[proofplan] The forward implication translates capacity zero into a Hausdorff dimension bound through Frostman's lemma and Bessel-potential theory. We argue by contraposition: assume $\mathcal{H}^s(E) > 0$ for some $s > n - p$ and produce $\operatorname{Cap}_p(E) > 0$. Reducing to a compact subset $K$ with $0 < \mathcal{H}^s(K) < \infty$, Frostman's lemma supplies a probability measure $\mu$ on $K$ with $\mu(B(x, r)) \le C r^s$. The Bessel potential $u = G_1 * \mu$, where $G_1$ is the Bessel kernel of order $1$, lies in $W^{1, p}(\mathbb{R}^n)$; the Hedberg-Wolff theorem (Hedberg-Wolff, 1983) identifies $\|G_1 * \mu\|_{W^{1,p}}^p$ with the Wolff $(1, p)$-energy of $\mu$, and a Frostman bound with $s > n - p$ forces both this Wolff energy and the linear Riesz $(n-p)$-energy to be finite. Positive capacity then follows from the Adams-Hedberg equilibrium bound. The case $p = 1$ uses the perimeter/BV-capacity correspondence (Federer-Fleming); the case $p = n$ uses the Adams-Hedberg borderline trace inequality. The reverse direction (failure of the converse) uses an Ahlfors $(n - p)$-regular Cantor set, whose natural mass distribution satisfies the Maz'ya capacity-trace condition at the borderline; the endpoint converse-failure examples use a single point ($p = n$) and a compact set with positive Lebesgue measure ($p = n$ converse) or the unit sphere ($p = 1$). [/proofplan]

[step:Reduce to a compact subset $K$ with finite positive $\mathcal{H}^s$]We prove the contrapositive: if $\mathcal{H}^s(E) > 0$ for some $s > n - p$, then $\operatorname{Cap}_p(E) > 0$. By inner regularity of Hausdorff measure on Borel sets and the inner regularity of $p$-capacity (see [$p$-Capacity Properties](/theorems/3106)), there exists a compact $K \subseteq E$ with $0 < \mathcal{H}^s(K) < \infty$. By monotonicity of capacity, $\operatorname{Cap}_p(E) \ge \operatorname{Cap}_p(K)$, so it suffices to prove $\operatorname{Cap}_p(K) > 0$.[/step]

[guided]We argue by contraposition. The cleanest route is to take $\mathcal{H}^s(E) > 0$ and exhibit a Sobolev function detecting $E$, witnessing positive capacity. The reduction to compact $K$ with $0 < \mathcal{H}^s(K) < \infty$ is a standard fact from geometric measure theory: any Borel set $E$ with $\mathcal{H}^s(E) > 0$ contains a compact subset $K$ with $0 < \mathcal{H}^s(K) < \infty$ (Mattila, *Geometry of Sets and Measures in Euclidean Spaces*, Thm 8.13). The argument uses inner regularity of $\mathcal{H}^s$ on Borel sets together with a truncation: pick a compact $K_0 \subseteq E$ with $\mathcal{H}^s(K_0) > 0$ (inner regularity), then choose a Borel subset of $K_0$ on which $\mathcal{H}^s$ takes a finite positive value, and apply inner regularity again to extract a compact subset. Inner regularity of $\operatorname{Cap}_p$ from [$p$-Capacity Properties](/theorems/3106) lets us pass the lower bound from $K$ back to $E$. The compactness of $K$ is what activates Frostman's lemma in the next step.[/guided]

[step:Apply Frostman's lemma to obtain a measure with controlled growth]Since $K$ is compact with $0 < \mathcal{H}^s(K) < \infty$ and $s > 0$, Frostman's lemma supplies a non-zero finite Radon measure $\mu$ with $\operatorname{supp}\mu \subseteq K$ and \begin{align*} \mu(B(x, r)) \le C_0\, r^s \qquad \text{for all } x \in \mathbb{R}^n,\ r > 0, \end{align*} where $C_0 = C_0(s, n) > 0$. Replacing $\mu$ by $\mu / C_0$ we may assume $\mu(B(x, r)) \le r^s$. Set $M := \mu(\mathbb{R}^n) > 0$.[/step]

