since $I^2 = I$ and $g \to g_I$ maps the set $\mathcal{M}$ to itself in a one-to-one and onto fashion.

Another and perhaps easier way to see the conformal invariance is to pull the problem up to the sphere via the stereographic projection, as discussed in Section 2.1. On the sphere the inversion $I$ takes the form of the reflection $(s_1, \dots, s_d, s_{d+1}) \to (s_1, \dots, s_d, -s_{d+1})$, which clearly leaves the functional on the sphere unchanged.

A second ingredient for the construction of the discrete symmetrization flow is the technique of ‘competing symmetries’, invented in [25]. Consider any nonnegative function $f \in \dot{H}^1(\mathbb{R}^d)$ and its counterpart $F \in H^1(\mathbb{S}^d)$ given by (8). Set

$$ (UF)(\omega) = F(\omega_1, \omega_2, \dots, \omega_{d+1}, -\omega_d), $$

which corresponds to a rotation by $\pi/2$ that maps the ‘north pole’ axis $(0, 0, \dots, 1)$ to $(0, \dots, 1, 0)$. Reversing (8) the function on $\mathbb{R}^d$ that corresponds to $UF$ is given by

$$ (28) \quad (Uf)(x) = \left( \frac{2}{|x - e_d|^2} \right)^{\frac{d-2}{2}} f \left( \frac{x_1}{|x - e_d|^2}, \dots, \frac{x_{d-1}}{|x - e_d|^2}, \frac{|x|^2 - 1}{|x - e_d|^2} \right), $$

where $e_d = (0, \dots, 0, 1) \in \mathbb{R}^d$. It follows that

$$ \mathcal{E}(Uf) = \mathcal{E}(f). $$

The operation $U$ is obviously linear, invertible and an isometry on $L^{2^*}(\mathbb{R}^d)$.

We also consider the symmetric decreasing rearrangement

$$ \mathcal{R}f(x) = f^*(x). $$

The most important properties are that $f$ and $f^*$ are equimeasurable and that $\|\nabla f^*\|_2 \le \|\nabla f\|_2$. For elementary properties of rearrangements the reader may consult [64]. Being equimeasurable, this map is also an isometry on $L^{2^*}(\mathbb{R}^d)$. It is when using the decreasing rearrangement that we use the fact that $f$ is a nonnegative function. For functions that change sign one conventionally defines their rearrangement as the rearrangement of their absolute value. Passing from a function to its absolute value does not alter the numerator of $\mathcal{E}(f)$ but may decrease the denominator so that other arguments are needed.

On $\mathbb{R}^d$, let

$$ (29) \qquad g_*(x) := |\mathbb{S}^d|^{-\frac{d-2}{2d}} \left( \frac{2}{1 + |x|^2} \right)^{\frac{d-2}{2}}. $$
APPENDIX A. SOME PROPERTIES OF CONTINUOUS REARRANGEMENT

In this subsection we discuss several aspects of the continuous rearrangement and prove some of its properties.

Brock's continuous Steiner rearrangement is based on the following operation for functions of one real variable that are finite union of disjoint characteristic functions $\sum_{k=1}^{N} \chi_{(-a_k, a_k)}(x - b_k)$. Replace this function by $\sum_{k=1}^{N} \chi_{(-a_k, a_k)}(x - e^{-t} b_k)$ where $t$ varies from 0 to $\infty$. As $t$ increases, the intervals start moving closer and as soon as any two intervals touch one stops the process and redefines the set of intervals by joining the two that touched. Then one restarts the process and keeps repeating it until all of them are joined into one. The movement stops once this interval is centered at the origin. By the outer regularity of Lebesgue measure the level sets of a measurable function can be approximated by open sets and, since in one dimension this is a countable union of open intervals, one can further approximate the level set by a finite number of open disjoint intervals for which one uses the sliding argument explained above.

As mentioned before, this procedure can be generalized to higher dimensions by considering Steiner symmetrization with respect to a hyperplane. One considers any hyperplane $H$ through the origin and then rearranges the function symmetrically about the hyperplane along each line perpendicular to $H$, resulting in a function denoted by $f^{*H}$. For more information see [57]. In this fashion one obtains a continuous rearrangement $f \to f_{\tau}^{H}$, $\tau \in [0, \infty]$, which was studied in detail by Brock [15, 16]. We shall refer to the statements in those papers.

To pass from Steiner symmetrization to the symmetric decreasing rearrangement we consider a sequence of continuous Steiner symmetrizations and chain them with a new continous parameter $la$ Bucur-Henrot. Inspired by [17, 27], we proceed as follows. Given a function $f \in L^p(\mathbb{R}^d)$ for some $1 \le p < \infty$ there is a sequence $(H_n)_{n \in \mathbb{N}}$ of hyperplanes such that, defining recursively with $f_0 = f$,

$$f_n := f_{n-1}^{*H_n}, \quad n = 1, 2, \dots,$$

we have

$$f_n \to f^* \quad \text{in } L^p(\mathbb{R}^d) \quad \text{as} \quad n \to \infty.$$

In fact, it is shown in [70, Theorem 4.3] that this holds for 'almost every' (in an appropriate sense) choice of hyperplanes. It is also of interest that this sequence can actually be chosen in a universal fashion (that is, independent of $f$ and $p$); see [68, Theorem 5.2].

Given $f$ and the sequence $(f_n)_{n \in \mathbb{N}}$ as above, we set for any $n = 0, 1, 2, \dots$

$$\phi_n(\tau) := e^{\frac{\tau-n}{n+1-\tau}} - 1, \quad \tau \in [n, n+1],$$

and define

$$f_{\tau} := f_{n, \phi_{n}(\tau)}, \qquad (66)$$

where the right side denotes Brock's continuous Steiner symmetrization with respect to the hyperplane $H_n$ with parameter $\phi_n(\tau)$ applied to $f_n$. As $\tau$ runs from $n$ to $n+1$, $\phi_n(\tau)$ runs from 0 to $\infty$, so $f_{\tau}$ is well defined even for $\tau \in \mathbb{N}_0$.

From the properties of Brock's flow, see, in particular, [16, Lemma 4.1], we obtain the following properties for our flow.

**Proposition A.1.** Let $d \ge 1$, $1 \le p < \infty$ and let $0 \le f \in L^p(\mathbb{R}^d)$. Then, for any $\tau \in [0, \infty]$, the function $f_{\tau}$ defined by (66) is in $L^p(\mathbb{R}^d)$ and $\|f_{\tau}\|_p = \|f\|_p$. Moreover, for any $\tau \in [0, \infty]$ and any sequence $(\tau_n)_{n \in \mathbb{N}}$ with $\lim_{n \to \infty} \tau_n = \tau$,

$$\lim_{n \to \infty} \|f_{\tau_n} - f_{\tau}\|_p = 0.$$

The following fact is important for us.
Note that $||g_*||_{2^*} = 1$ because it is obtained as the stereographic projection of the constant function on $\mathbb{S}^d$ with $2^*$-norm equal to 1. The following theorem was proved in [25].

**Theorem 3.2.** Let $f \in L^{2^*}(\mathbb{R}^d)$ be a nonnegative function. Consider the sequence $(f_n)_{n \in \mathbb{N}}$ of functions

$$ (30) \qquad f_n = (\mathcal{R}U)^n f \quad \forall n \in \mathbb{N}. $$

Then

$$ \lim_{n \to \infty} \|f_n - h_f\|_{2^*} = 0 $$

where $h_f = \|f\|_{2^*} g_* \in \mathcal{M}$. Moreover, if $f \in \dot{\mathbb{H}}^1(\mathbb{R}^d)$, then $(\|\nabla f_n\|_{2^*})_{n \in \mathbb{N}}$ is a nonincreasing sequence.

It does not seem clear whether the functional $\mathcal{E}(f)$ decreases or increases under rearrangement. The next lemma helps to explain this point. Define $\mathcal{M}_1$ to be the set of the elements in $\mathcal{M}$ with $2^*$-norm equal to 1.

**Lemma 3.3.** For any $f \in \dot{\mathbb{H}}^1(\mathbb{R}^d)$, we have

$$ \mathrm{dist}(f, \mathcal{M})^2 = \inf_{g \in \mathcal{M}} \|\nabla f - \nabla g\|_2^2 = \|\nabla f\|_2^2 - S_d \sup_{g \in \mathcal{M}_1} (f, g^{2^*-1})^2. $$

Here and in the sequel, $(\cdot, \cdot)$ is the $L^2(\mathbb{R}^d)$ inner product or, more precisely, the duality pairing between $L^{2^*}(\mathbb{R}^d)$ and $L^{(2^*)'}(\mathbb{R}^d)$.

*Proof.* Let $g$ be any Aubin-Talenti function. The function $g$ is an optimizer of the Sobolev inequality, i.e., $\|\nabla g\|_2^2 = S_d \|g\|_{2^*}^2 = S_d$ and is a solution of the Sobolev equation

$$ (31) \qquad - \Delta g = S_d \frac{g^{2^*-1}}{\|g\|_{2^*}^{2^*-2}} = S_d g^{2^*-1}. $$

Hence for any nonnegative constant $c$, if $\|g\|_{2^*} = 1$, we find

$$ \begin{aligned} \|\nabla(f - c g)\|_2^2 &= \|\nabla f\|_2^2 - 2c(\nabla f, \nabla g) + c^2 \|\nabla g\|_2 \\ &= \|\nabla f\|_2^2 - 2c S_d (f, g^{2^*-1}) + S_d c^2 \end{aligned} $$

and minimizing with respect to $c$ we find the lower bound $\|\nabla f\|_2^2 - S_d (f, g^{2^*-1})^2$, which proves the lemma. $\square$
**Lemma A.2.** Let $d \ge 3$ and $0 \le f \in L^{2^*}(\mathbb{R}^d)$. The function

$$ \tau \mapsto \sup_{u \in \mathcal{M}_1} \left( f_{\tau}, u^{2^*-1} \right)^2 $$

with $f_{\tau}$ defined by (66) is continuous.

*Proof.* We use the fact, shown in Proposition A.1, that

$$ \lim_{\tau_1 \to \tau_2} \|f_{\tau_1} - f_{\tau_2}\|_{2^*} = 0. $$

Fix $\varepsilon > 0$. There exists $u_1 \in \mathcal{M}_1$ such that $\sup_{u \in \mathcal{M}_1} |(f_{\tau_1}, u^{2^*-1})| \le |(f_{\tau_1}, u_1^{2^*-1})| + \varepsilon$ and hence

$$ \sup_{u \in \mathcal{M}_1} |(f_{\tau_1}, u^{2^*-1})| - \sup_{u \in \mathcal{M}_1} |(f_{\tau_2}, u^{2^*-1})| \le |(f_{\tau_1}, u_1^{2^*-1})| + \varepsilon - |(f_{\tau_2}, u_1^{2^*-1})| \\ \le |(f_{\tau_1}, u_1^{2^*-1}) - (f_{\tau_2}, u_1^{2^*-1})| + \varepsilon, $$

which by Hölder's inequality is bounded above by

$$ \|f_{\tau_1} - f_{\tau_2}\|_{2^*} \|u_1^{2^*-1}\|_q + \varepsilon = \|f_{\tau_1} - f_{\tau_2}\|_{2^*} + \varepsilon $$

with $q = \frac{2^*}{2^*-1}$. Hence

$$ \limsup_{\tau_2 \to \tau_1} \left( \sup_{u \in \mathcal{M}_1} |(f_{\tau_1}, u^{2^*-1})| - \sup_{u \in \mathcal{M}_1} |(f_{\tau_2}, u^{2^*-1})| \right) \le \varepsilon. $$

There exists $u_2 \in \mathcal{M}_1$ such that $\sup_{u \in \mathcal{M}_1} |(f_{\tau_2}, u^{2^*-1})| \le |(f_{\tau_2}, u_2^{2^*-1})| + \varepsilon$ and hence

$$ \sup_{u \in \mathcal{M}_1} |(f_{\tau_1}, u^{2^*-1})| - \sup_{u \in \mathcal{M}_1} |(f_{\tau_2}, u^{2^*-1})| \ge |(f_{\tau_1}, u_2^{2^*-1})| - |(f_{\tau_2}, u_2^{2^*-1})| - \varepsilon, $$

which is greater or equal to

$$ - |(f_{\tau_1}, u_2^{2^*-1}) - (f_{\tau_2}, u_2^{2^*-1})| - \varepsilon \ge - \|f_{\tau_1} - f_{\tau_2}\|_{2^*} - \varepsilon. $$

Hence

$$ \liminf_{\tau_2 \to \tau_1} \left( \sup_{u \in \mathcal{M}_1} |(f_{\tau_1}, u^{2^*-1})| - \sup_{u \in \mathcal{M}_1} |(f_{\tau_2}, u^{2^*-1})| \right) \ge -\varepsilon. $$

This proves the claimed continuity. $\square$

<figure><img src="image_8.png" /></figure>

We now consider the behavior of the gradient under the rearrangement flow. The following proposition is closely related to [16, Theorems 3.2 and 4.1], but there inhomogeneous Sobolev spaces are considered, which leads to some minor changes. For the sake of simplicity we provide the details.

**Proposition A.3.** Let $0 \le f \in \dot{H}^1(\mathbb{R}^d)$. Then $f_{\tau}$ defined by (66) is in $\dot{H}^1(\mathbb{R}^d)$ and $\tau \mapsto \|\nabla f_{\tau}\|_2$ is a nonincreasing, right-continuous function.

*Proof.* By construction, it suffices to prove these properties for Brock's flow. Since the latter has the semigroup property $(f_{\sigma})_{\tau} = f_{\sigma+\tau}$ for all $\sigma, \tau \ge 0$, it suffices to prove monotonicity and right-continuity at $\tau = 0$.

