Cauchy Problem for Incompressible Neo-Hookean Materials

Abstract

In this paper we consider the Cauchy problem for neo-Hookean incompressible elasticity in spatial dimension $d \geqq 2$. The Cauchy problem can be formulated in terms of maps $x(t,\cdot )$ with domain a reference space ${\mathbb {R}}^d_\xi$, and with values in space ${\mathbb {R}}^d_x$. Initial data consists of initial deformation $\phi (\xi ) = x(0,\xi )$ and velocity $\psi (\xi ) = \partial x(t,\xi )/\partial t |_{t=0}$. We consider the initial deformations of the form $x(0, \xi ) = A \xi + \varphi (\xi )$, where A is a constant $SL(d, {\mathbb {R}})$ matrix. We assume that $\varphi$ and $\psi$ are in Sobolev spaces $(\varphi , \psi ) \in H^{s+1}({\mathbb {R}}^d)\times H^{s}({\mathbb {R}}^d)$. If $s>s_{crit}= d/2+1$, well-posedness is well-known. We are here interested primarily in the low regularity case, $s \le s_{crit}$. For $d = 2, 3$, we prove existence and uniqueness for $s_0 < s\le s_{crit}$, and we can prove the well-posedness, but for a smaller range, $s_1 < s \le s_{crit}$, where, if $d = 2$, $s_0 = 7/4$ and $s_1 = 7/4 + (\sqrt{65}-7)/8$, and if $d=3$, then $s_0=2$ and $s_1 = 1 + \sqrt{3/2}$. For the full range (in s) results, as indicated above, we need additional Hölder regularity assumptions on certain combinations of second order derivatives of $\varphi$. A key observation in the proof is that the equations of evolution for the vorticities decompose into a first-order hyperbolic system, for which a Strichartz estimate holds, and a coupled transport system. This allows one to set up a bootstrap argument to prove local existence and uniqueness. Continuous dependence on initial data is proved using an argument inspired by Bona and Smith, and Kato and Lai, with a modification based on new estimates for Riesz potentials. The results of this paper should be compared to what is known for the ideal fluid equations, where, as shown by Bourgain and Li, the requirement $s > s_{crit}$ is necessary.

Appendices

Function Spaces

For the convenience of the reader, we include some background on the function spaces used in this paper. All spaces we use belong to the scales of Besov spaces $B^s_{p, q}({{\mathbb {R}}}^d)$ and Lizorkin–Triebel spaces $F^s_{p, q}({{\mathbb {R}}}^d)$. Also, we use the homogenous versions of these spaces. For their definition and basic properties we rely on [36].

The $L^p({{\mathbb {R}}}^d)$ norm of a function, f, is denoted $\Vert f\Vert _p$ or, if it is convenient, in one of the following forms:

$$\begin{aligned} \Vert f\;\big |\;L^p\Vert = \Vert f(x)\;\big |\;L^p(\textrm{d}x)\Vert . \end{aligned}$$

Similar forms are used for the norms in other function spaces. The integral $\int$ is the Lebesgue integral over ${{\mathbb {R}}}^d$. ${{{\mathcal {S}}}} = {{{\mathcal {S}}}}({{\mathbb {R}}}^d)$ is the Schwartz space of rapidly decreasing test functions, and ${{{\mathcal {S}}}}^\prime = {{{\mathcal {S}}}}^\prime ({{\mathbb {R}}}^d)$ is its dual, the space of tempered distributions. The pairing between ${{{\mathcal {S}}}}^\prime$ and ${{{\mathcal {S}}}}$ is denoted $\langle f, g\rangle$ and, for a regular distribution $f\in L^1\subset {{{\mathcal {S}}}}^\prime$, $\langle f, g\rangle = \int f(x)\,g(x)\,\textrm{d}x$.

• The Fourier transform ${{{\mathcal {F}}}}$ is defined as

  $$\begin{aligned} {{{\mathcal {F}}}}_{x\rightarrow \kappa } f = {{{\hat{f}}}}(\kappa ) = \int e^{-i x \kappa } f(x)\,\textrm{d}x, \end{aligned}$$

  with the inverse

  

  where   .

• Notation for the Riesz and Bessel potentials: $D^s = \left( -\Delta \right) ^{s/2} = {{{\mathcal {F}}}}^{-1}_{\kappa \rightarrow \cdot } |\kappa |^s {{{\mathcal {F}}}}_{x\rightarrow \kappa }$ and $J^s = \left( 1 - \Delta \right) ^{s/2} = {{{\mathcal {F}}}}^{-1}_{\kappa \rightarrow \cdot }\left( 1 + |\kappa |^2\right) ^{s/2} {{{\mathcal {F}}}}_{x\rightarrow \kappa }$, with $s\in {{\mathbb {R}}}$.

• In this paper, all function spaces are subspaces of ${{{\mathcal {S}}}}^\prime$. If $A_1$ and $A_2$ are such Banach spaces, their intersection, $A_1 \cap A_2$, is viewed as a Banach space with the norm $\Vert f\,\big |\,A_1 \cap A_2\Vert = \max \left( \Vert f\,\big |\,A_1\Vert ,\,\Vert f\,\big |\,A_2\Vert \right)$.

1.1 Littlewood–Paley decomposition

Pick a smooth function $\psi _0: [0, +\infty )\rightarrow [0, 1]$ such that $\psi _0(s) = 1$ if $s \le 1$, $\psi _0(s) = 0$ if $s \ge 2$, and $0< \psi _0(s) < 1$ if $1< s < 2$. Set $\varphi _0(s) = \psi _0(s) - \psi _0(2\,s)$. Then $\hbox {supp}\,\varphi _0 = [ 2^{-1}, 2]$. For $n\in {{\mathbb {Z}}}$, define $\psi _n(t) = \psi _0(2^{-n}t)$ and $\varphi _n(t) = \varphi _0(2^{-n}t)$. Then, for all integer n,

$$\begin{aligned} \varphi _{n-1}(t) + \varphi _m(t) + \varphi _{n+1}(t) = 1\quad \text {when}\quad t\in \hbox {supp}\; \varphi _n = \{2^{n-1} \le s \le 2^{n+1}\}\, \end{aligned}$$

and

$$\begin{aligned} \sum _{n = -\infty }^{+\infty } \varphi _n(t) = 1,\quad \forall t > 0. \end{aligned}$$

Also, for all $N\in {{\mathbb {Z}}}$ and for all $t\ge 0$, we have

$$\begin{aligned} \begin{aligned}&\psi _N(t) + \sum _{n = N+1}^\infty \varphi _n(t) = 1, \\&\psi _0(t) + \sum _{n=1}^N \varphi _n(t) = \psi _N(t). \end{aligned} \end{aligned}$$

(A.1)

Abusing the notation we write $\varphi _n(\kappa )$ instead of $\varphi _n(|\kappa |)$, where $\kappa \in {{\mathbb {R}}}^d$, and similarly understood are $\psi _n(\kappa )$. Also, we write $\varphi _n(D)$, $\psi _n(D)$, etc. for the corresponding pseudodifferential operators, i.e.,

Thanks to the first identity in (A.1), any tempered distribution $f\in {{{\mathcal {S}}}}^\prime$ can be expanded as (the Littlewood–Paley decomposition)

$$\begin{aligned} f = {{{\mathcal {F}}}}^{-1}\psi _N {{{\mathcal {F}}}} f + \sum _{n=N+1}^\infty {{{\mathcal {F}}}}^{-1}\varphi _n {{{\mathcal {F}}}} f\;= \psi _N(D) f +\sum _{n=N+1}^\infty \varphi _n(D) f, \end{aligned}$$

(A.2)

where the series converges in ${{{\mathcal {S}}}}^\prime$. We will abbreviate sometimes $f_n = \varphi _n (D) f$.

1.2 Homogeneous Besov and Lizorkin–Triebel spaces

Following Triebel [36], define ${{{\mathcal {Z}}}} = {{{\mathcal {Z}}}}({{\mathbb {R}}}^d)$ as the subspace of ${{\mathcal {S}}}$ consisting of those test functions $\eta$ which satisfy the condition $\int x^\alpha \,\eta (x)\,\textrm{d}x = 0$ for all multiindices $\alpha = (\alpha _1,\dots , \alpha _d)\in {{\mathbb {Z}}}^d$ with all $\alpha _j\ge 0$. Equivalently, $\eta \in {{{\mathcal {Z}}}}$ iff ($\eta \in {{{\mathcal {S}}}}$ and) $\partial ^\alpha {{{\hat{\eta }}}}(0) = 0$ for all nonnegative multiindices $\alpha$. With the topology inherited from ${{\mathcal {S}}}$, ${{\mathcal {Z}}}$ is a complete locally convex space. Polynomials when viewed as elements of ${{{\mathcal {S}}}}^\prime$, annihilate ${{\mathcal {Z}}}$: If P is a polynomial, $P(x) = \sum c_\alpha x^\alpha$, and if $\eta \in {{{\mathcal {Z}}}}$, then

$$\begin{aligned} \langle P, \eta \rangle = \langle {{{\hat{P}}}}, {{{\hat{\eta }}}}\rangle = \sum _\alpha c_\alpha \,\langle (i \partial )^\alpha \,\delta , {{{\hat{\eta }}}}\rangle = \sum _\alpha c_\alpha \,(-i)^\alpha \,\langle \delta , \partial ^\alpha {{{\hat{\eta }}}}\rangle = 0. \end{aligned}$$

Conversely, any tempered distribution f that annihilates ${{{\mathcal {Z}}}}$ is a polynomial. Indeed, if $\langle f, \eta \rangle = 0$ for all $\eta \in {{{\mathcal {Z}}}}$, then, in particular, $\langle {{{\hat{f}}}}, {{{\hat{\eta }}}}\rangle = 0$ for every $\eta$ with $0\notin \hbox {supp}\,{{{\hat{\eta }}}}$. Hence, $\hbox {supp}\,{{{\hat{f}}}} = \{0\}$. Therefore, f is a polynomial. Denote by ${{{\mathcal {Z}}}}^\prime$ the topological dual of ${{\mathcal {Z}}}$. If $\ell$ is a linear continuous functional on ${{{\mathcal {Z}}}}$, then there exist constants $C\ge 0$ and $K\in {{\mathbb {Z}}}$, $K\ge 0$, such that

$$\begin{aligned} |\ell (\eta )|\le \,C\,\sum _{|\alpha |\le K, |\beta |\le K}\sup _x |x^\alpha \,\partial ^\beta \eta (x)|. \end{aligned}$$

By the Hahn-Banach theorem, there exists a linear extension of $\ell$ from ${{{\mathcal {Z}}}}$ to ${{\mathcal {S}}}$ with the same inequality valid for all $\eta$ in ${{\mathcal {S}}}$. As elements of ${{{\mathcal {S}}}}^\prime$, any two such extensions must differ by a polynomial. This leads to identification of ${{{\mathcal {Z}}}}^\prime$ with the quotient space of ${{{\mathcal {S}}}}^\prime$ by the subspace $P\subset {{{\mathcal {S}}}}^\prime$ of all polynomials: ${{{\mathcal {Z}}}}^\prime \simeq {{{\mathcal {S}}}}^\prime /P$. The following Littlewood–Paley decomposition applies to distributions in ${{{\mathcal {Z}}}}^\prime$:

$$\begin{aligned} f = \sum _{n=-\infty }^\infty \varphi _n(D)\,f, \end{aligned}$$

(A.3)

which really means that for every $f\in {{{\mathcal {S}}}}^\prime$ there exist an integer $K\ge 0$, a sequence of polynomials $p_N(x)$ of degree not greater than K, and a polynomial $p_\infty (x)$ such that

$$\begin{aligned} \sum _{n = - N}^\infty \varphi _n(D)\,f + p_N \underset{N\rightarrow +\infty }{\longrightarrow }\ f + p_\infty \quad \text {in}\;\;{{{\mathcal {S}}}}^\prime \, \end{aligned}$$

(A.4)

(see [28]).

