On Certain Commutator Estimates for Vector Fields

Abstract

A unifying approach for proving certain commutator estimates involving smooth, not-necessarily divergence-free vector fields is introduced and implemented in the scales of weighted Triebel–Lizorkin and Besov spaces and certain variable exponent Triebel–Lizorkin and Besov spaces. Such commutator estimates are motivated by the study of well-posedness results for some models in incompressible fluid mechanics.

Appendix

Proof of Remark 1

Let $\psi _j$, $\Delta _j$, and $\Phi _j^N$ be as defined in the previous sections; assume without loss of generality that $\psi (0)\ne 0.$ Let $f\in \dot{F}_{p}^{s,q}(w)\cap {\mathcal {S}}({{\mathbb R}^n})$. We have

$$\begin{aligned}&\left\| f\right\| _{ \dot{F}_{p}^{s,q}(w)}\ge \left| \int _{\mathbb R^n}f(y)\mathrm{d}y\right| \left( \int _{\mathbb R^n}\left( \sum _{j<0}\left[ 2^{sj}|\psi _j(x)|\right] ^q\right) ^{\frac{p}{q}}w(x)\,\mathrm{d}x\right) ^{\frac{1}{p}}\nonumber \\&\quad -\left( \int _{\mathbb R^n}\left( \sum _{j<0}2^{sqj}\left| \int _{\mathbb R^n}(\psi _j(x-y)-\psi _j(x))f(y)\mathrm{d}y\right| ^q\right) ^{\frac{p}{q}}w(x)\,\mathrm{d}x\right) ^{\frac{1}{p}}. \end{aligned}$$

Note that for $j<0,$

$$\begin{aligned} \left| \int _{\mathbb R^n}(\psi _j(x-y)-\psi _j(x))f(y)\mathrm{d}y\right| \lesssim 2^j\Phi _j^n(x)\le \frac{2^{j}}{(2^{-j}+|x|)^n}\le \frac{2^{j}}{(1+|x|)^n}, \end{aligned}$$

and hence we have

$$\begin{aligned}&A:=\int _{\mathbb R^n}\left( \sum _{j<0}2^{sqj}\left| \int _{\mathbb R^n}(\psi _j(x-y)-\psi _j(x))f(y)\mathrm{d}y\right| ^q\right) ^{\frac{p}{q}}w(x)\mathrm{d}x\\&\quad \lesssim \left( \sum _{j<0}2^{(1+s)qj} \right) ^{\frac{p}{q}}\int _{\mathbb R^n}\frac{w(x)}{(1+|x|)^{pn}}\,\mathrm{d}x<\infty , \end{aligned}$$

where we have used that $s+1>0$ and $w\in A_p.$ Since $\eta :=|\psi (0)|>0,$ there exists $0<\delta <1$ such that $|\psi _j(x)|\ge 2^{jn}\eta /2$ for $2^j|x|<\delta$. Then

$$\begin{aligned}&\int _{\mathbb R^n}\left( \sum _{j<0}\left[ 2^{sj}|\psi _j(x)|\right] ^q\right) ^{\frac{p}{q}}w(x)\mathrm{d}x\ge \int _{\mathbb R^n}\sup _{j<0}\left( 2^{sj}|\psi _{j}(x)|\right) ^pw(x)\,\mathrm{d}x\\&\quad \ge (\eta /2)^p\sup _{j<0}2^{(n+s)pj}\int _{|x|<2^{-j}\delta }w(x)\,\mathrm{d}x\\&\quad =(\eta /2)^p\delta ^{(n+s)p}\sup _{j<0}\frac{1}{(\delta 2^{-j})^n}\int _{|x|<2^{-j}\delta }w(x)(\delta 2^{-j})^{\left( \frac{1}{p}-\frac{n+s}{n}\right) pn}\,\mathrm{d}x\\&\quad \ge (\eta /2)^p\delta ^{(n+s)p}\sup _{j<0}\frac{1}{(\delta 2^{-j})^n}\int _{|x|<2^{-j}\delta }w(x)|x|^{\left( \frac{1}{p}-\frac{n+s}{n}\right) pn}\,\mathrm{d}x=\infty , \end{aligned}$$

where we have used (5.4). It follows that

$$\begin{aligned} \left| \int _{\mathbb R^n}f(y)\mathrm{d}y\right| ^p\left[ \int _{\mathbb R^n}\left( \sum _{j<0}\left[ 2^{sj}|\psi _j(x)|\right] ^q\right) ^{\frac{p}{q}}w(x)\,\mathrm{d}x\right] \lesssim ||f||_{\dot{F}_p^{s,q}(w)}^p+A<\infty . \end{aligned}$$

(6.4)

The term in the square brackets on the left-hand side of (6.4) is infinite. Hence the only way to avoid a contradiction is if f has integral zero.

