The Muskat problem with surface tension and equal viscosities in subcritical $L_p$-Sobolev spaces

Abstract

In this paper we establish the well-posedness of the Muskat problem with surface tension and equal viscosities in the subcritical Sobolev spaces $W^s_p(\mathbb {R})$, where ${p\in (1,2]}$ and ${s\in (1+1/p,2)}$. This is achieved by showing that the mathematical model can be formulated as a quasilinear parabolic evolution problem in $W^{\overline{s}-2}_p(\mathbb {R})$, where ${\overline{s}\in (1+1/p,s)}$. Moreover, we prove that the solutions become instantly smooth and we provide a criterion for the global existence of solutions.

1 Introduction

In this paper we study the evolution equation

$$\begin{aligned} \partial _tf(t,x)=\displaystyle \frac{k}{2\pi \mu }\mathop \mathrm{PV}\nolimits \int _{\mathbb R}\frac{y+\partial _xf(t,x)\delta _{[x,y]}f(t)}{ y^2+\left( \delta _{[x,y]}f(t)\right) ^2 }\partial _x(\sigma \kappa (f)-\Delta _\rho f)(t,x-y)\, dy, \end{aligned}$$

(1a)

which is defined for $t>0$ and $x\in {\mathbb R}$. The function f is assumed to be known at time $t=0$, that is

$$\begin{aligned} f(0,\cdot )= f_0. \end{aligned}$$

(1b)

The evolution problem (1a) is the contour integral formulation of the Muskat problem with surface tension and with/without gravity effects, see [24, 25] for an equivalence proof of (1a) to the classical formulation of the Muskat problem [29]. The problem (1a) describes the two-dimensional motion of two fluids with equal viscosities $\mu _-=\mu _+=\mu$ and general densities $\rho _-$ and $\rho _+$ in a vertical/horizontal homogeneous porous medium which is identified with ${\mathbb R}^2$. The fluids occupy the entire plane, they are separated by the sharp interface ${\{y=f(t,x)+Vt\}}$, and they move with constant velocity (0, V), where ${V\in {\mathbb R}}$. The fluid denoted by $+$ is located above this moving interface. We use ${g\in [0,\infty )}$ to denote the Earth’s gravity, $k>0$ is the permeability of the homogeneous porous medium, and $\sigma >0$ is the surface tension coefficient at the free boundary. Moreover, to shorten the notation we have set $\Delta _\rho :=g(\rho _--\rho _+)\in {\mathbb R}$ and

$$\begin{aligned} \delta _{[x,y]} f:=f(x)-f(x-y),\qquad x,\, y\in {\mathbb R}. \end{aligned}$$

Finally, $\kappa (f(t))$ is the curvature of $\{y=f(t,x)+tV\}$ and $\mathop \mathrm{PV}\nolimits$ denotes the principal value.

The Muskat problem with surface tension has received much interest in the recent years. Besides the fundamental well-posedness issue also other important features like the stability of stationary solutions [13, 14, 16, 20, 26, 28, 33, 36], parabolic smoothing properties [24,25,26], the zero surface tension limit [8, 31], and the degenerate limit when the thickness of the fluid layers (or a certain nondimensional parameter) vanishes [15, 22, 37] have been investigated in this context. We also refer to [19, 21] for results on the Hele–Shaw problem with surface tension effects, which is the one-phase version of the Muskat problem, and to [34, 35, 38] for results on the related Verigin problem with surface tension.

Concerning the well-posedness of the Muskat problem with surface tension effects, this property has been investigated in bounded (layered) geometries in [14, 16, 17, 33, 36] where abstract parabolic theories have been employed in the analysis, the approach in [23] relies on Schauder’s fixed-point theorem, and in [10] the authors use Schauder’s fixed-point theorem in a setting which allows for a sharp corner point of the initial geometry.

The results on the Muskat problem with surface tension in the unbounded geometry considered in this paper (and possibly in the general case of fluids with different viscosities) are more recent, cf. [8, 24,25,26, 30, 31, 39]. While in [8, 39] the initial data are chosen from ${H^s(\mathbb {T})}$, with $s\ge 6$, the regularity of the initial data has been decreased in [24,25,26] to $H^{2+\varepsilon }({\mathbb R})$, with $\varepsilon \in (0,1)$ arbitrarily small. Finally, the very recent references [30, 31] consider the problem with initial data in $H^{1+\frac{d}{2}+\varepsilon }({\mathbb R}^d)$, with $d\ge 1$ and $\varepsilon >0$ arbitrarily small, covering all subcritical $L_2$-based Sobolev spaces in all dimensions.

It is the aim of this paper to study the Muskat problem (1a) in the subcritical $L_p$-based Sobolev spaces $W^s_p({\mathbb R})$ with $p\in (1,2]$ and $s\in (1+1/p,2)$. This issue is new in the context of (1a) (see [2, 12] for results in the case when $\sigma =0$). To motivate why $W^{1+1/p}_p({\mathbb R})$ is a critical space for (1a) we first emphasize that the surface tension is the dominant factor for the dynamics as it contains the highest spatial derivatives of f. Besides, if we set $g=0$, then it is not difficult to show that if f is a solution to (1a), then, given $\lambda >0$, the function

$$\begin{aligned} f_\lambda (t,x):=\lambda ^{-1}f\left( \lambda ^3 t,\lambda x\right) \end{aligned}$$

also solves (1a). This scaling identifies $W^{1+1/p}_p({\mathbb R})$ as a critical space for (1a). The main result Theorem 1.1 establishes the well-posedness of (1a) in $W^s_p({\mathbb R})$. This is achieved by showing that (1a) can be recast as a quasilinear parabolic evolution equation, so that abstract results available for this type of problems, cf. [3, 4, 6, 28], can be applied in our context. A particular feature of the Muskat problem (1a) is the fact that the equations have to be interpreted in distributional sense as they are realized in the Sobolev space $W^{\overline{s}-2}_p({\mathbb R})$, where $\overline{s}$ is chosen such that $1+1/p<\overline{s}<s<2$. Additionally to well-posedness, Theorem 1.1 provides two parabolic smoothing properties, showing in particular that (1a) holds pointwise, and a criterion for the global existence of solutions.

Theorem 1.1

Let $p\in (1,2]$ and $1+1/p<\overline{s}<s<2$. Then, the Muskat problem (1a) possesses for each $f_0\in W^s_p({\mathbb R})$ a unique maximal solution $f:=f(\,\cdot \,; f_0)$ such that

$$\begin{aligned} f\in \mathrm{C}\left( [0,T^+), W^s_p({\mathbb R})\right) \cap \mathrm{C}\left( (0,T^+ ), W^{\overline{s}+1}_p({\mathbb R})\right) \cap \mathrm{C}^1\left( (0,T^+), W^{\overline{s}-2}_p({\mathbb R})\right) , \end{aligned}$$

with $T^+=T^+(f_0)\in (0,\infty ]$ denoting the maximal time of existence. Moreover, the following properties hold true:

• (i) The solution depends continuously on the initial data;

• (ii) Given $k\in {\mathbb N}$, we have $f\in \mathrm{C}^\infty ((0,T^+ )\times {\mathbb R},{\mathbb R})\cap \mathrm{C}^\infty ((0,T^+), W^k_p({\mathbb R}));$

• (iii) The solution is global if

  $$\begin{aligned} \sup _{[0,T^+)\cap [0,T]}\Vert f(t)\Vert _{ W^s_p({\mathbb R})}<\infty \qquad \text {for all}\quad T>0. \end{aligned}$$

We emphasize that some of the arguments use in an essential way the fact that $p\in (1,2]$. More precisely, we employ several times the Sobolev embedding ${W^s_p({\mathbb R})\hookrightarrow W^{t}_{p'}({\mathbb R})},$ where $p'$ is the adjoint exponent of p, that is $p^{-1}+{p'}^{-1}=1$ (this notation is used in the entire paper). This provides the restriction $p\in (1,2]$.

Additionally, we expect that the assertion (ii) of Theorem 1.1 can be improved to real-analyticity instead of smoothness. However, this would require to establish real-analytic dependence of the right-hand side of (1a) on f in the functional analytic framework considered in Sect. 3, which is much more involved than showing the smooth dependence (see [25, Proposition 5.1] for a related proof of real-analyticity).

1.1 Notation

Given $k\in {\mathbb N}$ and $p\in (1,\infty ),$ we let $W^k_p({\mathbb R})$ denote the standard $L_p$-based Sobolev space with norm

$$\begin{aligned} \Vert f\Vert _{W^k_p}:=\left( \sum _{\ell =0}^k\left\| f^{(\ell )}\right\| _p^p\right) ^{1/p}. \end{aligned}$$

Given $0<s\not \in {\mathbb N}$ with $s=[s]+\{s\}$, where ${[s]\in {\mathbb N}}$ and $\{s\}\in (0,1)$, the Sobolev space $W^s_p({\mathbb R})$ is the subspace of $W^{[s]}_p({\mathbb R})$ that consists of functions for which the seminorm

$$\begin{aligned}{}[f]_{W^{s}_p}^p:=\int _{{\mathbb R}^2}\frac{\left| f^{([s])}(x)-f^{([s])}(y)\right| ^p}{|x-y|^{1+\{s\}p}}\, d(x,y)=\int _{{\mathbb R}}\frac{\left\| f^{([s])} -\tau _\xi f^{([s])}\right\| _p^p}{|\xi |^{1+\{s\}p}}\, d\xi \end{aligned}$$

is finite. Here $\{\tau _\xi \}_{\xi \in {\mathbb R}}$ is the group of right translations and $\Vert \cdot \Vert _q:=\Vert \cdot \Vert _{L_q({\mathbb R})}$, $q\in [1,\infty ].$ The norm on $W^s_p({\mathbb R})$ is defined by

$$\begin{aligned} \Vert f\Vert _{W^s_p}:=\left( \Vert f\Vert _{W^{[s]}_p}^p+[f]_{W^{s}_p}^p\right) ^{1/p}. \end{aligned}$$

For $s<0$, $W^s_p({\mathbb R})$ is defined as the dual space of $W^{-s}_{p'}({\mathbb R})$.

The following properties can be found e.g. in [40].

• (i) $\mathrm{C}^\infty _0({\mathbb R})$ lies dense in $W^s_p({\mathbb R})$ for all $s\in {\mathbb R}$. Moreover, $W^s_p({\mathbb R})\hookrightarrow \mathrm{C}^{s-1/p}({\mathbb R})$ holds provided that $0<s-1/p\not \in {\mathbb N}$.

• (ii) $W^s_p({\mathbb R})$ is an algebra for $s>1/p$.

• (iii) If $\rho >\max \{s, -s\},$ then $f\in \mathrm{C}^\rho ({\mathbb R})$ is a pointwise multiplier for $W^s_p({\mathbb R})$, that is

  $$\begin{aligned} \Vert fg\Vert _{W^s_p}\le C\Vert f\Vert _{\mathrm{C}^\rho }\Vert g\Vert _{W^s_p} \qquad \text {for all}\; g\in W^s_p({\mathbb R}), \end{aligned}$$

  (2)

  with C independent of f and g.

• (iv) Given $\theta \in (0,1)$ and $p\in (1,\infty )$, let $(\cdot ,\cdot )_{\theta ,p}$ denote the real interpolation functor of exponent $\theta$ and parameter $p\in (1,\infty ).$ Given $s_0,\, s_1\in {\mathbb R}$ with $(1-\theta )s_0+\theta s_1\not \in {\mathbb Z}$, it holds that

  $$\begin{aligned} \left( W^{s_0}_{p}({\mathbb R}), W^{s_1}_{p}({\mathbb R})\right) _{\theta ,p}=W^{(1-\theta )s_0+\theta s_1}_{p}({\mathbb R}). \end{aligned}$$

  (3)

• (v) $W^s_p({\mathbb R})\hookrightarrow W^{t}_q({\mathbb R}))$ if $1<p\le q<\infty$ and $s-1/p\ge t-1/q.$

Besides, we need also the following properties.

• (a) Given $r\in [0,1)$ and $p\in (1,\infty )$, there exists a constant $C>0$ such that

  $$\begin{aligned} \Vert gh\Vert _{W^{r}_p}\le 2\Vert g\Vert _\infty \Vert h\Vert _{W^{r}_p}+C\Vert g\Vert _{W^{r+1}_p}\Vert h\Vert _p,\quad g\in W^{r+1}_p({\mathbb R}), h\in W^{r}_p({\mathbb R}). \end{aligned}$$

  (4)

• (b) Given $r\in (1/p,1)$ and $p\in (1,\infty )$, there exists a constant $C>0$ such that

  $$\begin{aligned} \Vert gh\Vert _{W^{r}_p}\le 2\left( \Vert g\Vert _\infty \Vert h\Vert _{W^{r}_p}+\Vert h\Vert _\infty \Vert g\Vert _{W^{r}_p}\right) ,\quad g,\, h\in W^{r}_p({\mathbb R}). \end{aligned}$$

  (5)

• (c) Given $p\in (1,2]$, $r\in (1/p,1)$, and $\rho \in (0,\min \{r-1/p,1-r\}),$ there exists a constant $C>0$ such that

  $$\begin{aligned} \Vert gh\Vert _{W^{1-r}_{p'}}\le 2\Vert g\Vert _\infty \Vert h\Vert _{W^{1-r}_{p'}}+C\Vert g\Vert _{W^{r}_p}\Vert h\Vert _{W^{1-r-\rho }_{p'}} \end{aligned}$$

  (6)

  for all $g\in W^{r}_p({\mathbb R})$ and $h\in W^{1-r}_{p'}({\mathbb R}),$ and

  $$\begin{aligned} \Vert \varphi h\Vert _{W^{r-1}_p}\le 5\Vert \varphi \Vert _\infty \Vert h\Vert _{W^{r-1}_p}+C\frac{ \Vert \varphi \Vert _{W^{r}_{p}}^{1+2r/\rho }}{\Vert \varphi \Vert _{\infty }^{2r/\rho }}\Vert h\Vert _{W^{r-1-\rho }_p} \end{aligned}$$

  (7)

  for all $h\in W^{r-1}_p({\mathbb R})$ and $0\ne \varphi \in W^r_p({\mathbb R})$.

The estimates (4) and (5) are straightforward consequences of the properties (i)-(v) listed above. The inequalities (6) and (7) are established in Appendix 2. Let us point out that (6) implies in particular that the multiplications

$$\begin{aligned} \begin{aligned}&[(g,h)\mapsto gh]: W^{r}_{p}({\mathbb R})\times W^{1-r}_{p'}({\mathbb R})\rightarrow W^{1-r}_{p'}({\mathbb R}),\\&[(g,h)\mapsto gh]: W^{r}_{p}({\mathbb R})\times W^{r-1}_{p}({\mathbb R})\rightarrow W^{r-1}_{p}({\mathbb R}) \end{aligned} \end{aligned}$$

(8)

are both continuous if $p\in (1,2]$ and $r\in (1/p,1)$. The continuity of (8)$_1$ is a straightforward consequence of (6), while the continuity of (8)$_2$ follows by a standard duality argument.