• The homogeneous Besov space ${\dot{B}}^s_{p, q} = {\dot{B}}^s_{p, q}({{\mathbb {R}}}^d)$ with the parameters $s\in {{\mathbb {R}}}$, $1\le p\le \infty$, and $1\le q < \infty$, is the subspace of ${{{\mathcal {Z}}}}^\prime$ composed of those $f\in {{{\mathcal {S}}}}^\prime$ for which the norm

  $$\begin{aligned} \Vert f\;\big |\; {\dot{B}}^s_{p, q}\Vert = \left( \sum _{n=-\infty }^\infty 2^{s n q} \Vert \varphi _n (D) f\Vert _p^q\right) ^{1/q} \end{aligned}$$

  (A.5)

  is finite. If $q = \infty$, then

  $$\begin{aligned} \Vert f\;\big |\; {\dot{B}}^s_{p, \infty }\Vert = \sup _{n\in {{\mathbb {Z}}}} 2^{s n} \Vert \varphi _n(D) f\Vert _p \end{aligned}$$

  (A.6)

• For $s\in {{\mathbb {R}}}$, $1\le p < \infty$, and $1\le q < \infty$, the homogeneous Lizorkin–Triebel space ${\dot{F}}^s_{p, q} = {\dot{F}}^s_{p, q}({{\mathbb {R}}}^d)$ is the subspace of ${{{\mathcal {Z}}}}^\prime$ composed of those $f\in {{{\mathcal {S}}}}^\prime$ for which the norm

  $$\begin{aligned} \Vert f\;\big |\; {\dot{F}}^s_{p, q}\Vert = \Vert \left( \sum _{n=0}^\infty 2^{s n q}\,|\varphi _n(D) f|^q \right) ^{1/q}\;\big |\;L^p\Vert \end{aligned}$$

  is finite. A modification as above is needed in the case $q = \infty$. (The case $p=\infty$ requires a special treatment, see [36].)

• The homogeneous Sobolev space ${\dot{H}}^s_p = {\dot{H}}^s_p({{\mathbb {R}}}^d)$ is the space of all $f\in {{{\mathcal {Z}}}}^\prime$ such that the norm

  $$\begin{aligned} \Vert f\;\big |\; {\dot{H}}^s_{p}\Vert = \Vert \sum _{n=-\infty }^\infty D^s \varphi _n(D)f\;\big |\;L^p\Vert \end{aligned}$$

  (A.7)

  is finite (the range of parameters is $-\infty< s < +\infty$, $1\le p\le \infty$). When $p = 2$, we write ${\dot{H}}^s$ instead of ${\dot{H}}^s_2$.

• Basic embeddings.

  $$\begin{aligned} \begin{aligned}&{\dot{B}}^s_{p, q_1}\subset {\dot{B}}^s_{p, q_2},\quad {\dot{F}}^s_{p, q_1}\subset {\dot{F}}^s_{p, q_2},\;\text {if}\; 1\le q_1\le q_2\le \infty \\&{\dot{B}}^s_{p, \min (p,q)}\subset {\dot{F}}^s_{p, q}\subset {\dot{B}}^s_{p, \max (p,q)},\\&{\dot{B}}^{s_1}_{p_1, q_1} \subset {\dot{B}}^{s_2}_{p_2, q_2}\;\text {if}\;1\le p_1\le p_2\le \infty , \; 1\le q_1\le q_2\le \infty , \;s_2 - \frac{d}{p_2} = s_1 - \frac{d}{p_1} \\&{\dot{F}}^{s_1}_{p_1, q_1} \subset {\dot{F}}^{s_2}_{p_2, q_2}\;\text {if}\;1\le p_1< p_2 < \infty , \; 1\le q_1,\, q_2\le \infty , \;s_2 - \frac{d}{p_2} = s_1 - \frac{d}{p_1} \end{aligned} \end{aligned}$$

  (A.8)

• For all $s, r\in {{\mathbb {R}}}$, the operator $D^r = (-\Delta )^{r/2} = {{{\mathcal {F}}}}^{-1}_{\kappa \rightarrow \cdot }|\kappa |^r{{{\mathcal {F}}}}_{x\rightarrow \kappa }$ is an isomorphism between ${\dot{B}}^{s+r}_{p, q}$ and ${\dot{B}}^{s}_{p, q}$ if $p, q\in [1,\infty ]$, and between ${\dot{F}}^{s+r}_{p, q}$ and ${\dot{F}}^{s}_{p, q}$, when $1\le p < \infty$, $1\le q\le \infty$, see Theorem 5.2.3.1 in [36].

• The topological dual of ${\dot{B}}^s_{p,q}$ is ${\dot{B}}^{-s}_{p^\prime ,q^\prime }$ and the topological dual of ${\dot{F}}^s_{p,q}$ is ${\dot{F}}^{-s}_{p^\prime ,q^\prime }$, where $s\in {{\mathbb {R}}}$, $1\le q <\infty$, and $1\le p <\infty$. (As usual, $1/p+1/p^\prime = 1/q+1/q^\prime = 1$.)

• Interpolation inequalities. If $0< \theta < 1$, $s = (1-\theta ) s_0 + \theta s_1$, and if

  $$\begin{aligned} \frac{1}{p} = \frac{1-\theta }{p_0} + \frac{\theta }{p_1},\quad \frac{1}{q} = \frac{1-\theta }{q_0} + \frac{\theta }{q_1}, \end{aligned}$$

  where $1\le p_0, p_1\le \infty$ and $1\le q_0, q_1\le \infty$, then

  $$\begin{aligned} \Vert f\;\big |\; {\dot{B}}^s_{p, q}\Vert \lesssim \Vert f\;\big |\; {\dot{B}}^{s_0}_{p_0, q_0}\Vert ^{1-\theta }\;\Vert f\;\big |\; {\dot{B}}^{s_1}_{p_1, q_1}\Vert ^{\theta }. \end{aligned}$$

  This is due to the fact that ${\dot{B}}^s_{p, q} = \left[ {\dot{B}}^{s_0}_{p_0, q_0}, {\dot{B}}^{s_1}_{p_1, q_1}\right] _\theta$, the complex interpolation. The analogous result is true for the Lizorkin–Triebel spaces:

  $$\begin{aligned} \Vert f\;\big |\; {\dot{F}}^s_{p, q}\Vert \lesssim \Vert f\;\big |\; {\dot{F}}^{s_0}_{p_0, q_0}\Vert ^{1-\theta }\;\Vert f\;\big |\; {\dot{F}}^{s_1}_{p_1, q_1}\Vert ^{\theta }. \end{aligned}$$

• Isomorphisms between spaces.

  $$\begin{aligned} \begin{aligned}&{\dot{F}}^0_{p,2} \simeq L^p,\quad {\dot{F}}^s_{p,2}\simeq {\dot{H}}^s_p,\; 1< p < \infty , s\in {{\mathbb {R}}}\\&{\dot{F}}^s_{p,p} \simeq {\dot{B}}^s_{p,p} \end{aligned} \end{aligned}$$

  (A.9)

  It is known ([36]) that, for $r\in (0, 1)$, the ${\dot{B}}^r_{\infty , \infty }$-seminorm is equivalent to the homogeneous Hölder ${\dot{C}}^r$ seminorm:

  $$\begin{aligned} \sup _{n\in {{\mathbb {Z}}}} 2^{rn}\;\Vert {{{\mathcal {F}}}}^{-1}\varphi _n {{{\mathcal {F}}}} f\Vert _\infty \quad \simeq \quad \{f\}_r = \sup _{x \ne y} \frac{|f(x) - f(y)|}{|x - y|^r}, \end{aligned}$$

  and, for $1\le p < \infty$, the ${\dot{B}}^r_{p, p}$-seminorm is equivalent to the Gagliardo seminorm

  $$\begin{aligned} \left[ f\;\big |\;{\dot{B}}^r_{p, p}\right] _* = \left( \int \int \frac{|f(x) - f(y)|^p}{|x - y|^{r p + d}} \right) ^{1/p} \end{aligned}$$

  An equivalent seminorm in ${\dot{H}}^s$ is

  

• The nonhomogeneous Besov and Lizorkin–Triebel spaces are made of tempered distributions with the norms

  $$\begin{aligned} \Vert f\;\big |\; B^s_{p, q}\Vert = \Vert \psi _0(D)\,f\Vert _p + \left( \sum _{n=0}^\infty 2^{s n q} \Vert \varphi _n (D) f\Vert _p^q\right) ^{1/q} \end{aligned}$$

  and

  $$\begin{aligned} \Vert f\;\big |\; F^s_{p, q}\Vert = \Vert \psi _0(D)\,f\Vert _p + \Vert \left( \sum _{n=0}^\infty 2^{s n q}\,|\varphi _n(D) f|^q \right) ^{1/q}\;\big |\;L^p\Vert , \end{aligned}$$

  respectively. Equivalent norms are obtained when the part $\Vert \psi _0(D)\,f\Vert _p$ is replaced with $\Vert f\Vert _p$. For all $s, r\in {{\mathbb {R}}}$, the operator $J^r = (1-\Delta )^{r/2} = {{{\mathcal {F}}}}^{-1}_{\kappa \rightarrow \cdot }(1+|\kappa |^2)^{r/2}{{{\mathcal {F}}}}_{x\rightarrow \kappa }$ is an isomorphism between the nonhomogeneous spaces ${ B}^{s+r}_{p, q}$ and ${ B}^{s}_{p, q}$ if $p, q\in [1,\infty ]$, and between ${ F}^{s+r}_{p, q}$ and ${ F}^{s}_{p, q}$, when $1\le p < \infty$, $1\le q\le \infty$. The nonhomogeneous spaces are monotone with respect to the parameter S:

  $$\begin{aligned} B^{s_1}_{p, q} \subset B^{s_2}_{p, q}\quad \text {and}\quad F^{s_1}_{p, q} \subset F^{s_2}_{p, q} \end{aligned}$$

  when $s_1\ge s_2$. The corresponding homogeneous spaces are not monotone with respect to s. However, the following result is easy to prove.

Lemma A.1

Let $s > 0$, $1\le m\le p\le \infty$, and $1\le q\le \infty$. Then $L^m\cap {\dot{B}}^s_{p, q} = L^m\cap { B}^s_{p, q}$. If $s_1> s_2 > 0$, then $L^m\cap {\dot{B}}^{s_1}_{p, q}\subset L^m\cap {\dot{B}}^{s_2}_{p, q}$. Moreover, if $s_1 \ge s_2 > 0$, $1\le m\le p_2\le p_1\le \infty$, and

$$\begin{aligned} s_1 - \frac{d}{p_1} \ge s_2 - \frac{d}{p_2}, \end{aligned}$$

then $L^m\cap {\dot{B}}^{s_1}_{p_1, q} \subset L^m\cap {\dot{B}}^{s_2}_{p_2, q}$ (for any $1\le q\le \infty$).

