Some Notes on Weighted Sobolev Spaces

Weighted Sobolev spaces turns out to be useful in many applications: it saw a lot of use in the study of the constraint equation for general relativity, and more generally elliptic problems on noncompact domains (some key names include Choquet-Bruhat and Christodoulou, as well as Bartnik, and Isenberg). More recently they also turn out to be useful in the study of hyperbolic partial differential equations (see my paper and references therein). The basic Sobolev-type inequalities for these spaces are well-known, and can be easily derived by transforming the standard Sobolev inequalities on $\mathbb{R}^d$. However, for purposes of studying hyperbolic PDEs it is sometimes useful to have "correctly scaled" versions. The prototype of the these inequalities are the Gagliardo-Nirenberg-Sobolev interpolation inequalities. For example, the basic Sobolev embedding gives that, on $\mathbb{R}^3$, \[ \|f\|_{L^3} \lesssim \|f\|_{W^{1,2}}.\] But this inequality is not scale invariant. The correctly scaled, interpolated inequality is instead \[ \|f\|_{L^3} \lesssim \|f\|_{L^2}^{1/2} \|f\|_{\mathring{W}^{1,2}}^{1/2}. \] Additionally, for applications in hyperbolic PDEs, specifically using the "hyperboloidal foliation method", we would like the definition of the Sobolev spaces to be based on the Lorentz boost operators, and not the coordinate differentiation operators.

Below I summarize the Sobolev inequalities for such Sobolev spaces. Some portions of the estimates are previously known (see the book by LeFloch and Ma, as well as my paper cited above); some others, to my knowledge, are not recorded in publications. (Though I am in the process of including them in my lecture notes for my graduate course in dispersive and wave equations.) The main goal of this post is to document these inequalities to make a reference easily available. For a more detailed discussion of the proofs, you can consult the course material I posted for my grad course.

Definitions

Throughout we will consider functions defined on $\mathbb{R}^d$ with $d \geq 2$. We will use $L^p$ to denote the standard Lebesgue spaces. $\newcommand{\norm}[1]{\|#1\|}\newcommand{\jb}[1]{\langle #1 \rangle}$ By $\jb{x}$ we refer to $\jb{x} = \sqrt{1 + |x|^2}$. We will use $L^i$ to denote the differential operator $\jb{x} \frac{\partial}{\partial x^i}$.

$\newcommand{\Leb}{\mathcal{L}}\newcommand{\Sob}{\mathcal{W}}\newcommand{\HSob}{\mathring{\mathcal{W}}}$ The function spaces we will consider are based on the weighted Lebesgue spaces \begin{equation} \norm{f}_{\Leb^p_s} := \norm{\jb{\cdot}^s f}_{L^p}. \end{equation} On top of them, we build Sobolev-like spaces using the $L^i$ operators: \begin{equation} \norm{f}_{\HSob_s^{k,p}} := \sum_{i = 1}^d \norm{L^i f}_{\Leb^p_s}, \qquad \norm{f}_{\Sob_s^{k,p}} := \sum_{j = 1}^k \norm{f}_{\HSob_s^{j,p}}. \end{equation}

We remark that the standard $L^p = \Leb^p_0$, and the standard $\mathring{W}^{1,p} = \HSob_{-1}^{1,p}$.

Noting that $L^i \jb{x} = x^i$ and $L^i x^j = \delta^{ij} \jb{x}$ we find that \[ | L^i L^j \cdots L^k \jb{y} | \leq \jb{y} \] and hence \begin{equation} \norm{\jb{\cdot}^s f}_{\Sob_t^{k,p}} \approx \norm{f}_{\Sob_{s+t}^{k,p}}. \end{equation} Note however that the same comparison result does not hold for the homogeneous versions of the spaces.

Basic inequalities

The following inequalities can be proven by the classical Hölder inequality; I omit the proofs here.