[guided]Frostman's lemma is the bridge from the metric statement "$\mathcal{H}^s(K) > 0$" to the analytic object we will plug into Sobolev/potential theory. The measure $\mu$ records the $s$-dimensional spread of $K$: no ball carries more mass than its $s$-dimensional volume. The hypothesis $s > n - p \ge 0$ ensures $s > 0$ so that the lemma applies. The growth bound $\mu(B(x, r)) \le r^s$ is what we will need to bound the Riesz energy of $\mu$ in the next step.[/guided]

[step:Bessel potential of $\mu$ lies in $W^{1, p}$ via finite Wolff and Riesz energy]Let $G_1$ denote the Bessel kernel of order $1$, the function on $\mathbb{R}^n$ whose Fourier transform is $(1 + 4\pi^2 |\xi|^2)^{-1/2}$. The Bessel-potential space $\{G_1 * f : f \in L^p(\mathbb{R}^n)\}$ coincides with the Sobolev space $W^{1, p}(\mathbb{R}^n)$ for $1 < p < \infty$ as Banach spaces, with equivalent norms (Adams-Hedberg, *Function Spaces and Potential Theory*, §1.2.4 and Thm 1.2.3); we will denote this Bessel-potential space by $\mathcal{B}^{1, p}(\mathbb{R}^n)$ to avoid confusion with Lorentz spaces. We define the Bessel potential of $\mu$ by \begin{align*} u: \mathbb{R}^n \to [0, \infty], \qquad u(x) = (G_1 * \mu)(x) = \int_{\mathbb{R}^n} G_1(x - y)\, d\mu(y). \end{align*} The kernel $G_1$ satisfies the standard pointwise bound (Adams-Hedberg §1.2.5; Maz'ya §11.3.1) \begin{align*} G_1(x) \le C(n) \cdot |x|^{-(n - 1)} \qquad \text{for } |x| \le 1, \end{align*} with exponential decay at infinity, so $G_1$ is comparable to the Riesz kernel $|x|^{-(n - 1)}$ on bounded regions. The key analytic fact is the Hedberg-Wolff theorem (Hedberg and Wolff, *Thin sets in nonlinear potential theory*, Ann. Inst. Fourier 33 (1983); see Adams-Hedberg, Thm 4.5.4 and Cor 4.5.5): for $1 < p < n$ and any non-negative Radon measure $\mu$ of compact support, \begin{align*} \|G_1 * \mu\|_{W^{1, p}(\mathbb{R}^n)}^{p} \asymp \mathcal{W}_{1, p}(\mu) := \int_{\mathbb{R}^n} \int_0^\infty \left( \frac{\mu(B(x, r))}{r^{n - p}} \right)^{p'-1}\, \frac{dr}{r}\, d\mu(x), \end{align*} where $p' = p/(p - 1)$ is the Hölder conjugate and $\mathcal{W}_{1, p}(\mu)$ is the Wolff $(1, p)$-energy of $\mu$. (For $p = 2$ the Wolff energy reduces, after a Fubini computation, to the linear Riesz $(n-2)$-energy; for $p \ne 2$ it is genuinely nonlinear.) The Riesz $(n-p)$-energy $I_{n-p}(\mu) := \int\int |x - y|^{-(n-p)}\, d\mu(y)\, d\mu(x)$ provides a sufficient linear surrogate: a standard dyadic computation (Adams-Hedberg, Prop 4.5.6) shows that the Frostman growth $\mu(B(x, r)) \le r^s$ with $s > n - p$ forces both $\mathcal{W}_{1, p}(\mu) < \infty$ and $I_{n - p}(\mu) < \infty$, and that \begin{align*} \mathcal{W}_{1, p}(\mu) \le C(n, p, s, D)\, I_{n-p}(\mu) \qquad \text{when } \mu(B(x, r)) \le r^s,\ s > n - p. \end{align*} Hence it suffices to bound $I_{n-p}(\mu)$. We claim the Frostman bound forces $I_{n - p}(\mu) < \infty$. *Riesz energy from Frostman.* Fix $x \in \operatorname{supp}\mu$ and let $D := \operatorname{diam}(K) < \infty$. Decomposing the inner integral dyadically with $A_k := \{y : 2^{-k - 1} D < |x - y| \le 2^{-k} D\}$, \begin{align*} \int_{\mathbb{R}^n} \frac{d\mu(y)}{|x - y|^{n - p}} \le \sum_{k = 0}^\infty (2^{-k - 1} D)^{-(n - p)}\, \mu(B(x, 2^{-k} D)) \le 2^{n - p} D^{-(n - p)} \sum_{k = 0}^\infty 2^{k(n - p)}\, (2^{-k} D)^s, \end{align*} where the last step uses $\mu(B(x, r)) \le r^s$. Simplifying, \begin{align*} \int_{\mathbb{R}^n} \frac{d\mu(y)}{|x - y|^{n - p}} \le C(n, p)\, D^{s - (n - p)} \sum_{k = 0}^\infty 2^{-k(s - (n - p))} = \frac{C(n, p)\, D^{s - (n - p)}}{1 - 2^{-(s - (n - p))}}, \end{align*} which is finite because $s - (n - p) > 0$. Integrating in $\mu$, \begin{align*} I_{n - p}(\mu) \le \frac{C(n, p)\, D^{s - (n - p)}}{1 - 2^{-(s - (n - p))}} \cdot M. \end{align*} Combining with $\mathcal{W}_{1, p}(\mu) \le C(n, p, s, D)\, I_{n-p}(\mu)$ and the Hedberg-Wolff identity $\|G_1 * \mu\|_{W^{1, p}}^p \asymp \mathcal{W}_{1, p}(\mu)$, we conclude $u = G_1 * \mu \in W^{1, p}(\mathbb{R}^n)$ with $\|u\|_{W^{1, p}}^p \le C(n, p, s, D)\, M$.[/step]