Proof

Assume $f\in L^m\cap {\dot{B}}^s_{p, q}$. Then, for any $g\in L^{p^\prime }$,

$$\begin{aligned} \langle g, \psi _0(D) f\rangle = \int h(x - y)\,f(y)\,g(x)\,\textrm{d}x\,\textrm{d}y, \end{aligned}$$

where

By Young’s convolution inequality,

$$\begin{aligned} |\langle g, \psi _0(D) f\rangle | \le \Vert h\Vert _\ell \;\Vert f\Vert _m\,\Vert g\Vert _{p^\prime }, \end{aligned}$$

if

$$\begin{aligned} \frac{1}{\ell }= 1 + \frac{1}{p} - \frac{1}{m}. \end{aligned}$$

Since $m \le p$, this equality defines $\ell$ so that $1\le \ell < \infty$. Clearly, $h\in L^\ell$, since $\psi _0$ is smooth and has compact support. Thus,

$$\begin{aligned} \Vert \psi _0(D) f\Vert _p \lesssim \Vert f\Vert _m. \end{aligned}$$

This proves that $f\in L^m\cap { B}^s_{p, q}$ and $L^m\cap {\dot{B}}^s_{p, q} \subset L^m\cap { B}^s_{p, q}$. If $s > 0$, then ${ B}^s_{p, q}\subset {\dot{B}}^s_{p, q}$. Together, these observations prove the isomorphism of spaces: $L^m\cap {\dot{B}}^s_{p, q} = L^m\cap { B}^s_{p, q}$. The remaining statements follow from the corresponding statements for non-homogeneous Besov spaces.

$\square$

1.3 Gagliardo–Nirenberg inequality and Runst’s lemma

Recall the classical Gagliardo–Nirenberg inequality:

$$\begin{aligned} \Vert D^j g\Vert _p \lesssim \Vert D^m g\Vert _{p_1}^\alpha \,\Vert g\Vert _{p_2}^{1 - \alpha }, \end{aligned}$$

(A.10)

where j and m are integers such that $0< j < m$, $1 \le p_1 < \infty$, $1 \le \,p_2 \le \infty$, and

$$\begin{aligned} \alpha = \frac{j}{m},\quad \frac{1}{p} = \frac{\alpha }{p_1} + \frac{1 - \alpha }{p_2}. \end{aligned}$$

For more general forms/versions of this inequality see [7, 34].

Runst’s inequality is a type of a Gagliardo–Nirenberg inequality stated in terms of the Lizorkin–Triebel spaces. Its interesting feature is that there is no restriction on the parameters $q_1$ and q within the range $(0, +\infty ]$ (though the constants depend on their choice).

Lemma A.2

Let $\alpha \in (0, 1)$ and $0< p < \infty$, and $0 < q_1, q \le \infty$, $r > 0$. Then, for any $g\in L^\infty \cap F^s_{p, q_1}$,

$$\begin{aligned} \Vert g\;\big |\; F^{\alpha r}_{p/\theta , q}\Vert \lesssim \Vert g\;\big |\; F^r_{p, q_1}\Vert ^\alpha \,\Vert g\Vert _\infty ^{1 - \alpha }. \end{aligned}$$

(A.11)

Also, for any $g\in L^\infty \cap {\dot{F}}^r_{p, q_1}$,

$$\begin{aligned} \Vert g\;\big |\; {\dot{F}}^{\alpha r}_{p/\theta , q}\Vert \lesssim \Vert g\;\big |\; {\dot{F}}^r_{p, q_1}\Vert ^\alpha \,\Vert g\Vert _\infty ^{1 - \alpha }. \end{aligned}$$

(A.12)

Proof

The proof in the nonhomogeneous case, (A.11), is given by Runst, see Lemma 1, Section 5.2 of [30], and also Lemma 1, Section 5.3.7 [31]). It relies on Oru’s lemma, [7, Lemma 3.7]. For the homogeneous spaces one needs a slight generalization of Oru’s lemma, namely,

Lemma A.3

If $-\infty< s_1< s_2 < +\infty$, $0< q < \infty$, $0< \alpha < 1$, and $s = \alpha s_1 + (1 - \alpha ) s_2$, then

$$\begin{aligned} \Vert 2^{s j} a_j\Vert _{\ell ^q} \lesssim \Vert 2^{s_1j} a_j\Vert _{\ell ^\infty }^{\alpha }\;\Vert 2^{s_2j} a_j\Vert _{\ell ^\infty }^{1 - \alpha } \end{aligned}$$

(A.13)

for any sequence $\{a_j\}_{j=-\infty }^\infty$.

We leave its proof to the reader and continue with the proof of (A.12).

Denote $g_n = {{{\mathcal {F}}}}^{-1} \varphi _n {{{\mathcal {F}}}}g$. With $s = \alpha r$, $s_1 = r$, and $s_2 = 0$, it follows from Lemma A.3 that

$$\begin{aligned}{} & {} \left( \sum _n 2^{s n q} |g_n|^q\right) ^{1/q} \lesssim \left( \sup _n 2^{s_1 n} |g_n|\right) ^{\alpha }\;\left( \sup _n 2^{s_2 n} |g_n|\right) ^{1 - \alpha }\\{} & {} = \left( \sup _n 2^{r n} |g_n|\right) ^{\alpha }\;\left( \sup _n |g_n|\right) ^{1 - \alpha } \end{aligned}$$

and, consequently,

$$\begin{aligned} \left( \sum _n 2^{\alpha r n q} |g_n|^q\right) ^{1/q} \lesssim \left( \sup _n 2^{r n} |g_n|\right) ^{\alpha }\;\Vert \sup _n |g_n|\Vert _\infty ^{1 - \alpha }. \end{aligned}$$

Since $\sup _n 2^{r n} |g_n| \le \left( \sum _{n\in {{\mathbb {Z}}}} 2^{r n q_1} |g_n|^{q_1} \right) ^{1/q_1}$ for any $q_1 > 0$, and since $\Vert \sup _n |g_n|\Vert _\infty \lesssim \Vert g\Vert _\infty$, we have

$$\begin{aligned} \left( \sum _n 2^{\alpha r n q} |g_n|^q\right) ^{1/q} \lesssim \left( \sum _{n\in {{\mathbb {Z}}}} 2^{r n q_1} |g_n|^q \right) ^{\alpha /q_1}\;\Vert g\Vert _\infty ^{1 - \alpha }. \end{aligned}$$

Take the $L^{p/\alpha }$ norm of both sides to obtain (A.12). $\square$

Norms of Products and Compositions of Functions

1.1 Norms of products

There are several results (sometimes called the fractional Leibniz rule) on the Sobolev/Lizorkin–Triebel norms of products of functions.

Lemma B.1

If $s > 0$ and $1< p < \infty$, then

$$\begin{aligned} \begin{aligned} \Vert D^s (f_1\cdot f_2)\Vert _p \lesssim \Vert D^s f_1\Vert _{q_1}\,\Vert f_2\Vert _{q_2} + \Vert f_1\Vert _{q_3}\,\Vert D^s f_2\Vert _{q_4}, \end{aligned} \end{aligned}$$

(B.1)

provided that

$$\begin{aligned} 1 < q_1, q_2, q_3, q_4\le \infty , \quad \frac{1}{q_1} + \frac{1}{q_2} = \frac{1}{q_3} + \frac{1}{q_4} = \frac{1}{p}. \end{aligned}$$

The same inequality is true with $D^s$ replaced by the operator $J^s = (1 - \Delta )^{s/2}$, with the same restrictions on the parameters:

$$\begin{aligned} \begin{aligned} \Vert f_1\cdot f_2\;\big |\; F^s_{p, 2}\Vert \lesssim \Vert f_1\;\big |\; F^s_{p\,q_1, 2}\Vert \,\Vert f_2\Vert _{p q_2} + \Vert f_1\Vert _{p q_3}\,\Vert f_2\;\big |\; F^s_{p\,q_4, 2}\Vert . \end{aligned} \end{aligned}$$

(B.2)

As a corollary, if $s > 0$ and $1< p < \infty$, then

$$\begin{aligned} \Vert f_1\cdot \cdots \cdot f_N\;\big |\;F^s_{p, 2}\Vert \le C\,\sum _{j = 1}^N \Vert f_j\;\big |\; F^s_{p\,p_j, 2}\Vert \,\cdot \,\prod _{i\ne j} \Vert f_i\Vert _{p\,p_i} \end{aligned}$$

(B.3)

provided that

$$\begin{aligned} \sum _1^N \frac{1}{p_j} = 1, \end{aligned}$$

For the proofs of (B.1) and (B.2) see [12, Theorem 1] and [4, Theorem1.1]. Some more general inequalities are established in [31]. For earlier results see the Christ and Weinstein paper [8, Prop. 3.3] and [35, Prop. 2.1.1].

1.2 Norms of compositions

Lemma B.1 is used, in particular, to estimate the $H^s$ norms of compositions f(u), where f is a sufficiently smooth functions and $u\in H^s\cap L^\infty$. The simplest result deals with the case $0 < s \le 1$.

Lemma B.2

Suppose that $f: {{\mathbb {R}}}\rightarrow {{\mathbb {R}}}$ is a locally Lipschitz function such that $f(0) = 0$. Then, if $0 < s \le 1$ and $1< p < \infty$,

$$\begin{aligned} \Vert f(u(\cdot ))\,\big |\;F^s_{p, 2}({{\mathbb {R}}}^d)\Vert \le C_f(\Vert u\Vert _\infty )\;\Vert u\,\big |\;F^s_{p, 2}({{\mathbb {R}}}^d)\Vert \end{aligned}$$

for any function $u\in F^s_{p, 2}({{\mathbb {R}}}^d)$ (recall that $F^s_{p, 2}({{\mathbb {R}}}^d) = H^{s, p}({{\mathbb {R}}}^d)$). Here

$$\begin{aligned} C_f(\Vert u\Vert _\infty ) = \inf \{C:\;|f(z_1) - f(z_2)|\le C \,|z_1 - z_2|,\;\forall z_1, z_2:\;|z_{1,2}|\le \Vert u\Vert _\infty \}. \end{aligned}$$

The proof of Lemma B.2 relies on the fact that $|f(u(x)) - f(u(y))|\le C_f(\Vert u\Vert _\infty )\,|u(x) - u(y)|$, see [35, Prop. 2.4.1]. Next, consider larger s. In the main body of the paper we need only the case of the Sobolev scale $H^s$.

