Maximizers for Gagliardo–Nirenberg inequalities and related non-local problems

Abstract

In this paper we study the existence of maximizers for two families of interpolation inequalities, namely a generalized Gagliardo–Nirenberg inequality and a new inequality involving the Riesz energy. Two basic tools in our argument are a generalization of Lieb’s Translation Lemma and a Riesz energy version of the Brézis–Lieb lemma.

Appendix A: Proof of Proposition 2.1

In this appendix we extend the method from [4] to derive Proposition 2.1. Let $d\ge 1, s>0$ and $0<\lambda <d$ be fixed. First note that if $d>2s$ and $p=d/(d-2s)$, then the claimed inequality is the Sobolev inequality.

Therefore, in the following we assume that $p\ne d/(d-2s)$. Note that this implies, in particular, that $\lambda \ne 4s$. A tedious but straightforward computation shows that the restriction on $p$ in the proposition (in the case $p\ne d/(d=2s)$) is equivalent to the fact that

$$\begin{aligned} \theta = \frac{2d-2pd+p\lambda }{d-2ps-pd+p\lambda }. \end{aligned}$$

(7.1)

satisfies the bounds

$$\begin{aligned} \frac{d-\lambda }{d+2s -\lambda }\le \theta < 1. \end{aligned}$$

(7.2)

Note that $\theta$ is always well-defined in ${\mathbb R}\cup \{\pm \infty \}$, since numerator and denominator do not vanish simultaneously (since $\lambda \ne 4s$).

We now recall a version of the fractional Gagliardo–Nirenberg inequality.

Lemma 6.3

Let $p\in (1,\infty )$ and define $\theta$ by (7.1). If $\theta$ satisfies (7.2), then

$$\begin{aligned} \Vert D^{\frac{d-\lambda }{2}} \psi \Vert _{L^p}\le C \Vert \psi \Vert _{L^2}^{1-\theta } \Vert D^{s+\frac{d-\lambda }{2}} \psi \Vert _{L^\frac{2p}{p+1}}^\theta . \end{aligned}$$

(7.3)

If $d<2s$, then (7.3) remains valid for $p=\infty$.

For $p<\infty$ this is explicitly stated in [35], but we use the opportunity to show how it can be deduced from the older results of [23]. As similar argument works for inequality (1.2). An advantage of our reduction to [23] is that it clearly shows why $p=(d+4s-\lambda )/(d+2s-\lambda )$ is an endpoint.

Proof

The case $p<\infty$ can be easily deduced from [23]. More precisely, for any $1<p<\infty$ and any $0\le \theta \le 1$, we have the convexity inequality

$$\begin{aligned} \Vert \psi \Vert _{L^2}^{1-\theta } \Vert D^{s+\frac{d-\lambda }{2}} \psi \Vert _{L^\frac{2p}{p+1}}^\theta \ge c \Vert D^{\theta (1+\frac{d-\lambda }{2})}\psi \Vert _{L^{\frac{2p}{p+\theta }}} \end{aligned}$$

with some $c>0$ [23, Lemma A.1]. If $p=(d+4s-\lambda )/(d+2s-\lambda )$, then $\theta$ defined in (7.1) equals $(d-\lambda )/(d+2s-\lambda )$ and the lemma follows. Otherwise, for the choice (7.1) satisfying (7.2) we can apply [23, Lemma A.2] to deduce that

$$\begin{aligned} \Vert D^{\theta (1+\frac{d-\lambda }{2})}\psi \Vert _{L^{\frac{2p}{p+\theta }}} \ge c' \Vert D^{\frac{d-\lambda }{2}} \psi \Vert _{L^p}, \end{aligned}$$

which completes the proof of the lemma for $p<\infty$.