We begin with the proof of monotonicity, which we first prove under the additional assumption that $f \in L^2(\mathbb{R}^d)$. This is shown in [16, Theorem 3.2], but we give an alternative proof. We proceed as in the proof of [57, Lemma 1.17]. Extending [15, Corollary 2] to the sequence of Steiner symmetrizations we find for three nonnegative functions $f, g, h$ that

$$ \iint_{\mathbb{R}^d \times \mathbb{R}^d} f_{\tau}(x) g_{\tau}(x-y) h_{\tau}(y) dx dy \ge \iint_{\mathbb{R}^d \times \mathbb{R}^d} f(x) g(x-y) h(y) dx dy. $$
In the above proof, notice that the optimal value of c is such that $c = (f, g^{2*-1}) \le \|f\|_{L^{2^*(\mathbb{R}^d)}}$ and elementary considerations on $M_1$ show that $c > 0$ if $f \ne 0$ is nonnegative, so that $\text{dist}(f, M)^2 < \|\nabla f\|_2^2$. With the notation of (1) and $g_{a,b}(x) := (a|\mathbb{S}^d|^{1/d})^{-(d-2)/2} \bar{g}((x-b)/a)$, we find that $(f, g_{a,b}^{2*-1})$ converges to 0 as $a + a^{-1} + |b| \to +\infty$. A sequence $(a_n, b_n, c_n)$ such that

$$ \lim_{n \to +\infty} \|\nabla(f - c_n g_{a_n, b_n})\|_{L^2(\mathbb{R}^d)}^2 = \text{dist}(f, M)^2 $$

is therefore relatively compact in $(0, +\infty) \times \mathbb{R}^d \times \mathbb{R}$, which proves that $\text{dist}(f, M)$ is attained at some Aubin-Talenti function of the form (1).

We note that, under the decreasing rearrangement, the term $\|\nabla f\|_2^2$ does not increase whereas the term $\sup_{g \in M_1} (f, g^{2*-1})^2$ increases. To see this, note that $\text{dist}(f, M)$ is attained at some Aubin-Talenti function which is a strictly symmetric decreasing function about some point $b \in \mathbb{R}^d$. Replacing $f$ by its symmetric decreasing rearrangement about that point increases $(f, g^{2*-1})^2$, in fact strictly unless $f$ is already symmetric decreasing about the point $b$. Thus, while the numerator in $\mathcal{E}(f)$ decreases under rearrangements, so does the denominator and there are no direct conclusions to be drawn from this. The next lemma summarizes what we have shown.

**Lemma 3.4.** For the sequence $(f_n)_{n \in \mathbb{N}}$ in Theorem 3.2 we have that $n \mapsto \sup_{g \in M_1} (f_n, g^{2*-1})^2$ is strictly increasing, $n \mapsto \inf_{g \in M} \|\nabla f_n - \nabla g\|_{2^*}^2$ is strictly decreasing and, with $h_f = \|f\|_{2^*} g_*$ as in Theorem 3.2,

$$ \liminf_{n \to \infty} \|\nabla f_n - \nabla g\|_2^2 = \lim_{n \to \infty} \|\nabla f_n\|_2^2 - S_d \|h_f\|_{2^*}^2 \\ = \lim_{n \to \infty} \|\nabla f_n\|_2^2 - S_d \|f\|_{2^*}^2. $$

*Proof.* From

$$ \inf_{g \in M} \|\nabla f_n - \nabla g\|_2^2 = \|\nabla f_n\|_2^2 - S_d \sup_{g \in M_1} (f_n, g^{2*-1})^2 $$

we see that the first term converges since $(\|\nabla f_n\|_2^2)_{n \in \mathbb{N}}$ is a nonincreasing sequence. For the second term, which is strictly increasing, we have by Hölder's inequality

$$ \sup_{g \in M_1} (f_n, g^{2*-1})^2 \le \|f_n\|_{2^*}^2 = \|f\|_{2^*}^2 $$

and since $g_*$ as defined in (29) is in $M_1$ we have

$$ \liminf_{n \to \infty} \sup_{g \in M_1} (f_n, g^{2*-1})^2 \ge \liminf_{n \to \infty} (f_n, g_*^{2*-1})^2 = \|f\|_{2^*}^2 $$
If we choose $g(x-y)$ to be the standard heat kernel, i.e., $g(x-y) = e^{\Delta t}(x-y)$, then $g_{\tau}(x-y) = g(x-y)$ and hence

$$ \iint_{\mathbb{R}^d \times \mathbb{R}^d} f_{\tau}(x) e^{\Delta t} (x-y) f_{\tau}(y) dx dy \geq \iint_{\mathbb{R}^d \times \mathbb{R}^d} f(x) e^{\Delta t} (x-y) f(y) dx dy. $$

Since $\|f_{\tau}\|_2 = \|f\|_2$ by the equimeasurability of rearrangement,

$$ \frac{1}{t} (\|f_{\tau}\|_2^2 - (f_{\tau}, e^{\Delta t} f_{\tau})) \leq \frac{1}{t} (\|f\|_2^2 - (f, e^{\Delta t} f)) $$

and letting $t \to 0$ yields the first claim under the additional assumption $f \in L^2(\mathbb{R}^d)$.

For general $0 \le f \in \dot{H}^1(\mathbb{R}^d)$ we apply the above argument to the functions $(f - \epsilon)_+$, $\epsilon > 0$. They belong to $L^2(\mathbb{R}^d)$ since $f$ vanishes at infinity and belongs to $L^{2^*}(\mathbb{R}^d)$. We obtain

$$ \|\nabla((f - \epsilon)_+)\|_2 \le \|\nabla(f - \epsilon)_+\|_2 \le \|\nabla f\|_2. \qquad (67) $$

We claim that $f_{\tau} \in \dot{H}^1(\mathbb{R}^d)$ and $\nabla((f - \epsilon)_+)_{\tau} \to \nabla f_{\tau}$ in $L^2(\mathbb{R}^d)$ as $\epsilon \to 0^+$. Once this is shown, the claimed inequality follows from (67) by the weak lower semicontinuity of the $L^2$ norm.

To prove the claimed weak convergence, note that by (67), $\nabla((f - \epsilon)_+)_{\tau}$ is bounded in $L^2(\mathbb{R}^d)$ as $\epsilon \to 0^+$ and therefore has a weak limit point. Let $F \in L^2(\mathbb{R}^d)$ be any such limit point. Since $(f - \epsilon)_+ \to f$ in $L^{2^*}(\mathbb{R}^d)$, the nonexpansivity of the rearrangement [15, Lemma 3] implies that $((f - \epsilon)_+)_{\tau} \to f_{\tau}$ in $L^{2^*}(\mathbb{R}^d)$. Thus, for any $\Phi \in C_c^1(\mathbb{R}^d)$,

$$ \int_{\mathbb{R}^d} (\nabla \cdot \Phi) f_{\tau} dx \leftarrow \int_{\mathbb{R}^d} (\nabla \cdot \Phi) ((f - \epsilon)_+)_{\tau} dx = - \int_{\mathbb{R}^d} \Phi \cdot \nabla ((f - \epsilon)_+)_{\tau} dx \to - \int_{\mathbb{R}^d} \Phi \cdot F dx $$

as $\epsilon \to 0^+$. This proves that $f_{\tau}$ is weakly differentiable with $\nabla f_{\tau} = F$. In particular, $f_{\tau} \in \dot{H}^1(\mathbb{R}^d)$ (note that $f_{\tau}$ vanishes at infinity since $f$ does and since these functions are equimeasurable) and the limit point $F$ is unique. This concludes the proof of the first part of the proposition.

Let us now show the right-continuity at $\tau = 0$. It follows from Proposition A.1 that $f_{\tau} \to f$ in $L^{2^*}(\mathbb{R}^d)$ as $\tau \to 0^+$. This implies that $\nabla f_{\tau} \to \nabla f$ in $L^2(\mathbb{R}^d)$ as $\tau \to 0^+$. (Indeed, the argument is similar to the one used in the first part of the proof. The family $\nabla f_{\tau}$ is bounded in $L^2(\mathbb{R}^d)$ as $\tau \to 0^+$ and, if $F$ denotes any weak limit point in $L^2(\mathbb{R}^d)$, then the convergence in $L^{2^*}(\mathbb{R}^d)$ and the definition of weak derivatives implies that $F = \nabla f$.) By weak lower semicontinuity, we deduce that

$$ \|\nabla f\|_2 \le \liminf_{\tau \to 0^+} \|\nabla f_{\tau}\|_2. $$

This, together with the reverse inequality, which was established in the first part of the proof, proves the claimed right continuity. $\square$

We note that the proposition remains valid for $0 \le f \in \dot{W}^{1,p}(\mathbb{R}^d)$ with $1 \le p < d$. If $p \ne 2$, the monotonicity for the gradient for $f \in W^{1,p}(\mathbb{R}^d)$ is proved in [16, Theorem 3.2]. The remaining arguments above carry over to $p \ne 2$.

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by Theorem 3.2.

**Lemma 3.5.** Assume that $0 \le f \in \dot{H}^1(\mathbb{R}^d) \setminus \mathcal{M}$ satisfies

$$ \inf_{g \in \mathcal{M}} \|\nabla f - \nabla g\|_2^2 \ge \delta \|\nabla f\|_2^2 $$

and let $(f_n)_{n \in \mathbb{N}}$ be the sequence defined by (30). Then one of the following alternatives holds:

(a) for all $n = 0, 1, 2, \dots$ we have

$$ \inf_{g \in \mathcal{M}} \|\nabla f_n - \nabla g\|_2^2 \ge \delta \|\nabla f_n\|_2^2, $$

(b) there is a natural number $n_0$ such that

$$ \inf_{g \in \mathcal{M}} \|\nabla f_{n_0} - \nabla g\|_2^2 \ge \delta \|\nabla f_{n_0}\|_2^2 $$

and

$$ \inf_{g \in \mathcal{M}} \|\nabla f_{n_0+1} - \nabla g\|_2^2 < \delta \|\nabla f_{n_0+1}\|_2^2. $$

**Proof.** Assume that alternative (a) does not hold. Then there is a largest value $n_0 \ge 0$ such that $\inf_{g \in \mathcal{M}} \|\nabla f_{n_0} - \nabla g\|_2^2 \ge \delta \|\nabla f_{n_0}\|_2^2$. □

**Lemma 3.6.** Assume that $0 \le f \in \dot{H}^1(\mathbb{R}^d) \setminus \mathcal{M}$ satisfies

$$ \inf_{g \in \mathcal{M}} \|\nabla f - \nabla g\|_2^2 \ge \delta \|\nabla f\|_2^2 $$

and suppose that in Lemma 3.5 alternative (a) holds for the sequence $(f_n)_{n \in \mathbb{N}}$ defined by (30). Then

$$ \mathcal{E}(f) \ge \delta. $$

**Proof.** We have

$$ (32) \quad \mathcal{E}(f) = \frac{\|\nabla f\|_2^2 - S_d \|f\|_2^2}{\inf_{g \in \mathcal{M}} \|\nabla f - \nabla g\|_2^2} \ge \frac{\|\nabla f\|_2^2 - S_d \|f\|_2^2}{\|\nabla f\|_2^2} \\ \ge \frac{\|\nabla f_n\|_2^2 - S_d \|f\|_2^2}{\|\nabla f_n\|_2^2}, $$

where the second inequality is a consequence of $\|\nabla f_n\|_2^2 \le \|\nabla f\|_2^2$ for all $n = 0, 1, 2, \dots$ proved in Theorem 3.2. By the assumption that alternative (a) holds and by Lemma 3.4, we learn that
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$$ \lim_{n \to \infty} \|\nabla f_n\|_2^2 \le \frac{1}{\delta} \lim_{n \to \infty} \inf_{g \in \mathcal{M}} \|\nabla f_n - \nabla g\|_2^2 \\ = \frac{1}{\delta} \left( \lim_{n \to \infty} \|\nabla f_n\|_2^2 - S_d \|f\|_{2^*}^2 \right). $$

Since

$$ \lim_{n \to \infty} \|\nabla f_n\|_2^2 - S_d \|f\|_{2^*}^2 \ge \delta \lim_{n \to \infty} \|\nabla f_n\|_2^2 \\ \ge \delta S_d \lim_{n \to \infty} \|f_n\|_{2^*}^2 = \delta S_d \|f\|_{2^*}^2 > 0, $$

we can take the limit as $n \to \infty$ on the right side of (32) and compute the limit of the quotient as the quotient of the limits. This proves the lemma. □

**3.1.2. Continuous rearrangement.** Next, we analyze the case where the alternative (b) in Lemma 3.5 holds. We recall that $\mathcal{I}(\delta)$ was defined in (27).

**Lemma 3.7.** For any $\delta \in (0, 1]$, we have $\mathcal{I}(\delta) \le 1$.

*Proof.* By Lemma 3.3, we have

$$ \inf_{g \in \mathcal{M}} \|\nabla f - \nabla g\|_2^2 = \|\nabla f\|_2^2 - S_d \sup_{g \in \mathcal{M}_1} (f, g^{2^*-1})^2 $$

and it follows from Hölder's inequality that

$$ \sup_{g \in \mathcal{M}_1} (f, g^{2^*-1})^2 \le \|f\|_{2^*}^2. $$

Thus, the denominator in $\mathcal{E}(f)$ that enters the definition of $\mathcal{I}(\delta)$ is at least as large as the numerator, so the quotient is at most 1. □

Our goal in this subsection is to prove the following lower bound on $\mathcal{E}(f)$.