Lemma B.3

Let $s > 0$. Let $f: {{\mathbb {R}}}\rightarrow {{\mathbb {R}}}$ be $r =\lfloor s \rfloor$ times continuously differentiable function such that its r th derivative is locally Lipschitz, and $f(0) = 0$. Then there is a continuous, nondecreasing function $C_{f, s, d}: {{\mathbb {R}}}_+\rightarrow {{\mathbb {R}}}_+$ such that

$$\begin{aligned} \Vert f(u(\cdot ))\,\big |\;{H^s({{\mathbb {R}}}^d)} \Vert \le C_{f, s, d}(\Vert u\Vert _\infty )\;\Vert u\Vert _{H^s({{\mathbb {R}}}^d)} \end{aligned}$$

(B.4)

for any function $u\in H^s({{\mathbb {R}}}^d)\cap L^\infty ({{\mathbb {R}}}^d)$.

Proof

For infinitely differential functions f, there is an elegant and short proof of (B.4) in Hörmander’s book [14, Theorem 8.5.1]. However, having in mind finitely differentiable f, we present a different argument. It can be used for other purposes as well (we use it to prove regularity of the inverse map in Lemma 3.1).

For s an integer, to prove (B.4) we follow Moser’s argument on p. 273 of paper [27]. Let $s = r$ be an integer greater than 1. Observe that $|f(u)|\le C_f(|u|)\,|u|$, and hence $\Vert f(u)\Vert \le C_f(\Vert u\Vert _\infty )\,\Vert u\Vert$. Now it suffices to consider the derivatives of the order r of f(u(x)). Schematically,

$$\begin{aligned} \partial ^r f(u) = \sum _{k \le r} f^{(k)}(u) \sum _{|\alpha | = k} C_{k \alpha } (\partial u)^{\alpha _1}(\partial ^2 u)^{\alpha _2}\dots (\partial ^r u)^{\alpha _r}, \end{aligned}$$

(B.5)

where $\alpha _1 + 2\alpha _2 + \dots + r\alpha _r = r$ and $C_{k \alpha }$ are non-negative constants. We have that

$$\begin{aligned} \Vert \partial ^r f(u)\Vert \le \sum _{k \le r} \Vert f^{(k)}(u)\Vert _\infty \sum _{|\alpha | = k} C_{k \alpha }\, \prod _{m=1}^r\Vert (\partial ^m u)^{\alpha _m}\Vert _{2 p_m}, \end{aligned}$$

where $1\le p_m\le \infty$ and $\sum _m 1/p_m = 1$. The right choice of $p_m$ is

$$\begin{aligned} p_m = \frac{r}{m\,\alpha _m}, \end{aligned}$$

because then

$$\begin{aligned} \Vert (\partial ^m u)^{\alpha _m}\Vert _{2 p_m} = \Vert \partial ^m u\Vert _{2 \alpha _m p_m}^{\alpha _m} \end{aligned}$$

and

$$\begin{aligned} \Vert \partial ^m u\Vert _{2 \alpha _m p_m} = \Vert \partial ^m u\Vert _{2 r/m} \lesssim \Vert u\Vert _{H^r}^{m/r}\,\Vert u\Vert _\infty ^{1 - m/r}, \end{aligned}$$

by the Gagliardo–Nirenberg inequality. Collecting all the terms, we obtain

$$\begin{aligned} \Vert \partial ^r f(u)\Vert _2 \lesssim \Vert u\Vert _{H^r}\;\sum _{k \le r} C_{f^{(k)}}(\Vert u\Vert _\infty )\,\Vert u\Vert _\infty ^{k-1} \sum _{|\alpha | = k}C_{k\alpha }. \end{aligned}$$

Now consider the case of fractional s. Assume $s = r + \gamma$, $r \ge 1$ is an integer and $0< \gamma < 1$. Since we have the $H^r$ norm of f(u) already bounded, it remains to show that the $H^\gamma$ norm of each term of the form

$$\begin{aligned} g(u)\,(\partial u)^{\alpha _1}(\partial ^2 u)^{\alpha _2}\dots (\partial ^r u)^{\alpha _r} \end{aligned}$$

(B.6)

is bounded (see (B.5)). First, consider the case g(u) is not a constant. Use the product estimate (B.2) to obtain that

$$\begin{aligned} \begin{aligned}&\Vert g(u)\,\prod _{k = 1}^r (\partial ^k u)^{\alpha _k}\Vert _{H^\gamma } \lesssim \Vert g(u)\Vert _\infty \,\Vert \prod _{k = 1}^r (\partial ^k u)^{\alpha _k}\Vert _{H^\gamma }\\&+ \Vert \prod _{k = 1}^r (\partial ^k u)^{\alpha _k}\Vert _{2 q}\,\Vert g(u)\,\big |\; F^\gamma _{2q^\prime , 2}\Vert , \end{aligned} \end{aligned}$$

(B.7)

where

$$\begin{aligned} q = \frac{s}{r},\quad q^\prime = \frac{s}{s - r}. \end{aligned}$$

(B.8)

Since $\Vert g(u)\Vert _\infty \lesssim C_g(\Vert u\Vert _\infty )\,\Vert u\Vert _\infty$, the first term on the right will be treated later (with the case $g(\cdot ) = const$). Thus, look at the second term. The choice of q is dictated by the following computation. First, use Hölder’s inequality,

$$\begin{aligned} \Vert \prod _{m = 1}^r (\partial ^m u)^{\alpha _m}\Vert _{2 q} \le \prod _{m = 1}^r\,\Vert \partial ^m u\Vert _{2 q \alpha _m p_m}^{\alpha _m}. \end{aligned}$$

Then,

$$\begin{aligned} \Vert \partial ^m u\Vert _{2 q \alpha _m p_m} \le \Vert u\;\big |\; F^m_{2 q \alpha _m p_m, 2}\Vert \end{aligned}$$

The norm $\Vert u\;\big |\; F^m_{2 q \alpha _m p_m, 2}\Vert$ will be bounded using (A.11) as follows:

$$\begin{aligned} \Vert u\;\big |\; F^m_{2 q \alpha _m p_m, 2}\Vert \le \Vert u\;\big |\;F^s_{2, 2}\Vert ^{\theta _m}\,\Vert u\Vert _\infty ^{1 - \theta _m}. \end{aligned}$$

This means the indices should satisfy

$$\begin{aligned} m = s\,\theta _m,\quad 2 q\,\alpha _m p_m = \frac{2}{\theta _m} \end{aligned}$$

and $1\le p_m \le \infty$,

$$\begin{aligned} \sum _m \frac{1}{p_m} = 1. \end{aligned}$$

Consequently, we must have that

$$\begin{aligned} \theta _m = \frac{m}{s} \end{aligned}$$

Then

$$\begin{aligned} 2 q \alpha _m p_m = \frac{2s}{m} \quad \Leftrightarrow \quad m\,\alpha _m = \frac{2s}{2 q}\,\frac{1}{p_m} \end{aligned}$$

Since $\sum m \alpha _m = r$ and we want $\sum 1/p_m = 1$, we must have $q = s/r$.

To estimate the norm $\Vert g(u)\,\big |\; F^\gamma _{2q^\prime , 2}\Vert$ We apply Lemma B.2 to get that

$$\begin{aligned} \Vert g(u)\,\big |\; F^\gamma _{2q^\prime , 2}\Vert \lesssim C(\Vert u\Vert _\infty )\,\Vert u\,\big |\; F^\gamma _{2q^\prime , 2}\Vert \end{aligned}$$

By Runst’s Lemma A.2,

$$\begin{aligned} \Vert u\,\big |\; F^\gamma _{2q^\prime , 2}\Vert \lesssim \Vert u\;\big |\; F^s_{2, 2}\Vert ^{1 - \frac{r}{s}}\,\Vert u\Vert _\infty ^{\frac{r}{s}}. \end{aligned}$$

Collecting the estimates, we get that

$$\begin{aligned} \begin{aligned}&\Vert \prod _{m = 1}^r (\partial ^m u)^{\alpha _m}\Vert _{2 q}\,\Vert g(u)\,\big |\; F^\gamma _{2q^\prime , 2}\Vert \lesssim \\&C(\Vert u\Vert _\infty )\,\Vert u\,\big |\;H^s\Vert ^{1 - \frac{r}{s}}\,\Vert u\Vert _\infty ^{\frac{r}{s}}\;\prod _{m=1}^r \Vert u\,\big |\;H^s\Vert ^{\alpha _m \theta _m}\,\Vert u\Vert _\infty ^{\alpha _m (1 - \theta _m)} = \\&C(\Vert u\Vert _\infty )\,\Vert u\Vert _\infty ^{\sum \alpha _m }\;\Vert u\Vert _{H^s}, \end{aligned} \end{aligned}$$

where we have used that, by construction,

$$\begin{aligned} \sum _{m=1}^r \alpha _m\,\theta _m = \frac{r}{s}. \end{aligned}$$

Finally, consider the case $g(u) \equiv const$. To estimate $\Vert \prod _m (\partial ^m u)^{\alpha _m}\;\big |\;{H^\gamma }\Vert$ we use (B.3) to get that

$$\begin{aligned} \Vert \prod _m (\partial ^m u)^{\alpha _m}\Vert _{H^\gamma } \lesssim \sum _m \Vert (\partial ^m u)^{\alpha _m}\;\big |\; F^\gamma _{2 q_m, 2}\Vert \,\prod _{j\ne m} \Vert (\partial ^j u)^{\alpha _j}\;\big |\; L^{2q_j}\Vert , \end{aligned}$$

where $q_1, \dots , q_r$ will be chosen later (and will satisfy $\sum _j 1/q_j = 1$). The factors in the product are bounded first as

$$\begin{aligned} \Vert (\partial ^j u)^{\alpha _j}\;\big |\; L^{2q_j}\Vert \lesssim \Vert u\;\big |\;F^j_{2\alpha _j q_j, 2}\Vert ^{\alpha _j}, \end{aligned}$$

and then Runst’s lemma is applied to get that

$$\begin{aligned} \Vert u\;\big |\;F^j_{2\alpha _j q_j, 2}\Vert \lesssim \Vert u\;\big |\;F^s_{2,2}\Vert ^{\beta _j}\,\Vert u\Vert _\infty ^{1 - \beta _j}. \end{aligned}$$

Lemma A.2 imposes the following restrictions:

$$\begin{aligned} j = \beta _j\,s,\quad 2 \beta _j\,\alpha _j\,q_j = 2; \end{aligned}$$

i.e.,

$$\begin{aligned} \beta _j = \frac{j}{s},\quad \frac{1}{q_j} = \alpha _j\,\frac{j}{s}. \end{aligned}$$

Thus,

$$\begin{aligned} \Vert (\partial ^j u)^{\alpha _j}\;\big |\; L^{2q_j}\Vert \lesssim \Vert u\;\big |\;H^s\Vert ^{j\,\alpha _j/s}\,\Vert u\Vert _\infty ^{\alpha _j(1 - j/s)}\, \end{aligned}$$

for $j\ne m$.