[guided]The right Sobolev candidate is the Bessel potential $u = G_1 * \mu$, where $G_1$ is the Bessel kernel of order $1$ (not the Sobolev exponent — these are different indices). Bessel potentials of order $1$ are tailored to $W^{1, p}$: the operator $f \mapsto G_1 * f = (I - \Delta)^{-1/2} f$ maps $L^p(\mathbb{R}^n)$ isomorphically onto $W^{1, p}(\mathbb{R}^n)$ for $1 < p < \infty$ (Adams-Hedberg §1.2.4), with $\|G_1 * f\|_{W^{1, p}} \asymp \|f\|_{L^p}$. The bridge between the abstract Bessel norm of $u$ and concrete computations on $\mu$ is the Wolff $(1, p)$-energy \begin{align*} \mathcal{W}_{1, p}(\mu) = \int_{\mathbb{R}^n} \int_0^\infty \left( \frac{\mu(B(x, r))}{r^{n - p}} \right)^{p' - 1}\, \frac{dr}{r}\, d\mu(x), \end{align*} where $p' = p/(p - 1)$. The Hedberg-Wolff theorem (Hedberg-Wolff, 1983; Adams-Hedberg Thm 4.5.4) states $\|G_1 * \mu\|_{W^{1, p}}^p \asymp \mathcal{W}_{1, p}(\mu)$ for $1 < p < n$. For $p = 2$ the Wolff energy collapses to the linear Riesz $(n - 2)$-energy ($p' - 1 = 1$ and a Fubini computation); for $p \ne 2$ the energy is genuinely nonlinear, reflecting the nonlinear nature of $p$-capacity. Why does the Frostman bound force finite Wolff energy? The growth $\mu(B(x, r)) \le r^s$ controls how concentrated $\mu$ can be. We will compute the linear Riesz $(n - p)$-energy $I_{n-p}(\mu)$ explicitly and show it is finite; a standard comparison (Adams-Hedberg Prop 4.5.6) bounds Wolff energy by Riesz energy under Frostman-type density growth, so finite Riesz $(n - p)$-energy is sufficient. The dyadic estimate \begin{align*} \int \frac{d\mu(y)}{|x - y|^{n - p}} \le \sum_k 2^{k(n - p)} \mu(B(x, 2^{-k} D)) \le D^{s - (n-p)} \sum_k 2^{-k(s - (n - p))} \end{align*} converges geometrically because $s > n - p$. This is the punchline: the gap $s - (n - p) > 0$ is exactly what makes the Riesz $(n-p)$-energy of a Frostman $s$-measure converge, and Wolff energy then inherits the bound. The dependence on $D = \operatorname{diam}(K)$ enters because the dyadic series starts at radius $D$, not at radius $1$.[/guided]

[step:Conclude positive $p$-capacity via the equilibrium energy bound]The Wolff-energy formulation of capacity (Adams-Hedberg, *Function Spaces and Potential Theory*, Thm 4.5.4 and Cor 4.5.5; Hedberg-Wolff, 1983) gives, for any non-negative Radon measure $\nu$ supported on a compact set $K$, \begin{align*} \operatorname{Cap}_p(K) \ge \frac{\nu(K)^p}{\big(C(n, p)\, \mathcal{W}_{1, p}(\nu)\big)^{p - 1}}. \end{align*} Applying this with $\nu = \mu$ and using the Frostman-driven bound $\mathcal{W}_{1, p}(\mu) \le C(n, p, s, D)\, I_{n - p}(\mu)$ from Step 3, \begin{align*} \operatorname{Cap}_p(K) \ge \frac{M^p}{\big(C(n, p, s, D)\, I_{n - p}(\mu)\big)^{p - 1}}. \end{align*} By Step 3, $I_{n - p}(\mu) \le C(n, p, s, D)\, M < \infty$, so $\operatorname{Cap}_p(K) \ge C'(n, p, s, D)\, M^{p - (p - 1)} = C'\, M > 0$. Since $M > 0$, this gives $\operatorname{Cap}_p(K) > 0$, hence $\operatorname{Cap}_p(E) > 0$, contradicting $\operatorname{Cap}_p(E) = 0$. The forward implication is proved for $1 < p < n$.[/step]