We next check that $w\ge u^\epsilon$ for some $u\in A_1$ and $0<\epsilon <1-p\frac{n+s}{n}$ is sufficient for (5.4). Since $u\in A_1$, it follows that $u(x)\gtrsim \mathcal M(u)(x)\gtrsim (1+|x|)^{-n}$. Then $w(x)\gtrsim (1+|x|)^{-n\epsilon }$ and

$$\begin{aligned} \frac{1}{R^n}\int _{B(0,R)}w(x)|x|^{pn(\frac{1}{p}-\frac{n+s}{n})}\mathrm{d}x&\gtrsim \frac{1}{R^n}\int _{1<|x|<R} |x|^{-n\epsilon }|x|^{pn(\frac{1}{p}-\frac{n+s}{n})}\mathrm{d}x \end{aligned}$$

which tends to infinity as R tends to infinity since $0<\epsilon <1-p\frac{n+s}{n}.$ $\square$

Proof of Lemma 6.1

The proof follows the ideas from [4, Theorems 1 and 2].

We first prove (6.2). Let ${p(\cdot )},$ q, and $f=\{f_j\}_{j\in \mathbb {Z}}$ be as in the assumptions. We will use the fact that $\rho _{p(\cdot )}\left( F/\left\| F\right\| _{L^{p(\cdot )}}\right) =1$ if $F\in L^{p(\cdot )}$ and $\left\| F\right\| _{L^{p(\cdot )}}\ne 0.$ (See [8, Proposition 2.21] for a proof.) It is enough to show that

$$\begin{aligned} \left\| f\right\| _{L^{p(\cdot )}(\ell ^q)}\le \sup _{\{h_j\}\in \mathcal {U}_{{p'(\cdot )}, {q'}}}\int _{{\mathbb R}^n}\sum _{j\in \mathbb {Z}} \left| f_j(x)h_j(x) \right| \, \mathrm{{d}}x\end{aligned}$$

(6.5)

since the reverse estimate is a consequence of Hölder’s inequality.

Suppose first that $1\le q<\infty$ and $0<\left\| f\right\| _{L^{p(\cdot )}(\ell ^q)}<\infty ;$ define

$$\begin{aligned} h_j(x)=\text {sign}(f_j(x))\left| f_j(x) \right| ^{q-1}\left( \sum _{k\in \mathbb {Z}}\left| f_k(x) \right| ^q\right) ^{\frac{p(x)-q}{q}}\left\| f\right\| _{L^{p(\cdot )}(\ell ^q)}^{1-p(x)}. \end{aligned}$$

We have that $\rho _{p'(\cdot )}\left( \left( \sum _{j\in \mathbb {Z}}\left| h_j \right| ^{q'}\right) ^{\frac{1}{q'}}\right) =\rho _{p(\cdot )}\left( \frac{\left( \sum _{j\in \mathbb {Z}}\left| f_j \right| ^{q}\right) ^\frac{1}{q}}{\left\| f\right\| _{L^{p(\cdot )}(\ell ^q)}}\right) =1,$ where corresponding changes in notation apply if $q'=\infty .$ By the definition of the $L^{p'(\cdot )}$ norm, we obtain that $\left\| \{h_j\}_{j\in \mathbb {Z}}\right\| _{L^{p'(\cdot )}(\ell ^{q'})}\le 1.$ Moreover,

$$\begin{aligned} \int _{{{\mathbb R}^n}}\sum _{j\in \mathbb {Z}} \left| f_j(x)h_j(x) \right| \, \mathrm{{d}}x&=\left\| f\right\| _{L^{p(\cdot )}(\ell ^q)}, \end{aligned}$$

from which (6.5) follows.

When $q=\infty$ and $0<\left\| f\right\| _{L^{p(\cdot )}(\ell ^\infty )}<\infty ,$ suppose first that $f_j=0$ for $\left| j \right| \ge N_0$ for some $N_0\in \mathbb {N}.$ For $\varepsilon >0$ and $x\in {{\mathbb R}^n},$ set $F_\varepsilon (x)=\left\{ j\in \mathbb {Z}:\left| f_j(x) \right| >\frac{1}{1+\varepsilon }\sup _{k\in \mathbb {Z}}\left| f_k(x) \right| \right\}$ and

$$\begin{aligned} h_{j,\varepsilon }(x)=\text {sign}(f_j(x))g_{j,\varepsilon }(x)\left( \sup _{k\in \mathbb {Z}}\left| f_k(x) \right| \right) ^{p(x)-1}\left\| f\right\| _{L^{p(\cdot )}(\ell ^\infty )}^{1-p(x)}, \end{aligned}$$

where $g_{j,\varepsilon }(x)=\frac{\chi _{F_\varepsilon (x)}(j)}{\#(F_\varepsilon (x))}$ if $\#(F_\varepsilon (x))\ne 0$ and $g_{j,\varepsilon }(x)=0$ otherwise. It follows that $\left\| \{h_{j,\varepsilon }\}_{j\in \mathbb {Z}}\right\| _{L^{p'(\cdot )}(\ell ^1)}\le 1$ and that

$$\begin{aligned} \int _{{{\mathbb R}^n}}\sum _{j\in \mathbb {Z}}\left| f_j(x)h_{j,\varepsilon }(x) \right| \,\mathrm{d}x&\ge \frac{1}{1+\varepsilon }\left\| f\right\| _{L^{p(\cdot )}(\ell ^\infty )}. \end{aligned}$$