**Lemma 3.8.** Assume that $0 \le f \in \dot{\mathrm{H}}^1(\mathbb{R}^d) \setminus \mathcal{M}$ satisfies

$$ \inf_{g \in \mathcal{M}} \|\nabla f - \nabla g\|_2^2 \ge \delta \|\nabla f\|_2^2 $$

for some $\delta \in (0, 1/2)$ and suppose that in Lemma 3.5 alternative (b) holds for the sequence $(f_n)_{n \in \mathbb{N}}$ of Theorem 3.2 defined by (30). Then, with $\mathcal{I}(\delta)$ defined by (27), we have

$$ \mathcal{E}(f) \ge \delta \mathcal{I}(\delta). $$
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For the proof of this lemma we introduce a continuous rearrangement flow that interpolates between a function and its symmetric decreasing rearrangement. The basic ingredient for this flow is similar to a flow that Brock introduced [16, 17] and that interpolates between a function and its Steiner symmetrization with respect to a given hyperplane. Brock’s construction, in turn, is based on ideas of Rogers [71] and Brascamp-Lieb-Luttinger [12]. Our flow is obtained by glueing together infinitely many copies of Brock’s flows with respect to a sequence of judiciously chosen hyperplanes. A similar construction was performed by Bucur and Henrot [18]; see also [31].

More specifically, for a given hyperplane $H$, Brock’s flow interpolates between a given function $f$ and $f^{*H}$, the Steiner symmetrized function with respect to $H$. The family that interpolates between $f$ and $f^{*H}$ is denoted by $f_{\tau}^{H}$, $\tau \in [0, \infty]$, and we have

$$f_0 = f, \quad f_{\infty}^{H} = f^{*H}.$$

Further, for any $\tau$, $f_{\tau}^{H}$ and $f$ are equimeasurable, i.e.,

$$|\{x \in \mathbb{R}^d : f_{\tau}^{H}(x) > t\}| = |\{x \in \mathbb{R}^d : f(x) > t\}| \quad \forall t > 0.$$

Moreover, if $f \in L^p(\mathbb{R}^d)$ for some $1 \le p < \infty$, then $\tau \mapsto f_{\tau}^{H}$ is continuous in $L^p(\mathbb{R}^d)$.

By choosing a sequence of hyperplanes we construct another flow $\tau \mapsto f_{\tau}$ that has the same properties but interpolates between $f$ and $f^*$, the symmetric decreasing rearrangement. In Appendix A we explain this in more detail and prove the following properties that are important for our proof, assuming $f \in \dot{H}^1(\mathbb{R}^d)$. From the $L^2^*(\mathbb{R}^d)$ continuity of the flow we will deduce that

$$(33) \qquad \lim_{\tau \to \tau_0} \sup_{g \in \mathcal{M}_1} (f_{\tau}, g)^2 = \sup_{g \in \mathcal{M}_1} (f_{\tau_0}, g)^2.$$

Concerning the gradient we prove the monotonicity

$$\|\nabla f_{\tau_2}\|_2 \le \|\nabla f_{\tau_1}\|_2, \quad 0 \le \tau_1 \le \tau_2 \le \infty,$$

and the right continuity

$$(34) \qquad \lim_{\tau_2 \to \tau_1^+} \|\nabla f_{\tau_2}\|_2 = \|\nabla f_{\tau_1}\|_2, \quad 0 \le \tau_1 < \infty.$$

*Proof of Lemma 3.8.* We begin by motivating and explaining the strategy of the proof. As before, we bound
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(J. Dolbeault, M. J. Esteban) CEREMADE (CNRS UMR No. 7534), PSL UNIVERSITY, UNIVERSITÉ PARIS-DAUPHINE, PLACE DE LATTRE DE TASSIGNY, 75775 PARIS 16, FRANCE.

E-mails: dolbeaul@ceremade.dauphine.fr, esteban@ceremade.dauphine.fr

(A. Figalli) MATHEMATICS DEPARTMENT, ETH ZÜRICH, RAMISTRASSE 101, 8092 ZÜRICH, SWITZERLAND.

E-mail: alessio figalli@math.ethz.ch

(R. L. Frank) DEPARTMENT OF MATHEMATICS, LMU MUNICH, THERESIENSTR. 39, 80333 MÜNCHEN, GERMANY, AND MUNICH CENTER FOR QUANTUM SCIENCE AND TECHNOLOGY, SCHELLINGSTR. 4, 80799 MÜNCHEN, GERMANY, AND MATHEMATICS 253-37, CALTECH, PASADENA, CA 91125, USA.

E-mail: r.frank@lmu.de

(M. Loss) SCHOOL OF MATHEMATICS, GEORGIA INSTITUTE OF TECHNOLOGY ATLANTA, GA 30332, UNITED STATES OF AMERICA.

E-mail: loss@math.gatech.edu
$$ (35) \quad \mathcal{E}(f) = \frac{\|\nabla f\|_2^2 - S_d \|f\|_{2^*}^2}{\inf_{g \in \mathcal{M}} \|\nabla f - \nabla g\|_2^2} \ge \frac{\|\nabla f\|_2^2 - S_d \|f\|_{2^*}^2}{\|\nabla f\|_2^2} \\ \ge \frac{\|\nabla f_{n_0}\|_2^2 - S_d \|f_{n_0}\|_{2^*}^2}{\|\nabla f_{n_0}\|_2^2}. $$

We could bound the right side further from below by replacing $f_{n_0}$ by $f_{n_0+1}$. This bound, however, might be too crude for our purposes and we proceed differently. The move from $f_{n_0}$ to $f_{n_0+1}$ consists of two steps, namely first applying a conformal rotation and second applying symmetric decreasing rearrangement. The first step leaves all terms on the right side invariant and we do carry out this step. The second step leaves the $2^*$-norm invariant, while the gradient term does not go up. In fact, the gradient term might go down too far. Therefore, we replace the application of the rearrangement by a continuous rearrangement flow. We denote by $f_\tau$, $n_0 \le \tau < n_0+1$, the continuous rearrangement starting at $f_{n_0} := Uf_{n_0}$, where $U$ denotes the conformal rotation (28). The 'time' variable $\tau$ has been reparametrized so that at $\tau = n_0+1$ we have arrived at the symmetric decreasing rearrangement of $f_{n_0}$, that is,

$$ (36) \qquad f_{n_0+1} = (f_{n_0})^* = f_{n_0+1}. $$

Ideally, we would like to find $\tau_0 \in [n_0, n_0+1)$ such that

$$ \inf_{g \in \mathcal{M}} \|\nabla f_{\tau_0} - \nabla g\|_2^2 = \delta \|\nabla f_{\tau_0}\|_2^2. $$

Then the right side of (35) is equal to

$$ 1 - S_d \frac{\|f_{n_0}\|_{2^*}^2}{\|\nabla f_{n_0}\|_2^2} \ge 1 - S_d \frac{\|f_{\tau_0}\|_{2^*}^2}{\|\nabla f_{\tau_0}\|_2^2} = \delta \frac{\|\nabla f_{\tau_0}\|_2^2 - S_d \|f_{\tau_0}\|_{2^*}^2}{\inf_{g \in \mathcal{M}} \|\nabla f_{\tau_0} - \nabla g\|_2^2}, $$

which can be bounded from below by $\delta \mathcal{I}(\delta)$, since $f_{\tau_0}$ is admissible in the infimum (27). This would prove the desired bound.

The problem with this argument is that the existence of such a $\tau_0 \in [n_0, n_0+1)$ is in general not clear, since neither of the terms $\inf_{g \in \mathcal{M}} \|\nabla f_\tau - \nabla g\|_2^2$ and $\|\nabla f_\tau\|_2^2$ needs to be continuous in $\tau$. Nevertheless, we will be able to adapt the above argument to yield the same conclusion.

We now turn to the details of the argument. Recalling that

$$ \inf_{g \in \mathcal{M}} \|\nabla f_0 - \nabla g\|_2^2 \ge \delta \|\nabla f_0\|_2^2, $$
we define

$$ \tau_0 := \inf \left\{ \tau \in (n_0, n_0 + 1) : \inf_{g \in \mathcal{M}} \|\nabla f_{\tau} - \nabla g\|_2^2 < \delta \|\nabla f_{\tau}\|_2^2 \right\} $$

with the convention that $\inf \emptyset = n_0 + 1$. If $\tau < \tau_0 \in (n_0, n_0 + 1]$, similarly as before, the right side of (35) is equal to

$$ \frac{\|\nabla f_{n_0}\|_2^2 - S_d \|f_{n_0}\|_{2^*}^2}{\|\nabla f_{n_0}\|_2^2} = 1 - S_d \frac{\|f_{n_0}\|_{2^*}^2}{\|\nabla f_{n_0}\|_2^2} \ge \frac{\|\nabla f_{\tau}\|_2^2 - S_d \|f_{\tau_0}\|_{2^*}^2}{\|\nabla f_{\tau}\|_2^2} \\ \ge \delta \frac{\|\nabla f_{\tau}\|_2^2 - S_d \|f_{\tau_0}\|_{2^*}^2}{\inf_{g \in \mathcal{M}} \|\nabla f_{\tau} - \nabla g\|_2^2}, $$

where the last inequality arises from $\inf_{g \in \mathcal{M}} \|\nabla f_{\tau} - \nabla g\|_2^2 \ge \delta \|\nabla f_{\tau}\|_2^2$ for any $\tau \in [n_0, \tau_0)$. Taking the limit inferior as $\tau \to \tau_0^-$, we obtain

$$ (37) \qquad \frac{\|\nabla f_{n_0}\|_2^2 - S_d \|f_{n_0}\|_{2^*}^2}{\|\nabla f_{n_0}\|_2^2} \ge \delta \frac{\lim_{\tau \to \tau_0^-} \|\nabla f_{\tau}\|_2^2 - S_d \|f_{\tau_0}\|_{2^*}^2}{\liminf_{\tau \to \tau_0^-} \inf_{g \in \mathcal{M}} \|\nabla f_{\tau} - \nabla g\|_2^2}. $$

Note that the denominator appearing here does not vanish. Indeed, we have

$$ \inf_{g \in \mathcal{M}} \|\nabla f_{\tau} - \nabla g\|_2^2 \ge \delta \|\nabla f_{\tau}\|_2^2 \ge \delta S_d \|f_{\tau}\|_{2^*}^2 = \delta S_d \|f\|_{2^*}^2 > 0 \quad \forall \tau \in [n_0, \tau_0) $$

and, as a consequence,

$$ \liminf_{\tau \to \tau_0^-} \inf_{g \in \mathcal{M}} \|\nabla f_{\tau} - \nabla g\|_2^2 \ge \delta S_d \|f\|_{2^*}^2 > 0. $$

The same inequality (37) remains valid if $\tau_0 = n_0$ and if we interpret $\lim_{\tau \to \tau_0^-}$ and $\liminf_{\tau \to \tau_0^-}$ as evaluating at $\tau_0 = n_0$.

At this point we find it convenient to apply Lemma 3.3 and use the representation

$$ \inf_{g \in \mathcal{M}} \|\nabla f_{\tau} - \nabla g\|_2^2 = \|\nabla f_{\tau}\|_2^2 - S_d \sup_{g \in \mathcal{M}_1} (f_{\tau}, g^{2^*-1})^2. $$

Using (33), that is, the continuity of $\tau \mapsto \sup_{g \in \mathcal{M}_1} (f_{\tau}, g^{2^*-1})^2$, we see that

$$ \liminf_{\tau \to \tau_0^-} \inf_{g \in \mathcal{M}} \|\nabla f_{\tau} - \nabla g\|_2^2 = \lim_{\tau \to \tau_0^-} \|\nabla f_{\tau}\|_2^2 - S_d \sup_{g \in \mathcal{M}_1} (f_{\tau_0}, g^{2^*-1})^2. $$
Thus, the relevant quotient is equal to

$$ (38) \qquad \lim_{\tau \to \tau_0^-} \frac{\|\nabla f_\tau\|_2^2 - S_d \|f_{\tau_0}\|_{2^*}^2}{\lim_{\tau \to \tau_0^-} \|\nabla f_\tau\|_2^2 - S_d \sup_{g \in \mathcal{M}_1} (f_{\tau_0}, g^{2^*-1})^2}. $$

Our goal in the remainder of this proof is to show that this quotient is larger or equal than $\mathcal{I}(\delta)$. We will use the fact that

$$ (39) \qquad \sup_{g \in \mathcal{M}_1} (f_{\tau_0}, g^{2^*-1})^2 \le \|f_{\tau_0}\|_{2^*}^2, $$

which follows from Hölder's inequality. We also note that equality holds here if and only if $f_{\tau_0} \in \mathcal{M}$.

Let us first handle the case where $f_{\tau_0} \in \mathcal{M}$. Then by (3.1.2) and because of equality in (39), the quotient (38) is equal to 1, which by Lemma 3.7 can be further bounded from below by $\mathcal{I}(\delta)$, leading to the claimed bound. This completes the proof in the case $f_{\tau_0} \in \mathcal{M}$ and in what follows we assume

$$ f_{\tau_0} \notin \mathcal{M}. $$

As a consequence of this assumption and (39), we have

$$ (40) \qquad \|\nabla f_{\tau_0}\|_2^2 > S_d \|f_{\tau_0}\|_{2^*}^2 \ge S_d \sup_{g \in \mathcal{M}_1} (f_{\tau_0}, g^{2^*-1})^2. $$

Next, we observe that for $\alpha > \beta$ the function $x \mapsto (x-\alpha)/(x-\beta)$ is monotone increasing on the interval $(\beta, \infty)$. This, together with the strict inequality in (40), implies that the quotient (38) can be bounded from below by

$$ (41) \qquad \lim_{\tau \to \tau_0^-} \frac{\|\nabla f_\tau\|_2^2 - S_d \|f_{\tau_0}\|_{2^*}^2}{\lim_{\tau \to \tau_0^-} \|\nabla f_\tau\|_2^2 - S_d \sup_{g \in \mathcal{M}_1} (f_{\tau_0}, g^{2^*-1})^2} \ge \frac{\|\nabla f_{\tau_0}\|_2^2 - S_d \|f_{\tau_0}\|_{2^*}^2}{\|\nabla f_{\tau_0}\|_2^2 - S_d \sup_{g \in \mathcal{M}_1} (f_{\tau_0}, g^{2^*-1})^2}. $$

We now claim that

$$ (42) \qquad \inf_{g \in \mathcal{M}} \|\nabla f_{\tau_0} - \nabla g\|_2^2 \le \delta \|\nabla f_{\tau_0}\|_2^2. $$

Once this is proved, we can bound the right side of (41) from below by $\mathcal{I}(\delta)$. This inequality is the claimed inequality after taking into account (37).
To prove (42), we first note that it is verified if $\tau_0 = n_0 + 1$. Indeed, $f_{n_0+1} = f_{n_0+1}$ by (36) and therefore, by assumption of alternative (b), $\inf_{g \in \mathcal{M}} \|\nabla f_{n_0+1} - \nabla g\|_2^2 < \delta \|\nabla f_{n_0+1}\|_2^2$.