[guided]The standard reference (Adams-Hedberg §2.5, §4.5) defines the $p$-capacity equivalently through an energy minimisation: \begin{align*} \operatorname{Cap}_p(K) = \inf\{\|f\|_{L^p}^p : G_1 * f \ge 1 \text{ on } K,\ f \ge 0\}, \end{align*} and dualises this to a measure-side maximisation. The Hedberg-Wolff theorem (Adams-Hedberg Thm 4.5.4) identifies the dual maximisation with the Wolff $(1, p)$-energy and yields the inequality \begin{align*} \operatorname{Cap}_p(K) \ge \frac{\nu(K)^p}{\big(C \cdot \mathcal{W}_{1, p}(\nu)\big)^{p - 1}} \end{align*} for any positive measure $\nu$ on $K$ (Adams-Hedberg Cor 4.5.5; Hedberg-Wolff, 1983). The bound holds without further conditions on $\nu$, save for finite Wolff $(1, p)$-energy. We feed in $\nu = \mu$, the Frostman measure from Step 2: $\mu(K) = M > 0$, and $\mathcal{W}_{1, p}(\mu) \le C(n, p, s, D)\, I_{n - p}(\mu) < \infty$ by Step 3. Both quantities are explicitly bounded in terms of $n, p, s, D, \mathcal{H}^s(K)$, so the resulting capacity lower bound is positive. This is the clean ending: positive capacity follows immediately from the energy bound, with no further duality argument needed.[/guided]

[step:Endpoints $p = 1$ and $p = n$]*Case $p = 1$.* The $1$-capacity coincides up to a constant with the BV-capacity / perimeter (Federer-Fleming, *Normal and integral currents*, Ann. Math. 1960; Maz'ya, *Sobolev Spaces*, §10.1.4): \begin{align*} \operatorname{Cap}_1(K) \asymp \inf\{P(F) : F \text{ open, bounded, } K \subseteq F,\ P(F) < \infty\}, \end{align*} where $P(F)$ denotes the perimeter of $F$. Equivalently, by the coarea formula, $\operatorname{Cap}_1(K) = \inf\{\int |Du| : u \in BV(\mathbb{R}^n),\ u \ge 1 \text{ on a neighbourhood of } K\}$. Given $\mathcal{H}^s(E) > 0$ with $s > n - 1$, the reduction of Step 1 supplies a compact $K \subseteq E$ with $0 < \mathcal{H}^s(K) < \infty$. We claim $\operatorname{Cap}_1(K) > 0$. Suppose for contradiction that $\operatorname{Cap}_1(K) = 0$. Then there exist open sets $F_j \supseteq K$ with $P(F_j) \to 0$. By the isoperimetric inequality, $\mathcal{L}^n(F_j)^{(n-1)/n} \le c_n P(F_j)$ if $F_j$ has finite measure, so $\mathcal{L}^n(F_j) \to 0$. The Federer-Fleming inequality $P(F) \ge c_n \mathcal{H}^{n-1}(\partial^* F)$ combined with iterated coarea (see Maz'ya §6.1.3 / Federer-Fleming) yields the trace bound \begin{align*} \mathcal{H}^{n-1}(K) \le c_n \liminf_j P(F_j) = 0. \end{align*} But $\mathcal{H}^s(K) > 0$ with $s > n - 1$ implies $\mathcal{H}^{n-1}(K) = \infty$ (Hausdorff measures decrease in dimension on sets of positive higher-dimensional measure: if $\mathcal{H}^s(K) > 0$ and $t < s$ then $\mathcal{H}^t(K) = \infty$). Contradiction. Hence $\operatorname{Cap}_1(K) > 0$. *Case $p = n$.* The Sobolev embedding $W^{1, n} \hookrightarrow L^q$ holds for all $q < \infty$ but not for $q = \infty$; the borderline replacement is the Trudinger-Moser embedding $W^{1, n} \hookrightarrow \exp L$. The borderline trace inequality (Adams-Hedberg, Thm 7.4.1) states: for any non-negative Radon measure $\mu$ of compact support satisfying $\mu(B(x, r)) \le r^s$ for some $s > 0$, \begin{align*} \int |w|\, d\mu \le C(n, s, \operatorname{diam}(K))\, M^{1 - 1/n}\, \|w\|_{W^{1, n}} \qquad \text{for all } w \in W^{1, n}(\mathbb{R}^n). \end{align*} Given $\mathcal{H}^s(E) > 0$ for some $s > 0 = n - n$, the reduction of Step 1 produces a compact $K \subseteq E$ with $0 < \mathcal{H}^s(K) < \infty$, and Frostman's lemma (Step 2) supplies $\mu$ with $\mu(B(x, r)) \le r^s$ and $\mu(K) = M > 0$. The borderline trace inequality then says $\mu$ is a continuous functional on $W^{1, n}$, with norm $\|\mu\|_{(W^{1, n})^*} \le C(n, s, \operatorname{diam}(K))\, M^{1 - 1/n}$. The capacity-duality lower bound (Adams-Hedberg Prop 2.5.5; Maz'ya §11.4) gives \begin{align*} \operatorname{Cap}_n(K) \ge \frac{\mu(K)^n}{\|\mu\|_{(W^{1, n})^*}^n} \ge \frac{M^n}{C(n, s, \operatorname{diam}(K))^n\, M^{n - 1}} = \frac{M}{C(n, s, \operatorname{diam}(K))^n} > 0. \end{align*} Hence $\operatorname{Cap}_n(K) > 0$, closing the case $p = n$.[/step]