Since $\varepsilon$ is arbitrary, we get the desired result. We next remove the assumption that $f_j=0$ if $\left| j \right| \ge N_0;$ for $N\in \mathbb {N},$ set $f_{j,N}=f_j$ if $\left| j \right| \le N$ and $f_{j,N}=0$ if $\left| j \right| >N.$ By the previous case, we have

$$\begin{aligned} \left\| \{f_{j,N}\}_{j\in \mathbb {Z}}\right\| _{L^{p(\cdot )}(\ell ^\infty )}\le \sup _{\{h_j\}\in \mathcal {U}_{{p'(\cdot )},1}}\int _{{\mathbb R}^n}\sum _{j\in \mathbb {Z}} \left| f_j(x)h_j(x) \right| \, \mathrm{{d}}x\quad \forall N\in \mathbb {N}. \end{aligned}$$

Since $\sup _{j\in \mathbb {Z}}\left| f_{j,N}(x) \right| \uparrow \sup _{j\in \mathbb {Z}}\left| f_j(x) \right|$ as $N\rightarrow \infty$ and for all $x\in {{\mathbb R}^n},$ the Monotone Convergence Theorem in the setting of variable Lebesgue spaces (see [8, Theorem 2.59]) and the above estimate imply (6.5).

Suppose now that $\left\| f\right\| _{L^{p(\cdot )}(\ell ^q)}=\infty ;$ for each $N\in \mathbb {N}$ define $I_{N}=\{j\in \mathbb {Z}:\left| j \right| \le N\}\times \{x\in {{\mathbb R}^n}:\left| x \right| \le N\}$ and $f^{N}_j(x)=f_j(x)\chi _{I_{N}}(j,x)$ if $\left| f_j(x)\chi _{I_{N}}(j,x) \right| <N$ and $f^{N}_j(x)=N,$ otherwise. Since $f^{N}=\{f_j^{N}\}_{j\in \mathbb {Z}}$ satisfies $\left\| f^{N}\right\| _{L^{p(\cdot )}(\ell ^q)}<\infty ,$ then

$$\begin{aligned} \left\| f^{N}\right\| _{L^{p(\cdot )}(\ell ^q)}\le \sup _{\{h_j\}\in \mathcal {U}_{{p'(\cdot )}, {q'}}}\int _{{\mathbb R}^n}\sum _{j\in \mathbb {Z}} \left| f_j(x)h_j(x) \right| \, \mathrm{{d}}x. \end{aligned}$$

Since $\left\| \{f_j^{N}(x)\}_{j\in \mathbb {Z}}\right\| _{\ell ^q}\uparrow \left\| \{f_j(x)\}_{j\in \mathbb {Z}}\right\| _{\ell ^q}$ as $N\rightarrow \infty$ for $x\in {{\mathbb R}^n},$ the Monotone Convergence Theorem for variable Lebesgue spaces and the last inequality imply (6.5).

The proof of (6.3) is similar; we briefly indicate the corresponding main changes. As in the proof of (6.2), it is enough to show the analogous inequality to (6.5).

In the case $1\le q<\infty$ and $0<\left\| f\right\| _{\ell ^q(L^{p(\cdot )})}<\infty$ define

$$\begin{aligned} h_j(x)=\text {sign}(f_j(x))\left| f_j(x) \right| ^{p(x)-1}\left\| f_j\right\| _{L^{p(\cdot )}}^{q-p(x)}\left\| f\right\| _{\ell ^q(L^{p(\cdot )})}^{1-q}. \end{aligned}$$

For the case $q=\infty$ and $0<\left\| f\right\| _{\ell ^\infty (L^{p(\cdot )})}<\infty$, assume first that $f_j=0$ for $\left| j \right| \ge N_0$ for some $N_0\in \mathbb {N}.$ Given $\varepsilon >0,$ define $F_\varepsilon =\left\{ j\in \mathbb {Z}:\left\| f_j\right\| _{L^{p(\cdot )}}>\frac{1}{1+\varepsilon }\left\| f\right\| _{\ell ^\infty (L^{p(\cdot )})}\right\}$ and set

$$\begin{aligned} h_{j,\varepsilon }(x)=\text {sign}(f_j(x))\frac{\chi _{F_\varepsilon }(j)}{\#(F_\varepsilon )} \left| f_j(x) \right| ^{p(x)-1}\left\| f_j\right\| _{L^{p(\cdot )}}^{1-p(x)}. \end{aligned}$$

The limiting arguments used to remove the assumption $f_j=0$ for $\left| j \right| \ge N_0$ in the case $q=\infty$ and for the case $\left\| f\right\| _{\ell ^q(L^{p(\cdot )})}=\infty$ are analogous to the ones used for (6.5). $\square$