Next, consider $\Vert (\partial ^m u)^{\alpha _m}\;\big |\; F^\gamma _{2 q_m, 2}\Vert$. Assuming $\alpha _m > 0$, first use (B.3) with equal exponents to get that

$$\begin{aligned} \Vert (\partial ^m u)^{\alpha _m}\;\big |\; F^\gamma _{2 q_m, 2}\Vert \lesssim {}&\sum _{k = 1}^{\alpha _m} \Vert (\partial ^m u)\;\big |\; F^\gamma _{2\alpha _m q_m, 2}\Vert \, \prod _{j = 1}^{\alpha _m - 1}\Vert (\partial ^m u)\;\big |\;L^{2\alpha _m q_m}\Vert \\ ={}&\alpha _m \,\Vert (\partial ^m u)\;\big |\; F^\gamma _{2\alpha _m q_m, 2}\Vert \,\Vert (\partial ^m u)\;\big |\;L^{2\alpha _m q_m}\Vert ^{\alpha _m -1}, \end{aligned}$$

and continue as follows:

$$\begin{aligned} \lesssim \Vert u\;\big |\; F^{m +\gamma }_{2\alpha _m q_m, 2}\Vert \,\Vert u \;\big |\;F^m_{2\alpha _m q_m}\Vert ^{\alpha _m -1} \end{aligned}$$

Now each norm is bounded using (A.11). We have

$$\begin{aligned} \Vert u\;\big |\; F^{m +\gamma }_{2\alpha _m q_m, 2}\Vert \lesssim \Vert u\,\big |\; H^s\Vert ^\lambda \,\Vert u\Vert _\infty ^{1 - \lambda },. \end{aligned}$$

where

$$\begin{aligned} \lambda = \frac{m + \gamma }{s} \end{aligned}$$

and $q_m$ must be chosen so that

$$\begin{aligned} \frac{1}{q_m} = \alpha _m\,\frac{m + \gamma }{s}. \end{aligned}$$

Similarly,

$$\begin{aligned} \Vert u \;\big |\;F^m_{2\alpha _m q_m}\Vert \lesssim \Vert u\,\big |\; H^s\Vert ^{\lambda _m}\,\Vert u\Vert _\infty ^{1 - \lambda _m}, \end{aligned}$$

where $q_m$ is as above and

$$\begin{aligned} \lambda _m = \frac{m}{s}. \end{aligned}$$

Bringing the estimates together, we obtain

$$\begin{aligned} \Vert \prod _m (\partial ^m u)^{\alpha _m}\;\big |\;{H^\gamma }\Vert \lesssim {{{\tilde{C}}}}(\Vert u\Vert _\infty )\,\Vert u\Vert _{H^s}, \end{aligned}$$

as claimed.

$\square$

Estimates on Riesz Transforms

The Riesz transform operators ${{{\mathcal {R}}}}_j$, $j = 1, \dots , d$, will be defined by the formula

$$\begin{aligned} {{{\mathcal {R}}}}_j f = {{{\mathcal {F}}}}^{-1} \frac{\kappa ^j}{|\kappa |} {{{\mathcal {F}}}} f. \end{aligned}$$

It is well known (see [26, Theorem 10.2.1]) that ${{{\mathcal {R}}}}_j$ is bounded in ${\dot{B}}^s_{\infty , \infty }$, when $s > 0$, but not in ${ B}^s_{\infty , \infty }$. The Riesz transform is a special case of a more general class of pseudo-differential operators which we shall now discuss.

Denote by $S^0_{ph}$ the space of all smooth functions $a: {{\mathbb {R}}}^d{\setminus }\{0\}\rightarrow {{\mathbb {R}}}$ which are positively homogeneous of degree 0 and satisfy

$$\begin{aligned} |\partial ^\alpha _\kappa \,a(\kappa )|\le C_\alpha \,|\kappa |^{-|\alpha |} \end{aligned}$$

for any multiindex $\alpha$. Each symbol $a\in S^0_{ph}$ gives rise to a pseudodifferential operator ${{\mathfrak {a}}}:\,f\mapsto {{\mathfrak {a}}}[f]$, where

Any finite composition of the Riesz transforms has its symbol in $S^0_{ph}$. An observation: if, as usual, q and $q^\prime$ are conjugate exponents, and $1\le q^\prime \le 2\le q \le \infty$, then (change of variables)

(C.1)

In particular, the integrals represent smooth functions with uniformly in n bounded $L^1$ norms.

Lemma C.1

Let the symbol $a\in S^0_{ph}$ be given.

• 1. For any $r \in {{\mathbb {R}}}$ and any $p\in [2, +\infty ]$,

  $$\begin{aligned} \Vert {{\mathfrak {a}}}[f]\;\big |\; {\dot{B}}^r_{p, p}\Vert \lesssim \Vert f\;\big |\; {\dot{B}}^r_{p, p}\Vert . \end{aligned}$$

  (C.2)

• 2. Assume the parameters r, p, and q satisfy the conditions $1\le p, q \le \infty$ and $r > \frac{d}{p}$. Then there exists a constant $C_1$ such that

  $$\begin{aligned} \Vert {{\mathfrak {a}}}[f]\Vert _\infty \le C_1\; [f]_{-1}^{\gamma _1}\;\Vert f\;\big |\;{\dot{B}}^r_{p, q}\Vert ^{1 - \gamma _1} \end{aligned}$$

  (C.3)

  for every $f\in {\dot{H}}^{-1}\cap {\dot{B}}^r_{p, q}$, where

  $$\begin{aligned} \gamma _1 = \gamma _1(d, r, p) = \frac{r - \frac{d}{p}}{r + 1 + d\,\frac{p-2}{2p}}. \end{aligned}$$

  (C.4)

• 3. Assume the parameters r, p, and q satisfy the conditions $1\le p, q \le \infty$ and $r > \frac{d}{p} - 1$. Then there exists a constant $C_2 > 0$ such that

  $$\begin{aligned} \Vert D^{-1}\,{{\mathfrak {a}}}[f]\Vert _\infty \le C_2\; [f]_{-1}^{\gamma _2}\;\Vert f\;\big |\;{\dot{B}}^r_{p, q}\Vert ^{1 - \gamma _2} \end{aligned}$$

  (C.5)

  for every $f\in {\dot{H}}^{-1}\cap {\dot{B}}^r_{p, q}$, where

  $$\begin{aligned} \gamma _2 = \gamma _2(d, r, p) = \frac{r + 1 - \frac{d}{p}}{r + 1 + d\,\frac{p-2}{2p}} \end{aligned}$$

  (C.6)

Proof

To prove the first claim we need to show that

$$\begin{aligned} \sum _{n\in {{\mathbb {Z}}}}\,2^{r n p}\,\Vert \varphi _n(D) {{\mathfrak {a}}}[f]\Vert _p^p \lesssim \sum _{m\in {{\mathbb {Z}}}}\,2^{r m p}\,\Vert f_m\Vert _p^p, \end{aligned}$$

where $f_m = \varphi _m(D) f$. This follows from the estimate

$$\begin{aligned} \Vert \varphi _n(D) {{\mathfrak {a}}}[f]\Vert _p \lesssim \Vert f_n\Vert _p, \end{aligned}$$

(C.7)

which is easy to prove. Indeed,

$$\begin{aligned} \left( \varphi _n(D) {{\mathfrak {a}}}[f]\right) (x) = \sum _{\ell = - 1}^{1} \left( \varphi _{n+\ell }(D) {{\mathfrak {a}}}[\varphi _{n}(D)f]\right) (x) \end{aligned}$$

and

Then (see (C.1))

Thus we have (C.7), and this implies (C.2).

To prove the remaining two claims, consider the expansion (A.2) for f, $f = g_N + h_N$, where

$$\begin{aligned} g_N = {{{\mathcal {F}}}}^{-1}\psi _N(\kappa )\,{{{\hat{f}}}}(\kappa )\quad \text {and}\quad h_N = {{{\mathcal {F}}}}^{-1}\sum _{n = N+1}^\infty \varphi _n(\kappa ) \,{{{\hat{f}}}}(\kappa ). \end{aligned}$$

the integer N will be chosen later. We have

$$\begin{aligned} \Vert {{\mathfrak {a}}}[g_N]\Vert _\infty \lesssim \Vert \psi (2^{-N}\kappa )\,a(\kappa )\,{{{\hat{f}}}}(\kappa )\Vert _{L^{1}(\textrm{d}\kappa )}\le [f]_{-1}\;\Vert \psi (2^{-N}\kappa )\,a(\kappa )\,|\kappa |\Vert _{L^{2}(\textrm{d}\kappa )} \end{aligned}$$

and

$$\begin{aligned} \Vert \psi (2^{-N}\kappa )\,a(\kappa )\,|\kappa |\Vert _{L^{2}(\textrm{d}\kappa )} = 2^{N(1 + \frac{d}{2})}\,\Vert \psi (\kappa )\,a(\kappa )\,|\kappa |\Vert _{L^{2}(\textrm{d}\kappa )}. \end{aligned}$$

Therefore,

$$\begin{aligned} \Vert {{\mathfrak {a}}}[g_N]\Vert _\infty \lesssim 2^{N(1 + \frac{d}{2})}\,[f]_{-1}. \end{aligned}$$

(C.8)

Next,

$$\begin{aligned} \begin{aligned}&{{\mathfrak {a}}}[h_N] = {{{\mathcal {F}}}}^{-1}\sum _{n = N+1}^\infty \varphi _n(\kappa )\,a(\kappa ) \,{{{\hat{f}}}}(\kappa ) = {{{\mathcal {F}}}}^{-1}\sum _{n = N+1}^\infty \sum _{\ell = -1}^1 \varphi _{n+\ell }(\kappa )\,a(\kappa ) \, \varphi _{n}(\kappa )\,{{{\hat{f}}}}(\kappa ) \\&\quad = \sum _{\ell = -1}^1 \sum _{n = N+1}^\infty {{{\mathcal {F}}}}^{-1}\left[ \varphi _{n+\ell }(\kappa )\,a(\kappa ) \, {{{\hat{f}}}}_{n}(\kappa )\right] = \sum _{\ell = -1}^1 \sum _{n = N+1}^\infty {{{\mathcal {F}}}}^{-1} \varphi _{n+\ell } a\,{{{\mathcal {F}}}} f_{n}, \end{aligned} \end{aligned}$$

and hence,

$$\begin{aligned} \Vert {{\mathfrak {a}}}[h_N]\Vert _\infty \le \sum _{\ell = -1}^1 \sum _{n = N+1}^\infty \Vert {{{\mathcal {F}}}}^{-1} \varphi _{n+\ell }\, a\,{{{\mathcal {F}}}} f_{n}\Vert _\infty . \end{aligned}$$

Pick a $p\ge 1$ and observe that

Applying (C.1), we obtain

$$\begin{aligned} \Vert {{\mathfrak {a}}}[h_N]\Vert _\infty \lesssim _p \sum _{n = N}^\infty 2^{n \frac{d}{p}} \;\Vert f_n\Vert _p. \end{aligned}$$

(C.9)

If $r > \frac{d}{p},$ then, for any $q\ge 1$,

$$\begin{aligned}&\sum _{n = N}^\infty 2^{n \frac{d}{p}} \;\Vert f_n\Vert _p = \sum _{n = N}^\infty 2^{n (\frac{d}{p} - r)} \;2^{n r}\Vert f_n\Vert _p\\ {}&\le \left( \sum _{n = N}^\infty 2^{n (\frac{d}{p} - r) q^\prime }\right) ^{1/q^\prime }\;\left( \sum _{m = N}^\infty 2^{n r q}\Vert f_n\Vert ^q_p\right) ^{1/q} \\ ={}&2^{N (\frac{d}{p} - r)}\,\left( 1 - 2^{(\frac{d}{p} - r) q^\prime }\right) ^{1/q^\prime } \;\Vert f\;\big |\; {\dot{B}}^r_{p, q}\Vert \end{aligned}$$