Now let $\tau_0 < n_0 + 1$. We argue by contradiction and assume that

$$ (43) \qquad \inf_{g \in \mathcal{M}} \|\nabla f_{\tau_0} - \nabla g\|_2^2 > \delta \|\nabla f_{\tau_0}\|_2^2. $$

Because of this strict inequality and the definition of $\tau_0$, for any $k \in \mathbb{N}$ there are $\sigma_k \in (\tau_0, n_0 + 1)$ with $\lim_{k \to \infty} \sigma_k = \tau_0$ such that $\inf_{g \in \mathcal{M}} \|\nabla f_{\sigma_k} - \nabla g\|_2^2 < \delta \|\nabla f_{\sigma_k}\|_2^2$, that is,

$$ \|\nabla f_{\sigma_k}\|_2^2 - S_d \sup_{g \in \mathcal{M}_1} (f_{\sigma_k}, g^{2*-1})^2 < \delta \|\nabla f_{\sigma_k}\|_2^2 \quad \forall k \in \mathbb{N}. $$

Letting $k \to \infty$ and using (33) as well as the right continuity of $\|\nabla f_{\tau}\|_2^2$, see (34), we deduce that

$$ \|\nabla f_{\tau_0}\|_2^2 - S_d \sup_{g \in \mathcal{M}_1} (f_{\tau_0}, g^{2*-1})^2 \le \delta \|\nabla f_{\tau_0}\|_2^2. $$

This is the same as $\inf_{g \in \mathcal{M}} \|\nabla f_{\tau_0} - \nabla g\|_2^2 \le \delta \|\nabla f_{\tau_0}\|_2^2$ and contradicts (43). This proves (42) and completes the proof of the lemma. $\square$

**Remark 3.9.** The above argument would be simpler if $\tau \mapsto \|\nabla f_{\tau}\|_2^2$ were continuous for an appropriate choice of hyperplanes (see Appendix A) in the definition of the flow. Since the flow is weakly continuous in $\dot{\mathbb{H}}^1(\mathbb{R}^d)$, continuity of the norm is equivalent to (strong) continuity of the flow in $\dot{\mathbb{H}}^1(\mathbb{R}^d)$. Thus, for continuity of the norm for an appropriate choice of hyperplanes, it is necessary that there is such a choice for which the Steiner symmetrizations approximate $f^*$ in $\dot{\mathbb{H}}^1(\mathbb{R}^d)$. According to a theorem of Burchard [19] this holds if and only if $f$ is co-area regular, i.e, if and only if the distribution function

$$ h \mapsto |\{x \in \mathbb{R}^d : f(x) > h, \nabla f(x) = 0\}| $$

has no absolutely continuous component. As shown by Almgren and Lieb [1], both co-area regular and co-area irregular functions are dense for $d \ge 2$.

**3.1.3. Proof of Theorem 3.1.** It is now easy to prove the main result of this section, Theorem 3.1. Let $\delta \in (0, 1/2)$ and assume that $0 \le f \in \dot{\mathbb{H}}^1(\mathbb{R}^d) \setminus \mathcal{M}$ satisfies

$$ \inf_{g \in \mathcal{M}} \|\nabla f - \nabla g\|_2^2 \ge \delta \|\nabla f\|_2^2. $$
By Lemma 3.5 either alternative (a) or (b) holds. In the first case, we apply Lemmas 3.6 and 3.7, and in the second case, we apply Lemma 3.8. This completes the proof. □

## 3.2. From nonnegative functions to arbitrary functions

We recall that $\mathcal{C}_{d, BE}$ denotes the optimal constant in (2). Similarly, we denote by $\mathcal{C}_{d, BE}^{\text{pos}}$ the optimal constant in (2) when restricted to nonnegative functions $f$. Thus, $\mathcal{C}_{d, BE}^{\text{pos}} \ge \mathcal{C}_{d, BE}$. We do not know whether these two constants coincide or not. The main result in this section will be to prove the following lower bound on $\mathcal{C}_{d, BE}$ in terms of $\mathcal{C}_{d, BE}^{\text{pos}}$.

**Proposition 3.10.** For any $d \ge 3$,

$$ \mathcal{C}_{d, BE} \ge \min \left\{ \frac{1}{2} \mathcal{C}_{d, BE}^{\text{pos}}, 1 - 2^{-\frac{2}{d}} \right\}. $$

*Proof.* To simplify the notation, given a function $v \in \dot{H}^1(\mathbb{R}^d)$, we introduce the deficit

$$ \mathrm{D}(v) := \|\nabla v\|_{\mathrm{L}^2(\mathbb{R}^d)}^2 - S_d \|v\|_{\mathrm{L}^{2^*}(\mathbb{R}^d)}^2 = \|v\|_{\mathrm{L}^{2^*}(\mathbb{R}^d)}^2 \delta_{\mathrm{Sob}}(v), $$

where $\delta_{\mathrm{Sob}}$ is defined in Section 1 (see for instance (2)). Also, we set $\alpha_d := \frac{2}{2^*} = 1 - \frac{2}{d} < 1$,

$$ h(p) := p^{\alpha_d} + (1-p)^{\alpha_d-1}, \quad \text{and} \quad h_d := h(\frac{1}{2}) = 2^{1-\alpha_d} - 1 = 2^{\frac{2}{d}} - 1. $$

Let us consider a function $u \in \dot{H}^1(\mathbb{R}^d)$. By homogeneity we can assume that $\|u\|_{\mathrm{L}^{2^*}(\mathbb{R}^d)} = 1$. Let $u_{\pm}$ denote the positive and negative parts of $u$, set

$$ m := \|u_{-}\|_{\mathrm{L}^{2^*}(\mathbb{R}^d)}^{2^*}, $$

and assume (without loss of generality) that

$$ (44) \qquad m \in [0, 1/2]. $$

Note that $\|u_+\|_{\mathrm{L}^{2^*}(\mathbb{R}^d)}^{2^*} = 1-m$ and $\|\nabla u\|_{\mathrm{L}^2(\mathbb{R}^d)}^2 = \|\nabla u_-\|_{\mathrm{L}^2(\mathbb{R}^d)}^2 + \|\nabla u_+\|_{\mathrm{L}^2(\mathbb{R}^d)}^2$. Hence, we have

$$ (45) \qquad \mathrm{D}(u) = \|\nabla u\|_{\mathrm{L}^2(\mathbb{R}^d)}^2 - S_d = \mathrm{D}(u_+) + \mathrm{D}(u_-) + S_d h(m). $$
Since the function $p \mapsto h(p)$ is monotone increasing and concave on $[0, 1/2]$, we have

$$ (46) \qquad 2 h_d p \leq h(p). $$

Also, if we set $\xi_d := 2(1 - 2^{-\alpha_d})$, the function $f(p) := (1-p)^{\alpha_d} - 1 + \xi_d p$ satisfies $f(0) = f(1/2) = 0$ and $f''(p) \leq 0$, so that $f(p) \geq 0$ for all $p \in [0, 1/2]$. Hence, by (44), we have

$$ (1 - p)^{\alpha_d} \geq 1 - \xi_d p, $$

which, by the definition of $h(p)$, yields

$$ h(p) \geq p^{\alpha_d} - \xi_d p. $$

Combining this bound with (46), this gives

$$ \left(1 + \frac{\xi_d}{2h_d}\right) h(p) \geq p^{\alpha_d}. $$

Therefore, recalling (45) and noticing that $D(u_-) + S_d m^{\alpha_d} = \|\nabla u_\-\|_{L^2(\mathbb{R}^d)}^2$, we get

$$ \begin{aligned} D(u) &\geq D(u_+) + D(u_-) + S_d \frac{2h_d}{2h_d + \xi_d} m^{\alpha_d} \\ &\geq D(u_+) + \frac{2h_d}{2h_d + \xi_d} \|\nabla u_-\|_{L^2(\mathbb{R}^d)}^2. \end{aligned} $$

By definition, we have

$$ D(u_+) \geq \mathcal{C}_{d,\text{BE}}^{\text{pos}} \inf_{g \in \mathcal{M}} \|\nabla u_+ - \nabla g\|_{L^2(\mathbb{R}^d)}^2. $$

As a consequence, if $g_+ \in \mathcal{M}$ is optimal for $u_+$, we obtain

$$ \begin{aligned} D(u) &\geq \mathcal{C}_{d,\text{BE}}^{\text{pos}} \|\nabla u_+ - \nabla g_+\|_{L^2(\mathbb{R}^d)}^2 + \frac{2h_d}{2h_d + \xi_d} \|\nabla u_-\|_{L^2(\mathbb{R}^d)}^2 \\ &\geq \min \left\{ \mathcal{C}_{d,\text{BE}}^{\text{pos}}, \frac{2h_d}{2h_d + \xi_d} \right\} \|\nabla u_+ - \nabla g_+\|_{L^2(\mathbb{R}^d)}^2 \\ &\quad + \min \left\{ \mathcal{C}_{d,\text{BE}}^{\text{pos}}, \frac{2h_d}{2h_d + \xi_d} \right\} \|\nabla u_-\|_{L^2(\mathbb{R}^d)}^2 \end{aligned} $$
$$ \geq \frac{1}{2} \min \left\{ \mathcal{C}_{d,\text{BE}}^{\text{pos}}, \frac{2h_d}{2h_d + \xi_d} \right\} \|\nabla u - \nabla g_+\|_{L^2(\mathbb{R}^d)}^2 . $$

Since $2h_d + \xi_d = 2 \cdot 2^{\frac{2}{d}} - 2 + 2 - 2^{1-\alpha_d} = 2^{\frac{2}{d}}$ we get

$$ \frac{h_d}{2h_d + \xi_d} = 2^{-\frac{2}{d}} (2^{\frac{2}{d}} - 1) = 1 - 2^{-\frac{2}{d}}, $$

which concludes the proof.

<figure><img src="image_4.png" /></figure>

### 3.3. Stability of the Sobolev inequality: Proof of Theorem 1.1

We now combine the results from the previous three sections and deduce in this way the main result of this paper.

*Proof.* We recall that the constant $\mathcal{C}_{d,\text{BE}}^{\text{pos}}$ was defined in the previous subsection and that $\mathcal{I}(\delta)$ was defined in (27). Then, as a consequence of Theorem 3.1, we have

$$ \mathcal{C}_{d,\text{BE}}^{\text{pos}} \geq \sup_{0<\delta \leq 1} \delta \mathcal{I}(\delta). $$

(Indeed, for any $\delta \in (0, 1/2)$, if $f$ satisfies $\|\nabla f - \nabla g\|_2^2 \geq \delta \|\nabla f\|^2$, then $\mathcal{E}(f) \geq \delta \mathcal{I}(\delta)$, while if $\|\nabla f - \nabla g\|_2^2 \leq \delta \|\nabla f\|^2$, then $\mathcal{E}(f) \geq \mathcal{I}(\delta) \geq \delta \mathcal{I}(\delta)$.) Thus, it remains to bound $\mathcal{I}(\delta)$ for a suitable $\delta \in (0, 1/2)$.

We let $\epsilon_0, \tilde{\delta} \in (0, 1/2)$ be as in Theorem 2.1. We will bound $\mathcal{I}(\delta)$ with $\delta = \frac{\tilde{\delta}}{1+\tilde{\delta}} \in (0, \frac{1}{2})$. Thus, let $0 \leq f \in \dot{H}^1(\mathbb{R}^d)$ with

$$ \inf_{g \in \mathcal{M}} \|\nabla f - \nabla g\|_2^2 \leq \frac{\tilde{\delta}}{1+\tilde{\delta}} \|\nabla f\|_2^2. $$

The infimum on the left side is attained for the reasons given in Section 3.1. After a translation, a dilation and multiplication by a constant, we may assume that it is attained at $g = (2/(1+|x|^2))^{(d-2)/2}$. We now pass to the sphere using the stereographic projection as in Section 2.1. Let $0 \leq u \in \dot{H}^1(\mathbb{S}^d)$ be the function associated to $f$. The function 1 is associated to $g$ and we set $r := u - 1$. The fact that the distance is attained at 1 implies that $r$ satisfies the orthogonality conditions (10). Moreover, with $A$ given by (7), we have

$$ \int_{\mathbb{S}^d} (|\nabla r|^2 + A r^2) d\mu \leq \frac{\tilde{\delta}}{1+\tilde{\delta}} \int_{\mathbb{S}^d} (|\nabla u|^2 + A u^2) d\mu \\ = \frac{\tilde{\delta}}{1+\tilde{\delta}} \left( A + \int_{\mathbb{S}^d} (|\nabla r|^2 + A r^2) d\mu \right), $$
so

$$ \int_{S^d} (|\nabla r|^2 + A r^2) d\mu \leq \tilde{\delta} A. $$

By the Sobolev inequality, this implies

$$ \left( \int_{S^d} r^q d\mu \right)^{2/q} \leq \tilde{\delta}, $$

and therefore we are in the situation of Theorem 2.1. We deduce that

$$ \int_{S^d} (|\nabla u|^2 + A u^2) d\mu - A \left( \int_{S^d} u^q d\mu \right)^{2/q} \geq \theta \epsilon_0 \int_{S^d} (|\nabla r|^2 + A r^2) d\mu. $$

Translating this result back to $\mathbb{R}^d$, we have shown that

$$ \mathcal{I} \left( \frac{\tilde{\delta}}{1+\tilde{\delta}} \right) \geq \theta \epsilon_0 = \frac{4\epsilon_0}{d-2}, $$

and therefore

$$ \mathcal{C}_{d, BE}^{\text{pos}} \geq \frac{\tilde{\delta}}{1+\tilde{\delta}} \frac{4\epsilon_0}{d-2}, $$

where we recall that $0 < \epsilon_0 < \frac{1}{3}$ is fixed and $\tilde{\delta}$ depends on $\epsilon_0$, but not on $d$. This constant has the claimed $d^{-1}$ behavior.