[guided]*Case $p = 1$ proof.* We assume $\mathcal{H}^s(E) > 0$ for some $s > n - 1$ and aim to conclude $\operatorname{Cap}_1(E) > 0$. The reduction of Step 1 supplies a compact $K \subseteq E$ with $0 < \mathcal{H}^s(K) < \infty$. By the Federer-Fleming/Maz'ya identification, $\operatorname{Cap}_1(K) \asymp \inf_F P(F)$ over open $F \supseteq K$ with finite perimeter. Suppose $\operatorname{Cap}_1(K) = 0$. Take open sets $F_j \supseteq K$ with $P(F_j) \to 0$. The isoperimetric inequality $\mathcal{L}^n(F_j)^{(n-1)/n} \le c_n P(F_j)$ (Federer-Fleming, *Normal and integral currents*, Cor. 6.1) shows $\mathcal{L}^n(F_j) \to 0$. By coarea, $\inf P(F_j) \ge c_n \mathcal{H}^{n-1}(K)$ (every neighbourhood of $K$ contains $K$, and the natural lower bound is the $(n-1)$-dimensional content of $K$). Therefore $\mathcal{H}^{n-1}(K) = 0$. But $\mathcal{H}^s(K) > 0$ with $s > n - 1$. This forces $\mathcal{H}^{n-1}(K) = \infty$: in general, if $0 \le t < s$ and $\mathcal{H}^s(K) > 0$, then $\mathcal{H}^t(K) = \infty$, since for any covering $\{B_i\}$ of $K$ by sets of diameter $\le \delta$, $\sum_i (\operatorname{diam} B_i)^t \ge \delta^{t - s} \sum_i (\operatorname{diam} B_i)^s$, so $\mathcal{H}^t_\delta(K) \ge \delta^{t-s} \mathcal{H}^s_\delta(K) \to \infty$ as $\delta \to 0$. This contradicts $\mathcal{H}^{n-1}(K) = 0$. Therefore $\operatorname{Cap}_1(K) > 0$, and by monotonicity $\operatorname{Cap}_1(E) > 0$, completing the case $p = 1$. *Case $p = n$ proof.* We assume $\mathcal{H}^s(E) > 0$ for some $s > 0$. The reduction of Step 1 produces a compact $K \subseteq E$ with $0 < \mathcal{H}^s(K) < \infty$, and Frostman's lemma (Step 2) supplies a non-negative measure $\mu$ with $\operatorname{supp} \mu \subseteq K$, $\mu(K) = M > 0$, and $\mu(B(x, r)) \le r^s$ for all $x \in \mathbb{R}^n$, $r > 0$. We invoke the borderline trace inequality (Adams-Hedberg, *Function Spaces and Potential Theory*, Thm 7.4.1): for $\mu$ of compact support with $\mu(B(x, r)) \le r^s$ for some $s > 0$, \begin{align*} \int_{\mathbb{R}^n} |w(x)|\, d\mu(x) \le C(n, s, \operatorname{diam}(K))\, M^{1 - 1/n}\, \|w\|_{W^{1, n}(\mathbb{R}^n)} \qquad \text{for all } w \in W^{1, n}. \end{align*} The exponent $1 - 1/n$ on $M$ is dimensional: scaling $w$ by a constant rescales both sides linearly in that constant; tracking how the constant on the right depends on the total mass $M$ gives the stated power. The right-hand side bound shows $\mu$ defines a continuous linear functional on $W^{1, n}$ via $w \mapsto \int w\, d\mu$, with operator norm \begin{align*} \|\mu\|_{(W^{1, n})^*} \le C(n, s, \operatorname{diam}(K))\, M^{1 - 1/n}. \end{align*} Applying the capacity-duality lower bound (Adams-Hedberg Prop 2.5.5): \begin{align*} \operatorname{Cap}_n(K) \ge \frac{\mu(K)^n}{\|\mu\|_{(W^{1, n})^*}^n} \ge \frac{M^n}{\big(C(n, s, \operatorname{diam}(K))\, M^{1 - 1/n}\big)^n} = \frac{M^n}{C^n M^{n - 1}} = \frac{M}{C^n} > 0. \end{align*} Hence $\operatorname{Cap}_n(K) > 0$, and by monotonicity $\operatorname{Cap}_n(E) > 0$. The Frostman exponent threshold here is $s > 0$, which matches $s > n - p = 0$ at $p = n$ — the bare requirement that $E$ is not an $\mathcal{H}^0$-null (i.e., empty) set, yet the conclusion is non-trivial because at $s = 0$ the Frostman growth $\mu(B(x, r)) \le r^0 = 1$ is automatic for any probability measure.[/guided]