1.2 Outline

In Sect. 2 we introduce some multilinear singular integral operators and study their properties. These operators are then used in Sect. 3 to formulate (1a) as a quasilinear evolution problem, cf. (27)–(28). Subsequently, we show in Theorem 3.5 that (27) is of parabolic type and we complete the section with the proof of Theorem 1.1. In Appendixes 1 and 2 we prove some technical results used in the analysis.

2 Some singular integral operators

In this section we investigate a family of multilinear singular integral operators which play a key role in the analysis of the Muskat problem (and also of the Stokes problem [27]). Given ${n,\,m\in {\mathbb N}}$ and Lipschitz continuous functions ${a_1,\ldots , a_{m},\, b_1, \ldots , b_n:\mathbb {R}\rightarrow \mathbb {R}}$ we set

$$\begin{aligned} B_{n,m}(a_1,\ldots , a_m)[b_1,\ldots ,b_n,\overline{\omega }](x):=\mathop \mathrm{PV}\nolimits \int _\mathbb {R} \frac{\overline{\omega }(x-y)}{y}\frac{\prod _{i=1}^{n}\big (\delta _{[x,y]} b_i /y\big )}{\prod _{i=1}^{m}\big [1+\big (\delta _{[x,y]} a_i /y\big )^2\big ]}\, dy. \end{aligned}$$

(9)

For brevity we write

$$\begin{aligned} B^0_{n,m}(f)[\overline{\omega }]:=B_{n,m}(f,\ldots , f)[f,\ldots ,f,\overline{\omega }]. \end{aligned}$$

(10)

The relevance of these operators in the context of (1) is enlightened by the fact that (1a) can be recast, at least at formal level, in a compact form as

$$\begin{aligned} \frac{d f(t)}{dt}=\frac{k}{2\mu }\mathbb {B}(f(t))\left[ (\sigma \kappa (f(t))-\Delta _\rho f(t))'\right] ,\quad t>0, \end{aligned}$$

(11)

where $\mathbb {B}(f)$ is defined by

$$\begin{aligned} \mathbb {B}(f)&:= \pi ^{-1}\left( B_{0,1}^0(f)+f'B_{1,1}^0(f)\right) \end{aligned}$$

(12)

and $f':=df/dx$. A key observation that we exploit in our analysis is the quasilinear structure of the curvature operator. Indeed, it holds that

$$\begin{aligned} \kappa (f)=\kappa (f)[f], \end{aligned}$$

where

$$\begin{aligned} \kappa (f)[h]:=\frac{h''}{\left( 1+f'^2\right) ^{3/2}}. \end{aligned}$$

(13)

Given $f\in W^s_p({\mathbb R})$ and $h\in W^{s+1}_p({\mathbb R})$, with $p\in (1,\infty )$ and $s\in (1+1/p,2),$ Lemma 3.1 ensures that in (11) the term

$$\begin{aligned} \left( \sigma \kappa (f)[h]-\Delta _\rho h\right) ', \end{aligned}$$

belongs to $W^{s-2}_p({\mathbb R})$. Therefore, it is natural to ask weather $\mathbb {B}(f)\in \mathcal {L}(W^{s-2}_p({\mathbb R}))$. The proof of this boundedness property, which enables us to view (11) as an evolution equation in ${W^{s-2}_p({\mathbb R})}$, see Sect. 3, is the main goal of this section. As already mentioned, some of the arguments require $p\in (1,2]$.

We first recall the following result.

Lemma 2.1

Let $p\in (1,\infty )$, $n,\,m \in {\mathbb N}$, and let $a_1,\ldots , a_{m},\, b_1, \ldots , b_n:{\mathbb R}\rightarrow {\mathbb R}$ be Lipschitz continuous. Then, there exists a constant $C=C(n,\, m,\,\max _{i=1,\ldots , m}\Vert a_i'\Vert _{\infty } )$ such that

$$\begin{aligned} \left\| B_{n,m}(a_1,\ldots , a_m)[b_1,\ldots ,b_n,\,\cdot \,]\right\| _{\mathcal {L}(L_p({\mathbb R}))}\le C\prod _{i=1}^{n} \Vert b_i'\Vert _{\infty }. \end{aligned}$$

Moreover,¹$B_{n,m}\in \mathrm{C}^{1-}((W^1_\infty ({\mathbb R}))^{m},\mathcal {L}^n_\mathrm{sym}(W^1_\infty ({\mathbb R}),\mathcal {L}( L_p({\mathbb R}))).$

Proof

See [2, Lemma 2].

We point out that, given Lipschitz continuous functions $a_1,\ldots , a_m, \,\widetilde{a}_1,\ldots , \widetilde{a}_m,\, b_1, \ldots , b_n$, we have

$$\begin{aligned} \begin{aligned}&B_{n,m}(\widetilde{a}_{1}, \ldots , \widetilde{a}_{m})\left[ b_1,\ldots , b_{n},\cdot \right] - B_{n,m}\left( a_1, \ldots , a_{m}\right) \left[ b_1,\ldots , b_{n},\cdot \right] \\&=\sum _{i=1}^{m} B_{n+2,m+1}\left( \widetilde{a}_{1},\ldots , \widetilde{a}_{i},a_i,\ldots \ldots , a_{m}\right) \left[ b_1,\ldots , b_{n},a_i+\widetilde{a}_{i}, a_i-\widetilde{a}_{i},\cdot \right] . \end{aligned} \end{aligned}$$

(14)

The formula (14) was used to establish the Lipschitz continuity property (denoted by $\mathrm{C}^{1-}$) in Lemma 2.1 and is also of importance for our later analysis.

The strategy is as follows. In Lemma 2.4 we show that, given $f\in W^s_p({\mathbb R})$, with $p\in (1,2]$ and $s\in (1+1/p,2)$, we have $B_{n,m}^0(f)\in \mathcal {L}(W^{r}_{p'}({\mathbb R}))$ for all $r\in [0,1-1/p)$ (in particular also for $r=2-s$). Lemma 2.2 below provides the key argument in the proof of Lemma 2.4. The desired mapping property $B_{n,m}^0(f)\in \mathcal {L}(W^{s-2}_p({\mathbb R}))$ stated in Lemma 2.5, follows then from Lemma 2.4 via a duality argument. Lemma 2.5 and the fact that $f'\in W^{s-1}_p({\mathbb R})$ is a pointwise multiplier for $W^{s-2}_p({\mathbb R})$, see (8), provide the desired property ${\mathbb {B}(f)\in \mathcal {L}(W^{s-2}_p({\mathbb R}))}$.

Lemma 2.2

Given $p\in (1,2]$, $s\in (1 +1/p,2)$, $r\in (0, 1-1/p)$, $n,\, m\in {\mathbb N}$, $n\ge 1$, and ${a_1,\ldots , a_m \in W^s_p({\mathbb R})}$, there exists a positive constant $C$, which depends only on $n,\, m,\, s,\,r,\,p,\,\max _{1\le i\le m}\Vert a_i\Vert _{W^s_p}$ ,  such that

$$\begin{aligned} \left\| B_{n,m}(a_1,\ldots , a_{m})[b_1,\ldots , b_n,\overline{\omega }]\right\| _{p'}\le C \Vert b_1 \Vert _{W^{s+1-r-2/p}_{p'}}\Vert \overline{\omega }\Vert _{W^{r}_{p'}}\prod _{i=2}^{n}\Vert b_i\Vert _{W^s_p} \end{aligned}$$

(15)

for all $b_1,\ldots , b_n\in W^s_p({\mathbb R})$ and $\overline{\omega }\in W^{r}_{p'}({\mathbb R}).$

Proof

Without loss of generality we may assume that $\overline{\omega }\in \mathrm{C}^\infty _0({\mathbb R}).$ Using the relation

$$\begin{aligned} \frac{\partial }{\partial y}\left( \frac{\delta _{[x,y]} b_1}{y}\right) =\frac{ b_1'(x-y)}{y}- \frac{\delta _{[x,y]} b_1}{y^2}, \end{aligned}$$

algebraic manipulations lead us to

$$\begin{aligned}&B_{n,m}(a_1,\ldots , a_{m})[b_1,\ldots , b_n,\overline{\omega }](x)\\&=\left( \overline{\omega }B_{n-1,m}(a_1,\ldots , a_{m})[b_2,\ldots , b_n,b_1']\right) (x)+\int _{\mathbb R}K(x,y)\, dy\\&-\overline{\omega }(x)\int _{\mathbb R}\frac{\prod _{i=2 }^{n}\left( \delta _{[x,y]} b_i /y\right) }{\prod _{i=1}^{m}\left[ 1+\left( \delta _{[x,y]} a_i /y\right) ^2\right] } \frac{\partial }{\partial y}\left( \frac{\delta _{[x,y]} b_1 }{y}\right) \, dy \end{aligned}$$

for $x\in {\mathbb R}$, where

$$\begin{aligned} K(x,y):=-\frac{\prod _{i=1 }^{n}\left( \delta _{[x,y]} b_i /y\right) }{\prod _{i=1}^{m}\left[ 1+\left( \delta _{[x,y]} a_i /y\right) ^2\right] } \frac{\delta _{[x,y]} \overline{\omega }}{y},\qquad x\in {\mathbb R}, y\ne 0. \end{aligned}$$

The last term on the right side of the previous identity vanishes if ${(n-1)^2+m^2=0}$. Otherwise, we use integration by parts and arrive at

$$\begin{aligned} \begin{aligned}&B_{n,m}(a_1,\ldots , a_{m})[b_1,\ldots , b_n,\overline{\omega }](x)\\&=\left( \overline{\omega }B_{n-1,m}(a_1,\ldots , a_{m})[b_2,\ldots , b_n,b_1']\right) (x)+\int _{\mathbb R}K(x,y)\, dy\\&-\overline{\omega }(x)\sum _{j=2}^n\int _{\mathbb R}K_{1,j}(x,y)\, dy+\overline{\omega }(x)\sum _{j=1}^m\int _{\mathbb R}K_{2,j}(x,y)\, dy, \end{aligned} \end{aligned}$$

(16)

where, given $x\in {\mathbb R}$ and $y\ne 0$, we set

$$\begin{aligned} K_{1,j}(x,y)&:= \frac{\prod _{i=1, i\ne j }^{n}\left( \delta _{[x,y]} b_i /y\right) }{\prod _{i=1}^{m}\left[ 1+\left( \delta _{[x,y]} a_i /y\right) ^2\right] }\frac{\delta _{[x,y]}b_j-yb_j'(x-y)}{y^2},\\ K_{2,j}(x,y)&:= 2\frac{ \delta _{[x,y]} a_j /y}{1+\left( \delta _{[x,y]} a_j /y\right) ^2} \frac{\prod _{i=1 }^{n}\left( \delta _{[x,y]} b_i /y\right) }{\prod _{i=1}^{m}\left[ 1+\left( \delta _{[x,y]} a_i /y\right) ^2\right] } \frac{\delta _{[x,y]}a_j-ya_j'(x-y)}{y^2}. \end{aligned}$$

We estimate the $L_{p'}$-norm of the four terms on the right of (16) separately. Let $q\in ( p',\infty )$ be defined as the solution to

$$\begin{aligned} \frac{1}{q}+\frac{1}{1/r}=\frac{1}{p'}. \end{aligned}$$

Term 1. We note that ${\overline{\omega }\in W^{r}_{p'}({\mathbb R})\hookrightarrow L_q({\mathbb R})}$, ${b_1'\in W^{s-2/p}_{p'}({\mathbb R})\hookrightarrow W^{s-r-2/p}_{p'}({\mathbb R})}$, and additionally we have $W^{s-r-2/p}_{p'}({\mathbb R})\hookrightarrow L_{1/r}({\mathbb R})$. Hölder’s inequality together with Lemma 2.1 (with $p=1/r$) then yields

$$\begin{aligned} \begin{aligned} \left\| \overline{\omega }B_{n-1,m}(a_1,\ldots , a_{m})\left[ b_2,\ldots , b_n,b_1'\right] \right\| _{p'}&\le \Vert \overline{\omega }\Vert _q\left\| B_{n-1,m}(a_1,\ldots , a_{m})\left[ b_2,\ldots , b_n,b_1'\right] \right\| _{1/r}\\&\le C\Vert \overline{\omega }\Vert _{q}\Vert b_1'\Vert _{1/r}\left( \prod _{i=2}^{n}\Vert b_i'\Vert _{\infty }\right) \\&\le C\Vert \overline{\omega }\Vert _{W^{r}_{p'}}\Vert b_1 \Vert _{W^{s+1-r-2/p}_{p'}}\left( \prod _{i=2}^{n}\Vert b_i\Vert _{W^s_p}\right) . \end{aligned} \end{aligned}$$

(17)

Term 2. Let $s_0\in (1+1/p, s]$ be chosen such that $s_0-r-1/p<1$. Taking advantage of Minkowski’s integral inequality, of Hölder’s inequality, and of the embedding property $b_1\in W^{s+1-r-2/p}_{p'}({\mathbb R})\hookrightarrow \mathrm{C}^{s_0-r-1/p}({\mathbb R})$  we get

$$\begin{aligned} \left( \int _{\mathbb R}\left| \int _{\mathbb R}K(x,y)\, dy \right| ^{p'}\, dx\right) ^{1/p'}&\le \left( \int _{\{|y|<1\}}+\int _{\{|y|>1\}}\right) \left( \int _{\mathbb R}|K(x,y)|^{p'}\, dx \right) ^{1/p'}\, dy\\&\le 4\Vert b_1\Vert _\infty \left( \prod _{i=2}^n \Vert b_i'\Vert _\infty \right) \Vert \overline{\omega }\Vert _{p'}\\& + [b_1]_{ s_0-r-1/p}\left( \prod _{i=2}^n \Vert b_i'\Vert _\infty \right) \int _{\{|y|<1\}}\frac{\Vert \overline{\omega }-\tau _y\overline{\omega }\Vert _{p'}}{|y|^{2- s_0+r+1/p}}\, dy, \end{aligned}$$

where

$$\begin{aligned} \int _{\{|y|<1\}}\frac{\Vert \overline{\omega }-\tau _y\overline{\omega }\Vert _{p'}}{|y|^{2- s_0+r+1/p}}\, dy&\le \Vert \overline{\omega }\Vert _{W^{r}_{p'}}\left( \int _{\{|y|<1\}}\frac{1}{|y|^{(1-s_0+2/p)p}}\, dy\right) ^{1/p}\le C\Vert \overline{\omega }\Vert _{W^{r}_{p'}}. \end{aligned}$$

We arrive at

$$\begin{aligned} \left( \int _{\mathbb R}\left| \int _{\mathbb R}K(x,y)\, dy \right| ^{p'}\, dx\right) ^{1/p'} \le C\Vert \overline{\omega }\Vert _{W^{r}_{p'}} \Vert b_1\Vert _{W^{s+1-r-2/p}_{p'}} \left( \prod _{i=2}^{n}\Vert b_i\Vert _{W^s_p}\right) . \end{aligned}$$

