Polynomial differential operators and Besov spaces

1. A regularity estimate for polynomial differential operator.
2. Review of Besov spaces \( B^{s,p} \).
3. Proof of the local estimate.

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We say that \( P \) is a polynomial differential operator of type \( (n,k) \) if \( P \) is of the form \[ P(F) = \sum c_{\alpha_1,\dots, \alpha_\nu} (x, F(x)) D^{\alpha_1} F^{a_1} \dots D^{\alpha_\nu} F^{a_{\nu}} \] where the coefficients \( c_{\alpha_1,\dots,\alpha_nu} \) depend smoothly and nonlinearly on \( x \) and \( F \) and \( \alpha_i\in \mathbb{R}^N \) are indices with the weighted norm \( \|\alpha_i\| \leq k \) and \( \sum \|\alpha_i\| \leq n \).

On \( M\times [\alpha,\omega] \) the tension field \( \tau(F) := -\Delta F^\alpha + g^{ij}\Gamma'^\alpha_{\beta\gamma}(F) F^\beta_i F^\gamma_j \) is a polynomial differential operator of type (2,2). The quadratic term alone is of type (2,1).

1 A regularity estimate for polynomial differential operator.

Our goal in this part is to prove the following estimate for polynomial differential operator, in which \( X \) will be \( M\times[\alpha,\omega] \).

Let \( X \) be a compact Riemannian manifold, \( B\subset \mathbb{R}^N \) is a large Euclidean ball and \( P \) be a polynomial differential operator of type \( (n,k) \) on \( X \). Suppose that

\begin{equation} \label{eq:cond:thm:reg-poly-diff} r\geq 0,\quad p,q\in (1,\infty),\quad r+k < s, \quad \frac{1}{p}> \frac{r+n}{s} \frac{1}{q}. \end{equation}

Then for all \( F\in C (X,B) \cap W^{s,q}(X) \), one has \( PF\in W^{r,p}(X) \) and \[ \|P F \|_{W^{r,p}} \leq C\left(1 + \|F\|_{W^{s,q}}\right)^{q/p}. \] where \( C \) is a constant independent of \( F \).

We will prove that the result is local, in a sense to be defined. Then we will prove the local statement using Besov spaces.

Let \(\{ \varphi_i: U_i \longrightarrow V_i \}\) be an atlas of \( M \). We denote a point in \( U_i \) by \( x \) and its coordinates in \( V_i \) by \( \xi \). Let \( \sum \psi_i = 1 \) be a partition of unity subordinated to \( \{U_i\} \) and \( \tilde \psi_i \) be smooth functions supported in \( U_i \) with \( 0\leq \tilde \psi_i \leq 1 \) and \( \tilde \psi_i = 1 \) in the support of \( \psi_i \), as in the definition of Sobolev spaces on manifold. We suppose the following local statement is true:

Let \( P \) be a polynomial differential operator of type \( (n,k) \) and coefficients \( c_{\alpha_1,\dots,\alpha_\nu}(x,F) \) are smooth and vanish when \( x\in \mathbb{R}^{\dim X} \) is outside of a compact. Let \( B\subset \mathbb{R}^N \) be a large Euclidean ball and \( r,p,q,s\) as in \eqref{eq:cond:thm:reg-poly-diff}. Then for all compactly supported \( F\in C (\mathbb{R}^{\dim X},B) \cap W^{s,q}(\mathbb{R}^{\dim X}) \), one has \[ \|P F \|_{W^{r,p}} \leq C\left(1 + \|F\|_{W^{s,q}}\right)^{q/p} \] where the constant \( C \) depends only on \( B \) and the support of \( F \), and not on \( F \).

One has \[ \| PF \|_{W^{r,p}} := \sum_i \|\psi_i P F\|_{W^{r,p}} \] where viewed in the chart \( U_i \), each \( \psi_i(x) PF(x) \) is \( \sum_\alpha\psi_i(\xi). c_\alpha (\xi,g_i). D^\alpha g_i \) where \( g_i = f_i\circ \varphi_i^{-1} \) is \( f_i \) viewed in the chart. Since \( \tilde \psi_i = 1 \) in the support of \( \psi_i \), one has \[ \psi_i(\xi). c_\alpha (\xi,g_i). D^\alpha g_i = \psi_i(\xi). c_\alpha(\xi,\tilde \psi_i g_i) D^\alpha (\tilde\psi_i g_i) \] hence by the local statement: \[ \| \psi_i(\xi). c_\alpha (\xi,g_i). D^\alpha g_i \|_{W^{r,p}} \leq C \left( 1 + \|\tilde \psi_i g_i\|_{W^{s,q}} \right)^{q/p} \leq C \left( 1 + \|F \|_{W^{s,q}}\right)^{q/p}. \] Therefore \( \|PF\|_{W^{r,p}} \leq m C \left( 1 + \|F \|_{W^{s,q}}\right)^{q/p} \) where \( m \) is the number of charts we used to cover \( M \).