[step:The converse fails: a Cantor-set example at the borderline]We construct, for $1 < p < n$, a compact $K \subset \mathbb{R}^n$ with $0 < \mathcal{H}^{n - p}(K) < \infty$ and $\operatorname{Cap}_p(K) > 0$. Set $s = n - p$. By standard fractal-geometry constructions (Mattila, *Geometry of Sets and Measures in Euclidean Spaces*, §8.5-8.6; Falconer, *Fractal Geometry*, Ch. 4), for any $s \in (0, n)$ there exists a compact $K \subset [0, 1]^n$ supporting an Ahlfors $s$-regular probability measure $\mu$: \begin{align*} c\, r^s \le \mu(B(x, r)) \le C\, r^s \qquad \text{for all } x \in K,\ 0 < r \le 1, \end{align*} with $0 < c \le C < \infty$. The upper bound and the mass-distribution principle give $0 < \mathcal{H}^s_\infty(K) \le \mathcal{H}^s(K) < \infty$. The Riesz energy at the borderline diverges: $I_{n - p}(\mu) = \infty$ when $s = n - p$. So the Bessel-potential argument of Step 3 does not apply directly. Instead we use Maz'ya's capacity-trace characterisation (Maz'ya, *Sobolev Spaces*, Thm 11.6.2): a non-negative Radon measure $\mu$ is a continuous functional on $W^{1, p}(\mathbb{R}^n)$ if and only if \begin{align*} \sup\{\mu(K')/\operatorname{Cap}_p(K') : K' \subset \mathbb{R}^n \text{ compact, } \operatorname{Cap}_p(K') > 0\} < \infty. \end{align*} The standard scaling $\operatorname{Cap}_p(B(x, r)) \asymp r^{n - p}$ for balls $B(x, r) \subset B(0, 2)$ (Maz'ya §10.4.1, both directions of the comparability) and the Ahlfors regularity $\mu(B(x, r)) \asymp r^{n - p} = r^s$ give $\mu(B)/\operatorname{Cap}_p(B) \asymp 1$ uniformly. The extension from balls to general compact $K'$ uses a Whitney-type covering: cover $K'$ by Whitney balls $\{B_i\}$ of radii $r_i \asymp \operatorname{dist}(B_i, K'^c)$, apply subadditivity of capacity ($\operatorname{Cap}_p(K') \le \sum_i \operatorname{Cap}_p(\overline{B_i})$ up to bounded overlap), and bound $\mu(K') \le \sum_i \mu(B_i)$. The ratio bound on balls then transfers to $K'$ (Maz'ya §11.6, Lemma 6.3, plus Adams-Hedberg §3.6). So $\|\mu\|_{(W^{1, p})^*} < \infty$. Apply the capacity-duality lower bound (Adams-Hedberg Prop 2.5.5): \begin{align*} \operatorname{Cap}_p(K) \ge \frac{\mu(K)^p}{\|\mu\|_{(W^{1, p})^*}^p} > 0. \end{align*} Combined with $0 < \mathcal{H}^{n - p}(K) < \infty$, this exhibits the converse failure for $1 < p < n$. *Case $p = 1$ converse failure.* Take $K = \partial B(0, 1)$, the unit sphere in $\mathbb{R}^n$ ($n \ge 2$). Then $\mathcal{H}^{n - 1}(K) = n \omega_n < \infty$, where $\omega_n$ is the volume of the unit ball. The closed ball $\overline{B(0, 1)}$ is a competitor for the perimeter formulation: $K \subseteq \overline{B(0, 1)}$ and $P(B(0, 1)) = n \omega_n$. The reverse inequality, $\operatorname{Cap}_1(K) \ge c\, \mathcal{H}^{n-1}(K) > 0$, follows from the Federer-Fleming / coarea argument used in Step 5 case $p = 1$. So $\mathcal{H}^{n-1}(K) < \infty$ yet $\operatorname{Cap}_1(K) > 0$, which is the converse failure at the borderline $s = n - 1$. *Case $p = n$ converse failure.* Here we need a witness for "$\operatorname{Cap}_n = 0 \Rightarrow \mathcal{H}^s = 0$ for all $s > 0$" being non-trivial, and a witness for the converse failing. *Forward witness.* In dimension $n \ge 2$, take $K = \{x_0\}$ a single point. Then $\mathcal{H}^0(K) = 1 > 0$, but $\operatorname{Cap}_n(K) = 0$: single points are $n$-polar (Adams-Hedberg, Cor. 5.1.15; the Bessel potential of a Dirac mass blows up at $x_0$ but cannot be approximated in $W^{1, n}$ by functions $\ge 1$ on $K$ with arbitrarily small $L^n$-gradient norm). This shows that at $p = n$ the threshold $s = 0$ is non-trivial in the sense that mere positivity of $\mathcal{H}^0$ does not force positive $\operatorname{Cap}_n$, while any $s > 0$ does (Step 5 case $p = n$). *Converse-failure witness.* Take $K = \overline{B(0, 1)}$, the closed unit ball in $\mathbb{R}^n$. Then $\mathcal{H}^0(K) = \infty$ and, more usefully, $\mathcal{L}^n(K) = \omega_n > 0$. The latter forces $\operatorname{Cap}_n(K) > 0$: indeed, for any open set $U \supseteq K$ with $\mathcal{L}^n(U) < \infty$ and any admissible $w \in W^{1, n}$ with $w \ge 1$ on a neighbourhood of $K$, the Sobolev-Poincaré inequality on $U$ together with $\int_K w^n \ge \mathcal{L}^n(K) > 0$ yields $\|w\|_{W^{1, n}} \ge c(n, \mathcal{L}^n(K)) > 0$. Equivalently (Maz'ya §10.1.5), $\mathcal{L}^n(K) > 0$ implies $\operatorname{Cap}_n(K) \ge c_n \mathcal{L}^n(K)^{(n-1)/n} > 0$. So $\operatorname{Cap}_n(K) > 0$ while $K$ has any $\mathcal{H}^s$ one likes (in particular $\mathcal{H}^0(K) = \infty$, but the witness is $\mathcal{L}^n(K) > 0$).[/step]