(18)

Terms 3 & 4. Given $2\le j\le n$, Hölder inequality and Minkowski’s integral inequality yield

$$\begin{aligned} \begin{aligned} \left( \int _{\mathbb R}\left| \overline{\omega }(x)\int _{\mathbb R}K_{1,j}(x,y)\, dy\right| ^{p'}\, dx\right) ^{1/p'}&\le \Vert \overline{\omega }\Vert _q \left( \int _{\mathbb R}\left| \int _{\mathbb R}K_{1,j}(x,y)\, dy\right| ^{\frac{1}{r}}\, dx\right) ^{r}\\&\le \Vert \overline{\omega }\Vert _{q}\int _{\mathbb R}\left( \int _{\mathbb R}|K_{1,j}(x,y)|^{\frac{1}{r}}\, dx\right) ^{r}\, dy. \end{aligned} \end{aligned}$$

To estimate the integral term we choose $s_0\in (1+1/p, s]$ with $s_0-r-1/p<1.$ Since $b_1\in W^{s+1-r-2/p}_{p'}({\mathbb{R}})\hookrightarrow W^1_{1/r}({\mathbb{R}})\cap {\mathrm{C}}^{s_0-r-1/p}({\mathbb{R}})$, we get

$$\begin{aligned}&\int _{\mathbb R}\left( \int _{\mathbb R}|K_{1,j}(x,y)|^{1/r}\, dx\right) ^{r}\, dy\\&\le 4\Vert b_1\Vert _{1/r}\left( \prod _{i=2 }^{n}\Vert b_i\Vert _{W^{s}_p}\right) \\&+[b_1]_{s_0-r-1/p}\left( \prod _{i=2, i\ne j}^{n}\Vert b_i'\Vert _{\infty }\right) \int _{\{|y|<1\}}\frac{\Vert b_j-\tau _y b_j-y\tau _y b_j'\Vert _{1/r}}{|y|^{3-s_0+r+1/p}} \, dy \end{aligned}$$

for $2\le j\le n.$ Since $b_j\in W^{s_0}_p({\mathbb R})\hookrightarrow W^{s_0+r-\frac{1}{p} }_{1/r}({\mathbb R})$, $2\le j\le n$, Minkowski’s integral inequality, Hölder’s inequality, and a change of variables lead to

$$\begin{aligned}\int \limits _{\{|y|<1\}}&\frac{\Vert b_j-\tau _y b_j-y\tau _y b_j'\Vert _{1/r}}{|y|^{3-s_0+r+1/p}} \, dy \\&=\int _{\{|y|1\}}\frac{1 }{|y|^{2- s_0+r+1/p}} \left( \int _{\mathbb R}\left| \int _0^1 [b_j'(x-(1-t )y)- b_j'(x-y)]\, dt\right| ^{1/r}\, dx\right) ^{r}\, dy\\&\le \int _0^1 \int _{\{|y|<1\}}\frac{\Vert b_j'- \tau _{-t y}b_j'\Vert _{1/r}}{|y|^{2- s_0+r+1/p}}\,dy\, dt\\&\le \left( \int _{\{|y|<1\}}\frac{1 }{|y|^{(3-2s_0-r+2/p)/(1-r)}}\, dy\right)^{1-r}\left(\int _0^1 t^{1- s_0+r+1/p}\, dt\right) \Vert b_j'\Vert _{W^{ s_0+r-1-\frac{1}{p} }_{1/r}}\\&\le C\Vert b_j\Vert _{W^{s}_p}. \end{aligned}$$

Consequently, given $2\le j\le n$, we have

$$\begin{aligned} \left( \int _{\mathbb R}\left| \overline{\omega }(x)\int _{\mathbb R}K_{1,j}(x,y) \, dy\right| ^{p'}\, dx\right) ^{1/p'}\le C\Vert \overline{\omega }\Vert _{W^{r}_{p'}} \Vert b_1\Vert _{W^{ s+1-r-2/p}_{p'}} \left( \prod _{i=2}^{n}\Vert b_i\Vert _{W^s_p}\right) \end{aligned}$$

(19)

and, similarly,

$$\begin{aligned} \left( \int _{\mathbb R}\left| \overline{\omega }(x)\int _{\mathbb R}K_{2,j}(x,y)\, dy\right| ^{p'}\, dx\right) ^{1/p'}\le C\Vert \overline{\omega }\Vert _{W^{r}_{p'}} \Vert b_1\Vert _{W^{ s+1-r-2/p}_{p'}} \left( \prod _{i=2}^{n}\Vert b_i\Vert _{W^s_p}\right) . \end{aligned}$$

(20)

The desired claim follows now from (16) to (20).

Lemma 2.3 below is used to prove Lemma 2.4 (see also [2, Lemma 7] for a related result).

Lemma 2.3

Let $p\in (1,\infty )$, $n\in {\mathbb N}$, and $n<t'<t<n+1$. Then, there exists $C>0$ such that

$$\begin{aligned} \int _{\mathbb R}\frac{\left\| b-\tau _\xi b\right\| _{W^{t'}_p}^p}{|\xi |^{1+(t-t')p}}\, d\xi \le C\Vert b\Vert _{W^{t}_p}^p\qquad \text {for all} \quad b\in W^{t}_p({\mathbb R}). \end{aligned}$$

Proof

We have

$$\begin{aligned} \left\| b-\tau _\xi b\right\| _{W^{t'}_p}^p=\sum _{k=0}^n\left\| b^{(k)}-\tau _\xi b^{(k)}\right\| _{p}^p+\left[ b-\tau _\xi b\right] _{W^{t'}_p}^p \end{aligned}$$

and, given $k\in \{0,\ldots ,n\}$,

$$\begin{aligned} \int _{\mathbb R}\frac{\left\| b^{(k)}-\tau _\xi b^{(k)}\right\| _{p}^p}{|\xi |^{1+(t-t')p}}\, d\xi&=\left[ b^{(k)}\right] _{W^{t-t'}_p}^p\le C\Vert b\Vert _{W^{t}_p}^p. \end{aligned}$$

Moreover, since ${\mathbb R}^2=\overline{\{|\xi |<|h|\}}\cup \{|\xi |>|h|\}$ and

$$\begin{aligned} \int _{\mathbb R}\frac{[b-\tau _\xi b]_{W^{t'}_p}^p}{|\xi |^{1+(t-t')p}}\, d\xi&=\int _{{\mathbb R}^2}\frac{\left\| b^{(n)}-\tau _\xi b^{(n)}-\tau _h(b^{(n)}-\tau _\xi b^{(n)})\right\| _{p}^p}{|\xi |^{1+(t-t')p}|h|^{1+(t'-n)p}}\, dh\, d\xi , \end{aligned}$$

with

$$\begin{aligned}&\int \limits _{\{|\xi |<|h|\}}\frac{\left\| b^{(n)}-\tau _\xi b^{(n)}-\tau _h(b^{(n)}-\tau _\xi b^{(n)})\right\| _{p}^p}{|\xi |^{1+(t-t')p}|h|^{1+(t'-n)p}}\, dh\, d\xi \\&\le 2^{p}\int _{\mathbb R}\frac{\left\| b^{(n)}-\tau _\xi b^{(n)}\right\| _p^p}{|\xi |^{1+(t-t')p}} \left( \int _{\{|\xi |<|h|\}}\frac{1}{|h|^{1+(t'-n)p}}\, dh\right) \, d\xi \le C[b]^p_{W^t_p},\\&\int _{\{|h|<|\xi |\}}\frac{\left\| b^{(n)}-\tau _\xi b^{(n)}-\tau _h(b^{(n)}-\tau _\xi b^{(n)})\right\| _{p}^p}{|\xi |^{1+(t-t')p}|h|^{1+(t'-n)p}}\, dh\, d\xi \\&\le 2^{p}\int _{\mathbb R}\frac{\left\| b^{(n)}-\tau _h b^{(n)}\right\| _p^p}{|h|^{1+(t'-n)p}} \left( \int _{\{|h|<|\xi |\}}\frac{1}{|\xi |^{1+(t-t')p}}\, d\xi \right) \, dh\le C[b]^p_{W^t_p}, \end{aligned}$$

the desired estimate is immediate.

We are now in a position to prove that $B_{n,m}^0(f)\in \mathcal {L}(W^{r}_{p'}({\mathbb R}))$ for all $r\in [0,1-1/p)$.

Lemma 2.4

Given $p\in (1,2]$, $s\in (1 +1/p,2)$, $n,\, m\in {\mathbb N}$, ${a_1,\ldots , a_m \in W^s_p({\mathbb R})}$, and ${r\in [0,1-1/p)}$, there exists a constant $C=C(n,\, m,\,s,\,r,\,p,\, \max _{1\le i\le m}\Vert a_i\Vert _{W^s_p})$ such that

$$\begin{aligned} \left\| B_{n,m}(a_1,\ldots , a_{m})\left[ b_1,\ldots , b_n,\overline{\omega }\right] \right\| _{W^{r}_{p'}}\le C \Vert \overline{\omega }\Vert _{W^{r}_{p'}}\prod _{i=1}^{n}\Vert b_i\Vert _{W^{s}_p} \end{aligned}$$

(21)

for all $b_1,\ldots , b_n\in W^s_p({\mathbb R})$ and $\overline{\omega }\in W^{r}_{p'}({\mathbb R}).$

Moreover, $B_{n,m}\in \mathrm{C}^{1-}((W^s_p({\mathbb R}))^m,\mathcal {L}^{n}_\mathrm{sym}( W_p^{s}({\mathbb R}) , \mathcal {L}(W^{r}_{p'}({\mathbb R})))).$

Proof

Let $B_{n,m}:=B_{n,m}(a_1,\ldots , a_{m})[b_1,\ldots , b_n,\cdot ].$ Recalling Lemma 2.1 (with $p=p'$), we get

$$\begin{aligned} \Vert B_{n,m}[\overline{\omega }]\Vert _{p'}\le C \Vert \overline{\omega }\Vert _{p'}\prod _{i=1}^{n}\Vert b_i\Vert _{W^s_p}, \end{aligned}$$

which proves (21) for $r=0$. Let now ${r\in (0,1-1/p)}$. It remains to estimate the quantity

$$\begin{aligned} \left[ B_{n,m}[\overline{\omega }]\right] _{W^{r}_{p'}}^{p'}&= \int _{{\mathbb R}}\frac{\Vert B_{n,m}[\overline{\omega }]-\tau _\xi B_{n,m}[\overline{\omega }]\Vert _{p'}^{p'}}{|\xi |^{1+r p'}}\, d\xi . \end{aligned}$$

Taking advantage of (14), we write

$$\begin{aligned} B_{n,m}[\overline{\omega }]-\tau _\xi B_{n,m}[\overline{\omega }] = T_1(\cdot ,\xi )+T_2(\cdot ,\xi )-T_3(\cdot ,\xi ),\quad \xi \in {\mathbb R}, \end{aligned}$$

with

$$\begin{aligned} T_1(\cdot ,\xi )&:=B_{n,m}\left( a_1,\ldots , a_{m}\right) \left[ b_1,\ldots , b_n,\overline{\omega }-\tau _\xi \overline{\omega }\right] ,\\ T_2(\cdot ,\xi )&:=\sum _{i=1}^nB_{n,m}\left( a_1,\ldots , a_{m}\right) \left[ \tau _\xi b_1,\ldots ,\tau _\xi b_{i-1}, b_i-\tau _\xi b_i, b_{i+1},\ldots b_n,\tau _\xi \overline{\omega }\right] ,\\ T_3(\cdot ,\xi )&:=\sum _{i=1}^mB_{n+2,m+1}\left( a_1,\ldots ,a_{i},\tau _\xi a_i,\ldots , \tau _\xi a_{m}\right) \left[ \tau _\xi b_1,\ldots \tau _\xi b_n,a_i+\tau _\xi a_i,a_i-\tau _\xi a_i, \tau _\xi \overline{\omega }\right] . \end{aligned}$$

Hence,

$$\begin{aligned}{}[B_{n,m}[\overline{\omega }]]_{W^{r}_{p'}}^{p'}\le 3^{p'}\sum _{\ell =1}^3\int _{{\mathbb R}}\frac{\Vert T_\ell (\cdot ,\xi )\Vert _{p'}^{p'}}{|\xi |^{1+rp'}}\, d\xi \end{aligned}$$

(22)

and Lemma 2.1 (with $p=p'$) yields

$$\begin{aligned} \int _{{\mathbb R}}\frac{\Vert T_1(\cdot ,\xi )\Vert _{p'}^{p'}}{|\xi |^{1+rp'}}\, d\xi&\le C^p\left( \prod _{i=1}^{n}\Vert b_i'\Vert _{\infty }^{p'} \right) \int _{{\mathbb R}}\frac{\Vert \overline{\omega }-\tau _\xi \overline{\omega }\Vert _{p'}^{p'}}{|\xi |^{1+rp'}}\, d\xi \le \left( C[\overline{\omega }]_{W^{r}_{p'}}\prod _{i=1}^{n}\Vert b_i'\Vert _{\infty }\right) ^{p'}. \end{aligned}$$

(23)

Furthermore, using (15), Lemma 2.3 (with $p=p'$, $t=s+1-2/p$, and $t'=s+1-r-2/p$), and the embedding $W^s_p({\mathbb R})\hookrightarrow W^{s+1-2/p}_{p'}({\mathbb R})$, we deduce that

$$\begin{aligned} \begin{aligned} \int _{{\mathbb R}}\frac{\Vert T_2(\cdot ,\xi )\Vert _{p'}^{p'}}{|\xi |^{1+rp'}}\, d\xi&\le C^p\Vert \overline{\omega }\Vert _{W^{r}_{p'}}^{p'}\sum _{i=1}^n \left( \int _{{\mathbb R}}\frac{\Vert b_i-\tau _\xi b_i\Vert _{W^{t'}_{p'}}^{p'}}{|\xi |^{1+(t-t')p'}}\, d\xi \right) \prod _{j=1,j\ne i}^n\Vert b_j\Vert _{W^{s}_p}^{p'} \\&\le \left( C\Vert \overline{\omega }\Vert _{W^{r}_{p'}}\prod _{i=1}^{n}\Vert b_i\Vert _{W^{s}_p} \right) ^{p'} \end{aligned} \end{aligned}$$

(24)

and, by similar arguments,

$$\begin{aligned} \begin{aligned} \int _{{\mathbb R}}\frac{\Vert T_3(\cdot ,\xi )\Vert _{p'}^{p'}}{|\xi |^{1+rp'}}\, d\xi \le \left( C\Vert \overline{\omega }\Vert _{W^{r}_{p'}}\prod _{i=1}^{n}\Vert b_i\Vert _{W^{s}_p} \right) ^{p'}. \end{aligned} \end{aligned}$$

(25)

The relations (22)–(25) lead to the desired estimate. The local Lipschitz continuity property follows from (14) and (21).