We turn now to the case of general, not necessarily nonnegative functions. By Proposition 3.10

$$ \mathcal{C}_{d, BE} \geq \min \left\{ \frac{1}{2} \mathcal{C}_{d, BE}^{\text{pos}}, 1 - 2^{-\frac{2}{d}} \right\}. $$

Using $1 - 2^{-\frac{2}{d}} \geq (2 \ln 2)/d$ together with the result for $\mathcal{C}_{d, BE}^{\text{pos}}$ we obtain also in the general case the claimed $d^{-1}$ behavior. As constant in Theorem 2.1 we get

$$ (47) \qquad \beta = \min \left\{ \frac{2 \epsilon_0 \tilde{\delta}}{1+\tilde{\delta}}, 2 \ln 2 \right\}, $$

which is computable, since $\tilde{\delta}$ depends in a complicated, yet explicit way on $\epsilon_0$. □

**Remark 3.11.** The constant given by (47) is a lower estimate of $d \mathcal{C}_{d, BE}$, which for large $d$ is of the same order as the strict upper estimate obtained
from (4). If we apply Proposition 2.4 instead of Theorem 2.1 in the above argument, we obtain

$$ \mathcal{C}_{d,\text{BE}}^{\text{pos}} \geq \sup_{0<\delta<1/2} \delta \mathcal{I}(\delta) \geq \sup_{0<\delta<1} \frac{\delta}{1+\delta} \mathbf{m}(\tilde{\delta}^{1/2}) = \sup_{0<\delta<1/2} \delta \mathbf{m}\left(\sqrt{\frac{\delta}{1-\delta}}\right) $$

with $\mathbf{m}$ given by (11). As explained in Remark 2.8, this lower bound is not very good for large dimensions. In the above expression, it corresponds to a right-hand side of the order of $2^{-d} d^{-(d+2)/2}$ as $d \to +\infty$, but for $d = 3, 4, 5$, 6 it gives decent numerical lower bounds on $\mathcal{C}_{d,\text{BE}}^{\text{pos}}$.

## 4. The large-dimensional limit: Proof of Corollary 1.2

Assume that $d \geq 3$ and consider the stability estimate for Sobolev's inequality

$$ (48) \qquad \|\nabla f\|_{L^2(\mathbb{R}^d)}^2 - S_d \|f\|_{L^{2^*}(\mathbb{R}^d)}^2 \geq \frac{\beta(d)}{d} \inf_{g \in \mathcal{M}} \|\nabla f - \nabla g\|_{L^2(\mathbb{R}^d)}^2, $$

for all $0 \leq f \in \dot{H}^1(\mathbb{R}^d)$, where $\beta(d) = d \mathcal{C}_{d,\text{BE}}^{\text{pos}} > 0$ denotes the optimal stability constant for nonnegative functions. Theorem 1.1 (also see Theorem 3.1) provides us with an explicit lower estimate of $\beta(d)$ and shows that

$$ (49) \qquad \beta_{\star} = \liminf_{d \to +\infty} \beta(d) > 0. $$

As noted for instance in [6], to obtain the logarithmic Sobolev inequality as a limit of the Sobolev inequality when $d \to +\infty$, an important step is to perform a rescaling depending on $d$. In order to do this, let $u$ be a nonnegative Lipschitz function of compact support in $\mathbb{R}^N$ and consider the ansatz

$$ (50) \qquad f(x) := u(x_1, \dots, x_N) f_*(x), $$

where $f_*$ is a Sobolev optimizer in dimension $d \geq N$. Specifically, we choose

$$ f_*(x) = Z_d^{\frac{2-d}{2d}} \left(1 + \frac{1}{r_d^2} |x|^2\right)^{1-\frac{d}{2}} \quad \forall x \in \mathbb{R}^d, $$

with

$$ r_d = \sqrt{\frac{d}{2\pi}}. $$
The normalization constant $Z_d$ is chosen to render $\|f_*\|_{L^{2^*}(\mathbb{R}^d)} = 1$. Note that $f_*(x) = r_d^{1-d/2} g_*(x/r_d)$, with $g_*$ given by (29), solves the Sobolev equation (31) with sharp Sobolev constant $S_d = d(d-2)r_d^{-2}Z_d^{2/d}$ and

$$ (51) \qquad Z_d = \left(\frac{d}{2}\right)^{\frac{d}{2}} \frac{\Gamma\left(\frac{d}{2}\right)}{\Gamma(d)} = \left(\frac{d}{8\pi}\right)^{\frac{d}{2}} |\mathbb{S}^d| = \frac{r_d^d}{2^d} |\mathbb{S}^d|. $$

It is also easy to see that

$$ (52) \qquad \lim_{d \to +\infty} Z_d^{\frac{d}{2}} = \frac{e}{4}. $$

By integration by parts, using the fact that $f_*$ is a Sobolev optimizer, we find

$$ (53) \qquad \begin{aligned} \|\nabla f\|_{L^2(\mathbb{R}^d)}^2 &= \int_{\mathbb{R}^d} |\nabla u|^2 f_*^2 dx - \int u^2 f_* \Delta f_* dx \\ &= \int_{\mathbb{R}^d} |\nabla u|^2 f_*^2 dx + \frac{d(d-2)}{r_d^2} Z_d^{\frac{d}{2}} \int_{\mathbb{R}^d} u^2 f_*^{2*} dx. \end{aligned} $$

It follows that the left-hand side of the stability inequality (48), written for $f = u f_*$, is

$$ \int_{\mathbb{R}^d} |\nabla u|^2 f_*^2 dx + \frac{d(d-2)}{r_d^2} Z_d^{\frac{d}{2}} \int_{\mathbb{R}^d} u^2 f_*^{2*} dx - \frac{d(d-2)}{r_d^2} Z_d^{\frac{d}{2}} \left( \int_{\mathbb{R}^d} u^{2*} f_*^{2*} dx \right)^{2/2*}, $$

which can be written as

$$ Z_d^{\frac{d}{2}} \int_{\mathbb{R}^d} |\nabla u|^2 \left(1 + \frac{1}{r_d^2} |x|^2\right)^2 d\mu_d - 2\pi(d-2) Z_d^{\frac{d}{2}} \left( \left( \int_{\mathbb{R}^d} u^{2*} d\mu_d \right)^{2/2*} - \int_{\mathbb{R}^d} u^2 d\mu_d \right), $$

where $d\mu_d = f_*^{2*}(x) dx$ is a probability measure.

Let us write $x = (y, z) \in \mathbb{R}^N \times \mathbb{R}^{d-N} \approx \mathbb{R}^d$, for some integer $N$ such that $1 \le N < d$. With $|x|^2 = |y|^2 + |z|^2$ and

$$ 1 + \frac{1}{r_d^2} |x|^2 = 1 + \frac{1}{r_d^2} (|y|^2 + |z|^2) = \left(1 + \frac{1}{r_d^2} |y|^2\right) \left(1 + \frac{|z|^2}{r_d^2 + |y|^2}\right), $$

we can integrate over the $z$ variable to obtain
$$ (54) \quad \int_{\mathbb{R}^{d-N}} \frac{dz}{\left(1 + \frac{1}{r_d^2}(|y|^2 + |z|^2)\right)^d} \\ = \frac{r_d^{d-N}}{\left(1 + \frac{1}{r_d^2} |y|^2\right)^{\frac{N+d}{2}}} \int_{\mathbb{R}^{d-N}} \frac{d\zeta}{(1 + |\zeta|^2)^d} = \frac{\Gamma\left(\frac{d+N}{2}\right) \left(\frac{d}{2}\right)^{\frac{d-N}{2}}}{\Gamma(d) \left(1 + \frac{1}{r_d^2} |y|^2\right)^{\frac{N+d}{2}}}. $$

By taking into account the limits

$$ (55) \quad \lim_{d \to +\infty} \left(1 + \frac{1}{r_d^2} |y|^2\right)^{-\frac{N+d}{2}} = e^{-\pi |y|^2} \quad \text{and} \\ \lim_{d \to +\infty} \frac{r_d^{d-N}}{Z_d} \int_{\mathbb{R}^{d-N}} \frac{d\zeta}{(1 + |\zeta|^2)^d} = \lim_{d \to +\infty} \frac{\Gamma\left(\frac{d+N}{2}\right)}{Z_d \Gamma(d)} \left(\frac{d}{2}\right)^{\frac{d-N}{2}} = 1, $$

we obtain

$$ (56) \quad \lim_{d \to +\infty} \int_{\mathbb{R}^d} |u(y)|^2 d\mu_d = \int_{\mathbb{R}^N} u^2 d\gamma $$

where $d\gamma(y) := e^{-\pi|y|^2} dy$ is a Gaussian probability measure. A similar computation shows that

$$ \lim_{d \to +\infty} \int_{\mathbb{R}^d} |\nabla u|^2 \left(1 + \frac{1}{r_d^2} |x|^2\right)^2 d\mu_d = 4 \int_{\mathbb{R}^N} |\nabla u|^2 d\gamma, $$

because

$$ \lim_{d \to +\infty} \frac{1}{Z_d} \int_{\mathbb{R}^{d-N}} \left(1 + \frac{1}{r_d^2} (|y|^2 + |z|^2)\right)^{2-d} dz = 4. $$

On the other hand, let $\varepsilon := 1/(d-2)$ and write

$$ (d-2) \left[ \left( \int_{\mathbb{R}^N} u^{2*} d\gamma \right)^{2/2*} - \int_{\mathbb{R}^N} u^2 d\gamma \right] \\ = \frac{1}{\varepsilon} \left[ \left( \int_{\mathbb{R}^N} u^{2+4\varepsilon} d\gamma \right)^{\frac{1}{1+2\varepsilon}} - \int_{\mathbb{R}^N} u^2 d\gamma \right]. $$

As a consequence, we obtain

$$ \lim_{d \to +\infty} (d-2) \left[ \left( \int_{\mathbb{R}^N} u^{2*} d\gamma \right)^{2/2*} - \int_{\mathbb{R}^N} u^2 d\gamma \right] $$
$$ = \left. \frac{d}{d\varepsilon} \right|_{\varepsilon=0} \left( \int_{\mathbb{R}^N} u^2 (1+2\varepsilon) d\gamma \right)^{\frac{1}{1+2\varepsilon}} = 2 \int_{\mathbb{R}^N} u^2 \ln \left( \frac{u^2}{\int_{\mathbb{R}^N} u^2 d\gamma} \right) d\gamma. $$

Altogether, we find that

$$ \frac{1}{4} \lim_{d \to +\infty} \left[ \int_{\mathbb{R}^d} |\nabla u|^2 \left(1 + \frac{1}{r_d^2} |x|^2\right)^2 d\mu_d \right. \\ \left. - 2\pi(d-2) \left( \left( \int_{\mathbb{R}^d} u^{2*} d\mu_d \right)^{2/2*} - \int_{\mathbb{R}^d} u^2 d\mu_d \right) \right] \\ = \int_{\mathbb{R}^N} |\nabla u|^2 d\gamma - \pi \int_{\mathbb{R}^N} u^2 \ln \left( \frac{u^2}{\int_{\mathbb{R}^N} u^2 d\gamma} \right) d\gamma. $$

Using (52), we have proved

**Lemma 4.1.** Let $f$ be given by (50) where $u$ is a nonnegative Lipschitz function of compact support in $\mathbb{R}^N$. Then the limit of the left-hand side of the stability inequality (48) as $d \to +\infty$ is

$$ \lim_{d \to +\infty} \|\nabla f\|_{L^2(\mathbb{R}^d)}^2 - S_d \|f\|_{L^{2^*}(\mathbb{R}^d)}^2 \\ = e \left[ \int_{\mathbb{R}^N} |\nabla u|^2 d\gamma - \pi \int_{\mathbb{R}^N} u^2 \ln \left( \frac{u^2}{\int_{\mathbb{R}^N} u^2 d\gamma} \right) d\gamma \right]. $$

Next we deal with the large $d$ limit of the right side of (48).

**Lemma 4.2.** Let $f$ be given by (50) where $u$ is a nonnegative Lipschitz function of compact support in $\mathbb{R}^N$. Then

$$ \lim_{d \to +\infty} \frac{1}{d} \inf_{\substack{a>0, b \in \mathbb{R}^d \\ c \in \mathbb{R}}} \|\nabla f - c \nabla h_{a,b}(x)\|_{L^2(\mathbb{R}^d)}^2 \\ = \frac{\pi e}{2} \inf_{c \in \mathbb{R}, b' \in \mathbb{R}^N} \int_{\mathbb{R}^N} |u(y) - c e^{\pi b' \cdot y}|^2 d\gamma, $$

where $h_{a,b}(x) := |\mathbb{S}^d|^{-\frac{d-2}{2d}} \left( \frac{2a}{a^2 + |x-b|^2} \right)^{\frac{d-2}{2}}$ is, up to a multiplicative constant, any Sobolev optimizer.

