I would like to take the space of this post to prove some Sobolev space estimates that I have encountered a few times, but whose proofs I do not yet know. I decided to spend last night working through the details and proving them myself.

The following estimates are a bit more tricky, because the compactness results used to derive them (by contradiction) fail at the Sobolev conjugate. However, the results themselves are still true; they just require a bit more care to prove.

Proposition: Let \(p \geq 1\), and take any ball \(B \subseteq \mathbb{R}^d\) and \(u \in W^{1, p}(B)\). Then the following endpoint Poincaré-Sobolev inequalities hold:

If \(1 \leq p < d\), then writing \(p^*\) to be the Sobolev conjugate (defined by \(\frac{1}{p^*} = \frac{1}{p} \ – \, \frac{1}{d}\), \(1 < p^* < d\)), we have $$\|u \ – \, u_B\|_{L^{p^*}(B)} \lesssim_{p, d} \|\nabla u\|_{L^p(B)}.$$

If \(p = d\), then $$\lambda^d(B)^{- \frac{1}{q}} \|u \ – \, u_B\|_{L^q(B)} \lesssim_{q, d} [u]_{\text{BMO}(B)} \lesssim_d \|\nabla u\|_{L^d(B)}$$ for all \(1 \leq q < \infty\). (However, the constant blows up as \(q\) grows large.)

If \(p > d\), then $$\|u \ – \, u_B\|_{L^{\infty}(B)} \lesssim_{p, d} \lambda^d(B)^{1 \ – \, \frac{d}{p}} \|\nabla u\|_{L^p(B)}.$$

To resolve the last one quickly, we simply use the supercritical embedding \(W^{1, p}(B) \hookrightarrow C^{0, 1 \ – \, \frac{d}{p}}(B)\) for this case. (Cf. the relevant Lieb and Loss exercise, where we showed this for all convex domains.) The second one follows by the bound $$\|u \ – \, u_{B’}\|_{L^1(B’)} \leq \|u \ – \, u_{B’}\|_{L^d(B)} (\lambda^d(B’))^{\frac{1}{d’}} \lesssim r_{B’}^{d \ – \, 1} (r_{B’} \|\nabla u\|_{L^d(B’)}),$$ where the last inequality follows by $$\frac{1}{\lambda^d(B’)} \int_{B’} |u(x) \ – \, u(y)| \, dy \leq \text{I}_0 + \text{I}_1 \\ = \frac{1}{\lambda^d(B’)} \text{diam}(B’) \int_{B’} \bigg( \int_{0}^{1 / 2} + \int_{1 / 2}^{1} \bigg) |\nabla u|((1 \ – \, \theta) x + \theta y) \, d\theta \, dy,$$ and for \(\text{I}_1\) we integrate in the existing variable \(y\) and apply Hölder, using that \(|\theta| \gtrsim 1\) to ensure the volume factor is nonsingular, so that we get a bound of $$\text{I}_1(x) \lesssim \text{diam}(B’) \, \lambda^d(B’)^{\frac{1}{p’} \ – \, 1} \|\nabla u\|_{L^p(B’)} \simeq \text{diam}(B’) \lambda^d(B)^{- \frac{1}{p}} \|\nabla u\|_{L^p(B’)}$$ on \(B’\) for \(p = d\). This shows $$\|\text{I}_1\|_{L^p(B’)} = \|\text{I}_1\|_{L^d(B’)} \lesssim \lambda^d(B’)^{\frac{1}{p}} \, \text{diam}(B’) \lambda^d(B)^{- \frac{1}{p}} \|\nabla u\|_{L^p(B’)} = r_{B’} \|\nabla u\|_{L^p(B’)} = r_{B’} \|\nabla u\|_{L^d(B’)},$$ as desired. For the first term, we use Minkowski’s integral inequality to take the \(L^d\) norm in \(x\), and using that \(|1 \ – \, \theta| \gtrsim 1\) to ensure that we obtain no singular factors when changing variables. For this we get $$\|\text{I}_0\|_{L^p(B’)} \lesssim \text{diam}(B’) \frac{1}{\lambda^d(B’)} \int_{0}^{1 / 2} \int_{B’} \|\nabla u\|_{L^p(B’)} \, dy \, d\theta \simeq r_{B’} \|\nabla u\|_{L^d(B’)},$$ which also recovers what we seek. (I first learned this trick, splitting at \(\theta = \frac{1}{2}\) in the integral, from a set of French lecture notes somewhere; the reference eludes me at the moment, unfortunately.)