The use of partition of unity in the last proof is to decompose \( PF = \sum \psi_i PF \) and not \( F = \psi_i F \) since we no longer have linearity of the operator \( P \) in \( F \).

2 Review of Besov spaces \( B^{s,p} \).

In this part, \( X = \mathbb{R}^n \) coordinated by \( (x_1,\dots,x_n) \) with weight \( (\sigma_1,\dots,\sigma_n) \). We define \[ T_j^v f(x_1,\dots,x_n) := f(x_1,\dots, x_j +v,\dots, x_n),\quad \Delta^v_j := T^v_j - {\rm Id} \] for \( f\in \mathcal{S}(X) \).

For the notation, we will denote the Besov spaces by \( B^{s,p} \) with \( s\in \mathbb{R}_{>0}\setminus \mathbb{Z} \) and \( p\in (1,\infty) \) so that they look similar to Sobolev space \( W^{s,p} \). In a more standard notation, our spaces \( B^{s,p} \) are denoted by \( B^{s}_{p,p} \)

We define \( B^{s,p} \) as the completion of \( \mathcal{S}(X) \) under the norm \[ \|f\|_{B^{s,p}} := \sum_{\|\gamma\| < s} \|D^\gamma f\|_{L^p} + \sum_{ s- \frac{\sigma}{\sigma_j} < \|\gamma\| < s}\sup_{v} \frac{\|\Delta^v_j D^\gamma f\|_{L^p}}{|v|^{(s - \|\gamma\|) \sigma_j/\sigma}} \]

We cite here some well-known facts

While Sobolev spaces with non-integral regularity are complex interpolation of integral ones, Besov spaces are their real interpolation.
Besov spaces \( B^{s,p}(X) \) are reflexive Banach spaces with their dual spaces being \( B^{-s,p'}(X) \) where \( \frac{1}{p} + \frac{1}{p'}=1 \).

If \( r < s \) then \[ W^{s,p}(X) \subset B^{s,p}(X) \subset W^{r,p}(X). \]

For \( f,g\in \mathcal{S}(X) \) and \( \begin{cases} 0<\alpha <1, \tilde p \leq p,\tilde q \leq q,\tilde r\leq r \\ \frac{1}{p} + \frac{1}{q} = \frac{1}{r}, \frac{1}{\tilde p} + \frac{1}{q} = \frac{1}{p} + \frac{1}{\tilde q} = \frac{1}{\tilde r} \end{cases} \), one has

\begin{align} \|fg\|_{B^{\alpha,\tilde r}} &\leq C \left( \|f\|_{B^{\alpha,\tilde p}}\|g\|_{L^q} + \|f\|_{L^p}\|g\|_{B^{\alpha,\tilde q}} \right) \label{eq:estimate-product-1} \\ \|fg\|_{L^r} &\leq \|f\|_{L^p}\|g\|_{L^q} \label{eq:estimate-product-2} \end{align}

Therefore by density \eqref{eq:estimate-product-1} is true for all \( f\in L^p\cap B^{\alpha,\tilde p}, g\in L^q\cap B^{\alpha,\tilde q} \) and \eqref{eq:estimate-product-2} is true for all \( f\in L^p, g\in L^q \).