Together with Lemma 2.4 (with $r=2-s\in (0,1-1/p)$) we obtain the following result.

Lemma 2.5

Given $p\in (1,2]$, $s\in (1 +1/p,2)$, $n,\, m\in {\mathbb N},$ and ${a_1,\ldots , a_m \in W^s_p({\mathbb R})}$, there exists a constant $C=C(n,\, m,\,s,\,p,\,\max _{1\le i\le m}\Vert a_i\Vert _{W^s_p})$ such that

$$\begin{aligned} \left\| B_{n,m}(a_1,\ldots , a_{m})\left[ b_1,\ldots , b_n,\overline{\omega }\right] \right\| _{W^{s-2}_{p}}\le C \Vert \overline{\omega }\Vert _{W^{s-2}_{p}}\prod _{i=1}^{n}\Vert b_i\Vert _{W^{s}_p} \end{aligned}$$

(26)

for all $b_1,\ldots , b_n\in W^s_p({\mathbb R})$ and $\overline{\omega }\in L_{p}({\mathbb R}).$

Moreover, $B_{n,m}\in \mathrm{C}^{1-}((W^s_p({\mathbb R}))^m,\mathcal {L}^{n}_\mathrm{sym}(W_p^{s}({\mathbb R}),\mathcal {L}( W^{s-2}_{p}({\mathbb R})))).$

Proof

We recall that $W^{s-2}_p({\mathbb R})=(W^{2-s}_{p'}({\mathbb R}))'$. Let ${B_{n,m}:=B_{n,m}(a_1,\ldots , a_{m})[b_1,\ldots , b_n,\cdot ]}$. It is not difficult to prove that the $L_2$-adjoint of $B_{n,m}$ is the operator $-B_{n,m}$. Therefore, given $\overline{\omega },\,\varphi \in \mathrm{C}^\infty _0({\mathbb R})$, we obtain, in view of Lemma 2.4,

$$\begin{aligned} \left| \langle B_{n,m}[\overline{\omega }]|\varphi \rangle _{W^{s-2}_p({\mathbb R})\times W^{2-s}_{p'}({\mathbb R})}\right|&=\left| \langle \overline{\omega }| B_{n,m}[\varphi ]\rangle _{W^{s-2}_p({\mathbb R})\times W^{2-s}_{p'}({\mathbb R})}\right| \\&\le C \Vert \overline{\omega }\Vert _{W^{s-2}_p}\Vert \varphi \Vert _{W^{2-s}_{p'}}\prod _{i=1}^{n}\Vert b_i\Vert _{W^{s}_p}. \end{aligned}$$

The estimate (26) follows via a standard density argument.The Lipschitz continuity property is a consequence of (26) and of (14).

3 A functional analytic framework for the Muskat problem

In this section we take advantage of the mapping properties established in Sect. 2 and formulate the Muskat problem (1a) as a quasilinear evolution problem in a suitable functional analytic setting, see (27)–(29). Afterwards, we show that the problem is of parabolic type. This enables us to employ theory for such evolution equations as presented in [6, 28] to establish our main result in Theorem 1.1. The quasilinear structure of (1a) is due to the quasilinearity of the curvature operator, the latter being established in Lemma 3.1.

Lemma 3.1

Given $p\in (1,\infty )$ and $s\in (1+1/p,2)$, the operator ${\kappa (\cdot )[\cdot ]}$ defined in (13) satisfies $\kappa \in \mathrm{C}^\infty (W^s_p({\mathbb R}),\mathcal {L}(W^{s+1}_p({\mathbb R}), W^{s-1}_p({\mathbb R}))).$

Proof

The arguments are similar to those presented in [27, Appendix C] and are therefore omitted.

The Muskat problem (1a) can thus be formulated as the evolution problem

$$\begin{aligned} \frac{df}{dt}(t)=\Phi (f(t))[f(t)],\quad t>0,\qquad f(0)=f_0, \end{aligned}$$

(27)

where

$$\begin{aligned} \Phi (f)[h]:=\frac{k}{2\mu }\mathbb {B}(f)\left[ \sigma (\kappa (f)[h])'-\Delta _\rho h'\right] , \end{aligned}$$

(28)

with $\mathbb {B}$ introduced in (12). Arguing as in [27, Appendix C], we may infer from Lemma 2.5 that, given $n,\, m\in {\mathbb N}$, $p\in (1,2]$, and $s\in (1+1/p,2),$ we have

$$\begin{aligned} \left[ f\mapsto B^0_{n,m}(f)\right] \in \mathrm{C}^\infty (W^s_p({\mathbb R}), \mathcal {L}(W^{s-2}_p({\mathbb R}))). \end{aligned}$$

This property and Lemma 3.1 combined yield

$$\begin{aligned} \Phi \in \mathrm{C}^\infty \left( W^s_p({\mathbb R}),\mathcal {L}\left( W^{s+1}_p({\mathbb R}), W^{s-2}_p({\mathbb R})\right) \right) \end{aligned}$$

(29)

for all $p\in (1,2]$ and $s\in (1+1/p,2).$

Let $p\in (1,2]$, $s\in (1+1/p,2)$, and $f\in W^s_p({\mathbb R})$ be fixed in the remaining of this section. The analysis below is devoted to showing that the linear operator $\Phi (f)$, viewed as an unbounded operator in $W^{s-2}_p({\mathbb R})$ and with definition domain $W^{s+1}_p({\mathbb R})$, is the generator of an analytic semigroup in $\mathcal {L}(W^{2-s}_p({\mathbb R}))$, which reads in the notation used in [7] as

$$\begin{aligned} -\Phi (f)\in \mathcal {H}\left( W^{s+1}_p({\mathbb R}), W^{s-2}_p({\mathbb R})\right) . \end{aligned}$$

(30)

This property is established in Theorem 3.5 below and it identifies the quasilinear evolution problem (27) as being of parabolic type. To start, we note that $\pi ^{-1}B_{0,0 }= H$, where H is the Hilbert transform, and therefore

$$\begin{aligned} \Phi (0)=\frac{k\sigma }{2 \mu }H\circ \frac{d^3}{dx^3}-\frac{k\Delta _\rho }{2 \mu }H\circ \frac{d}{dx}=-\frac{k\sigma }{2 \mu }\left( \frac{d^4}{dx^4}\right) ^{3/4}-\frac{k\Delta _\rho }{2 \mu }\left( -\frac{d^2}{dx^2}\right) ^{1/2}, \end{aligned}$$

(31)

where $(d^4/dx^4)^{3/4}$ denotes the Fourier multiplier with symbol $m(\xi ):=|\xi |^3$ and $(-d^2/dx^2)^{1/2}$ is the Fourier multiplier with symbol $m(\xi ):=|\xi |$. We shall locally approximate the operator $\Phi (\tau f)$, with $\tau \in [0,1]$, by certain Fourier multipliers ${\mathbb A}_{j,\tau }$. Therefore we choose for each ${\varepsilon \in (0,1)}$ a so-called finite $\varepsilon$-localization family, that is a set

$$\begin{aligned} \left\{ (\pi _j^\varepsilon ,x_j^\varepsilon )\,:\, -N+1\le j\le N\right\} \subset \mathrm{C}^\infty (\mathbb {R},[0,1])\times {\mathbb R}\end{aligned}$$

such that

$$\begin{aligned} \bullet \,\,\,\,&\sum _{j=-N+1}^N(\pi _j^\varepsilon )^2=1 \text { and } \Vert \pi _j^\varepsilon )^{(k)}\Vert _\infty \le C\varepsilon ^{-k} \text { for all } k\in {\mathbb N},\, -N+1\le j\le N; \\ \bullet \,\,\,\,&\mathop \mathrm{supp}\nolimits \pi _j^\varepsilon \text { is an interval of length }\varepsilon \text { for }|j|\le N-1, \, \, \mathop \mathrm{supp}\nolimits \pi _{N}^\varepsilon \subset \{|x|>1/\varepsilon \}; \\ \bullet \,\,\,\,&\pi _j^\varepsilon \cdot \pi _l^\varepsilon =0\text { if }[|j-l|\ge 2,\, \max \{|j|,\, |l|\}\le N-1]\text { or }[|l|\le N-2,\, j=N]; \\ \bullet \,\, \,\,&x^\varepsilon _j\in \mathop \mathrm{supp}\nolimits \pi _j^\varepsilon ,\; |j|\le N-1. \end{aligned}$$

The real number $x_N^\varepsilon$ plays no role in the analysis below. To each finite $\varepsilon$-localization family we associate a second family $\{\chi _j^\varepsilon \,:\, -N+1\le j\le N\}\subset \mathrm{C}^\infty (\mathbb {R},[0,1])$ such that

$$\begin{aligned} \bullet \,\,\,\,&\chi _j^\varepsilon =1 \text { on }\mathop \mathrm{supp}\nolimits \pi _j^\varepsilon , -N+1\le j\le N, \text { and }\mathop \mathrm{supp}\nolimits \chi _N^\varepsilon \subset \{|x|>1/\varepsilon -\varepsilon \}; \\ \bullet \,\,\,\,&\mathop \mathrm{supp}\nolimits \chi _j^\varepsilon \text { is an interval of length }3\varepsilon \text { with the same midpoint as } \mathop \mathrm{supp}\nolimits \pi _j^\varepsilon , |j|\le N-1. \end{aligned}$$

To each finite $\varepsilon$-localization family we associate a norm on $W^r_p({\mathbb R}),$ $r\in {\mathbb R}$, which is equivalent to the standard norm.

Lemma 3.2

Let $\varepsilon \in (0,1)$ and let $\{(\pi _j^\varepsilon ,x_j^\varepsilon )\,:\, -N+1\le j\le N\}$ be a finite $\varepsilon$-localization family. Given $p\in (1,\infty )$ and $r\in {\mathbb R}$, there exists $c=c(\varepsilon ,r,p)\in (0,1)$ such that

$$\begin{aligned} c\Vert f\Vert _{W^r_p}\le \sum _{j=-N+1}^N\Vert \pi _j^\varepsilon f\left\| _{W^r_p}\le c^{-1}\right\| f\Vert _{W^r_p},\qquad f\in W^r_p({\mathbb R}). \end{aligned}$$

Proof

The claim follows from the fact that $\pi _j^\varepsilon \in \mathrm{C}^\infty ({\mathbb R})$ is a pointwise multiplier for $W^r_p({\mathbb R})$.

The next result is the main step in the proof of (30).

Theorem 3.3

Let $p\in (1,2]$, $s\in (1+1/p,2)$, $\rho \in (0,\min \{(s-1-1/p)/2),2-s\})$, and $\nu >0$ be given. Then, there exist $\varepsilon \in (0,1)$, a $\varepsilon$-localization family $\{(\pi _j^\varepsilon ,x_j^\varepsilon )\,:\, -N+1\le j\le N\}$, a constant $K=K(\varepsilon )$, and bounded operators

$$\begin{aligned} {\mathbb A}_{j,\tau }\in \mathcal {L}\left( W^{s+1}_p({\mathbb R}), W^{s-2}_p({\mathbb R})\right) , \qquad j\in \{-N+1,\ldots ,N\} \text { and }\tau \in [0,1], \end{aligned}$$

such that

$$\begin{aligned} \left\| \pi _j^\varepsilon \Phi (\tau f) [h]-{\mathbb A}_{j,\tau }\left[ \pi ^\varepsilon _j h\right] \right\| _{W^{s-2}_p}\le \nu \left\| \pi _j^\varepsilon h\right\| _{W^{s+1}_p}+K\Vert h\Vert _{W^{ s+1-\rho }_p} \end{aligned}$$

(32)

for all $-N+1\le j\le N$, $\tau \in [0,1],$ and $h\in W^{s+1}_p({\mathbb R})$. The operators ${\mathbb A}_{j,\tau }$ are defined by

$$\begin{aligned} {\mathbb A}_{j,\tau }:=- \alpha _\tau (x_j^\varepsilon ) \left( \frac{d^4}{dx^4}\right) ^{3/4}, \quad |j|\le N-1, \qquad {\mathbb A}_{N,\tau }:= - \frac{k\sigma }{2 \mu } \left( \frac{d^4}{dx^4}\right) ^{3/4}, \end{aligned}$$

and $\alpha _\tau :=(k\sigma /(2\mu ))(1+\tau ^2f'^2)^{-3/2}$.

Proof

In this proof we denote by C constants that do not depend on $\varepsilon$ and we write K for constants that depend on $\varepsilon$.

Given $-N+1\le j\le N$, $\tau \in [0,1],$ and $h\in W^{s+1}_p({\mathbb R})$, Lemma 2.1 yields

$$\begin{aligned} \left\| \pi _j^\varepsilon \mathbb {B}(\tau f)[h']\right\| _{W^{s-2}_p}\le \left\| \pi _j^\varepsilon \mathbb {B}(\tau f)[h']\right\| _{p}\le C\Vert h'\Vert _{p}\le C\Vert h\Vert _{W^{s+1-\rho }_p}, \end{aligned}$$

(33)

while, using some elementary arguments, we get

$$\begin{aligned} \begin{aligned} \left\| \pi _j^\varepsilon (\kappa (f)[h])'\right\| _{W^{s-2}_p}&\le \left\| \pi _j^\varepsilon \kappa (f)[h]\right\| _{W^{s-1}_p}+K\left\| \kappa (f)[h]\right\| _{p}\\&\le C_0 \left\| \pi _j^\varepsilon h\right\| _{W^{s+1}_p}+K\Vert h\Vert _{W^{s+1-\rho }_p}. \end{aligned} \end{aligned}$$

(34)

Below we take advantage of (34) when considering the leading order term $\mathbb {B}(\tau f)[(\kappa (\tau f)[h])']$ of $~\Phi (\tau f) [h]$. Let $C_1:=C_0k\sigma /2\pi \mu .$

Step 1: The case $|j|\le N-1$. Given $|j|\le N-1$, we infer from Lemma 4.2 and (34) that

$$\begin{aligned} \begin{aligned}&\left\| \pi _j^\varepsilon B_{0,1}^0(\tau f) \left[ (\kappa (\tau f)[h])'\right] -\frac{1}{1+\tau ^2f'^2(x_j^\varepsilon )}B_{0,0}\left[ \pi ^\varepsilon _j (\kappa (\tau f)[h])'\right] \right\| _{W^{s-2}_p}\\&\le \frac{\nu }{4C_1} \left\| \pi _j^\varepsilon \left( \kappa (\tau f)[h]\right) '\right\| _{W^{s-2}_p}+K\left\| \left( \kappa (\tau f)[h]\right) '\right\| _{W^{ s-2-\rho }_p}\\&\le \frac{\nu C_0}{4C_1} \left\| \pi _j^\varepsilon h\right\| _{W^{s+1}_p}+K\Vert h\Vert _{W^{ s+1-\rho }_p} \end{aligned} \end{aligned}$$