Sobolev inequalities

The following basic representation formula for Schwartz functions is well-known: \begin{equation}\label{eq:rep} |f(x)| \lesssim \Big| \int_{\mathbb{R}^d} \frac{x-y}{|x-y|^d} \cdot \nabla f(y) ~dy\Big|. \end{equation} (It can be derived by using the fundamental theorem of calculus to integrate along all rays originating from $x$ and directed to infinity.) By the Hardy-Littlewood-Sobolev (fractional integration) inequality, and noting $\nabla f(y) = \jb{y}^{-1} Lf(y)$, we find immediately

Combining with the Interpolation Theorem above, we find that

As is well-known, the endpoint Sobolev $W^{1,d} \hookrightarrow L^\infty$ is false. What is true, however, is the embedding into any $p < \infty$. We have the similar result for the weighted spaces that we describe. The proof is not too difficult: start with the Gagliardo-Nirenberg-Sobolev inequality above, apply it to the function $f^{1+\gamma}$ for $\gamma \in (0,\infty)$. And if we set $\theta = \gamma / (1+\gamma)$, then we find the following:

Morrey type inequalities

When $p > d$, examining the representation formula \eqref{eq:rep} we see that we cannot apply Holder due to the fact that $|x|^{1-d}$ fails to be in $L^{p'}$ because of failure of integrability at infinity. For our purpose what we will do is truncate the function $f$ and pass the truncation over to get integrability.

Fix a bump function $\chi$ such that it is identically $1$ on the ball of radius 1, and identically $0$ outside the ball of radius 2. Denote by \[ \chi_{s,x}(y) = \chi(\frac1s(y-x)). \] If we apply the representation formula \eqref{eq:rep} to $f(y) \chi_{s,x}(y)$, evaluated at $y = x$, we find \[ |f(x)| \lesssim |I_1| + |I_2| \] where \begin{gather} I_1 = \int \chi_{s,x}(y) \frac{x-y}{|x-y|^d} \cdot \nabla f(y) ~dy \newline I_2 = \int \frac{x-y}{|x-y|^d} \cdot \nabla\chi_{s,x}(y) \; f(y) ~dy \end{gather}

The first integral is easy enough to control. Put $\nabla f$ in $L^p$ and the remainder in $L^{p'}$, using that the cut-off function restricts the integration to $|x-y| < 2s$, and that $p > d \implies p' < \frac{d}{d-1}$, we find \begin{equation} |I_1| \lesssim s^{1 - \frac{d}p} \norm{f}_{\HSob_{-1}^{1,p}}. \end{equation} The second integral can be treated in two ways¹. First, we may wish to simply place $f$ in $L^p$ (with no weight!). In this case, using that $\nabla\chi_{s,x}$ has size $\approx s^{-1}$ and is supported when $|x-y| \approx s$, this can be pretty simply estimated to give \begin{equation} |I_2| \lesssim s^{-\frac{d}p} \norm{f}_{\Leb^p_0}. \end{equation} On the other hand, we may wish to place $f \in \Leb^p_{-1}$, to give it the same weight as the gradient term. In this case, on the support of $\nabla\chi_{s,x}$ we have that \[ \jb{y} \lesssim \max(\jb{x}, \jb{s}) \] And this implies that we have \begin{equation} |I_2| \lesssim \max(\jb{x}, \jb{s}) s^{-\frac{d}p} \norm{f}_{\Leb^{p}_{-1}}. \end{equation}

Now, using the freedom of rescaling in $s$, we can optimize the inequalities. And we find finally that

Higher order inequalities

One can of course formulate Sobolev inequalities for higher derivatives, simply by applying the inequalities inductively. The most useful are the following.

Combining the various inequalities we have derived so far, we also have the following higher order Morrey inequality. I only list the case where $p = 2$ since that is the case most applicable to hyperbolic PDEs. For the Klein-Gordon type inequalities only a very specific weight is listed: generalizations to other weights are also possible, but those weights do not give improvements to the estimates in applications, since they do not exhibit the "one order of homogeneity" phenomenon.