The reason for which we use the Besov norm is the following estimate:

Let \( \Gamma(x,y) \) be a continuous, nonlinear function of variables \( x\in \mathbb{R}^n, y\in \mathbb{R}^N \). Suppose that \( \Gamma \) vanishes for all \( x \) outside of a compact in \( \mathbb{R}^n \) and \( \Gamma \) is \( C \)-Lipschitz in \( y \), and define \[ \Gamma f:= \left( x\longmapsto \Gamma(x,f(x)) \right). \] Then \[ \|\Gamma f\| \leq C\left( 1+ \|f\|_{B^{\alpha,p}}\right) \]

3 Proof of the local estimate.

Since \( B^{r+\epsilon,p}(X)\subset W^{r,p}(X) \), by increasing \( r \) a bit, we can suppose that \( r\not\in \mathbb{Z} \) and replace the \( W^{r,p} \) norm in the statement by the \( B^{r,p} \) norm, that is to estimate: \[ \|PF\|_{B^{r,p}} = \sum_{\|\gamma\| < r} \| D^\gamma(PF)\|_{L^p} + \sum_{ r-\sigma/\sigma_j< \|\gamma\| < r} \frac{\| \Delta^v_j D^\gamma (PF)\|_{L^p}}{|v|^{(r-\|\gamma\|)\sigma_j/\sigma}} \] where

\begin{equation} \label{eq:term-small} D^\gamma (PF) = \sum c_{\beta_1,\dots,\beta_\mu}(x,F) D^{\beta_1} f^{b_1}\dots D^{\beta_\mu}f^{b_\mu} \end{equation}

Using \( \Delta^v_j (fg) = \Delta^v_j f\ T^v_j g + f \Delta^v_j g \), one can see that \( \Delta^v_j D^\gamma (PF) \) is a sum of terms of 2 types:

\begin{equation} \label{eq:term-c} \Delta^v_j c_{\beta_1,\dots,\beta_\mu}\ T^v_j(D^{\beta_1}f^{b_1})\dots T^v_j (D^{\beta_\mu} f^{b_\mu}) \end{equation}

and

\begin{equation} \label{eq:term-f} c_{\beta_1,\dots,\beta_\mu}\ D^{\beta_1}f^{b_1}\ \dots \ D^{\beta_{i-1}}f^{b_{i-1}}\ \Delta^v_j (D^{\beta_i} f^{b_i})\ T^v_j(D^{\beta_{i+1}} f^{b_{i+1}})\ \dots\ T^v_j(D^{\beta_\mu} f^{b_\mu}) \end{equation}

Our strategy is to use Theorem thm:estimate-product to estimate the terms \eqref{eq:term-small}, \eqref{eq:term-c} and \eqref{eq:term-f} as follows, where we denote \( \|g\|_p:= \|g\|_{L^p} \)

\begin{equation} \label{eq:est-term-small} \left\| c_{\beta_1,\dots,\beta_\mu}(x,F)\ D^{\beta_1} f^{b_1}\dots D^{\beta_\mu}f^{b_\mu} \right\|_{p} \leq \|c_{\beta_1,\dots,\beta_\mu} \|_{\infty}.\|D^{\beta_1} f^{b_1}\|_{p_1}\dots \|D^{\beta_\mu} f^{b_\mu}\|_{p_\mu} \end{equation} \begin{equation} \label{eq:est-term-c} \left\| \Delta^v_j c_{\beta_1,\dots,\beta_\mu}\ T^v_j(D^{\beta_1}f^{b_1})\dots T^v_j (D^{\beta_\mu} f^{b_\mu}) \right\|_p \leq \|\Delta^v_j c_{\beta_1,\dots,\beta_\mu}\|_{\tilde p_0}. \| D^{\beta_1} f^{b_1} \|_{p_1}\dots \|D^{\beta_\mu} f^{b_\mu}\|_{p_\mu} \end{equation} \begin{multline} \label{eq:est-term-f} \left\| c_{\beta_1,\dots,\beta_\mu}\ D^{\beta_1}f^{b_1}\ \dots \ D^{\beta_{i-1}}f^{b_{i-1}}\ \Delta^v_j (D^{\beta_i} f^{b_i})\ T^v_j(D^{\beta_{i+1}} f^{b_{i+1}})\ \dots\ T^v_j(D^{\beta_\mu} f^{b_\mu}) \right\|_p \leq \\ \|c_{\beta_1,\dots,\beta_\mu}\|_\infty. \| D^{\beta_1}f^{b_1} \|_{p_1}\dots \| D^{\beta_{i-1}}f^{b_{i-1}}\|_{p_{i-1}}. \|\Delta^v_j (D^{\beta_i} f^{b_i}) \|_{\tilde p_i}. \| D^{\beta_{i+1}} f^{b_{i+1}} \|_{p_{i+1}}\dots \|D^{\beta_\mu} f^{b_\mu} \|_{p_\mu} \end{multline}