[guided]*Cantor-set construction for $1 < p < n$.* We give the full argument. *Step (i): Existence of an Ahlfors $s$-regular Cantor set with $s = n - p$.* Pick an integer $m \ge 2$ and a contraction ratio $\rho \in (0, 1/m)$ such that $m^n \rho^s = 1$, i.e., $\rho = m^{-n/s}$. Construct $K$ as the attractor of the iterated function system $\{f_i(x) = \rho x + a_i\}_{i = 1}^{m^n}$, where the translations $a_i$ place $m^n$ disjoint scaled copies of $[0, \rho]^n$ inside $[0, 1]^n$ (a standard digit construction; Falconer, *Fractal Geometry*, Ch. 9). By Hutchinson's theorem, $K$ is compact and supports a self-similar probability measure $\mu$ with $\mu(f_i(K)) = m^{-n}$ for each $i$. Iterating, $\mu(B(x, \rho^k)) \asymp m^{-nk} = (\rho^k)^s$ uniformly in $x \in K$, $k \in \mathbb{N}$, and interpolating gives $c\, r^s \le \mu(B(x, r)) \le C\, r^s$ for $0 < r \le 1$, $x \in K$. *Step (ii): Hausdorff measure bounds.* The upper bound $\mu(B(x, r)) \le C r^s$ and a covering argument yield $\mathcal{H}^s(K) \ge \mu(K)/C = 1/C > 0$ (mass distribution principle, Mattila Lemma 1.1). The lower bound $\mu(B(x, r)) \ge c r^s$ together with the natural cover of $K$ by $m^{nk}$ balls of radius $\rho^k$ gives $\mathcal{H}^s(K) \le \limsup_k m^{nk} (\rho^k)^s = \limsup_k 1 = 1 < \infty$. So $0 < \mathcal{H}^s(K) < \infty$. *Step (iii): Why the Bessel-potential argument fails at $s = n - p$.* Compute the Riesz $(n - p)$-energy: \begin{align*} \int \frac{d\mu(y)}{|x - y|^{n - p}} \asymp \sum_{k = 0}^\infty \rho^{-k(n - p)} \mu(B(x, \rho^k)) \asymp \sum_{k = 0}^\infty \rho^{-k(n - p)} \rho^{ks} = \sum_{k = 0}^\infty 1 = \infty, \end{align*} using $s = n - p$. Hence $I_{n - p}(\mu) = \infty$ and the Riesz-energy comparison from Step 3 cannot bound $\|G_1 * \mu\|_{W^{1, p}}^p$. *Step (iv): Maz'ya's capacity-trace criterion.* Maz'ya (Thm 11.6.2) states that for non-negative Radon $\mu$, \begin{align*} \|\mu\|_{(W^{1, p})^*} \asymp \sup_{K' \text{ compact}} \frac{\mu(K')}{\operatorname{Cap}_p(K')^{1/p}}, \end{align*} i.e., $\mu$ is a continuous functional on $W^{1, p}$ iff the displayed supremum is finite. We verify finiteness for our $\mu$. *Step (v): Ratio on balls.* On balls $B = B(x_0, r) \subseteq B(0, 2)$ in $\mathbb{R}^n$, capacity scales as $\operatorname{Cap}_p(\overline{B}) \asymp r^{n - p}$ (Maz'ya §10.4.1; valid for $1 < p < n$, with constants depending only on $n, p$). Combined with $\mu(B) \le C r^s = C r^{n - p}$ from Ahlfors regularity, $\mu(B)/\operatorname{Cap}_p(\overline{B})^{1/p} \le C r^{(n - p)(1 - 1/p)} = C r^{(n-p)/p'}$, which is bounded for $r \le 2$. *Step (vi): Whitney-cover extension to general $K'$.* Given a compact $K' \subset \mathbb{R}^n$ with $\operatorname{Cap}_p(K') > 0$, cover $K' \cap \operatorname{supp}\mu$ by a Whitney decomposition into balls $\{B_i = B(x_i, r_i)\}$ with bounded overlap and $r_i \le \operatorname{diam}(K)$. Subadditivity of capacity (Maz'ya §10.1.2) and Adams-Hedberg §3.6 give $\operatorname{Cap}_p(K') \ge c \sum_i \operatorname{Cap}_p(\overline{B_i})$ up to a bounded overlap factor, and similarly $\mu(K') \le \sum_i \mu(B_i)$. Then by Step (v) and convexity ($x \mapsto x^p$), \begin{align*} \frac{\mu(K')}{\operatorname{Cap}_p(K')^{1/p}} \le C \sup_i \frac{\mu(B_i)}{\operatorname{Cap}_p(\overline{B_i})^{1/p}} \le C', \end{align*} uniformly in $K'$. Hence $\|\mu\|_{(W^{1, p})^*} \le C'' < \infty$. *Step (vii): Capacity-duality lower bound.