(35)

provided that $\varepsilon$ is sufficiently small. Besides, we have

$$\begin{aligned} \left\| \pi _j^\varepsilon \tau f'B_{1,1}^0(\tau f) \left[ (\kappa (\tau f)[h])'\right] -\frac{\tau ^2f'^2(x_j^\varepsilon )}{1+\tau ^2f'^2(x_j^\varepsilon )}B_{0,0}\left[ \pi ^\varepsilon _j (\kappa (\tau f)[h])'\right] \right\| _{W^{s-2}_p}\le T_1+T_2+ T_3, \end{aligned}$$

where

$$\begin{aligned}T_1&:=\left\| \chi _j^\varepsilon \left( f'-f'(x_j^\varepsilon )\right) B_{1,1}^0(\tau f)[\pi _j^\varepsilon (\kappa (\tau f)[h])']\right\|_{W^{s-2}_p},\\T_2&:=\left\|\chi _j^\varepsilon \left( f'-f'(x_j^\varepsilon )\right) \left(\pi _j^\varepsilon B_{1,1}^0(\tau f)[(\kappa (\tau f)[h])']-B_{1,1}^0(\tau f)[\pi _j^\varepsilon (\kappa (\tau f)[h])']\right)\right\|_{W^{s-2}_p},\\T_3&:=\Vert f'\Vert_\infty \left\|\pi _j^\varepsilon B_{1,1}^0(\tau f) \left[ (\kappa (\tau f)[h])'\right] -\frac{\tau f'(x_j^\varepsilon )}{1+\tau ^2f'^2(x_j^\varepsilon )}B_{0,0}[ \pi ^\varepsilon _j (\kappa (\tau f)[h])'] \right\|_{W^{s-2}_p}.\end{aligned}$$

For $\varepsilon$ sufficiently small to guarantee that

$$\begin{aligned} \left\| \chi _j^\varepsilon (f'-f'(x_j^\varepsilon ))\right\| _\infty <\frac{\nu }{40C_1}\left( \max _{\tau \in [0,1]}\Vert B_{1,1}^0(\tau f)\Vert _{\mathcal {L}(W^{s-2}_p({\mathbb R}))}\right) ^{-1}, \end{aligned}$$

it follows from (7) (with $r=s-1$), Lemma 2.5, and (34) (if $\chi _j^\varepsilon (f'-f'(x_j^\varepsilon ))$ is not identically zero, otherwise the estimate is trivial) that

$$\begin{aligned} T_1\le \frac{\nu C_0}{8C_1} \left\| \pi _j^\varepsilon h\right\| _{W^{s+1}_p}+K\Vert h\Vert _{W^{ s+1-\rho }_p}. \end{aligned}$$

(36)

As $\chi _j^\varepsilon (f'-f'(x_j^\varepsilon ))\in W^{s-1}_p({\mathbb R})$ is a pointwise multiplier for $W^{s-2}_p({\mathbb R}),$ cf. (8), Lemma 4.1 yields

$$\begin{aligned} T_2\le K\left\| (\kappa (\tau f)[h])'\right\| _{W^{ s-2-\rho }_p}\le K\Vert h\Vert _{W^{ s+1-\rho }_p}. \end{aligned}$$

(37)

Finally, if $\varepsilon$ is sufficiently small, we may argue as in the derivation of (35) to get

$$\begin{aligned} T_3\le \frac{\nu C_0}{8C_1}\left\| \pi _j^\varepsilon h\right\| _{W^{s+1}_p}+K\Vert h\Vert _{W^{ s+1-\rho }_p}. \end{aligned}$$

(38)

Gathering (36)–(38), we conclude that

$$\begin{aligned} \begin{aligned}&\left\| \pi _j^\varepsilon \tau f'B_{1,1}^0(\tau f) \left[ (\kappa (\tau f)[h])'\right] -\frac{\tau ^2f'^2(x_j^\varepsilon )}{1+\tau ^2f'^2(x_j^\varepsilon )}B_{0,0}\left[ \pi ^\varepsilon _j (\kappa (\tau f)[h])'\right] \right\| _{W^{s-2}_p}\\&\le \frac{\nu C_0}{4C_1} \left\| \pi _j^\varepsilon h\right\| _{W^{s+1}_p}+K\Vert h\Vert _{W^{ s+1-\rho }_p}. \end{aligned} \end{aligned}$$

(39)

We now combine (33), (35), and (39) and obtain that

$$\begin{aligned} \left\| \pi _j^\varepsilon \Phi (\tau f) [h]-\frac{k\sigma }{2\pi \mu }B_{0,0}\left[ \pi ^\varepsilon _j (\kappa (\tau f)[h])'\right] \right\| _{W^{s-2}_p}\le \frac{\nu }{2} \left\| \pi _j^\varepsilon h\right\| _{W^{s+1}_p}+K\Vert h\Vert _{W^{ s+1-\rho }_p}. \end{aligned}$$

(40)

As a final step we show that, if $\varepsilon$ sufficiently small, then

$$\begin{aligned} \left\| B_{0,0}\left[ \pi ^\varepsilon _j (\kappa (\tau f)[h])'\right] -\frac{B_{0,0}\left[ \left( \pi ^\varepsilon _j h\right) '''\right] }{\left( 1+\tau ^2f'^2(x_j^\varepsilon )\right) ^{3/2}}\right\| _{W^{s-2}_p}\le \frac{\nu C_0}{2C_1} \left\| \pi _j^\varepsilon h\right\| _{W^{s+1}_p}+K\Vert h\Vert _{W^{ s+1-\rho }_p} \end{aligned}$$

(41)

for all $h\in W^{s+1}_p({\mathbb R})$, $\tau \in [0,1]$, and $|j|\le N-1.$ To start, we note that

$$\begin{aligned} B_{0,0}\left[ \pi ^\varepsilon _j \left( \kappa (\tau f)[h]\right) '\right] =B_{0,0}\left[ \left( \pi ^\varepsilon _j \kappa (\tau f)[h]\right) '\right] -B_{0,0}\left[ (\pi ^\varepsilon _j)' \kappa (\tau f)[h]\right] , \end{aligned}$$

and Lemma 2.5 yields for $-N+1\le j\le N$ that

$$\begin{aligned} \left\| B_{0,0}\left[ (\pi ^\varepsilon _j)' \kappa (\tau f)[h]\right] \right\| _{W^{s-2}_p}\le K\Vert h\Vert _{W^{ s+1-\rho }_p}. \end{aligned}$$

(42)

Moreover, letting $C_2:=\Vert B_{0,0}\Vert _{\mathcal {L}(W^{s-2}_p({\mathbb R}))}$, the algebra property of $W^{s-1}_p({\mathbb R})$ leads us to

$$\begin{aligned} \begin{aligned}&\left\| B_{0,0}[(\pi ^\varepsilon _j \kappa (\tau f)[h])']-\frac{B_{0,0}[(\pi _j^\varepsilon h)''']}{\left( 1+\tau ^2 f'^2(x_j^\varepsilon )\right) ^{3/2}}\right\| _{W^{s-2}_p}\\&\le C_2\left\| \frac{ \pi ^\varepsilon _j h''}{(1+\tau ^2f'^2)^{3/2}} -\frac{ (\pi _j^\varepsilon h) ''}{\left( 1+\tau ^2 f'^2(x_j^\varepsilon )\right) ^{3/2}}\right\| _{W^{s-1}_p}\\&\le C_2\left\| \left( \frac{1}{(1+\tau ^2f'^2)^{3/2}} -\frac{1}{\left( 1+\tau ^2 f'^2(x_j^\varepsilon )\right) ^{3/2}}\right) (\pi _j^\varepsilon h) ''\right\| _{W^{s-1}_p}+K \Vert h\Vert _{W^s_p}. \end{aligned} \end{aligned}$$

(43)

Using (5) together with the identity $\chi _j^\varepsilon \pi _j^\varepsilon =\pi _j^\varepsilon$, for $\varepsilon$ sufficiently small we get

$$\begin{aligned} \begin{aligned}&\left\| \left( \frac{1}{(1+\tau ^2f'^2)^{3/2}} -\frac{1}{\left( 1+\tau ^2 f'^2(x_j^\varepsilon )\right) ^{3/2}}\right) (\pi _j^\varepsilon h) ''\right\| _{W^{s-1}_p}\\&\le 2\left\| \chi _j^\varepsilon \left( \frac{1}{(1+\tau ^2f'^2)^{3/2}} -\frac{1}{(1+\tau ^2 f'^2(x_j^\varepsilon ))^{3/2}}\right) \right\| _\infty \Vert \pi _j^\varepsilon h\Vert _{W^{s+1}_p}+K\Vert h\Vert _{W^{s+1-\rho }_p}\\&\le \frac{\nu C_0}{2C_1C_2} \left\| \pi _j^\varepsilon h\right\| _{W^{s+1}_p}+K\Vert h\Vert _{W^{ s+1-\rho }_p}. \end{aligned} \end{aligned}$$

(44)

Gathering (42)–(44), we conclude that (41) holds true. The desired estimate (32) follows now, for $|j|\le N-1,$ by combining (40) and (41).

Step 2: The case $j=N$. Similarly to (35), we obtain from Lemma 4.5 and (34) that

$$\begin{aligned} \begin{aligned}&\left\| \pi _N^\varepsilon B_{0,1}^0(\tau f) \left[ (\kappa (\tau f)[h])'\right] -B_{0,0}\left[ \pi ^\varepsilon _N (\kappa (\tau f)[h])'\right] \right\| _{W^{s-2}_p}\\&\le \frac{\nu C_0}{4C_1} \left\| \pi _N^\varepsilon h\right\| _{W^{s+1}_p}+K\Vert h\Vert _{W^{ s+1-\rho }_p} \end{aligned} \end{aligned}$$

(45)

for all $h\in W^{s+1}_p({\mathbb R})$ and $\tau \in [0,1]$, provided that $\varepsilon$ is sufficiently small. Moreover,

$$\begin{aligned} \left\| \pi _N^\varepsilon \tau f'B_{1,1}^0(\tau f) \left[ (\kappa (\tau f)[h])'\right] \right\| _{W^{s-2}_p}\le T_a+T_b, \end{aligned}$$

where

$$\begin{aligned} \begin{aligned} T_a&:=\left\| \chi _N^\varepsilon f' B_{1,1}^0(\tau f)[\pi _N^\varepsilon (\kappa (\tau f)[h])']\right\| _{W^{s-2}_p},\\ T_b&:=\left\| \chi _N^\varepsilon f' \left( \pi _N^\varepsilon B_{1,1}^0(\tau f)[(\kappa (\tau f)[h])']-B_{1,1}^0(\tau f)[\pi _N^\varepsilon (\kappa (\tau f)[h])']\right) \right\| _{W^{s-2}_p}. \end{aligned} \end{aligned}$$

Since $\chi _N^\varepsilon f'\in W^{s-1}_p({\mathbb R})$ is a pointwise multiplier for $W^{s-2}_p({\mathbb R}),$ cf. (8), Lemma 4.1 yields

$$\begin{aligned} T_b\le K\left\| (\kappa (\tau f)[h])'\right\| _{W^{ s-2-\rho }_p}\le K\Vert h\Vert _{W^{ s+1-\rho }_p}. \end{aligned}$$

(46)

Because $f'\in W^{s-1}_p({\mathbb R})$ vanishes at infinity, for $\varepsilon$ sufficiently small to ensure that

$$\begin{aligned} \left\| \chi _N^\varepsilon f'\right\| _\infty <\frac{\nu }{20C_1}\left( \max _{\tau \in [0,1]}\Vert B_{1,1}^0(\tau f)\Vert _{\mathcal {L}(W^{s-2}_p({\mathbb R}))}\right) ^{-1}, \end{aligned}$$

it follows from (7) (with $r=s-1$), Lemma 2.5, and (34) (if $\chi _N^\varepsilon f'$ is not identically zero, otherwise the estimate is trivial) that

$$\begin{aligned} T_a\le \frac{\nu C_0}{4C_1} \left\| \pi _N^\varepsilon h\right\| _{W^{s+1}_p}+K\Vert h\Vert _{W^{ s+1-\rho }_p}. \end{aligned}$$

(47)

Gathering (33) and (45)–(47), we have shown that if $\varepsilon$ is sufficiently small, then

$$\begin{aligned} \left\| \pi _N^\varepsilon \Phi (\tau f) [h]-\frac{k\sigma }{2\pi \mu }B_{0,0}\left[ \pi ^\varepsilon _N (\kappa (\tau f)[h])'\right] \right\| _{W^{s-2}_p}\le \frac{\nu }{2} \left\| \pi _N^\varepsilon h\right\| _{W^{s+1}_p}+K\Vert h\Vert _{W^{ s+1-\rho }_p} \end{aligned}$$

(48)

for all $h\in W^{s+1}_p({\mathbb R})$ and $\tau \in [0,1]$. It remains to show that for $\varepsilon$ sufficiently small

$$\begin{aligned} \left\| B_{0,0}\left[ \pi ^\varepsilon _N (\kappa (\tau f)[h])'\right] -B_{0,0}\left[ (\pi ^\varepsilon _N h)'''\right] \right\| _{W^{s-2}_p}\le \frac{\nu C_0}{2C_1} \left\| \pi _N^\varepsilon h\right\| _{W^{s+1}_p}+K\Vert h\Vert _{W^{ s+1-\rho }_p} \end{aligned}$$

(49)

for all $h\in W^{s+1}_p({\mathbb R})$ and $\tau \in [0,1]$. Arguing as in the first step (see (43)), we find in view of (42) that

$$\begin{aligned}& \left\| B_{0,0}\left[ \pi ^\varepsilon _N( \kappa (\tau f)[h])'\right] - B_{0,0}\left[ (\pi _N^\varepsilon h)'''\right] \right\| _{W^{s-2}_p} \\&\le C_2\left\| \left( \frac{1}{((1+\tau ^2f'^2)^{3/2}} - 1\right) (\pi _N^\varepsilon h)''\right\| _{W^{s-1}_p}+K\Vert h\Vert _{W^{ s+1-\rho }_p}. \end{aligned}$$

Using (5) and the fact that $f'$ vanishes at infinity, for $\varepsilon$ sufficiently small, we obtain

$$\begin{aligned} \begin{aligned}&C_2\left\| \left( \frac{1}{((1+\tau ^2f'^2)^{3/2}} - 1\right) (\pi _N^\varepsilon h)''\right\| _{W^{s-1}_p}\\&\le 2C_2 \Vert \chi _N^\varepsilon \left( 1 -(1+\tau ^2f'^2)^{3/2} \right) \Vert _\infty \Vert \pi _N^\varepsilon h\Vert _{W^{s+1}_p}+K\Vert h\Vert _{W^{s+1-\rho }_p}\\&\le \frac{\nu C_0}{2C_1} \left\| \pi _N^\varepsilon h\Vert _{W^{s+1}_p}+K\Vert h\right\| _{W^{ s+1-\rho }_p}. \end{aligned} \end{aligned}$$

This proves (49). The claim (32) for $j= N$ follows now directly from (48) and (49).

