epsilon {\left|B\right|}^{\frac{d}{n}}}$ for any ball B centered in Ω of volume |B|≤1. Let ${\displaystyle H_p^s}$ denote the Bessel potential space on ${\displaystyle {ℝ}^n}$ 1 < p < ∞,s > 0, and let ${\displaystyle H_p^s[\Omega ]}$ be the linear space of restrictions of elements of ${\displaystyle H_p^s}$ to Ω endowed with the quotient space norm. We find sufficient conditions for the existence of a linear extension operator for ${\displaystyle H_p^s[\Omega ]}$, i.e., a bounded linear operator ${\displaystyle H_p^s[\Omega ]\to H_p^s}$ such that ${\displaystyle ext⨍{\mid }_{\Omega }\}=⨍}$ for all ⨍. The main result is that such an operator exists if (i) d > n-1 and s > (n-d)/min(p,2), or (ii) d≤n-1 and s-[s] > (n-d)/min(p,2). It is an open problem whether these assumptions are sharp.