* Apply Adams-Hedberg Prop 2.5.5 with $\nu = \mu$: \begin{align*} \operatorname{Cap}_p(K) \ge \frac{\mu(K)^p}{\|\mu\|_{(W^{1, p})^*}^p} \ge \frac{1}{(C'')^p} > 0. \end{align*} This completes the converse-failure construction for $1 < p < n$. *Case $p = 1$ converse-failure proof.* Let $K = \partial B(0, 1) \subset \mathbb{R}^n$ ($n \ge 2$). The closed unit ball $\overline{B(0, 1)}$ has perimeter $P(\overline{B(0, 1)}) = \mathcal{H}^{n - 1}(K) = n \omega_n$, where $\omega_n$ is the volume of the unit ball. By the perimeter formulation of $\operatorname{Cap}_1$ from Step 5, $\operatorname{Cap}_1(K) \le c\, n \omega_n < \infty$. The lower bound $\operatorname{Cap}_1(K) > 0$ follows from the Federer-Fleming contradiction argument of Step 5 case $p = 1$: if $\operatorname{Cap}_1(K) = 0$ then $\mathcal{H}^{n - 1}(K) = 0$, but $\mathcal{H}^{n - 1}(\partial B(0, 1)) = n \omega_n > 0$, contradiction. Therefore $\operatorname{Cap}_1(K) > 0$ while $\mathcal{H}^{n - 1}(K) < \infty$. (Note: this is converse failure at the *finite* level, i.e., $\mathcal{H}^{n-1}$ does not vanish despite being finite — the converse "$\mathcal{H}^{n - 1}(K) < \infty \Rightarrow \operatorname{Cap}_1(K) = 0$" fails because $\mathcal{H}^{n - 1}(K) > 0$ already gives positive capacity. The genuine borderline test is: positive but finite $\mathcal{H}^{n - 1}$ does not force $\operatorname{Cap}_1 = 0$.) *Case $p = n$ converse-failure proof.* The endpoint $s = 0$ is anomalous because $\mathcal{H}^0$ is counting measure, so positivity of $\mathcal{H}^s$ for any $s > 0$ already forces $\operatorname{Cap}_n > 0$ by Step 5. The converse-failure test at $s = 0$ asks: is there a compact $K$ with $\mathcal{H}^0(K) > 0$ (i.e., $K \ne \emptyset$) and $\operatorname{Cap}_n(K) > 0$? Take $K = \overline{B(0, 1)} \subset \mathbb{R}^n$. Then $\mathcal{H}^0(K) = \infty > 0$. To see $\operatorname{Cap}_n(K) > 0$: by Maz'ya §10.1.5, for any compact $K$ with $\mathcal{L}^n(K) > 0$, \begin{align*} \operatorname{Cap}_n(K) \ge c_n \mathcal{L}^n(K)^{(n - 1)/n} = c_n \omega_n^{(n - 1)/n} > 0. \end{align*} The proof: any admissible $w \in W^{1, n}$ with $w \ge 1$ on a neighbourhood of $K$ satisfies $\int_K |w|^n d\mathcal{L}^n \ge \mathcal{L}^n(K)$. By the Sobolev-Poincaré inequality on a fixed bounded domain containing $K$, $\|w\|_{L^n(K)} \le C_n \|\nabla w\|_{L^n(\mathbb{R}^n)}$ after subtracting an average — but more directly, the embedding $W^{1, n}_0(\Omega) \hookrightarrow L^n(\Omega)$ for $\Omega = B(0, 2)$ gives $\|w\|_{L^n(K)}^n \le C_n \|\nabla w\|_{L^n}^n$ provided $w$ vanishes outside $\Omega$ (which can be arranged by truncation without increasing capacity). Hence $\|\nabla w\|_{L^n}^n \ge C_n^{-1} \mathcal{L}^n(K)$, giving the lower bound. Conversely, that the threshold $s = 0$ is non-trivial — i.e., there exist compact $K$ with $\mathcal{H}^0(K) > 0$ but $\operatorname{Cap}_n(K) = 0$ — is witnessed by $K = \{x_0\}$ a single point in $\mathbb{R}^n$, $n \ge 2$: $\mathcal{H}^0(\{x_0\}) = 1$ yet $\operatorname{Cap}_n(\{x_0\}) = 0$ (Adams-Hedberg, Cor 5.1.15). This complements the converse-failure witness: the implication "$\mathcal{H}^0(K) > 0 \Rightarrow \operatorname{Cap}_n(K) > 0$" can fail (single point) and the converse "$\operatorname{Cap}_n(K) > 0 \Rightarrow \mathcal{H}^0(K) = 0$" obviously fails (closed ball). So at $p = n$ neither direction of the borderline implication holds at $s = 0$.[/guided]