We now consider a class of Fourier multipliers related to the multipliers from Theorem 3.3.

Lemma 3.4

Let $\eta \in (0,1)$. Given $\alpha \in [\eta ,1/\eta ]$, let

$$\begin{aligned} {\mathbb A}_{\alpha }:=- \alpha \left( \frac{d^4}{dx^4}\right) ^{3/4}. \end{aligned}$$

Then, there exits a constant $\kappa _0=\kappa _0(\eta )\ge 1$ such that

$$\begin{aligned} \bullet&\quad \lambda -{\mathbb A}_{\alpha }\in \mathrm{Isom}(W^{s+1}_p({\mathbb R}),W^{s-2}_p({\mathbb R}))\quad \forall \, \mathop \mathrm{Re}\nolimits \lambda \ge 1,\end{aligned}$$

(50)

$$\begin{aligned} \bullet&\quad \kappa _0\Vert (\lambda -{\mathbb A}_{\alpha })[h]\Vert _{W^{s-2}_p}\ge |\lambda |\cdot \Vert h\Vert _{W^{s-2}_p}+\Vert h\Vert _{W^{s+1}_p} \quad \forall \, h\in W^{s+1}_p({\mathbb R}),\, \mathop \mathrm{Re}\nolimits \lambda \ge 1. \end{aligned}$$

(51)

Proof

We first consider the realizations

$$\begin{aligned} {\mathbb A}_{\alpha }\in \mathcal {L}\left( W^2_p({\mathbb R}),W^{-1}_p({\mathbb R})\right) \quad \text {and}\quad {\mathbb A}_{\alpha }\in \mathcal {L}\left( W^3_p({\mathbb R}),L_p({\mathbb R})\right) . \end{aligned}$$

Since $W^k_p({\mathbb R})=H^k_p({\mathbb R})$, $k\in {\mathbb Z}$, Mikhlin’s multiplier theorem, cf. e.g. [1, Theorem 4.23], shows that the properties (50)–(51) (in the appropriate spaces) are valid for these realizations. Then, using the interpolation property (3), we obtain that (50)-(51) hold true.

The next result provides the generator property announced in (30).

Theorem 3.5

Given $p\in (1,2]$, $s\in (1+1/p,2)$, and $f\in W^{s}_p({\mathbb R})$ we have

$$\begin{aligned} -\partial \Phi (f)\in \mathcal {H}\left( W^{s+1}_p({\mathbb R}), W^{s-2}_p({\mathbb R})\right) . \end{aligned}$$

Proof

Fix $\rho \in (0,\min \{(s-1-1/p)/2),2-s\})$. Since $f'\in W^{s-1}_p({\mathbb R})$ is bounded, there exists a constant $\eta >0$ with the property that the function $\alpha _\tau$, $\tau \in [0,1]$, from Theorem 3.3 satisfies $\eta \le |\alpha _\tau |\le 1/\eta$ for all $\tau \in [0,1]$. Hence, regardless of $\varepsilon >0,$ the operators $A_{j,\tau },$ with $-N+1\le j\le N$ and $\tau \in [0,1],$ defined in Theorem 3.3 satisfy (50)-(51) with a constant $\kappa _0\ge 1.$

We set $\nu :=(2\kappa _0)^{-1}$. Theorem 3.3 provides an $\varepsilon \in (0,1)$, a $\varepsilon$-localization family, a constant $K=K(\varepsilon )>0$, and bounded operators ${\mathbb A}_{j,\tau }\in \mathcal {L}(W^{s+1}_p({\mathbb R}), W^{s-2}_p({\mathbb R}))$, ${-N+1\le j\le N}$ and $\tau \in [0,1],$ such that

$$\begin{aligned} 2\kappa _0\left\| \pi _j^\varepsilon \Phi (\tau f)[h]-{\mathbb A}_{j,\tau }\left[ \pi ^\varepsilon _j h\right] \right\| _{W^{s-2}_p}\le \left\| \pi _j^\varepsilon h\right\| _{W^{s+1}_p}+2\kappa _0 K\Vert h\Vert _{W^{s+1-\rho }_p} \end{aligned}$$

for all $-N+1\le j\le N$, $\tau \in [0,1],$ and $h\in W^{s+1}_p({\mathbb R})$. Besides, Lemma 3.4 yields

$$\begin{aligned} 2\kappa _0\left\| (\lambda -{\mathbb A}_{j,\tau })\left[ \pi ^\varepsilon _jh\right] \right\| _{W^{s-2}_p}\ge 2|\lambda |\cdot \left\| \pi ^\varepsilon _jh\right\| _{W^{s-2}_p}+ 2\left\| \pi ^\varepsilon _j h\right\| _{W^{s+1}_p} \end{aligned}$$

for all $-N+1\le j\le N$, $\tau \in [0,1],$ $\mathop \mathrm{Re}\nolimits \lambda \ge 1$, and $h\in W^{s+1}_p({\mathbb R})$. The latter inequalities imply

$$\begin{aligned} 2\kappa _0\left\| \pi _j^\varepsilon (\lambda -\Phi (\tau f))[h]\right\| _{W^{s-2}_p}\ge&2\kappa _0\left\| (\lambda -{\mathbb A}_{j,\tau })[\pi ^\varepsilon _j h]\right\| _{W^{s-2}_p}\\&-2\kappa _0\left\| \pi _j^\varepsilon \Phi (\tau f)[h]-{\mathbb A}_{j,\tau }[\pi ^\varepsilon _j h]\right\| _{W^{s-2}_p}\\ \ge&2|\lambda |\cdot \left\| \pi ^\varepsilon _j h\right\| _{W^{s-2}_p}+ \left\| \pi ^\varepsilon _j h\right\| _{W^{s+1}_p}-2\kappa _0K\Vert f\Vert _{W^{s-1-\rho }_p}. \end{aligned}$$

Summing up over j and using Lemma 3.2, the interpolation property (3), and Young’s inequality we find constants $\kappa =\kappa (f)\ge 1$ and $\omega =\omega ( f)>0$ such that

$$\begin{aligned} \kappa \left\| (\lambda -\Phi (\tau f ))[h]\right\| _{W^{s-2}_p}\ge |\lambda |\cdot \Vert h\Vert _{W^{s-2}_p}+ \Vert h\Vert _{W^{s+1}_p} \end{aligned}$$

(52)

for all $\tau \in [0,1],$ $\mathop \mathrm{Re}\nolimits \lambda \ge \omega$, and $h\in W^{s+1}_p({\mathbb R})$.

Recalling (31) and arguing as in Lemma 3.4 we may choose $\omega$ sufficiently large to guarantee that $\omega -\Phi (0) \in \mathrm{Isom}(W^{s+1}_p({\mathbb R}), W^{s-2}_p({\mathbb R}))$. The method of continuity together with (52) consequently yields

$$\begin{aligned} \omega -\Phi (f)\in \mathrm{Isom}\left( W^{s+1}_p({\mathbb R}), W^{s-2}_p({\mathbb R})\right) . \end{aligned}$$

(53)

The relations (52) (with $\tau =1$) and (53) imply the desired claim, cf. [7, Chapter I].

We next present the proof of the main result. The arguments rely to a large extent on the theory of quasilinear parabolic problems presented in [3, 4, 6] (see also [28]). Besides, in order to obtain the parabolic smoothing properties, we additionally employ a parameter trick which was successfully applied also to other problems, cf., e.g., [9, 11, 18, 32].

Proof of Theorem 1.1

Fix

$$\begin{aligned} 1+\frac{1}{p}<\overline{s}<s<2 \qquad \text {and}\qquad 0< \beta :=\frac{2}{3}<\alpha :=\frac{s-\overline{s}+2}{3}<1. \end{aligned}$$

We further set $E_1:=W^{\overline{s}+1}_p({\mathbb R})$, $E_0:=W^{\overline{s}-2}_p({\mathbb R})$, and $E_\theta :=(E_0,E_1)_{\theta ,p}$, with $\theta \in \{\alpha ,\, \beta \}$. Recalling the interpolation property (3), we have $E_\alpha =W^{s}_p({\mathbb R})$ and $E_\beta =W^{\overline{s}}_p({\mathbb R})$. In view of (29) and (30) (with ${s=\overline{s}}$), we then get

$$\begin{aligned} -\Phi \in \mathrm{C}^{\infty }\left( E_\beta ,\mathcal {H}(E_1, E_0)\right) , \end{aligned}$$

and the assumptions of [28, Theorem 1.1] are satisfied in the context of the Muskat problem (27). Applying [28, Theorem 1.1], we may conclude that (27) has for each $f_0\in W^s_p({\mathbb R})$ a unique maximal classical solution $f= f(\,\cdot \, ; f_0)$ such that

$$\begin{aligned} f\in \mathrm{C}\left( [0,T^+),W^s_p(\mathbb {R})\right) \cap \mathrm{C}\left( (0,T^+), W^{{\overline{s}}+1}_p(\mathbb {R})\right) \cap \mathrm{C}^1\left( (0,T^+), W^{{\overline{s}}-2}_p(\mathbb {R})\right) . \end{aligned}$$

(54)

Moreover, if the solution belongs to the set

$$\begin{aligned} \bigcup _{\eta \in (0,1)}\mathrm{C}^{\eta }\left( [0,T^+),W^{\overline{s}}_p(\mathbb {R})\right) , \end{aligned}$$

then it is also unique, cf. [28, Remark 1.2 (ii)]. We now prove that each solution to (27) that satisfies (54) belongs to $\mathrm{C}^{\eta }([0,T^+),W^{\overline{s}}_p(\mathbb {R}))$ with $\eta =(s-\overline{s})/(3+s-\overline{s})$. Indeed, since $f\in \mathrm{C}([0,T^+),W^s_p(\mathbb {R}))$, we infer from Lemma 3.6 that

$$\begin{aligned} \sup _{t\in (0,T]}\left\| \frac{df}{dt}(t)\right\| _{W^{\overline{s}-3}_p({\mathbb R})}<\infty \end{aligned}$$

for each $T\in (0,T^+)$. Since for $\theta :=3/(s-\overline{s}+3)$ we have $(W^{\overline{s}-3}_{p}({\mathbb R}), W^{s}_{p}({\mathbb R}))_{\theta ,p}=W^{\overline{s}}_{p}({\mathbb R}),$ cf. (3), (54) and the latter estimate now yield

$$\begin{aligned} \Vert f(t_1)-f(t_2)\Vert _{W^{\overline{s}}_p}\le C\Vert f(t_1)-f(t_2)\Vert _{W^{\overline{s}-3}_p}^{1-\theta } \le C |t_1-t_2|^{\theta(s-\overline{s})/3}, \quad t_1,\, t_2\in[0, T]. \end{aligned}$$

Therewith the existence and uniqueness claim is proven. Moreover, the assertion (i) follows from the abstract theory (see the proof of [28, Theorem 1.1]).

The parabolic smoothing property established at (ii) can be shown by arguing as in the more restrictive case considered in [25, Theorem 1.3].

Finally, in order to prove (iii), we assume that $f= f(\,\cdot \, ; f_0):[0,T^+)\rightarrow W^{s}_p({\mathbb R})$ is a maximal classical solution with $T^+<\infty$ and that

$$\begin{aligned} \sup _{t\in [0,T^+)}\Vert f(t)\Vert _{W^s_p({\mathbb R})}<\infty . \end{aligned}$$

Using again Lemma 3.6 and arguing as above, we conclude that $f:[0,T^+)\rightarrow W^{\overline{s}}_p(\mathbb {R})$ is uniformly continuous. Let now

$$\begin{aligned} 1+\frac{1}{p}<\widetilde{s}<\overline{s}<2 \qquad \text {and}\qquad 0< \beta :=\frac{2}{3}<\alpha ':=\frac{\overline{s}-\widetilde{s}+2}{3}<1. \end{aligned}$$

Choosing $F_1:=W^{\widetilde{s}+1}_p({\mathbb R})$, $F_0:=W^{\widetilde{s}-2}_p({\mathbb R})$, and setting $F_\theta :=(F_0,F_1)_{\theta ,p}$, $\theta \in \{\alpha ',\, \beta \}$, we have that $F_\alpha =W^{\overline{s}}_p({\mathbb R})$ and $F_\beta =W^{\widetilde{s}}_p({\mathbb R})$. Moreover (29) and (30) (with ${s=\widetilde{s}}$), yield

$$\begin{aligned} -\Phi \in \mathrm{C}^{\infty }\left( F_\beta ,\mathcal {H}(F_1, F_0)\right) . \end{aligned}$$

Thus, we may apply again [28, Theorem 1.1] (iv) ($\alpha$) to (27) and conclude that f can be extended to an interval $[0,{\widetilde{T}}^+)$ with ${\widetilde{T}}^+>T^+$ and such that

$$\begin{aligned} f\in \mathrm{C}\left( [0,\widetilde{T}^+),W^{\overline{s}}_p(\mathbb {R})\right) \cap \mathrm{C}\left( (0,\widetilde{T}^+), W^{{\widetilde{s}}+1}_p(\mathbb {R})\right) \cap \mathrm{C}^1\left( (0,\widetilde{T}^+), W^{{\widetilde{s}}-2}_p(\mathbb {R})\right) . \end{aligned}$$

Moreover, by (ii) (with $(s,\overline{s})=(\overline{s},\widetilde{s})$) we also have $f\in \mathrm{C}^1((0,{\widetilde{T}}^+), W^{3}_p(\mathbb {R})),$ and this contradicts the maximality of f. This proves the claim (iii) and the argument is complete.

We finish the section by presenting a result used in the proof of Theorem 1.1.

Lemma 3.6

Given $M>0$, there exists a constant $C=C(M)$ such that

$$\begin{aligned} \Vert \Phi (f)[f]\Vert _{W^{s-3}_p}\le C \end{aligned}$$

(55)

for all $f\in W^{s+1}_p({\mathbb R})$ with $\Vert f\Vert _{W^s_p}\le M$.