This gives a \(\text{BMO}\) estimate for \(u\) in \(B\), and by John-Nirenberg, we obtain the equivalence for all \(q\) and hence the bound.

It remains to show the subcritical range. I learned of the following argument through this answer on MSE: we begin with the simplest nontrivial Sobolev extension, $$\|w\|_{L^{p^*}(B_1)} \lesssim \|w\|_{L^p(B_1)} + \|\nabla w\|_{L^p(B_1)},$$ where \(B_1 = B(0, 1)\) denotes the unit ball at the origin. Then, by considering \(x \mapsto w(x_0 + r x)\), we recover $$r^{- \frac{d}{p^*}} \|w\|_{L^{p^*}(B)} \lesssim r^{- \frac{d}{p}} \|w\|_{L^p(B)} + r \cdot r^{- \frac{d}{p}} \|\nabla w\|_{L^p(B)},$$ for any \(B = B(x_0, r)\). In particular, if we substitute \(w = u \ – \, u_B\), we get $$\|u \ – \, u_B\|_{L^{p^*}(B)} \lesssim r_B^{- 1} \|u \ – \, u_B\|_{L^p(B)} + \|\nabla (u \ – \, u_B)\|_{L^p(B)} \lesssim r_B^{- 1} r_B \|\nabla u\|_{L^p(B)} + \|\nabla u\|_{L^p(B)},$$ which recovers what we seek.

We can also consider the weighted versions of these inequalities.

Theorem: Let \(1 \leq p < d\), and take \(w \in A_p(\mathbb{R}^d)\). Then for all balls \(B \subseteq \mathbb{R}^d\) and all \(f \in C^{\infty}(B)\), $$\|f \ – \, f_B\|_{L^p(B, w)} \lesssim r_B \|\nabla f\|_{L^p(B, w)}.$$

Let’s begin with the \(p > 1\) case separately. Then we have, for every \(x \in B\), $$f(x) \ – \, f_B = \frac{1}{\lambda^d(B)} \int_{- x + B} f(x) \ – \, f(x \ – \, h) \, dh = \frac{1}{\lambda^d(B)} \int_{- B + x} \int_{0}^{1} h \cdot (\nabla f)(x \ – \, \theta h) \, d\theta \, dh.$$ We further write this as $$\frac{1}{\lambda^d(B)} \int_{- B + x} \int_{0}^{1} h \cdot (\nabla f)(x \ – \, \theta h) \chi_B(x \ – \, \theta h) \, d\theta \, dh$$ (because, by convexity, \(x \ – \, h \in B\) implies \(x \ – \, \theta h\) for all \(\theta \in [0, 1]\), so only values of \(\nabla f\) with arguments in \(B\) matter), then estimate the size of this as $$\leq \frac{1}{\lambda^d(B)} \int_{0}^{2 r_B} \int_{S^{d \ – \, 1}} \int_{0}^{1} r |\nabla f|(x \ – \, \theta r \omega) \chi_B(x \ – \, \theta r \omega) \, d\theta \, d\sigma \, r^{d \ – \, 1} \, dr \\ = \frac{1}{\lambda^d(B)} \int_{0}^{2 r_B} \int_{S^{d \ – \, 1}} \int_{0}^{r} |\nabla f|(x \ – \, \theta \omega) \chi_B(x \ – \, \theta \omega) \, d\theta \, d\sigma \, r^{d \ – \, 1} \, dr,$$ and we control this by $$\leq \frac{1}{\lambda^d(B)} \bigg( \int_{0}^{2 r_B} r^{d \ – \, 1} \, dr \bigg) \int_{S^{d \ – \, 1}} \int_{0}^{\infty} \chi_B(x \ – \, \theta \omega) |\nabla f|(x \ – \, \theta \omega) \, d\theta \, d\sigma,$$ which is $$\lesssim_d \int_{S^{d \ – \, 1}} \int_{0}^{\infty} \theta^{- (d \ – \, 1)} (\chi_B |\nabla f|)(x \ – \, \theta \omega) \theta^{d \ – \, 1} \, d\theta \, d\sigma = \int_{\mathbb{R}^d} |y|^{- (d \ – \, 1)} (\chi_B |\nabla f|)(x \ – \, y) \, dy.$$