If $p=\infty$ and $d<2s$, then by the Cauchy–Schwarz inequality

$$\begin{aligned} \left| D^{\frac{d-\lambda }{2}}\psi (x)\right|&= (2\pi )^{-d/2}\left| \ \int _{{\mathbb R}^d} |\xi |^{\frac{d-\lambda }{2}} e^{i\xi \cdot x} \hat{\psi }(\xi ) \,dx \right| \\&\le (2\pi )^{-d/2} \int _{|\xi |<R} |\xi |^{\frac{d-\lambda }{2}} |\hat{\psi }(\xi )| \,d\xi + (2\pi )^{-d/2} \int _{|\xi |\ge R} |\xi |^{\frac{d-\lambda }{2}} |\hat{\psi }(\xi )| \,d\xi \\&\le C_1 R^{\frac{2d-\lambda }{2}} \left( \ \int _{|\xi |<R} |\hat{\psi }(\xi )|^2 \,d\xi \right) ^{1/2}\\&\quad +C_2 R^{-\frac{2s-d}{2}} \left( \ \int _{|\xi |<R} |\xi |^{2s+d-\lambda } |\hat{\psi }(\xi )|^2 \,d\xi \right) ^{1/2} \\&\le C_1 R^{\frac{2d-\lambda }{2}} \Vert \psi \Vert _2 + C_2 R^{-\frac{2s-d}{2}} \Vert D^{s+\frac{d-\lambda }{2}} \psi \Vert _{L^2}. \end{aligned}$$

Optimizing in $R$ yields the claimed inequality.$\square$

We combine Lemma 6.3 with the identity

$$\begin{aligned} \int \int _{{\mathbb R}^d\times {\mathbb R}^d} \frac{|\varphi (x)|^2 |\varphi (y)|^2}{|x-y|^{\lambda }}dxdy= C_{d,\lambda } \left\| D^{-\frac{d-\lambda }{2}} |\varphi |^2 \right\| _{L^2}^2 \end{aligned}$$

for some positive constant $C_{d,\lambda }$, and obtain

$$\begin{aligned} \left\| \varphi \right\| _{L^{2p}}^2 = \left\| |\varphi |^2 \right\| _{L^p}&\le C \left\| D^{-\frac{d-\lambda }{2}}|\varphi |^2 \right\| _{L^2}^{1-\theta } \left\| D^{s} |\varphi |^2 \right\| _{L^\frac{2p}{p+1}}^\theta \\&= C C_{d,\lambda }^{-\frac{1-\theta }{2}} \left( \,\ \int \int _{{\mathbb R}^d\times {\mathbb R}^d} \frac{|\varphi (x)|^2 |\varphi (y)|^2}{|x-y|^{\lambda }}dxdy \right) ^{\frac{1-\theta }{2}} \left\| D^{s} |\varphi |^2 \right\| _{L^\frac{2p}{p+1}}^\theta . \end{aligned}$$

By the fractional chain-rule (see, e.g., [10]), this implies

$$\begin{aligned} \Vert \varphi \Vert _{L^{2p}}^2\le C' \left( \,\ \int \int _{{\mathbb R}^d\times {\mathbb R}^d} \frac{|\varphi (x)|^2 |\varphi (y)|^2}{|x-y|^{\lambda }}dxdy \right) ^{\frac{1-\theta }{2}} \left\| D^s \varphi \right\| _{L^2}^\theta \Vert \varphi \Vert _{L^{2p}}^{\theta }. \end{aligned}$$

(Strictly speaking, [10] contains only the case $0<s<1$ and $p<\infty$. A proof with $(1+D^2)^{1/2}$ instead of $D$ is contained in [42, Prop. 1.1]; see also [20] and references therein for the general case.) To summarize, we have shown that

$$\begin{aligned} \Vert \varphi \Vert _{L^{2p}} \le \left( C'\right) ^{\frac{1}{2-\theta }} \left( \,\ \int \int _{{\mathbb R}^d\times {\mathbb R}^d} \frac{|\varphi (x)|^2 |\varphi (y)|^2}{|x-y|^{\lambda }}dxdy\right) ^{\frac{1-\theta }{4-2\theta }} \Vert \varphi \Vert _{\dot{H}^s}^\frac{\theta }{2-\theta }, \end{aligned}$$

which is the stated inequality.$\square$