Thus,

$$\begin{aligned} \Vert {{\mathfrak {a}}}[h_N]\Vert _\infty \lesssim _{p, q, d, r}\,2^{N (\frac{d}{p} - r)}\;\Vert f\;\big |\; {\dot{B}}^r_{p, q}\Vert \end{aligned}$$

(C.10)

Combine this with (C.8) to obtain

$$\begin{aligned} \Vert {{\mathfrak {a}}}[f]\Vert _\infty \lesssim 2^{N(1 + \frac{d}{2})}\,[f]_{-1} + 2^{N (\frac{d}{p} - r)}\;\Vert f\;\big |\; {\dot{B}}^r_{p, q}\Vert . \end{aligned}$$

(C.11)

Now choose N so that the two terms have (almost) the same magnitudes:

$$\begin{aligned} N = \lfloor \; \log _2 \left( \frac{\Vert f\;\big |\; {\dot{B}}^r_{p, q}\Vert }{[f]_{-1}} \right) ^{1/(1 + r + d \frac{p-2}{2p})}\;\rfloor \end{aligned}$$

( $\lfloor \cdot \rfloor$ is the floor function). The resulting inequality is

$$\begin{aligned} \Vert \rho [f]\Vert _\infty \lesssim [f]_{-1}^{\gamma _1(d, p, r)}\;\cdot \;\Vert f\;\big |\; {\dot{B}}^r_{p, q}\Vert ^{1 - \gamma _1(d, p, r)} \end{aligned}$$

(C.12)

with

$$\begin{aligned} \gamma _1(d, p, r) = \frac{r - \frac{d}{p}}{r + 1 + d\,\frac{p-2}{2p}}. \end{aligned}$$

The second part is proved similarly. First we have

$$\begin{aligned}{} & {} \Vert D^{-1}{{\mathfrak {a}}}[g_N]\Vert _\infty \lesssim \Vert \psi (2^{-N}\kappa )\,a(\kappa )\,\frac{1}{|\kappa |}\,{{{\hat{f}}}}(\kappa )\Vert _{L^{1}(\textrm{d}\kappa )}\\{} & {} \le [f]_{-1}\;\Vert \psi (2^{-N}\kappa )\,a(\kappa )\Vert _{L^{2}(\textrm{d}\kappa )}\lesssim 2^{N\,\frac{d}{2}}\,[f]_{-1}. \end{aligned}$$

After that, consider $\Vert D^{-1}{{\mathfrak {a}}}[h_N]\Vert _\infty$. We have

$$\begin{aligned} \Vert D^{-1}{{\mathfrak {a}}}[h_N]\Vert _\infty \le \sum _{\ell = -1}^1 \sum _{n = N+1}^\infty \Vert D^{-1}{{{\mathcal {F}}}}^{-1} \varphi _{n+\ell }\, a\,{{{\mathcal {F}}}} f_{n}\Vert _\infty . \end{aligned}$$

Now,

and

This leads to

$$\begin{aligned} \begin{aligned} \Vert D^{-1}\rho [h_N]\Vert _\infty \lesssim \sum _{n = N}^\infty 2^{n (\frac{d}{p} - 1)}\;\Vert f_{n}\Vert _p \end{aligned} \end{aligned}$$

and we proceed as in the first part. Now the restriction on r will be $r > \frac{d}{p} - 1$ and then

$$\begin{aligned} \sum _{n = N}^\infty 2^{n (\frac{d}{p} - 1)}\;\Vert f_{n}\Vert _p \le {}&\left( \sum _{n = N}^\infty 2^{n (\frac{d}{p} - 1 - r) q^\prime }\right) ^{1/q^\prime }\,\left( \sum _{m = N}^\infty 2^{n r q}\,\Vert f_{n}\Vert _p^q\right) ^{1/q}\\ ={}&2^{N (\frac{d}{p} - 1 - r)}\,\left( 1 - 2^{(\frac{d}{p} - 1 - r) q^\prime }\right) ^{1/q^\prime }\,\Vert f\;\big |\;{\dot{B}}^r_{p, q}\Vert \end{aligned}$$

This time we obtain

$$\begin{aligned} \Vert D^{-1} {{\mathfrak {a}}}[f]\Vert _\infty \lesssim 2^{N\,\frac{d}{2}}\,[f]_{-1} + 2^{N (\frac{d}{p} - 1 - r)}\,\Vert f\;\big |\;{\dot{B}}^r_{p, q}\Vert . \end{aligned}$$

Again, choose N so that

$$\begin{aligned} 2^{N(1 + r + d\,\frac{p-2}{2p})} \approx \frac{\Vert f\;\big |\;{\dot{B}}^r_{p, q}\Vert }{[f]_{-1}}; \end{aligned}$$

i.e.,

$$\begin{aligned} N = \lfloor \; \log _2\left( \Vert f\;\big |\;{\dot{B}}^r_{p, q}\Vert /[f]_{-1}\right) ^{1/(1 + r + d\, \frac{p-2}{2p})}\;\rfloor . \end{aligned}$$

Plugging this value into the inequality above we obtain

$$\begin{aligned} \Vert D^{-1} \rho [f]\,\Vert _\infty \lesssim [f]_{-1}^{\gamma _2}\;\Vert f\;\big |\;{\dot{B}}^r_{p, q}\Vert ^{1 - \gamma _2} \end{aligned}$$

with

$$\begin{aligned} \gamma _2 = \gamma _2(d, p, r)= \frac{1 + r - \frac{d}{p}}{1 + r + d \,\frac{p-2}{2p}}. \end{aligned}$$

$\square$

Strichartz Estimates

In this section we derive the homogeneous ${{\mathbb {R}}}^d$ version of the Strichartz estimates in Theorem 2 [17]. We work with functions of $x\in {{\mathbb {R}}}^d$ and the Fourier variables are $\kappa$. Note, that the estimates are applied in Section 5 to functions of $\xi$ with the dual Fourier variables k.

Consider the Cauchy problem

$$\begin{aligned} \partial _t w = i D w + f,\quad w(0) = w_0, \end{aligned}$$

(D.1)

where $D = \sqrt{- \Delta }$ in ${{\mathbb {R}}}^d$. Given $w_0\in {\dot{H}}^\theta$ and $f\in L^1_{loc}({{\mathbb {R}}}\rightarrow {\dot{H}}^\theta )$, there exists a unique solution w of equation (D.1) such that w(t) is a continuous functions of t with values in ${\dot{H}}^\theta$, and

$$\begin{aligned} \sup _{[0, T]} \Vert w(t)\;\big |\;{\dot{H}}^\theta \Vert \le C_\theta \,\left( \Vert w_0\;\big |\; {\dot{H}}^\theta \Vert + \int _0^T \Vert f(t)\;\big |\; {\dot{H}}^\theta \Vert \,\textrm{d}t\right) \end{aligned}$$

(D.2)

with the positive constant $C_\theta$ independent of the particular choice of $w_0$ and f.

Theorem D.1

Let $2\le q \le \infty$ and let p be such that if $d = 2$, then $2 \le p \le +\infty$, and if $d \ge 3$, then $2 \le p < \frac{2(d-1)}{d - 3}$. Assume the parameters r and $\theta$ satisfy the conditions

$$\begin{aligned} r\in {{\mathbb {R}}},\quad \theta = r + \frac{d+1}{4}\,\frac{p-2}{p}. \end{aligned}$$

(D.3)

There exists a constant $C > 0$ (dependent on d, r, and p) such that, for any $T > 0$, the following estimate is true for the solutions of (D.1):

$$\begin{aligned}{} & {} \left( \int _0^T \Vert w(t)\;\big |\;{\dot{B}}^r_{p, q}\Vert ^{\frac{4p}{(d-1)(p-2)}}\,\textrm{d}t\right) ^{\frac{(d-1)(p-2)}{4p}}\\{} & {} \le C\,\left( \Vert w_0\;\big |\; {\dot{H}}^\theta \Vert + \int _0^T \Vert f(t)\;\big |\; {\dot{H}}^\theta \Vert \,\textrm{d}t\right) \end{aligned}$$

(D.4)

Corollary D.2

Two special cases: If $d = 2$, then take $p = q = \infty$, $0< r < 1$, and $\theta = r + \frac{3}{4}$, and obtain

$$\begin{aligned} \left( \int _0^T \Vert w(t)\;\big |\;{\dot{B}}^r_{\infty , \infty }\Vert ^4\,\textrm{d}t\right) ^{1/4} \le C\,\left( \Vert w_0\;\big |\; {\dot{H}}^\theta \Vert + \int _0^T \Vert f(t)\;\big |\; {\dot{H}}^\theta \Vert \,\textrm{d}t\right) . \end{aligned}$$

(D.5)

If $d = 3$, take $2\le p = q < \infty$, and $\theta = r + \frac{p-2}{p}$, and obtain

$$\begin{aligned} \left( \int _0^T \Vert w(t)\;\big |\;{\dot{B}}^r_{p, p}\Vert ^{\frac{2p}{p-2}}\,\textrm{d}t\right) ^{\frac{p-2}{2p}} \le C\,\left( \Vert w_0\;\big |\; {\dot{H}}^\theta \Vert + \int _0^T \Vert f(t)\;\big |\; {\dot{H}}^\theta \Vert \,\textrm{d}t\right) \nonumber \\ \end{aligned}$$

(D.6)

Proof of Theorem D.1

For fixed $w_0$ and f, the solution of the problem (D.1) is given by the formula

$$\begin{aligned} w(t) = e^{i t\,D} w_0 + \int _0^t e^{i (t - \tau ) D} f(\tau )\,\textrm{d}\tau . \end{aligned}$$

Each term on the right can be analyzed separately. Take a $g\in {{{\mathcal {S}}}}$ and consider $e^{i t\,D}g$. In fact, we need to look at the dyadic pieces $\varphi _n(D)e^{i t\,D}g = e^{i t\,D}g_n$, where $g_n = \varphi _n(D) g$. Since

$$\begin{aligned} e^{i t\,D}g_n = \sum _{j = n-1}^{n+1} \varphi _{j}(D)\,e^{i t\,D}g_n, \end{aligned}$$

we examine the terms $\varphi _j(D)\,e^{i t\,D}g_n$. The first important observation in the Strichartz analysis is the following bound on the operator $\varphi _j(D)\,e^{i t\,D}$ as an operator from $L^{p^\prime }$ to $L^p$, $2\le p\le \infty$:

$$\begin{aligned} \Vert \varphi _j(D)\,e^{i t\,D}\Vert _{L^{p^\prime }({{\mathbb {R}}}^d) \rightarrow L^p({{\mathbb {R}}}^d)} \lesssim \frac{1}{|t|^{(d-1)\,\frac{p-2}{2p}}}\,2^{j (d+1)\,\frac{p-2}{2p}}. \end{aligned}$$

(D.7)

As a corollary,

$$\begin{aligned} \Vert e^{i t\,D}g_n \Vert _p \lesssim \frac{1}{|t|^{(d-1)\,\frac{p-2}{2p}}}\,2^{n (d+1)\,\frac{p-2}{2p}}\,\Vert g_n\Vert _{p^\prime }, \end{aligned}$$