Proof

In this proof the constants denoted by C depend only on M. Given $f\in W^{s+1}_p({\mathbb R})$ with $\Vert f\Vert _{W^s_p}\le M$, we have

$$\begin{aligned} \left\| (\kappa (f)[f])'\right\| _{W^{s-3}_p({\mathbb R})}=\left\| \left( \frac{f'}{(1+f'^2)^{1/2}}\right) ''\right\| _{W^{s-3}_p}\le C \left\| \frac{f'}{(1+f'^2)^{1/2}} \right\| _{W^{s-1}_p}\le C. \end{aligned}$$

(56)

Moreover, we infer from Lemma 2.1 that

$$\begin{aligned} \Vert \mathbb {B}(f)[f']\Vert _{W^{s-3}_p}\le \Vert \mathbb {B}(f)[f']\Vert _{p}\le C \end{aligned}$$

(57)

and it remains to prove that

$$\begin{aligned} \Vert \mathbb {B}(f)[(\kappa (f)[f])']\Vert _{W^{s-3}_p}\le C. \end{aligned}$$

(58)

To this end we note that the $L_2$-adjoint $\mathbb {B}^*(f)$ of $\mathbb {B}(f)$ is identified by the relation

$$\begin{aligned} \pi \mathbb {B}^*(f):=-(B_{0,1}(f)+B_{1,1}(f)[f,f'\cdot ]). \end{aligned}$$

Therefore, given $\overline{\omega },$ $\psi \in \mathrm{C}^\infty _0({\mathbb R})$, we have

$$\begin{aligned} \begin{aligned} \left| \langle \mathbb {B}(f)[\overline{\omega }]|\psi \rangle _{W^{s-3}_p({\mathbb R})\times W^{3-s}_{p'}({\mathbb R})}\right|&= \left| \langle \overline{\omega }|\mathbb {B}^*(f)[\psi ]\rangle _{W^{s-3}_p({\mathbb R})\times W^{3-s}_{p'}({\mathbb R})}\right| \\&\le \Vert \overline{\omega }\Vert _{W^{s-3}_p}\Vert \mathbb {B}^*(f)[\psi ]\Vert _{W^{3-s}_{p'}}. \end{aligned} \end{aligned}$$

(59)

We next show that

$$\begin{aligned} \Vert \mathbb {B}^*(f)[\psi ]\Vert _{W^{3-s}_{p'}}\le C\Vert \psi \Vert _{W^{3-s}_{p'}} \end{aligned}$$

(60)

for all $\psi \in W^{3-s}_{p'}({\mathbb R})$ and $f\in W^{s+1}_p({\mathbb R})$ with $\Vert f\Vert _{W^s_p}\le M$. Indeed, Lemma 2.1 (with $p=p'$), yields

$$\begin{aligned} \Vert \mathbb {B}^*(f)[\psi ]\Vert _{ p' }\le C \Vert \psi \Vert _{p'}. \end{aligned}$$

(61)

Additionally, we may argue as in the proof of [25, Lemma 3.5] to infer in view of Lemma 2.1 that ${\mathbb {B}^*(f)[\psi ]\in W^1_{p'}({\mathbb R})}$ with

$$\begin{aligned} \pi ( \mathbb {B}^*(f)[\psi ])'(x)&=\pi \mathbb {B}^*(f)[\psi '](x)-\mathop \mathrm{PV}\nolimits \int _{\mathbb R}\partial _x\left( \frac{y+f'(x-y)\delta _{[x,y]}f}{y^2+(\delta _{[x,y]} f)^2}\right) \psi (x-y) \, dy\\&=\pi \mathbb {B}^*(f)[\psi '](x)-\mathop \mathrm{PV}\nolimits \int _{\mathbb R}\partial _y\left( \frac{ \delta _{[x,y]}f\delta _{[x,y]}f'}{y^2+(\delta _{[x,y]} f)^2 }\right) \psi (x-y) \, dy\\&=-\pi \mathbb {B}(f)[\psi '](x). \end{aligned}$$

Invoking Lemma 2.4 (with $r=2-s\in (0,1-1/p)$), we get that

$$\begin{aligned} \Vert B_{0,1}^0(f)[\psi ']\Vert _{W^{2-s}_{p'}}+\Vert B_{1,1}^0(f)[\psi ']\Vert _{W^{2-s}_{p'}}\le C\Vert \psi \Vert _{W^{3-s}_{p'}}, \end{aligned}$$

(62)

and since $f'\in W^{s-1}_p({\mathbb R})$ is a pointwise multiplier for $W^{2-s}_{p'}({\mathbb R})$, cf. (8), we may conclude from (61) and (62) that

$$\begin{aligned} \Vert \mathbb {B}^*(f)[\psi ]\Vert _{W^{3-s}_{p'}}\le \Vert \mathbb {B}^*(f)[\psi ]\Vert _{p'}+\Vert (\mathbb {B}^*(f)[\psi ])'\Vert _{W^{2-s}_{p'}}\le C\Vert \psi \Vert _{W^{3-s}_{p'}}. \end{aligned}$$

This proves (60). Combining (59) and (60), a standard density argument leads us to

$$\begin{aligned} \Vert \mathbb {B}(f)[\overline{\omega }]\Vert _{W^{s-3}_p}\le C\Vert \overline{\omega }\Vert _{W^{s-3}_p}, \qquad \overline{\omega }\in W^{s-3}_p({\mathbb R}), \end{aligned}$$

and (58) follows via (56). Recalling the definition (28) of $\Phi$, the bound (55) is a straightforward consequence of (57) and (58).

Appendices

Freezing the kernels of singular operators in Sobolev spaces with negative exponent

In this section we establish some technical results which enable us to locally approximate the Fréchet derivative $\partial \Phi (\tau f)$, $\tau \in [0,1]$ and $f\in W^s_p({\mathbb R})$, by certain explicit Fourier multipliers, cf. Theorem 3.3. These results can be viewed as a generalization of the method of freezing the coefficients of elliptic differential operators. We extend this method to a particular class of singular integral operators, one of the main difficulties arising from the fact that the Sobolev space where the error is measured has negative exponent. As a first result we establish a commutator type estimate.

Lemma 4.1

Let $p\in (1,2]$, $s\in (1+1/p,2),$ $\rho \in (0,(s-1-1/p)/2)$, $n,\, m\in {\mathbb N},$ ${f\in W^s_p({\mathbb R})}$, and $\varphi \in \mathrm{C}^1({\mathbb R})$ with uniformly continuous derivative $\varphi '$ be given. Then, there exists a constant $K=K(n, m, s,p,\rho , \Vert \varphi \Vert _{\mathrm{C}^1}, \Vert f\Vert _{W^s_p})$ such that

$$\begin{aligned} \left\| \varphi B_{n,m}^0(f)[\overline{\omega }]-B_{n,m}^0(f)[\varphi \overline{\omega }]\right\| _{W^{s-2}_{p}}\le K \Vert \overline{\omega }\Vert _{W^{ s-2-\rho }_{p}}\quad \text {for all}\quad \overline{\omega }\in W^{s-2}_{p}({\mathbb R}). \end{aligned}$$

(63)

Proof

Let $\overline{\rho }:=(3-s)\rho /(1-\rho )$. We point out that $\overline{\rho }\in (\rho ,s-1-1/p)$ and that $\varphi \in \mathrm{C}^1({\mathbb R})$ is a pointwise multiplier for $W^{ s-2-\overline{\rho }}_p({\mathbb R})$. Lemma 2.5 (with ${s=s-\overline{\rho }}$) then yields

$$\begin{aligned} \left\| \varphi B_{n,m}^0(f)[\overline{\omega }]-B_{n,m}^0(f)[\varphi \overline{\omega }]\right\| _{W^{ s-2-\overline{\rho }}_{p}}\le K \Vert \overline{\omega }\Vert _{W^{ s-2-\overline{\rho }}_{p}},\qquad \overline{\omega }\in W^{ s-2-\overline{\rho }}_{p}({\mathbb R}). \end{aligned}$$

(64)

Besides, in view of [2, Lemma 12], we also have

$$\begin{aligned} \left\| \varphi B_{n,m}^0(f)[\overline{\omega }]-B_{n,m}^0(f)[\varphi \overline{\omega }]\right\| _{W^1_p}\le K \Vert \overline{\omega }\Vert _{p}, \qquad \overline{\omega }\in L_p({\mathbb R}). \end{aligned}$$

(65)

Noticing that $W^{(1-\rho )( s-2-\overline{\rho })}_p({\mathbb R})=W^{ s-2-\overline{\rho }}_p({\mathbb R})$ and $W^{(1-\rho )( s-2-\overline{\rho })+\rho }_p({\mathbb R})= W^{s-2}_p({\mathbb R})$, the interpolation property (3) together with (64) and (65) leads us to the desired claim.

The first important result of this section is the following lemma.

Lemma 4.2

Let $n,\, m \in {\mathbb N}$, $p\in (1,2],$ $s\in (1+1/p,2)$, $\rho \in (0,(s-1-1/p)/2)$, ${\nu \in (0,\infty )}$, and $f\in W^s_p({\mathbb R})$ be given. For sufficiently small $\varepsilon \in (0,1)$ there exists a constant K that depends on $\varepsilon ,$ n,  m,  s,  p,  $\rho ,$ and $\Vert f\Vert _{W^s_p}$ such that

$$\begin{aligned} \left\| \pi _j^\varepsilon B_{n,m}^0(f)[ \overline{\omega }]-\frac{\left( f'(x_j^\varepsilon )\right) ^n}{\left[ 1+\left( f'(x_j^\varepsilon )\right) ^2\right] ^m}B_{0,0}[\pi _j^\varepsilon \overline{\omega }]\right\| _{W^{ s-2}_p}\le \nu \Vert \pi _j^\varepsilon \overline{\omega }\Vert _{W^{ s-2}_p}+K\Vert \overline{\omega }\Vert _{W^{s-2-\rho }_p} \end{aligned}$$

(66)

for all $|j|\le N-1$ and $\overline{\omega }\in W^{s-2}_p({\mathbb R})$.

Proof

Taking advantage of the relation $\chi _j^\varepsilon \pi _j^\varepsilon =\pi _j^\varepsilon$, we have

$$\begin{aligned} \pi _j^\varepsilon B_{n,m}^0(f)[\overline{\omega }]-\frac{\left( f'(x_j^\varepsilon )\right) ^n}{\left[ 1+\left( f'(x_j^\varepsilon )\right) ^2\right] ^m}B_{0,0}[\pi _j^\varepsilon \overline{\omega }]=T_{a}+ T_{b}, \end{aligned}$$

where

$$\begin{aligned} T_{a}&:=\frac{\left( f'(x_j^\varepsilon )\right) ^n}{\left[ 1+\left( f'(x_j^\varepsilon )\right) ^2\right] ^m}\left( \chi _j^\varepsilon B_{0,0}\left[ \pi _j^\varepsilon \overline{\omega }\right] - B_{0,0}\left[ \chi _j^\varepsilon (\pi _j^\varepsilon \overline{\omega })\right] \right)\\ & - \chi _j^\varepsilon \left( B_{n,m}^0 (f)\left[ \pi _j^\varepsilon \overline{\omega }\right] - \pi _j^\varepsilon B_{n,m}^0 (f)[ \overline{\omega }]\right) ,\\ T_{b}&:= \chi _j^\varepsilon B_{n,m}^0 (f)[\pi _j^\varepsilon \overline{\omega }]-\frac{(f'(x_j^\varepsilon ))^n}{\left[ 1+\left( f'(x_j^\varepsilon )\right) ^2\right] ^m}\chi _j^\varepsilon B_{0,0}[\pi _j^\varepsilon \overline{\omega }]. \end{aligned}$$

Since $\pi _j^\varepsilon$ is a pointwise multiplier for $W^{s-2-\rho }_p({\mathbb R})$ and $\chi _j^\varepsilon$ is a pointwise multiplier for $W^{s-2}_p({\mathbb R})$, Lemma 4.1 leads to

$$\begin{aligned} \Vert T_{a}\Vert _{W^{ s-2}_p}\le K\Vert \overline{\omega }\Vert _{W^{ s-2-\rho }_p}. \end{aligned}$$

(67)

It remains to estimate $T_{b}.$ Using (14) and the relation ${f'(x_j^\varepsilon )=\delta _{[x,y]} (f'(x_j^\varepsilon )\mathop \mathrm{id}\nolimits _{{\mathbb R}})/y},$ we get

$$\begin{aligned} T_{b}&=\sum _{k=0}^{n-1}\left( f'(x_j^\varepsilon )\right) ^{n-k-1}\chi _j^\varepsilon B_{k+1,m}(f,\ldots ,f)\left[ f,\ldots ,f, f-f'(x_j^\varepsilon )\mathop \mathrm{id}\nolimits _{{\mathbb R}},\pi _j^\varepsilon \overline{\omega }\right] \\&- \sum _{k=0}^{m-1}\frac{\left( f'(x_j^\varepsilon )\right) ^{n}}{\left[ 1+ ( f'(x_j^\varepsilon ))^2\right] ^{m-k}}\chi _j^\varepsilon B_{2,k+1}(f,\ldots ,f)\left[ f-f'\left( x_j^\varepsilon \right) \mathop \mathrm{id}\nolimits _{{\mathbb R}}, f+f'\left( x_j^\varepsilon \right) \mathop \mathrm{id}\nolimits _{{\mathbb R}},\pi _j^\varepsilon \overline{\omega }\right] . \end{aligned}$$

If $\varepsilon$ is sufficiently small, Lemma 4.4 (with $\widetilde{s}=s-\rho$) implies that

$$\begin{aligned} \Vert T_b\Vert _{W^{s-2}_p}\le \nu \Vert \pi _j^\varepsilon \overline{\omega }\Vert _{W^{ s-2}_p}+K\Vert \overline{\omega }\Vert _{W^{ s-2-\rho }_p}. \end{aligned}$$

(68)

The estimate (66) follows from (67) and (68).

In Lemma 4.3 we gather some classical properties of mollifiers.

Lemma 4.3

Let $\eta _\delta (x):=\delta ^{-1}\eta (x/\delta )$, $x\in {\mathbb R}$ and $\delta >0$, where $\eta \in \mathrm{C}^\infty _0({\mathbb R})$ is a nonnegative function with

$$\begin{aligned} \int _{\mathbb R}\eta (x)\, dx=1. \end{aligned}$$

Given $\delta >0$ and $f\in L_p({\mathbb R})$, with $p\in [1,\infty )$, let $f_\delta :=f*\eta _\delta$. The following properties hold true.

• (i) Given $r\ge 0,$ it holds $\sup _{\delta >0} \big \Vert [f\mapsto f_\delta ]\Vert _{\mathcal {L}(W^r_p({\mathbb R}))}= 1;$

• (ii) Given $r\ge 0$, there exists a constant $C>0$ such that

  $$\begin{aligned}&\Vert f_\delta \Vert _{W^{r}_p}\le C\delta ^{r'-r}\Vert f\Vert _{W^{r'}_p}\qquad \text {for all}\quad {f\in W^{r'}_p({\mathbb R})}, r'\in [0,r], \delta \in (0,1]. \end{aligned}$$

  (69)

• (iii) There exists a constant $C>0$ such that

  $$\begin{aligned}&\Vert f_\delta -f\Vert _{W^{r'}_p}\le C\delta ^{r-r'}\Vert f\Vert _{W^r_p}\qquad \text {for all} {f\in W^r_p({\mathbb R})}, 0\le r'\le r\le 1, \delta >0. \end{aligned}$$

  (70)

Lemma 4.4 below provides the key estimate in the proof of Lemma 4.2.