Then continue by bounding the \( \Delta^v_j \) terms:

\begin{equation} \label{eq:est-del-c} \|\Delta^v_j c_{\beta_1,\dots,\beta_\mu}\|_{\tilde p_0} \leq |v|^{\theta \sigma_j/\sigma} C (1+ \|F\|_{B^{\theta,\tilde p_0}}) \leq |v|^{\theta \sigma_j/\sigma} C (1+ \|F\|_{W^{\theta,\tilde p_0}}) \end{equation}

using Theorem thm:compo-besov, where \( C \) is the Lipschitz constant of \( c_{\beta_1,\dots,\beta_\mu}(x,F) \) in \( F \), which exists because \( c_{\beta_1,\dots,\beta_\mu} \) is smooth and \( F \) always remains in a large Euclidean ball \( B \). The next \( \Delta^v_j \) term to bound is, using Theorem thm:besov-sobolev:

\begin{equation} \label{eq:est-del-f} \|\Delta^v_j (D^{\beta_i} f^{b_i}) \|_{\tilde p_i}\leq |v|^{\theta \sigma_j/\sigma} \|f^{b_i}\|_{B^{\|\beta_i\| +\theta, \tilde p_i}} \leq |v|^{\theta \sigma_j/\sigma} \|f^{b_i}\|_{W^{\|\beta_i\| +\theta, \tilde p_i}} \end{equation}

And finally plugging \eqref{eq:est-del-c} and \eqref{eq:est-del-f} in \eqref{eq:est-term-c} and \eqref{eq:est-term-f}, and noting that \( \|c_{\beta_1,\dots,\beta_\mu} \|_{\infty} \) in \eqref{eq:est-term-small} is bounded by a constant, it remains to estimate \( \| f^{b_i} \|_{W^{\|\beta_i\|, p_i}} \), \( \| f^{b_i} \|_{W^{\|\beta_i\| + \theta, \tilde p_i}} \) and \( \|F\|_{W^{\theta, \tilde p_0}} \) in term of \( \|F\|_{W^{s,q}} \), for which we will use the following consequence of Interpolation inequality.

Let \( 0\leq r\leq s\) and \( p,q\in (1, \infty) \) such that \( 0 < \frac{1}{p} - \frac{r}{s}\frac{1}{q} < 1-\frac{r}{s} \). Then for all compactly supported \( F\in C(X, B)\cap W^{s,q} \) where \( B\subset \mathbb{R}^N \) is a large Euclidean ball, one has \[ \|F\|_{W^{r,p}} \leq C \|F\|^{1-r/s}_\infty \|F\|^{r/s}_{W^{s,q}}\leq C' \|F\|^{r/s}_{W^{s,q}} \] where \( C,C' \) depend only on \( B \) and the support of \( F \), but not \( F \).

Since \( F \) is bounded, \( f^\alpha\in W^{s,q} \cap W^{0,v} \) for all \( v > 1 \). By Interpolation inequality \[ \|f^\alpha\|_{W^{r,p}} \leq 2 \|f^\alpha\|^{r/s}_{W^{s,q}} \|f^\alpha\|^{1-r/s}_{W^{0,v}} \] then choose \( v \) with \( (1 - \frac{r}{s})\frac{1}{v} = \frac{1}{p} - \frac{r}{s}\frac{1}{q} \).

To apply Lemma lem:loc-est-reg, we have to choose \( p_i,\tilde p_i, \tilde p_0,\theta \) such that

\begin{cases} 0< \frac{1}{p_i} - \frac{\|\beta_i\|}{s} \frac{1}{q} < 1 - \frac{\|\beta_i\|}{s}, \\ 0< \frac{1}{\tilde p_i} - \frac{\|\beta_i +\theta\|}{s}\frac{1}{q} < 1 - \frac{\|\beta_i +\theta\|}{s} \\ 0< \frac{1}{\tilde p_0} - \frac{\theta}{s}\frac{1}{q} < 1- \frac{\theta}{s} \end{cases}