(D.8)

for any $p\in [2, +\infty ]$. This estimate, implies the estimates in the homogeneous Besov spaces with

$$\begin{aligned} \Vert e^{i t\,D}g\;\big |\; {\dot{B}}^r_{p, q}\Vert \lesssim \frac{1}{|t|^{(d-1)\,\frac{p-2}{2p}}}\;\Vert g\;\big |\; {\dot{B}}^{r + (d+1)\,\frac{p-2}{2p}}_{p^\prime , q}\Vert , \end{aligned}$$

(D.9)

$r\in {{\mathbb {R}}}$ and $1\le q \le \infty$. The second important observation is the space-time estimate for the homogeneous term $e^{i t\,D} g$. The argument uses duality. Let h(t, x) be a sufficiently smooth function. Then

$$\begin{aligned} \begin{aligned}&\int _0^T\langle e^{i t\,D} g,\;h(t)\rangle \,\textrm{d}t = \int _0^T\langle e^{i t\,D} g,\;h(t)\rangle \,\textrm{d}t = \int _0^T\langle g,\;e^{- i t\,D} h(t)\rangle \,\textrm{d}t\\&= \langle g,\;\int _0^T e^{- i t\,D} h(t)\,\textrm{d}t\rangle \\&\quad \le \Vert g\;\big |\;{\dot{H}}^\theta \Vert \,\Vert D^{-\theta }\int _0^T e^{- i t\,D} h(t)\,\textrm{d}t\Vert _2. \end{aligned} \end{aligned}$$

Now,

$$\begin{aligned} \begin{aligned}&\Vert D^{-\theta }\int _0^T e^{- i t\,D} h(t)\,\textrm{d}t\Vert _2^2 = \int _0^T \int _0^T \langle e^{- i t\,D}D^{-\theta } h(t), e^{- i t^\prime \,D} D^{-\theta }h(t^\prime )\rangle \,\textrm{d}t^\prime \,\textrm{d}t = \\&\int _0^T \int _0^T \langle e^{- i (t^\prime - t)\,D} \,D^{-2\theta } h(t), h(t^\prime )\rangle \,\textrm{d}t^\prime \,\textrm{d}t \\&\quad \le \int _0^T \int _0^T \Vert e^{ i (t - t^\prime )\,D} \,D^{-2\theta } h(t)\;\big |\; {\dot{B}}^r_{p, q}\Vert \cdot \Vert h(t^\prime )\;\big |\;{\dot{B}}^{-r}_{p^\prime , q^\prime }\Vert \;\textrm{d}t^\prime \,\textrm{d}t \quad \text {(use}\, (D.9)) \\&\quad \lesssim \int _0^T \int _0^T \frac{1}{|t - t^\prime |^{(d-1)\,\frac{p-2}{2p}}}\;\Vert h(t)\;\big |\; {\dot{B}}^{r+(d+1)\,\frac{p-2}{2p} - 2\theta }_{p^\prime , q}\Vert \cdot \Vert h(t^\prime )\;\big |\;{\dot{B}}^{-r}_{p^\prime , q^\prime }\Vert \;\textrm{d}t^\prime \,\textrm{d}t. \end{aligned} \end{aligned}$$

Choose $\theta$ so that

$$\begin{aligned} r + (d+1)\,\frac{p-2}{2p} - 2\theta = - r, \end{aligned}$$

i.e.,

$$\begin{aligned} \theta = r + \frac{d+1}{4}\,\frac{p-2}{p}, \end{aligned}$$

(D.10)

and assume that

$$\begin{aligned} q \ge 2 \end{aligned}$$

(then $q^\prime \le 2 \le q$ and ${\dot{B}}^{-r}_{p^\prime , q^\prime } \subset {\dot{B}}^{-r}_{p^\prime , q}$). Then

$$\begin{aligned} \Vert D^{-\theta }\int _0^T e^{- i t\,D} h(t)\,\textrm{d}t\Vert _2^2 \lesssim {}&\int _0^T \int _0^T \frac{\Vert h(t)\;\big |\; {\dot{B}}^{- r}_{p^\prime , q^\prime }\Vert \cdot \Vert h(t^\prime )\;\big |\;{\dot{B}}^{-r}_{p^\prime , q^\prime }\Vert }{|t - t^\prime |^{(d - 1)\,\frac{p-2}{2p}}}\;\textrm{d}t^\prime \,\textrm{d}t \\ \lesssim {}&\left( \int _0^T\Vert h(t)\;\big |\; {\dot{B}}^{-r}_{p^\prime , q^\prime }\Vert ^m\right) ^{1/m} \end{aligned}$$

with

$$\begin{aligned} m = \frac{1}{1 - \frac{d-1}{4}\,\frac{p-2}{p}}. \end{aligned}$$

The last inequality is a consequence of the Hardy–Littlewood–Sobolev inequality (see [15, Theorem 4.5.3]). It requires the following restrictions on p:

$$\begin{aligned} 0 \le \frac{d-1}{2}\,\frac{p-2}{p} < 1, \end{aligned}$$

(D.11)

i.e., if $d = 2$, then $2 \le p \le +\infty$, and if $d \ge 3$, then $2 \le p < \frac{2(d-1)}{d - 3}$. In any case,

$$\begin{aligned} |\int _0^T\langle e^{i t\,D} g,\;h(t)\rangle \,\textrm{d}t|\lesssim \Vert g\;\big |\;{\dot{H}}^\theta \Vert \cdot \left( \int _0^T\Vert h(t)\;\big |\; {\dot{B}}^{-r}_{p^\prime , q^\prime }\Vert ^m\right) ^{1/m} \end{aligned}$$

(D.12)

Hence, by duality,

$$\begin{aligned} \left( \int _0^T \Vert e^{i t D} g\;\big |\; {\dot{B}}^r_{p, q}\Vert ^{\frac{4 p}{(d-1)(p-2)}} \right) ^{\frac{(d-1)(p-2)}{4 p}} \lesssim \Vert g\;\big |\; {\dot{H}}^\theta \Vert , \end{aligned}$$

(D.13)

where the exponent $\frac{4 p}{(d-1)(p-2)}$ is the conjugate of m. This proves estimate (D.4) for the homogeneous equation (D.1). The estimate on $\int _0^t e^{i (t - \tau ) D} f(\tau )\,\textrm{d}\tau$ is obtained as in the second part of the proof of Theorem 2 in [17] (with necessary slight modifications).

$\square$

Norms in Euler and Lagrange Coordinates

In this section $\xi \rightarrow x(\xi )$ is a $C^1$ volume preserving diffeomorphism of the form $x(\xi ) = A \xi + \varphi (\xi )$ satisfying the assumptions of Lemma 3.1. In particular, A is a constant $SL(d, {{\mathbb {R}}})$ matrix and $\varphi \in H^{s+1}({{\mathbb {R}}}^d_\xi )$ with $s > d/2$. This transformation from the Lagrangean coordinates, $\xi$, to the Eulerian coordinates, x, pushes back the functions f(x) to the functions ${{{\tilde{f}}}}( \xi ) = f(x(\xi ))$. This is a linear isometry from $L^p({{\mathbb {R}}}^d, \textrm{d}x)$ to $L^p({{\mathbb {R}}}^d, \textrm{d}\xi )$, $1\le p \le \infty$. To analyze other norms, we use the superscripts L and E on functions to indicate the coordinate system used. Denote by $v^i_a(x)$ the entries of the Jacobian matrix, $\partial x/\partial \xi$, expressed in Eulerian coordinates, i.e., as functions of x. We have $v^i_a(x) = A^i_a + u^i_a(x)$, where $u^i_a \in H^s({{\mathbb {R}}}^d_x)$. Notation $v_a$ or $u_a$ is used to represent a generic $v_a$ or $u_a$, or when in a norm, the maximal over a norm, e.g., $\Vert v_a\Vert _\infty = \max _a \Vert v_a\Vert _\infty$. For the norms in $L^p$ we skip the superscripts E and L.

1.1 Homogenous Besov and Sobolev spaces

The following proposition contains inequalities between the homogeneous Besov norms $\{g\}_{r,p}$ and between the homogeneous $L^2$ Sobolev norms $[g]_\theta$ in the Euler and Lagrange coordinates. The range of r, p, and $\theta$ is restricted to the demands of the main body of the paper. All the spaces are over ${{\mathbb {R}}}^d$, $d \ge 2$.

Lemma E.1

• (a) Assume $0< r < 1$, $0 \le \theta \le 1$, and $2\le p \le \infty$. Then, for all $g\in {{{\mathcal {S}}}}({{\mathbb {R}}}^d)$,

  $$\begin{aligned}&\{g^L\}_{r,p} \lesssim \Vert v_a\Vert _\infty ^r\;\{g^E\}_{r,p},\quad&\{g^E\}_{r,p} \lesssim \Vert v_a\Vert _\infty ^{(d-1)r}\;\{g^L\}_{r,p}, \end{aligned}$$

  (D.14)
  $$\begin{aligned}&[g^L]_\theta \lesssim \Vert v_a\Vert _\infty ^\theta \;[g^E]_\theta ,\quad&[g^E]_\theta \lesssim \Vert v_a\Vert _\infty ^{(d - 1)\theta }\;[g^L]_\theta \end{aligned}$$

  (E.1)

• (b) Assume $1<\theta < 2$, then

  $$\begin{aligned}{}[g^L]_\theta \lesssim \Vert v_a\Vert _\infty ^{\theta -1}\;\left( \Vert v_a\Vert _\infty + \Vert u_a\Vert _\infty ^{2 - \theta }\;\Vert u_a\,\big |\;{\dot{F}}^1_{d, 2}({{\mathbb {R}}}^d)\Vert ^{\theta - 1}\right) \;[g^E]_\theta . \end{aligned}$$

  (E.2)

Proof

We first prove the inequalities for Sobolev norms in part a). Compare the norms $\Vert g^L\;\big |\;{\dot{H}}^1({{\mathbb {R}}}^d_\xi )\Vert$ and $\Vert g^E\;\big |\;{\dot{H}}^1({{\mathbb {R}}}^d_x)\Vert$. We have

$$\begin{aligned} \Vert g^L\;\big |\;{\dot{H}}^1({{\mathbb {R}}}^d_\xi )\Vert ^2 ={}&\int |\nabla _\xi {{{\tilde{g}}}}(\xi )|^2\,\textrm{d}\xi \\ \le {}&\int |\frac{\partial f(t, x(\xi ))}{\partial x^j}\;\frac{\partial x^j(\xi )}{\partial \xi }|^2\,\textrm{d}\xi \\ \le {}&\Vert v_a(t)\Vert _\infty ^2\;\int |\frac{\partial g(x(\xi ))}{\partial x^j}|^2\,\textrm{d}\xi \\ ={}&\Vert v_a(t)\Vert _\infty ^2\;\int |\frac{\partial g(x)}{\partial x^j}|^2\,\textrm{d}x \\ ={}&\Vert v_a(t)\Vert _\infty ^2\;\Vert g^E\;\big |\;{\dot{H}}^1({{\mathbb {R}}}^d_x)\Vert ^2. \end{aligned}$$

If $0< \theta < 1$, then ${\dot{H}}^\theta$ is an interpolation space between $L^2$ and ${\dot{H}}^1$, ${\dot{H}}^\theta = (L^2, {\dot{H}}^1)_{\theta , 2}$ (see [2, Theorem 6.3.1]). This explains why $[g^L]_\theta \lesssim \Vert v_a\Vert _\infty ^\theta \,[g^E]_\theta$. Similarly, $[g^E]_\theta \lesssim \Vert v_a\Vert _\infty ^{(d - 1)\theta } \,[g^L]_\theta$, where $(d - 1)$ appears because of the $L^\infty$ bound on $\partial \xi ^a/\partial x^i$ in terms of $\Vert v_a\Vert _\infty$.