*Proof.* In the main part of this proof, using $(\cdot, \cdot)$ as in Lemma 3.3, we shall
show that

$$ (57) \quad \lim_{d \to +\infty} \sup_{a>0, b \in \mathbb{R}^d} \left( f, h_{a,b}^{\frac{d+2}{d-2}} \right) = \sup_{b' \in \mathbb{R}^N} \int_{\mathbb{R}^N} u(y) e^{-\frac{\pi}{2} |y|^2} e^{-\frac{\pi}{2} |y-b'|^2} dy. $$

Before proving (57), let us show that it implies the assertion of the lemma. As in Lemma 3.3 we can optimize the right side of (48) over c and find

$$ (58) \quad \inf_{a>0, b \in \mathbb{R}^d} \inf_{c \in \mathbb{R}} \| \nabla f - c \nabla h_{a,b} \|_{L^2(\mathbb{R}^d)}^2 \\ = \| \nabla f \|_{L^2(\mathbb{R}^d)}^2 - S_d \sup_{a>0, b \in \mathbb{R}^d} \left( f, h_{a,b}^{\frac{d+2}{d-2}} \right)_{L^2(\mathbb{R}^d)}^2, $$

where $h_{a,b}$ satisfies

$$ \int_{\mathbb{R}^d} h_{a,b}(x)^{\frac{2d}{d-2}} dx = 1. $$

Similarly, from

$$ \int_{\mathbb{R}^N} |u(y) - c e^{\pi b' \cdot y}|^2 d\gamma \\ = \int_{\mathbb{R}^N} |u(y)|^2 d\gamma + c^2 e^{\pi |b'|^2} - 2c \int_{\mathbb{R}^N} u(y) e^{\pi b' \cdot y} d\gamma $$

we deduce that

$$ \sup_{c \in \mathbb{R}} \int_{\mathbb{R}^N} |u(y) - c e^{\pi b' \cdot y}|^2 d\gamma \\ = \int_{\mathbb{R}^N} |u(y)|^2 d\gamma - e^{-\pi |b'|^2} \left( \int_{\mathbb{R}^N} u(y) e^{\pi b' \cdot y} d\gamma \right)^2 \\ = \int_{\mathbb{R}^N} |u(y)|^2 d\gamma - \left( \int_{\mathbb{R}^N} u(y) e^{-\frac{\pi}{2} |y|^2} e^{-\frac{\pi}{2} |y-b'|^2} dy \right)^2 $$

and, consequently,

$$ \inf_{c \in \mathbb{R}, b' \in \mathbb{R}^N} \int_{\mathbb{R}^N} |u(y) - c e^{\pi b' \cdot y}|^2 d\gamma \\ = \int_{\mathbb{R}^N} u^2 d\gamma - \sup_{b' \in \mathbb{R}^N} \left( \int_{\mathbb{R}^N} u(y) e^{-\frac{\pi}{2} |y|^2} e^{-\frac{\pi}{2} |y-b'|^2} dy \right)^2. $$
Now as before, using (53), we get

$$ \lim_{d \to +\infty} \frac{1}{d} \| \nabla f \|_{L^2(\mathbb{R}^d)}^2 = \frac{\pi e}{2} \int e^{-\pi |y|^2} |u(y)|^2 dy . $$

Inserting this together with the fact that $\lim_{d \to +\infty} S_d/d = \pi e/2$ into (58), shows that (57) implies the assertion of the lemma.

Thus, from now on we concentrate on proving (57). Clearly, we may assume $u \neq 0$. It is easy to see that for every $d$, there are $a_d > 0$ and $b_d \in \mathbb{R}^d$ such that

$$ \sup_{a>0, b \in \mathbb{R}^d} \left( f, h_{a,b}^{\frac{d+2}{d-2}} \right) = \left( f, h_{a_d, b_d}^{\frac{d+2}{d-2}} \right). $$

To pass to the limit in (57) as $d \to +\infty$, we have to study the asymptotic behavior of $a_d$ and $b_d$.

• The limit of $a_d$. We will derive a lower and an upper bound on $\left( f, h_{a_d, b_d}^{\frac{d+2}{d-2}} \right)$. For the lower bound we test the supremum defining this quantity with $a = r_d$ and $b = 0$, in which case $h_{r_d, 0} = f_*$. Arguing similarly as in (56) and recalling $u \neq 0$, we obtain

$$ (59) \quad \liminf_{d \to +\infty} \left( f, h_{a_d, b_d}^{\frac{d+2}{d-2}} \right) \ge \lim_{d \to +\infty} \left( f, h_{r_d, 0}^{\frac{d+2}{d-2}} \right) = \int_{\mathbb{R}^N} u d\gamma > 0. $$

To derive an upper bound we use the fact that $f_*$ and $h_{a_d, 0}$ are symmetric decreasing functions, which implies that

$$ \begin{aligned} 0 \le \left( f, h_{a_d, b_d}^{\frac{d+2}{d-2}} \right) &\le \|u\|_{L^\infty(\mathbb{R}^d)} \int_{\mathbb{R}^d} f_*(x) h_{a_d, 0}(x-b_d)^{\frac{d+2}{d-2}} dx \\ &\le \|u\|_{L^\infty(\mathbb{R}^d)} \int_{\mathbb{R}^d} f_* h_{a_d, 0}^{\frac{d+2}{d-2}} dx. \end{aligned} $$

By inserting the expression (51) of $Z_d$ and setting $\alpha_d = a_d/r_d$, we obtain

$$ \begin{aligned} \int_{\mathbb{R}^d} f_* h_{a_d, 0}^{\frac{d+2}{d-2}} dx &= \frac{2^d}{r_d^d |\mathbb{S}^d|} \int_{\mathbb{R}^d} \left(1 + \frac{|x|^2}{r_d^2}\right)^{-\frac{d-2}{2}} \left(\alpha_d + \frac{|x|^2}{\alpha_d r_d^2}\right)^{-\frac{d+2}{2}} dx \\ &= \frac{1}{|\mathbb{S}^d|} \int_{\mathbb{R}^d} \left(\frac{2}{1+|x|^2}\right)^{\frac{d-2}{2}} \left(\frac{2\alpha_d}{\alpha_d^2 + |x|^2}\right)^{\frac{d+2}{2}} dx \\ &= \frac{|\mathbb{S}^{d-1}|}{|\mathbb{S}^d|} \int_0^{+\infty} \left(\frac{2r}{1+r^2}\right)^{\frac{d-2}{2}} \left(\frac{2\alpha_d r}{\alpha_d^2 + r^2}\right)^{\frac{d+2}{2}} \frac{dr}{r}, \end{aligned} $$
where we scaled $x \mapsto r_d x$ and introduced radial coordinates. If we now set $\alpha_d = e^{s_d}$ and change variables to $r = e^t$, and then rescale according to $t \mapsto t/\sqrt{d}$, we find

$$ 
\begin{aligned}
\int_{\mathbb{R}^d} f_* h_{a_d, 0}^{\frac{d+2}{d-2}} dx &= \frac{|\mathbb{S}^{d-1}|}{|\mathbb{S}^d|} \int_{-\infty}^{\infty} (\cosh t)^{-\frac{d-2}{2}} (\cosh(t-s_d))^{-\frac{d+2}{2}} dt \\
&= \frac{|\mathbb{S}^{d-1}|}{\sqrt{d} |\mathbb{S}^d|} \int_{-\infty}^{\infty} (\cosh \frac{t}{\sqrt{d}})^{-\frac{d-2}{2}} (\cosh \frac{t-\sigma_d}{\sqrt{d}})^{-\frac{d+2}{2}} dt
\end{aligned}
$$

with $s_d = \sigma_d/\sqrt{d}$. By the elementary inequality $\cosh t \ge 1 + t^2/2$, we find the following bound for the integral on the right side for large $d$:

$$ \int_{-\infty}^{\infty} \left(1 + \frac{t^2}{2d}\right)^{-\frac{d-2}{2}} \left(1 + \frac{(t-\sigma_d)^2}{2d}\right)^{-\frac{d+2}{2}} dt \approx \int_{-\infty}^{\infty} e^{-\frac{t^2}{4}} e^{-\frac{(t-\sigma_d)^2}{4}} dt = \sqrt{2\pi} e^{-\frac{\sigma_d^2}{8}}. $$

Using $\lim_{d \to +\infty} \frac{|\mathbb{S}^{d-1}|}{\sqrt{d}|\mathbb{S}^d|} = \sqrt{2\pi}$, we finally conclude by combining the upper and the lower bound that

$$ 2\pi \liminf_{d \to +\infty} e^{-\frac{\sigma_d^2}{8}} \ge \liminf_{d \to +\infty} \left(f, h_{a_d, b_d}^{\frac{d+2}{d-2}}\right) \ge \int_{\mathbb{R}^N} u d\gamma > 0. $$

As a consequence, $|\sigma_d|$ is bounded and we deduce that

$$ (60) \qquad \lim_{d \to +\infty} \frac{a_d}{r_d} = \lim_{d \to +\infty} \alpha_d = \lim_{d \to +\infty} e^{s_d} = \lim_{d \to +\infty} e^{\frac{\sigma_d}{\sqrt{d}}} = 1. $$

• A uniform bound on $b_d$. We begin by noting that

$$ 
\begin{aligned}
\left(f, h_{a_d, b}^{\frac{d+2}{d-2}}\right) &= \iint_{\mathbb{R}^N \times \mathbb{R}^{d-N}} u(y) f_*(y, z) h_{a_d, 0}(y-b', z-b'')^{\frac{d+2}{d-2}} dy dz \\
&\le \int_{\mathbb{R}^N} u(y) \left( \int_{\mathbb{R}^{d-N}} f_*(y, z) h_{a_d, 0}(y-b', z)^{\frac{d+2}{d-2}} dz \right) dy
\end{aligned}
$$

with $b = (b', b'') \in \mathbb{R}^N \times \mathbb{R}^{d-N}$, because $u$ is nonnegative, and $z \mapsto f_*(y, z)$ and $z \mapsto h_{a_d, 0}(y, z)^{\frac{d+2}{d-2}}$ are symmetric decreasing functions. As a consequence, we can assume without loss of generality $b_d = (b'_d, 0) \in \mathbb{R}^N \times \mathbb{R}^{d-N}$.

Our task is to obtain a bound on $|b'_d|$. As before, we obtain this by deriving a lower and upper bound on $\left(f, h_{a_d, b_d}^{\frac{d+2}{d-2}}\right)$. As lower bound we use again (59). For the upper bound we write
$$ (61) \quad \left( f, h_{a_d, (b'_d, 0)}^{\frac{d+2}{d-2}} \right) = \frac{1}{Z_d \alpha_d^{\frac{d+2}{2}}} \iint_{\mathbb{R}^N \times \mathbb{R}^{d-N}} u(y) \left( 1 + \frac{1}{r_d^2} (|y|^2 + |z|^2) \right)^{-\frac{d-2}{2}} \times \left( 1 + \frac{1}{\alpha_d^2 r_d^2} (|y - b'_d|^2 + |z|^2) \right)^{-\frac{d+2}{2}} dy dz, $$

where $Z_d$ is given by (51). From Hölder's inequality we learn that

$$ \left( f, h_{a_d, (b'_d, 0)}^{\frac{d+2}{d-2}} \right) \le \left( \frac{1}{Z_d} \iint_{\mathbb{R}^N \times \mathbb{R}^{d-N}} u(y) \left( 1 + \frac{1}{r_d^2} (|y|^2 + |z|^2) \right)^{-d} dy dz \right)^{\frac{d-2}{2d}} \times \left( \frac{1}{Z_d \alpha_d^d} \iint_{\mathbb{R}^N \times \mathbb{R}^{d-N}} u(y) \left( 1 + \frac{1}{\alpha_d^2 r_d^2} (|y - b'_d|^2 + |z|^2) \right)^{-d} dy dz \right)^{\frac{d+2}{2d}}. $$

Let $R > 0$ be such that $u$ is supported in the centered ball $\overline{B}_R$ of radius $R > 0$ and assume that $|b'_d| > R$ (otherwise $|b'_d| \le R$ and we have the claimed bound). It follows that $|y - b'_d|^2 \ge (|b'_d| - R)^2$ in the support of $u$. Using the identity

$$ (62) \quad \int_{\mathbb{R}^{d-N}} (A^2 + \frac{1}{\lambda^2} |z|^2)^{-d} dz = \frac{\lambda^{d-N}}{A^{d+N}} \int_{\mathbb{R}^{d-N}} (1 + |\zeta|^2)^{-d} d\zeta $$

based on the change of variables $z = A \lambda \zeta$, and applying it with $A = \frac{1}{\alpha_d r_d} \sqrt{\alpha_d^2 r_d^2 + (|b'_d| - R)^2}$ and $\lambda = \alpha_d r_d$, we obtain

$$ \begin{aligned} & \frac{1}{Z_d \alpha_d^d} \iint_{\mathbb{R}^N \times \mathbb{R}^{d-N}} u(y) \left( 1 + \frac{1}{\alpha_d^2 r_d^2} (|y - b'_d|^2 + |z|^2) \right)^{-d} dy dz \\ & \le |B_R| \|u\|_{L^\infty(\mathbb{R}^d)} \frac{1}{\alpha_d^N} \left( 1 + \frac{(|b'_d|-R)^2}{\alpha_d^2 r_d^2} \right)^{-\frac{d+N}{2}} \frac{r_d^{d-N}}{Z_d} \int_{\mathbb{R}^{d-N}} (1 + |\zeta|^2)^{-d} d\zeta \\ & \le |B_R| \|u\|_{L^\infty(\mathbb{R}^d)} \frac{d \alpha_d^{2-N}}{(d+N) \pi (|b'_d| - R)^2} \frac{r_d^{d-N}}{Z_d} \int_{\mathbb{R}^{d-N}} (1 + |\zeta|^2)^{-d} d\zeta \end{aligned} $$

using the inequality $(1+t/k)^{-k} \le 1/t$ for all $t > 0$ with $k = (d+N)/2$. As in (56), using (55) and (60), this yields

$$ \liminf_{d \to +\infty} \left( f, h_{a_d, (b'_d, 0)}^{\frac{d+2}{d-2}} \right) \le \sqrt{\frac{|B_R| \|u\|_{L^\infty(\mathbb{R}^d)} \int_{\mathbb{R}^N} u d\gamma}{\pi \limsup_{d \to +\infty} (|b'_d| - R)^2}} $$
Taking the lower bound in (59) into account, we obtain

$$ \limsup_{d \to \infty} |b'_d| \le R + \sqrt{\frac{|B_R| \|u\|_{L^\infty(\mathbb{R}^d)}}{\pi \int_{\mathbb{R}^N} u d\gamma}} $$

This proves that $b'_d$ is uniformly bounded w.r.t. $d$.