We recognize this to be $$(| \cdot |^{- (d \ – \, 1)} * (\chi_B |\nabla f|))(x) = (I_1(\chi_B |\nabla f|))(x).$$

Now, suppose \(p > 1\); on \(B\), we have that $$I_1(\chi_B |\nabla f|) = | \cdot |^{- (d \ – \, 1)} * (\chi_B |\nabla f|) = (| \cdot |^{- (d \ – \, 1)} \chi_{2 B^*}) * (\chi_B |\nabla f|),$$ where if \(B = B(c_B, r_B)\), then \(B^* = B(0, r_B)\), and so \(2 B^* = B \ – \, B\). It follows from the radial majorant lemma that this is then $$|f \ – \, f_B| \lesssim r_B M(\chi_B |\nabla f|)$$ on \(B\), and hence $$\int_B |f \ – \, f_B|^p \, w \, d\lambda^d \lesssim \int_B r_B^p |M(\chi_B |\nabla f|)|^p \, w \, d\lambda^d \lesssim r_B^p \int_{\mathbb{R}^d} |\chi_B |\nabla f||^p \, w \, d\lambda^d = (r_B \|\nabla f\|_{L^p(B, w)})^p,$$ by the properties of Muckenhoupt weights for \(p > 1\).

For \(p = 1\), we simply note that $$\int_B |f \ – \, f_B| \, w \, d\lambda^d \lesssim \int_B I_1(|\nabla f|) \, w \, d\lambda^d = \int_B w(x) \int_B |\nabla f|(y) |x \ – \, y|^{- (d \ – \, 1)} \, dy \, dx,$$ and this is $$\int_B |\nabla f|(y) \int_B w(x) |y \ – \, x|^{- (d \ – \, 1)} \, dx \, dy = \int_B |\nabla f|(y) \int_{\mathbb{R}^d} w(x) \chi_{- B + y}(y \ – \, x) |y \ – \, x|^{- (d \ – \, 1)} \, dx \, dy,$$ which is $$\leq \int_B |\nabla f|(y) \int_{\mathbb{R}^d} w(x) \chi_{2 B^*}(y \ – \, x) |y \ – \, x|^{- (d \ – \, 1)} \, dx \, dy = \int_B |\nabla f|(y) (w * (| \cdot |^{- (d \ – \, 1)} \chi_{2 B^*}))(y) \, dy$$ by the radial majorant lemma once more, or $$\lesssim \int_B |\nabla f|(y) r_B (M w)(y) \, dy \lesssim r_B \int_B |\nabla f|(y) w(y) \, dy,$$ by the definition (\(M w \lesssim w\)) of \(A_1\) weights.

We also now note that $$|f_B \ – \, f_{B, w}| = \frac{1}{w(B)} \int_B |f \ – \, f_B| \, w \, d\lambda^d \leq \frac{1}{w(B)} \|f \ – \, f_B\|_{L^p(B, w)} w(B)^{\frac{1}{p’}},$$ and so $$\|f_B \ – \, f_{B, w}\|_{L^p(B, w)} \leq \|f \ – \, f_B\|_{L^p(B, w)}.$$ Hence we also get the alternative bound $$\|f \ – \, f_{B, w}\|_{L^p(B, w)} \lesssim r_B \|\nabla f\|_{L^p(B, w)}.$$ Moreover, from the doubling property of \(w \in A_p(\mathbb{R}^d)\), we can also see that the full-average \(f_{B, w} = \frac{1}{w(B)} \int_B f \, w \, d\lambda^d\) can be replaced by an average \(f_{B’, w}\) over any interior ball \(B’ \subseteq B\) with comparable size \(r_{B’} \gtrsim r_B\).