The Besov norm inequalities in part a) can be obtained by interpolation between the inequalities for ${\dot{B}}^r_{2, 2} \simeq {\dot{H}}^r$, which we already have, and the inequalities for ${\dot{B}}^r_{\infty , \infty } \simeq {\dot{C}}^r$, the homogeneous Hölder spaces. For the Hölder seminorms we have

$$\begin{aligned} \{g^L\}_r ={}&\sup _{\xi , \xi ^\prime } \frac{|g^L(\xi ) - g^L(\xi ^\prime )|}{|\xi - \xi ^\prime |^r} \\ ={}&\sup _{\xi , \xi ^\prime } \frac{|g^E( x(\xi )) - g^E( x(\xi ^\prime ))|}{|\xi - \xi ^\prime |^r} \\ ={}&\sup _{\xi , \xi ^\prime } \frac{|g^E(x(\xi )) - g^E(x(\xi ^\prime ))|}{|x(\xi ) - x(\xi ^\prime )|^r}\; \frac{|x(\xi ) - x(\xi ^\prime )|^r}{|\xi - \xi ^\prime |^r} \\ \lesssim {}&\sup _{x, x^\prime } \frac{|g^E(x) - g^E(x^\prime )|}{|x - x^\prime |^r}\; \Vert v_a\Vert _\infty ^r. \end{aligned}$$

Thus, $\{g^L\}_r \le \Vert v_a\Vert _\infty ^r\,\{g^E\}_r$. The norm $\{g^E\}_r$ is bounded similarly.

Now turn to part b). Assume $1< \theta < 2$ and proceed with

$$\begin{aligned} {[}g^L]_\theta \simeq {}&[\frac{\partial g^L}{\partial \xi }]_{\theta -1} \nonumber \\ \underset{(E.1)}{\lesssim } {}&\Vert v_a\Vert _\infty ^{\theta -1}\,[\left( \frac{\partial g^L}{\partial \xi }\right) ^E]_{\theta -1} \nonumber \\ \simeq {}&\Vert v_a\Vert _\infty ^{\theta -1}\, [\frac{\partial g^E}{\partial x} \cdot \frac{\partial x}{\partial \xi }]_{\theta -1} \nonumber \\ \lesssim {}&\Vert v_a\Vert _\infty ^{\theta -1}\; [\frac{\partial g^E}{\partial x} \cdot v_a]_{\theta -1}. \end{aligned}$$

(E.3)

Apply the fractional product rule (B.1):

$$\begin{aligned} \left[ \frac{\partial g^E}{\partial x} \cdot v_a^E\right] _{\theta -1} \lesssim \left[ \frac{\partial g^E}{\partial x}\right] _{\theta -1}\,\Vert v_a\Vert _\infty + \Vert \frac{\partial g^E}{\partial x}\Vert _{q_1}\; \Vert v_a^E\,\big |\; {\dot{F}}^{\theta -1}_{q_2, 2}\Vert . \end{aligned}$$

Note that the homogeneous Lizorkin–Triebel norms of $v_a$ and $u_a$ are the same. So, we have

$$\begin{aligned} \left[ \frac{\partial g^E}{\partial x} \cdot v_a^E\right] _{\theta -1} \lesssim \left[ \frac{\partial g^E}{\partial x}\right] _{\theta -1}\,\Vert v_a\Vert _\infty + \Vert \frac{\partial g^E}{\partial x}\Vert _{q_1}\;\Vert u_a^E\,\big |\; {\dot{F}}^{\theta -1}_{q_2, 2}\Vert . \end{aligned}$$

(E.4)

The parameters $q_1$ and $q_2$ must satisfy $2\le q_1, q_2 \le \infty$ and

$$\begin{aligned} \frac{1}{q_1} + \frac{1}{q_2} = \frac{1}{2}. \end{aligned}$$

We choose $q_1$ and $q_2$ as follows

$$\begin{aligned} \frac{1}{q_1} = \frac{1}{2} - \frac{\theta - 1}{d},\quad \frac{1}{q_2} = \frac{\theta -1}{d}. \end{aligned}$$

(E.5)

Then ${\dot{H}}^{\theta -1}({{\mathbb {R}}}^d)\subset L^{q_1}({{\mathbb {R}}}^d)$, and so

$$\begin{aligned} \Vert \frac{\partial g^E}{\partial x}\Vert _{q_1} \lesssim \left[ \frac{\partial g^E}{\partial x}\right] _{\theta -1} \lesssim [g^E]_\theta . \end{aligned}$$

As for the other factor, use (A.12):

$$\begin{aligned} \Vert u_a^E\,\big |\; {\dot{F}}^{\theta -1}_{q_2, 2}({{\mathbb {R}}}^d)\Vert \lesssim \Vert u_a\Vert _\infty ^{2 - \theta }\;\Vert u_a^E\,\big |\;{\dot{F}}^1_{d, 2}({{\mathbb {R}}}^d)\Vert ^{\theta - 1}. \end{aligned}$$

Collecting the pieces, we arrive at (E.3).

$\square$

1.2 Vorticities

Let v be a vectorfield on ${{\mathbb {R}}}^d$ such that $\hbox {div}\,v = 0$. Denote $\omega ^{mn} = \partial _m v^n - \partial _n v^m$. In terms of Fourier transform,

$$\begin{aligned} {{{\hat{\omega }}}}^{mn} = i\,\left( \kappa ^m {{{\hat{v}}}}^n - \kappa ^n {{{\hat{v}}}}^m\right) \quad \text {and}\quad \kappa ^n {{{\hat{v}}}}^n = 0. \end{aligned}$$

From $\omega ^{mn}$ one recovers v as follows:

$$\begin{aligned} {{{\hat{v}}}}^n = - i \;\frac{\kappa ^m}{|\kappa |^2}\,{{{\hat{\omega }}}}^{mn}. \end{aligned}$$

In our notation,

$$\begin{aligned} v^n = - i D^{-1}{{{\mathcal {R}}}}_m \,\omega ^{mn}. \end{aligned}$$

In dimension $d=2$, $\omega = \partial _1 v^2 - \partial _2 v^1$ and

$$\begin{aligned} {{{\hat{v}}}}^1 = i\,\frac{\kappa ^2}{|\kappa |^2}\,{{{\hat{\omega }}}},\quad {{{\hat{v}}}}^2 = - i\,\frac{\kappa ^1}{|\kappa |^2}\,{{{\hat{\omega }}}}. \end{aligned}$$

We also have pseudovelocities $v_a$ with the components $v^i_a = u^i_a + A^i_a$, and the corresponding pseudovorticities $\omega _a = \partial _1 v^2_a - \partial _2 v^1_a = \partial _1 u^2_a - \partial _2 u^1_a$.

Lemma E.2

• 1. In the case $d = 2$, assume that $u\in L^2$ and $\omega = \textrm{curl}\, u\in {\dot{C}}^r$ for some $r\in (0, 1)$. Then $u\in L^\infty$ and $\nabla u\in L^\infty$, and the following inequalities are true:

  $$\begin{aligned} \Vert u\Vert _\infty \lesssim \Vert u\Vert ^{(r + 1)/(r + 2)}\;\{\omega \}_r^{1/(r + 2)} \end{aligned}$$

  (E.6)

  and

  $$\begin{aligned} \Vert \nabla u\Vert _\infty \lesssim \Vert u\Vert ^{r/(r + 2)}\;\{\omega \}_r^{2/(r + 2)}. \end{aligned}$$

  (E.7)

• 2. In the case $d = 2$, assume that $u\in L^2$ and $\omega = \textrm{curl}\, u\in {\dot{H}}^\theta$ for some $\theta \in (0, 1)$. Then $u\in L^\infty$ and the following inequality is true:

  $$\begin{aligned} \Vert u\Vert _\infty \lesssim \Vert u\Vert ^{\theta /(\theta +1))}\;[\omega ]_\theta ^{1/(\theta +1)}. \end{aligned}$$

  (E.8)

  If $\omega \in {\dot{H}}^\theta$ with $\theta > 1$, then $\nabla u\in L^\infty$ and

  $$\begin{aligned} \Vert \nabla u\Vert _\infty \lesssim \Vert u\Vert ^{(\theta - 1)/(\theta + 1)}\;[\omega ]_\theta ^{2/(\theta +1)}. \end{aligned}$$

  (E.9)

• 3. In the case $d \ge 3$, assume that $u\in L^2$ and $\omega \in {\dot{B}}^r_{p, p}$, where

  $$\begin{aligned} r\in (0, 1),\;r > \frac{d}{p},\;\;1\le p \le \infty . \end{aligned}$$

  Then $u\in L^\infty$, $\nabla u\in L^\infty$, and

  $$\begin{aligned} \Vert u\Vert _\infty \lesssim \Vert u\Vert ^{(r + 1 - d/p)/(r + 1 - d/p + d/2)}\;\Vert \omega \;\big |\;{\dot{B}}^r_{p, p}\Vert ^{(d/2)/(r + 1 - d/p + d/2)}\nonumber \\ \end{aligned}$$

  (E.10)

  and

  $$\begin{aligned} \Vert \nabla u\Vert _\infty \lesssim \Vert u\Vert ^{(r - d/p)/(r + 1 - d/p + d/2)}\;\Vert \omega \;\big |\;{\dot{B}}^r_{p, p}\Vert ^{(1 + d/2)/(r + 1 - d/p + d/2)}.\nonumber \\ \end{aligned}$$

  (E.11)

• 4. In the case $d \ge 3$ assume that $u\in L^2$ and $\omega = \textrm{curl}\, u \in {\dot{H}}^\theta$. If $\theta > \frac{d}{2} - 1$, then $u\in L^\infty$ and

  $$\begin{aligned} \Vert u\Vert _\infty \lesssim \Vert u\Vert ^{(\theta + 1 - d/2)/(\theta + 1)}\;[\omega ]_\theta ^{d/2/(\theta + 1)}. \end{aligned}$$

  (E.12)

  If $\theta > \frac{d}{2}$, then $\nabla u\in L^\infty$ and

  $$\begin{aligned} \Vert \nabla u\Vert _\infty \lesssim \Vert u\Vert ^{(\theta - d/2)/(\theta + 1)}\,[\omega ]_\theta ^{(d/2 + 1)/(\theta + 1)}. \end{aligned}$$

  (E.13)

The inequalities follow immediately from Lemma C.1 part 2 and the observation that $[\omega ]_{-1} \simeq \Vert u\Vert$ and $\nabla u \simeq {{{\mathcal {R}}}} \omega$.