• The large dimensional limit. We are finally in position to prove (57). We first show that

$$ (63) \quad \limsup_{d \to +\infty} \sup_{a>0, b \in \mathbb{R}^d} \left( f, h_{a,b}^{\frac{d+2}{d-2}} \right) \le \sup_{b' \in \mathbb{R}^N} \int_{\mathbb{R}^N} u(y) e^{-\frac{\pi}{2} |y|^2} e^{-\frac{\pi}{2} |y-b'|^2} dy. $$

To do so, we consider a sequence of $d$'s along which the limsup is attained. Because of the uniform bound on $b'_d$ we may pass to a subsequence along which $b'_d$ converges to some $b'_\infty \in \mathbb{R}^N$. It then suffices to prove (63) where the limsup is taken along the chosen subsequence. In the following, we will always consider this subsequence, without displaying it in our notation.

It remains to identify a bound on $\limsup_{d \to +\infty} \left( f, h_{a_d, (b'_d, 0)}^{\frac{d+2}{d-2}} \right)$. Our starting point is (61). By Hölder's inequality, we obtain

$$ \begin{aligned} & \int_{\mathbb{R}^{d-N}} \left(1 + \frac{1}{r_d^2} (|y|^2 + |z|^2)\right)^{-\frac{d-2}{2}} \left(1 + \frac{1}{\alpha_d^2 r_d^2} (|y - b'_d|^2 + |z|^2)\right)^{-\frac{d+2}{2}} dz \\ & \le \left(\int_{\mathbb{R}^{d-N}} \left(1 + \frac{1}{r_d^2} (|y|^2 + |z|^2)\right)^{-d} dz\right)^{-\frac{d-2}{2d}} \\ & \times \left(\int_{\mathbb{R}^{d-N}} \left(1 + \frac{1}{\alpha_d^2 r_d^2} (|y - b'_d|^2 + |z|^2)\right)^{-d} dz\right)^{-\frac{d+2}{2d}} \\ & = \alpha_d^{d+2} r_d^{2d} (r_d^2 + |y|^2)^{-\frac{(d-2)(d+N)}{4d}} (\alpha_d^2 r_d^2 + |y - b'_d|^2)^{-\frac{(d+2)(d+N)}{4d}} \\ & \qquad \times \int_{\mathbb{R}^{d-N}} (1 + |\zeta|^2)^{-d} d\zeta \\ & = \alpha_d^{\frac{(d+2)(d-N)}{2d}} r_d^{d-N} (1 + \frac{1}{r_d^2} |y|^2)^{-\frac{(d-2)(d+N)}{4d}} \\ & \qquad \times (1 + \frac{1}{\alpha_d^2 r_d^2} |y - b'_d|^2)^{-\frac{(d+2)(d+N)}{4d}} \int_{\mathbb{R}^{d-N}} (1 + |\zeta|^2)^{-d} d\zeta. \end{aligned} $$

Here we used the change of variables identity (62), with $A = \frac{1}{r_d} \sqrt{r_d^2 + |y|^2}$ and $\lambda = r_d$ for the first integral in the above right-hand side, and $A =$
$\frac{1}{\alpha_d r_d} \sqrt{\alpha_d^2 r_d^2 + |y|^2}$ and $\lambda = \alpha_d r_d$ for the second integral. We learn from (55) and (60) that

$$ \limsup_{d \to +\infty} \left( f, h_{a_d, (b'_d, 0)}^{\frac{d+2}{d-2}} \right) \le \int_{\mathbb{R}^N} u(y) e^{-\frac{\pi}{2}|y|^2} e^{-\frac{\pi}{2}|y-b'_\infty|^2} dy \\ \le \sup_{b' \in \mathbb{R}^N} \int_{\mathbb{R}^N} u(y) e^{-\frac{\pi}{2}|y|^2} e^{-\frac{\pi}{2}|y-b'|^2} dy. $$

This proves (63).

The converse asymptotic inequality, namely

$$ (64) \quad \liminf_{d \to +\infty} \sup_{a>0, b \in \mathbb{R}^d} \left( f, h_{a,b}^{\frac{d+2}{d-2}} \right) \ge \sup_{b' \in \mathbb{R}^N} \int_{\mathbb{R}^N} u(y) e^{-\frac{\pi}{2}|y|^2} e^{-\frac{\pi}{2}|y-b'|^2} dy, $$

follows in a similar, but simpler fashion. Indeed, it is easy to see that the supremum on the right side is attained at some $b'_* \in \mathbb{R}^N$, which we can use to bound the supremum on the left side from below by $(f, h_{0,(b'_*,0)}^{\frac{d+2}{d-2}})$. Starting from (61) and using similar arguments as above it is easy to see that

$$ \lim_{d \to +\infty} \left( f, h_{0,(b'_*,0)}^{\frac{d+2}{d-2}} \right) = \int_{\mathbb{R}^N} u(y) e^{-\frac{\pi}{2}|y|^2} e^{-\frac{\pi}{2}|y-b'_*|^2} dy \\ = \sup_{b' \in \mathbb{R}^N} \int_{\mathbb{R}^N} u(y) e^{-\frac{\pi}{2}|y|^2} e^{-\frac{\pi}{2}|y-b'|^2} dy. $$

This proves (64) and consequently also (57). □

<figure><img src="image_5.png" /></figure>

Using Lemmata 4.1 and 4.2, with $b = \pi \tilde{b}$, for nonnegative Lipschitz functions $u$ with compact support, we have proved the following result.

**Proposition 4.3.** With $\beta_*$ given by (49), for all nonnegative $u \in H^1(\gamma)$,

$$ \int_{\mathbb{R}^N} |\nabla u|^2 d\gamma - \pi \int_{\mathbb{R}^N} u^2 \ln \left( \frac{u^2}{\|u\|_{L^2(\gamma)}^2} \right) d\gamma \\ \ge \frac{\beta_* \pi}{2} \inf_{b \in \mathbb{R}^N, c \in \mathbb{R}} \int_{\mathbb{R}^N} (u - c e^{b \cdot x})^2 d\gamma. $$

The extension to any nonnegative function $u \in H^1(\gamma)$ follows by a simple density argument, as the constants in Proposition 4.3 depend neither on the support nor on the bound on $|\nabla u|$. A crucial feature of Proposition 4.3 is that the stability constant $\beta_* \pi/2$ does not depend on $N$. It is worth pointing
out the constant $\beta_*$ in this bound comes from the (unknown) best stability constant for Sobolev's inequality for nonnegative functions. Any lower bound on this stability constant gives a lower bound on the constant $\beta_*$. In particular, we have $\beta_* \ge \beta$ with $\beta$ as in Theorem 1.1.

*Proof of Corollary 1.2.* We have to extend the result of Proposition 4.3 to the case of sign-changing functions. This part of the proof is a variation of the argument used in the proof of Proposition 3.10. We shall use the notation

$$D(u) := \int_{\mathbb{R}^N} |\nabla u|^2 d\gamma - \pi \int_{\mathbb{R}^N} u^2 \ln \left( \frac{u^2}{\|u\|_{L^2(\gamma)}^2} \right) d\gamma \quad \text{for } u \in H^1(\gamma).$$

By homogeneity we can assume $\|u\|_{L^2(\gamma)} = 1$. Replacing $u$ by $-u$ if necessary, we can also assume that

$$m := \|u_\pm\|_{L^2(\gamma)}^2 \in [0, \frac{1}{2}].$$

Then

$$D(u) = D(u_+) + D(u_-) + \pi h(m) \quad \text{with} \quad h(p) := -(p \ln p + (1-p) \ln(1-p)).$$

Since the function $p \mapsto h(p)$ is monotone increasing and concave on $[0, \frac{1}{2}]$,

$$h(p) \ge (2 \ln 2) p \quad \text{for all} \quad p \in [0, \frac{1}{2}].$$

Thus, with $\beta_*$ denoting the constant in (49),

$$\begin{aligned} D(u) &\ge D(u_+) + (2\pi \ln 2)m \\ &\ge \frac{\beta_* \pi}{2} \inf_{b \in \mathbb{R}^N, c \in \mathbb{R}} \|u_+ - c e^{b \cdot x}\|_{L^2(\gamma)}^2 + (2\pi \ln 2) \|u_-\|_{L^2(\gamma)}^2 \\ &\ge \frac{1}{2} \min \left\{ \frac{\beta_* \pi}{2}, 2\pi \ln 2 \right\} \inf_{b \in \mathbb{R}^N, c \in \mathbb{R}} \|u - c e^{b \cdot x}\|_{L^2(\gamma)}^2. \end{aligned}$$

This proves the inequality for the general case with

$$(65) \qquad \beta = \frac{1}{2} \min \{\beta_*, 4 \ln 2\}$$

and $\beta_*$ given by (49).

<figure><img src="image_6.png" /></figure>

Up to this point, we have stated the logarithmic Sobolev inequality in its version with respect to the normalized Gaussian measure. It has an equivalent
version with respect to the Euclidean measure. We set $u = e^{\pi|x|^2/2} v$ and obtain from Corollary 1.2 and Proposition 4.3

$$ \int_{\mathbb{R}^N} |\nabla v|^2 dx - \pi \int_{\mathbb{R}^N} v^2 \ln \left( \frac{v^2}{\|v\|_{\text{L}^2(\mathbb{R}^N)}^2} \right) dx - N \pi \|v\|_{\text{L}^2(\mathbb{R}^N)}^2 \\ \geq \frac{\beta \pi}{2} \inf_{b \in \mathbb{R}^N, c \in \mathbb{R}} \int_{\mathbb{R}^N} \left| v - c e^{-\frac{\pi}{2}|x-b|^2} \right|^2 dx $$

by a simple integration by parts. Writing $v(x) = \lambda^{N/2} w(\lambda x)$ with a parameter $\lambda > 0$, we obtain equivalently

$$ \lambda^2 \int_{\mathbb{R}^N} |\nabla w|^2 dy - \pi \int_{\mathbb{R}^N} w^2 \ln \left( \frac{w^2}{\|w\|_{\text{L}^2(\mathbb{R}^N)}^2} \right) dy - N \pi (1 + \ln \lambda) \|w\|_{\text{L}^2(\mathbb{R}^N)}^2 \\ \geq \frac{\beta \pi}{2} \inf_{b \in \mathbb{R}^N, c \in \mathbb{R}} \int_{\mathbb{R}^N} \left| w - c e^{-\frac{\pi}{2\lambda^2} |y-b|^2} \right|^2 dy. $$

We bound the right side from below by extending the infimum over all $\lambda > 0$ and then we optimize the left side with respect to $\lambda > 0$. In this way we obtain the following stability version of the Euclidean logarithmic Sobolev inequality.

**Corollary 4.4.** With $\beta > 0$ given by (65) we have for all $N \in \mathbb{N}$ and all $w \in H^1(\mathbb{R}^N)$,

$$ \|w\|_{\text{L}^2(\mathbb{R}^N)}^2 \ln \left( \frac{2}{N \pi e} \frac{\int_{\mathbb{R}^N} |\nabla w|^2 dx}{\|w\|_{\text{L}^2(\mathbb{R}^N)}^2} \right) - \frac{2}{N} \int_{\mathbb{R}^N} w^2 \ln \left( \frac{w^2}{\|w\|_{\text{L}^2(\mathbb{R}^N)}^2} \right) dx \\ \geq \frac{\beta}{N} \inf_{\lambda > 0, b \in \mathbb{R}^N, c \in \mathbb{R}} \int_{\mathbb{R}^N} \left| w - c e^{-\frac{\pi}{2\lambda^2} |y-b|^2} \right|^2 dy. $$

## Appendix A. Some properties of continuous rearrangement

In this subsection we discuss several aspects of the continuous rearrangement and prove some of its properties.

Brock's continuous Steiner rearrangement is based on the following operation for functions of one real variable that are finite unions of disjoint characteristic functions $\sum_{k=1}^N \chi_{(-a_k, a_k)}(x - b_k)$. Replace this function by $\sum_{k=1}^N \chi_{(-a_k, a_k)}(x - e^{-t} b_k)$ where $t$ varies from 0 to $\infty$. As $t$ increases, the intervals start moving closer and as soon as any two intervals touch one stops
the process and redefines the set of intervals by joining the two that touched. Then one restarts the process and keeps repeating it until all of them are joined into one. The movement stops once this interval is centered at the origin. By the outer regularity of Lebesgue measure the level sets of a measurable function can be approximated by open sets and, since in one dimension this is a countable union of open intervals, one can further approximate the level set by a finite number of open disjoint intervals for which one uses the sliding argument explained above.

As mentioned before, this procedure can be generalized to higher dimensions by considering Steiner symmetrization with respect to a hyperplane. One considers any hyperplane $H$ through the origin and then rearranges the function symmetrically about the hyperplane along each line perpendicular to $H$, resulting in a function denoted by $f^*H$. For more information see [64]. In this fashion one obtains a continuous rearrangement $f \to f_\tau^H, \tau \in [0, \infty]$, which was studied in detail by Brock [16, 17]. We shall refer to the statements in those papers.