We choose \(\frac{1}{p_i}\) just a bit bigger than \( \frac{\|\beta_i\|}{s}\frac{1}{q} \), \( \frac{1}{\tilde p_i}\) just a bit bigger than \(\frac{\|\beta_i +\theta\|}{s}\frac{1}{q} \) and \( \frac{1}{\tilde p_0}\) just a bit bigger than \(\frac{\theta}{s}\frac{1}{q} \). We will now come back to justify the estimates \eqref{eq:est-term-small}, \eqref{eq:est-term-c}, \eqref{eq:est-term-f}. Since \( F \) is bounded in \( B \) and compactly supported in an open set \( V \), we see that \( \|f^\alpha\|_p \leq C(B,V) \|f^\alpha\|_q \) if \( p\leq q \). Therefore,

For \eqref{eq:est-term-small}, it is sufficient to have \[ \frac{1}{p} \geq \frac{1}{p_1}+\dots + \frac{1}{p_\mu} \] which is true because the RHS is is a bit bigger than \( \frac{1}{qs}\sum \|\beta_i\| \leq \frac{n + \|\gamma\|}{qs} < \frac{n+r}{qs} < \frac{1}{p} \).
For \eqref{eq:est-term-c}, it is sufficient to have \[ \frac{1}{p} \geq \frac{1}{\tilde p_0} + \frac{1}{p_1}+\dots + \frac{1}{p_\mu} \] where the RHS is is a bit bigger than \( \frac{\theta}{s}\frac{1}{q}+ \frac{1}{qs}\sum \|\beta_i\| \leq \frac{n + \|\gamma\| + \theta}{qs} \).
For \eqref{eq:est-term-f}, it is sufficient to have \[ \frac{1}{p} \geq \frac{1}{p_1}+\dots + \frac{1}{\tilde p_i} + \dots + \frac{1}{p_\mu} \] where the RHS is is a bit bigger than \( \frac{\theta}{s}\frac{1}{q}+ \frac{1}{qs}\sum \|\beta_i\| \leq \frac{n + \|\gamma\| + \theta}{qs} \).

It is sufficient then to take \( \theta = r- \|\gamma\| \). Now the estimates \eqref{eq:est-term-small}, \eqref{eq:est-term-c}, \eqref{eq:est-term-f} can be continued as

\begin{align} RHS \eqref{eq:est-term-small} &\leq \prod_i \|f^{b_i}\|^{\|\beta_i\|/s}_{W^{s,q}} \leq \|F\|_{W^{s,q}}^{\frac{n+\|\gamma\|}{s}} \leq \|F\|_{W^{s,q}}^{q/p}\label{eq:fin-del-small}\\ RHS \eqref{eq:est-term-c} &\leq |v|^{\theta\sigma_j/\sigma}\left( 1 + \|F\|_{W^{s,q}}^{\theta/s} \right)\prod_i \|f^{b_i}\|^{\|\beta_i\|/s}_{W^{s,q}}\leq |v|^{\theta\sigma_j/\sigma}\left( 1 + \|F\|_{W^{s,q}}^{\theta/s} \right)\|F\|_{W^{s,q}}^{q/p} \label{eq:fin-del-c}\\ RHS \eqref{eq:est-term-f} &\leq |v|^{\theta\sigma_j/\sigma}\left( 1 + \|f^{b_i}\|_{W^{s,q}}^{\frac{\|\beta_i\| +\theta}{s}} \right)\prod_{u\ne i} \|f^{b_u}\|^{\|\beta_u\|/s}_{W^{s,q}}\leq |v|^{\theta\sigma_j/\sigma}\left( 1 + \|F\|_{W^{s,q}}^{\frac{\|\beta_i\| +\theta}{s}} \right)\|F\|_{W^{s,q}}^{q/p} \label{eq:fin-del-f} \end{align}

While \eqref{eq:fin-del-small} gives \( \|D^\gamma (PF)\|_p \leq C \|F\|_{W^{s,q}}^{q/p} \), the last two \eqref{eq:fin-del-c} and \eqref{eq:fin-del-f} give \[ \sum_{ s- \frac{\sigma}{\sigma_j} < \|\gamma\| < s}\sup_{v} \frac{\|\Delta^v_j D^\gamma (PF) \|_p}{|v|^{(r-\|\gamma\|)\sigma_j/\sigma}} \leq C \left( 1 + \|F\|^{(n+r)/s}_{W^{q,s}}\right) \] We proved the local statement Lemma lem:loc-reg-poly-diff.

Polynomial differential operators and Besov spaces

Table of Contents

1 A regularity estimate for polynomial differential operator.

2 Review of Besov spaces \( B^{s,p} \).

3 Proof of the local estimate.