To pass from Steiner symmetrization to the symmetric decreasing rearrangement we consider a sequence of continuous Steiner symmetrizations and chain them with a new continuous parameter $\lambda$ la Bucur-Henrot. Inspired by [18, 31], we proceed as follows. Given a function $f \in L^p(\mathbb{R}^d)$ for some $1 \le p < \infty$ there is a sequence $(H_n)_{n \in \mathbb{N}}$ of hyperplanes such that, defining recursively with $f_0 = f$,

$$f_n := f_{n-1}^{*H_n}, \quad n = 1, 2, \dots,$$

we have

$$f_n \to f^* \quad \text{in } L^p(\mathbb{R}^d) \quad \text{as} \quad n \to \infty.$$

In fact, it is shown in [79, Theorem 4.3] that this holds for 'almost every' (in an appropriate sense) choice of hyperplanes. It is also of interest that this sequence can actually be chosen in a universal fashion (that is, independent of $f$ and $p$); see [77, Theorem 5.2].

Given $f$ and the sequence $(f_n)_{n \in \mathbb{N}}$ as above, we set for any $n = 0, 1, 2, \dots$

$$\phi_n(\tau) := e^{\frac{\tau-n}{n+1-\tau}} - 1, \quad \tau \in [n, n+1],$$

and define

$$(66) \qquad f_{\tau} := (f_n)_{\phi_n(\tau)}^{H_{n+1}},$$

where the right side denotes Brock's continuous Steiner symmetrization with respect to the hyperplane $H_{n+1}$ with parameter $\phi_n(\tau)$ applied to $f_n$. As $\tau$
runs from $n$ to $n + 1$, $\phi_n(\tau)$ runs from 0 to $\infty$ and there is no ambiguity at $\tau \in \mathbb{N}$ since $f_n = f_{n-1}^{*H_n}$ by definition. Thus, $f_\tau$ is well defined for $\tau \in [0, \infty]$.

From the properties of Brock's flow, see, in particular, [17, Lemma 4.1], we obtain the following properties for our flow.

**Proposition A.1.** Let $d \ge 1$, $1 \le p < \infty$ and let $0 \le f \in L^p(\mathbb{R}^d)$. Then, for any $\tau \in [0, \infty]$, the function $f_\tau$ defined by (66) is in $L^p(\mathbb{R}^d)$ and $\|f_\tau\|_p = \|f\|_p$. Moreover, for any $\tau \in [0, \infty]$ and any sequence $(\tau_n)_{n \in \mathbb{N}}$ with $\lim_{n \to \infty} \tau_n = \tau$,

$$ \lim_{n \to \infty} \|f_{\tau_n} - f_\tau\|_p = 0. $$

The following fact is important for us.

**Lemma A.2.** Let $d \ge 3$ and $0 \le f \in L^{2^*}(\mathbb{R}^d)$. The function

$$ \tau \mapsto \sup_{u \in \mathcal{M}_1} (f_\tau, u^{2^*-1})^2 $$

with $f_\tau$ defined by (66) is continuous.

*Proof.* We use the fact, shown in Proposition A.1, that

$$ \lim_{\tau_1 \to \tau_2} \|f_{\tau_1} - f_{\tau_2}\|_{2^*} = 0. $$

Fix $\varepsilon > 0$. There exists $u_1 \in \mathcal{M}_1$ such that $\sup_{u \in \mathcal{M}_1} |(f_{\tau_1}, u^{2^*-1})| \le |(f_{\tau_1}, u_1^{2^*-1})| + \varepsilon$ and hence

$$ \sup_{u \in \mathcal{M}_1} |(f_{\tau_1}, u^{2^*-1})| - \sup_{u \in \mathcal{M}_1} |(f_{\tau_2}, u^{2^*-1})| \\ \le |(f_{\tau_1}, u_1^{2^*-1})| + \varepsilon - |(f_{\tau_2}, u_1^{2^*-1})| \\ \le |(f_{\tau_1}, u_1^{2^*-1}) - (f_{\tau_2}, u_1^{2^*-1})| + \varepsilon, $$

which by Hölder's inequality is bounded above by

$$ \|f_{\tau_1} - f_{\tau_2}\|_{2^*} \|u_1^{2^*-1}\|_q + \varepsilon = \|f_{\tau_1} - f_{\tau_2}\|_{2^*} + \varepsilon $$

with $q = \frac{2^*}{2^*-1}$. Hence

$$ \limsup_{\tau_2 \to \tau_1} \left( \sup_{u \in \mathcal{M}_1} |(f_{\tau_1}, u^{2^*-1})| - \sup_{u \in \mathcal{M}_1} |(f_{\tau_2}, u^{2^*-1})| \right) \le \varepsilon. $$
There exists $u_2 \in \mathcal{M}_1$ such that $\sup_{u \in \mathcal{M}_1} |(f_{\tau_2}, u^{2*-1})| \le |(f_{\tau_2}, u_2^{2*-1})| + \varepsilon$
and hence

$$ \sup_{u \in \mathcal{M}_1} |(f_{\tau_1}, u^{2*-1})| - \sup_{u \in \mathcal{M}_1} |(f_{\tau_2}, u^{2*-1})| \ge |(f_{\tau_1}, u_2^{2*-1})| - |(f_{\tau_2}, u_2^{2*-1})| - \varepsilon, $$

which is greater or equal to

$$ -|(f_{\tau_1}, u_2^{2*-1}) - (f_{\tau_2}, u_2^{2*-1})| - \varepsilon \ge -\|f_{\tau_1} - f_{\tau_2}\|_{2^*} - \varepsilon. $$

Hence

$$ \liminf_{\tau_2 \to \tau_1} \left( \sup_{u \in \mathcal{M}_1} |(f_{\tau_1}, u^{2*-1})| - \sup_{u \in \mathcal{M}_1} |(f_{\tau_2}, u^{2*-1})| \right) \ge -\varepsilon. $$

This proves the claimed continuity.

<figure><img src="image_7.png" /></figure>

We now consider the behavior of the gradient under the rearrangement flow. The following proposition is closely related to [17, Theorems 3.2 and 4.1], but there inhomogeneous Sobolev spaces are considered, which leads to some minor changes. For the sake of simplicity we provide the details.

**Proposition A.3.** Let $0 \le f \in \dot{H}^1(\mathbb{R}^d)$. Then $f_\tau$ defined by (66) is in $\dot{H}^1(\mathbb{R}^d)$ and $\tau \mapsto \|\nabla f_\tau\|_2$ is a nonincreasing, right-continuous function.

*Proof.* By construction, it suffices to prove these properties for Brock's flow. Since the latter has the semigroup property $(f_\sigma)_\tau = f_{\sigma+\tau}$ for all $\sigma, \tau \ge 0$, it suffices to prove monotonicity and right-continuity at $\tau = 0$.

We begin with the proof of monotonicity, which we first prove under the additional assumption that $f \in L^2(\mathbb{R}^d)$. This is shown in [17, Theorem 3.2], but we give an alternative proof. We proceed as in the proof of [64, Lemma 1.17]. Extending [16, Corollary 2] to the sequence of Steiner symmetrizations we find for three nonnegative functions $f, g, h$ that

$$ \iint_{\mathbb{R}^d \times \mathbb{R}^d} f_\tau(x) g_\tau(x-y) h_\tau(y) dx dy \ge \iint_{\mathbb{R}^d \times \mathbb{R}^d} f(x) g(x-y) h(y) dx dy. $$

If we choose $g(x-y)$ to be the standard heat kernel, i.e., $g(x-y) = e^{\Delta t}(x-y)$, then $g_\tau(x-y) = g(x-y)$ and hence

$$ \iint_{\mathbb{R}^d \times \mathbb{R}^d} f_\tau(x) e^{\Delta t} (x-y) f_\tau(y) dx dy \ge \iint_{\mathbb{R}^d \times \mathbb{R}^d} f(x) e^{\Delta t} (x-y) f(y) dx dy. $$
Since $\|f_{\tau}\|_2 = \|f\|_2$ by the equimeasurability of rearrangement,

$$ \frac{1}{t} (\|f_{\tau}\|_2^2 - (f_{\tau}, e^{\Delta t} f_{\tau})) \le \frac{1}{t} (\|f\|_2^2 - (f, e^{\Delta t} f)) $$

and letting $t \to 0$ yields the first claim under the additional assumption $f \in L^2(\mathbb{R}^d)$.

For general $0 \le f \in \dot{H}^1(\mathbb{R}^d)$ we apply the above argument to the functions $(f - \epsilon)_+, \epsilon > 0$. They belong to $L^2(\mathbb{R}^d)$ since $f$ vanishes at infinity and belongs to $L^{2^*}(\mathbb{R}^d)$. We obtain

$$ (67) \qquad \|\nabla((f - \epsilon)_+)\|_2 \le \|\nabla(f - \epsilon)_+\|_2 \le \|\nabla f\|_2. $$

We claim that $f_{\tau} \in \dot{H}^1(\mathbb{R}^d)$ and $\nabla((f - \epsilon)_+)_{\tau} \to \nabla f_{\tau}$ in $L^2(\mathbb{R}^d)$ as $\epsilon \to 0^+$. Once this is shown, the claimed inequality follows from (67) by the weak lower semicontinuity of the $L^2$ norm.

To prove the claimed weak convergence, note that by (67), $\nabla((f - \epsilon)_+)_{\tau}$ is bounded in $L^2(\mathbb{R}^d)$ as $\epsilon \to 0^+$ and therefore has a weak limit point. Let $F \in L^2(\mathbb{R}^d)$ be any such limit point. Since $(f - \epsilon)_+ \to f$ in $L^{2^*}(\mathbb{R}^d)$, the non-expansivity of the rearrangement [16, Lemma 3] implies that $((f - \epsilon)_+)_{\tau} \to f_{\tau}$ in $L^{2^*}(\mathbb{R}^d)$. Thus, for any $\Phi \in C_c^1(\mathbb{R}^d)$,

$$ \int_{\mathbb{R}^d} (\nabla \cdot \Phi) f_{\tau} dx \leftarrow \int_{\mathbb{R}^d} (\nabla \cdot \Phi) ((f - \epsilon)_+)_{\tau} dx \\ = - \int_{\mathbb{R}^d} \Phi \cdot \nabla ((f - \epsilon)_+)_{\tau} dx \to - \int_{\mathbb{R}^d} \Phi \cdot F dx $$

as $\epsilon \to 0^+$. This proves that $f_{\tau}$ is weakly differentiable with $\nabla f_{\tau} = F$. In particular, $f_{\tau} \in \dot{H}^1(\mathbb{R}^d)$ (note that $f_{\tau}$ vanishes at infinity since $f$ does and since these functions are equimeasurable) and the limit point $F$ is unique. This concludes the proof of the first part of the proposition.

Let us now show the right-continuity at $\tau = 0$. It follows from Proposition A.1 that $f_{\tau} \to f$ in $L^{2^*}(\mathbb{R}^d)$ as $\tau \to 0^+$. This implies that $\nabla f_{\tau} \to \nabla f$ in $L^2(\mathbb{R}^d)$ as $\tau \to 0^+$. (Indeed, the argument is similar to the one used in the first part of the proof. The family $\nabla f_{\tau}$ is bounded in $L^2(\mathbb{R}^d)$ as $\tau \to 0^+$ and, if $F$ denotes any weak limit point in $L^2(\mathbb{R}^d)$, then the convergence in $L^{2^*}(\mathbb{R}^d)$ and the definition of weak derivatives implies that $F = \nabla f$.) By weak lower semicontinuity, we deduce that

$$ \|\nabla f\|_2 \le \liminf_{\tau \to 0^+} \|\nabla f_{\tau}\|_2. $$
This, together with the reverse inequality, which was established in the first part of the proof, proves the claimed right continuity. □

We note that the proposition remains valid for $0 \le f \in \dot{W}^{1,p}(\mathbb{R}^d)$ with $1 \le p < d$. If $p \ne 2$, the monotonicity for the gradient for $f \in W^{1,p}(\mathbb{R}^d)$ is proved in [17, Theorem 3.2]. The remaining arguments above carry over to $p \ne 2$.

## Acknowledgements

The authors thank two referees for their careful reading and useful suggestions. © 2024 by the authors. Reproduction of this article by any means permitted for noncommercial purposes. CC-BY 4.0

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JEAN DOLBEAULT

CEREMADE (CNRS UMR No. 7534), PSL UNIVERSITY, UNIVERSITÉ

PARIS-DAUPHINE, PLACE DE LATTRE DE TASSIGNY, 75775 PARIS 16

FRANCE

*E-mail address: dolbeaul@ceremade.dauphine.fr*

MARIA J. ESTEBAN

CEREMADE (CNRS UMR No. 7534), PSL UNIVERSITY, UNIVERSITÉ

PARIS-DAUPHINE, PLACE DE LATTRE DE TASSIGNY, 75775 PARIS 16

FRANCE

*E-mail address: esteban@ceremade.dauphine.fr*
ALESSIO FIGALLI

MATHEMATICS DEPARTMENT, ETH ZÜRICH, RAMISTRASSE 101, 8092 ZÜRICH
SWITZERLAND

E-mail address: alessio figalli@math.ethz.ch

RUPERT L. FRANK

DEPARTMENT OF MATHEMATICS, LMU MUNICH, THERESIENSTR. 39, 80333
MÜNCHEN, GERMANY, AND MUNICH CENTER FOR QUANTUM SCIENCE AND
TECHNOLOGY, SCHELLINGSTR. 4, 80799 MÜNCHEN
GERMANY

AND

MATHEMATICS 253-37, CALTECH, PASADENA, CA 91125

UNITED STATES OF AMERICA

E-mail address: r.frank@lmu.de

MICHAEL LOSS

SCHOOL OF MATHEMATICS, GEORGIA INSTITUTE OF TECHNOLOGY ATLANTA,
GA 30332

UNITED STATES OF AMERICA

E-mail address: loss@math.gatech.edu