Lemma 4.4

Let $n,\, m \in {\mathbb N}$, $n\ge 1$, $p\in (1,2],$ $1+1/p<\widetilde{s}<s<2$, $\nu \in (0,\infty )$, and ${f\in W^s_p({\mathbb R})}$ be given. For sufficiently small $\varepsilon \in (0,1)$ there exists a positive constant $K=K(\varepsilon ,\, n,\, m,\, s,\, \widetilde{s}, \Vert f\Vert _{W^s_p})$ such that

$$\begin{aligned} \left\| \chi _j^\varepsilon B_{n,m}(f,\ldots ,f)\left[ f,\ldots ,f, f-f'(x_j^\varepsilon )\mathop \mathrm{id}\nolimits _{{\mathbb R}}, \pi _j^\varepsilon \overline{\omega }\right] \right\| _{W^{ s-2}_p}\le \nu \left\| \pi _j^\varepsilon \overline{\omega }\right\| _{W^{ s-2}_p}+K\Vert \overline{\omega }\Vert _{W^{ \widetilde{s}-2}_p} \end{aligned}$$

(71)

for all $|j|\le N-1$ and $\overline{\omega }\in W^{s-2}_p({\mathbb R})$.

Proof

Given $\varphi ,\, \overline{\omega }\in \mathrm{C}^\infty _0({\mathbb R})$, it holds in view of $\chi _j^\varepsilon \pi _j^\varepsilon =\pi _j^\varepsilon$ and of the fact that the $L_2$-adjoint of $B_{n,m}^0(f)$ is the operator $-B_{n,m}^0(f)$, that

$$\begin{aligned}& \left\langle \chi _j^\varepsilon B_{n,m}(f,\ldots ,f)\left[ f,\ldots ,f, f-f'\left( x_j^\varepsilon \right) \mathop \mathrm{id}\nolimits _{{\mathbb R}}, \pi _j^\varepsilon \overline{\omega }\right] \Big |\varphi \right\rangle _{W^{s-2}_p({\mathbb R})\times W^{2-s}_{p'}({\mathbb R})}\\&=-\left\langle \pi _j^\varepsilon \overline{\omega }|\chi _j^\varepsilon B_{n,m}(f,\ldots ,f)\left[ f,\ldots , f ,f-f'(x_j^\varepsilon )\mathop \mathrm{id}\nolimits _{{\mathbb R}},\chi _j^\varepsilon \varphi \right] \right\rangle _{W^{s-2}_p({\mathbb R})\times W^{2-s}_{p'}({\mathbb R})}. \end{aligned}$$

Let $T_{j,\varepsilon }[\varphi ]:=\chi _j^\varepsilon B_{n,m}(f,\ldots ,f)[f,\ldots , f ,f-f'(x_j^\varepsilon )\mathop \mathrm{id}\nolimits _{{\mathbb R}},\chi _j^\varepsilon \varphi ]$. We decompose below

$$\begin{aligned} T_{j,\varepsilon }[\varphi ] =T_{j,\varepsilon }^1[\varphi ] +T^2_{j,\varepsilon }[\varphi ], \end{aligned}$$

(72)

and we prove subsequently that, if $\varepsilon$ sufficiently small, then

$$\begin{aligned} \left\| T_{j,\varepsilon }^1[\varphi ]\right\| _{W^{2-s}_{p'}}\le \nu \Vert \varphi \Vert _{W^{2-s}_{p'}}\qquad \text {and}\qquad \left\| T_{j,\varepsilon }^2[\varphi ]\right\| _{W^{2-\widetilde{s}}_{p'}}\le K \Vert \varphi \Vert _{W^{2-s}_{p'}} \end{aligned}$$

(73)

for all $\varphi \in W^{2-s}_{p'}({\mathbb R}).$ Having established (73), we get

$$\begin{aligned} \left| \left\langle \pi _j^\varepsilon \overline{\omega }|T_{j,\varepsilon }[\varphi ]\right\rangle _{W^{s-2}_p({\mathbb R})\times W^{2-s}_{p'}({\mathbb R})}\right|&\le \Vert \pi _j^\varepsilon \overline{\omega }\Vert _{W^{s-2}_p}\Vert T_{j,\varepsilon }^1[\varphi ]\Vert _{W^{2-s}_{p'}} +\Vert \pi _j^\varepsilon \overline{\omega }\Vert _{W^{\widetilde{s}-2}_p}\Vert T_{j,\varepsilon }^2[\varphi ]\Vert _{W^{2-\widetilde{s}}_{p'}}\\&\le \left( \nu \Vert \pi _j^\varepsilon \overline{\omega }\Vert _{W^{s-2}_p}+K\Vert \overline{\omega }\Vert _{W^{\widetilde{s}-2}_{p}}\right) \Vert \varphi \Vert _{W^{2-s}_{p'}}, \end{aligned}$$

and the claim (71) follows via a standard density argument.

In order to define the terms in (72), let $\{\eta _\delta \}_{\delta >0}$ be a mollifier as in Lemma 4.3. We set

$$\begin{aligned} T_{j,\varepsilon }^1[\varphi ]&=\chi _j^\varepsilon B_{n,m}(f,\ldots ,f)\left[ f,\ldots , f, f-f'(x_j^\varepsilon )\mathop \mathrm{id}\nolimits _{{\mathbb R}},\chi _j^\varepsilon (\varphi -\varphi _\delta )\right] ,\\ T_{j,\varepsilon }^2[\varphi ]&= \chi _j^\varepsilon B_{n,m}^0(f)\left[ \chi _j^\varepsilon \varphi _\delta \right] -f'(x_j^\varepsilon )\chi _j^\varepsilon B_{n-1,m}^0(f)\left[ \chi _j^\varepsilon \varphi _\delta \right] , \end{aligned}$$

where $\delta =\delta (\varepsilon )\in (0,1]$ will be fixed later on and $\varphi _\delta :=\varphi *\eta _\delta$.

Since $2-\widetilde{s}\in [0,1-1/p)$, Lemma 2.4 (with $r=2-\widetilde{s}$) and the fact that ${\chi _j^\varepsilon \varphi _\delta \in W^{2-\widetilde{s}}_{p'}({\mathbb R})}$ yield $T_{j,\varepsilon }^2[\varphi ]\in W^{2-\widetilde{s}}_{p'}({\mathbb R}),$ and together with (69) we obtain the estimate

$$\begin{aligned} \Vert T_{j,\varepsilon }^2[\varphi ]\Vert _{W^{2-\widetilde{s}}_{p'}}\le K\Vert \chi _j^\varepsilon \varphi _\delta \Vert _{W^{2-\widetilde{s}}_{p'}}\le K\delta ^{\widetilde{s} -s}\Vert \varphi \Vert _{W^{2-s}_{p'}}. \end{aligned}$$

(74)

It remains to estimate $T_{j,\varepsilon }^1[\varphi ].$ Therefore, we first infer from Lemma 2.1 and (70) that

$$\begin{aligned} \left\| T_{j,\varepsilon }^1[\varphi ]\right\| _{p'}\le C \left\| \chi _j^\varepsilon (\varphi _\delta -\varphi )\right\| _{ p' }\le C\left\| \varphi _\delta -\varphi \right\| _{p'}\le C\delta ^{2-s}\left\| \varphi \right\| _{W^{2-s}_{p'}}. \end{aligned}$$

(75)

We now consider the seminorm $[T_{j,\varepsilon }^1[\varphi ]]_{W^{2-s}_{p'}}$. In view of (14) we write

$$\begin{aligned} T_{j,\varepsilon }^1[\varphi ]-\tau _\xi T_{j,\varepsilon }^1[\varphi ]=T_{1}(\xi )+T_{2}(\xi )+ T_{3}(\xi ),\qquad \xi \in {\mathbb R}, \end{aligned}$$

where

$$\begin{aligned} T_{1}(\xi )&:=\left( \chi _j^\varepsilon -\tau _\xi \chi _j^\varepsilon \right) \tau _\xi B_{n,m}(f,\ldots ,f)\left[ f,\ldots ,f, f-f'\left( x_j^\varepsilon \right) \mathop \mathrm{id}\nolimits _{{\mathbb R}},\chi _j^\varepsilon (\varphi -\varphi _\delta )\right] ,\\ T_{2}(\xi )&:= \chi _j^\varepsilon B_{n,m}(f,\ldots ,f)\left[ f,\ldots ,f, f-f'\left( x_j^\varepsilon \right) \mathop \mathrm{id}\nolimits _{{\mathbb R}},\chi _j^\varepsilon (\varphi -\varphi _\delta ) -\tau _\xi \left( \chi _j^\varepsilon (\varphi -\varphi _\delta )\right) \right] ,\\ T_{3}(\xi )&:= \chi _j^\varepsilon \sum _{j=1}^{n-1}B_{n,m}(f,\ldots ,f)\left[ \underset{(j-1)-\mathrm{times}}{\underbrace{\tau _\xi f,\ldots ,\tau _\xi f}},f-\tau _\xi f,f,\ldots , f, f-f'\left( x_j^\varepsilon \right) \mathop \mathrm{id}\nolimits _{{\mathbb R}},\tau _\xi \left( \chi _j^\varepsilon (\varphi -\varphi _\delta )\right) \right] \\&+\chi _j^\varepsilon B_{n,m}(f,\ldots ,f)\left[ \tau _\xi f,\ldots ,\tau _\xi f, f-\tau _\xi f,\tau _\xi \left( \chi _j^\varepsilon (\varphi -\varphi _\delta )\right) \right] \\&+\chi _j^\varepsilon \sum _{j=1}^m B_{n+2,m+1}^j \left[ \tau _\xi f,\ldots ,\tau _\xi f, \tau _\xi f-f'\left( x_j^\varepsilon \right) \mathop \mathrm{id}\nolimits _{{\mathbb R}},\tau _\xi f+f,\tau _\xi f-f,\tau _\xi \left( \chi _j^\varepsilon (\varphi -\varphi _\delta )\right) \right] , \end{aligned}$$

with

$$\begin{aligned} B_{n+2,m+1}^j:=B_{n+2,m+1}\left( \underset{j-\mathrm{times}}{\underbrace{f,\ldots ,f}},\tau _\xi f,\ldots ,\tau _\xi f \right) . \end{aligned}$$

Lemma 2.1 (with $p=p'$) together wit (70) yields

$$\begin{aligned} \Vert T_{1}(\xi )\Vert _{p'}\le C\left\| \chi _j^\varepsilon -\tau _\xi \chi _j^\varepsilon \right\| _{\infty }\left\| \chi _j^\varepsilon (\varphi -\varphi _\delta )\right\| _{p'}\le C\delta ^{2-s} \left\| \chi _j^\varepsilon -\tau _\xi \chi _j^\varepsilon \right\| _{W^1_{p'}}\left\| \varphi \right\| _{W^{2-s}_{p'}}. \end{aligned}$$

(76)

With regard to $T_3(\xi )$ we fix $\rho \in (s-\widetilde{s},\min \{2-\widetilde{s},2(s-\widetilde{s})\})$. Combining Lemma 2.2 (with $s=\widetilde{s}$ and $r=2-\widetilde{s}-\rho \in (0,1-1/p)$) and (70) we get

$$\begin{aligned} \begin{aligned} \Vert T_{3}(\xi )\Vert _{p'}&\le C\Vert f-\tau _\xi f\Vert _{W_{p'}^{2\widetilde{s}-1 -\frac{2}{p}+\rho }}\left\| \chi _j^\varepsilon (\varphi -\varphi _\delta )\right\| _{W^{2-\widetilde{s}-\rho }_{p'}}\\&\le K\delta ^{\rho +\widetilde{s}-s}\left\| f-\tau _\xi f\Vert _{W_{p'}^{2s-1-2/p}}\Vert \varphi \right\| _{W^{2-s}_{p'}}. \end{aligned} \end{aligned}$$

(77)

In remains to estimate the function $T_{2}(\xi ),$ $\xi \in {\mathbb R}.$ To this end we choose $a_{j,\varepsilon }\in {\mathbb R}$ such that ${\mathop \mathrm{supp}\nolimits \chi _j^\varepsilon \subset [a_{j,\varepsilon }-3\varepsilon /2,a_{j,\varepsilon }+3\varepsilon /2]}$ and denote by $F_\varepsilon$ the Lipschitz continuous function defined by $F_\varepsilon =f$ on ${[a_{j,\varepsilon }-2\varepsilon ,a_{j,\varepsilon }+2\varepsilon ]}$ and $F_\varepsilon '=f'(x_j^\varepsilon )$ on ${\mathbb R}\setminus [a_{j,\varepsilon }-2\varepsilon ,a_{j,\varepsilon }+2\varepsilon ]$. Given $\xi \in {\mathbb R}$ with $|\xi |\ge \varepsilon /2,$ it follows from Lemma 2.1 and (70) that

$$\begin{aligned} \Vert T_{2}(\xi )\Vert _{p'}\le C\Vert \varphi -\varphi _\delta \Vert _{p'}\le C\delta ^{2-s}\Vert \varphi \Vert _{W^{2-s}_{p'}} . \end{aligned}$$

(78)

If $|\xi |<\varepsilon /2$, then $\xi + \mathop \mathrm{supp}\nolimits \chi _j^\varepsilon \subset [a_{j,\varepsilon }-2\varepsilon ,a_{j,\varepsilon }+2\varepsilon ]$, and, since $f'\in \mathrm{C}^{s-1-1/p}({\mathbb R})$, Lemma 2.1 and the properties defining $F_\varepsilon$ lead to

$$\begin{aligned} \begin{aligned} \Vert T_{2}(\xi )\Vert _{p'}&= \left\| \chi _j^\varepsilon B_{n,m}(f,\ldots ,f)\left[ f,\ldots ,f, F_\varepsilon -f'\left( x_j^\varepsilon \right) \mathop \mathrm{id}\nolimits _{{\mathbb R}},\chi _j^\varepsilon (\varphi -\varphi _\delta ) -\tau _\xi \left( \chi _j^\varepsilon (\varphi -\varphi _\delta )\right) \right] \right\| _{p'}\\&\le C\left\| f'-f'\left( x_j^\varepsilon \right) \right\| _{L_\infty \left( (a_{j,\varepsilon }-2\varepsilon ,a_{j,\varepsilon }+2\varepsilon )\right) }\left\| \chi _j^\varepsilon (\varphi -\varphi _\delta ) -\tau _\xi \left( \chi _j^\varepsilon (\varphi -\varphi _\delta )\right) \right\| _{p'}\\&\le \frac{\nu }{24 }\left\| \chi _j^\varepsilon (\varphi -\varphi _\delta ) -\tau _\xi \left( \chi _j^\varepsilon (\varphi -\varphi _\delta )\right) \right\| _{p'}, \end{aligned} \end{aligned}$$

(79)

provided that $\varepsilon$ is sufficiently small.

Combining (76)–(79), we conclude that if $\varepsilon$ is sufficiently small, then

$$\begin{aligned} \begin{aligned} {\left[ T_{j,\varepsilon }^1[\varphi ]\right] _{W^{2-s}_{p'}}}&\le 3\left( \sum _{j=1}^3\int _{{\mathbb R}}\frac{\Vert T_i(\xi )\Vert _{p'}^{p'}}{|\xi |^{1+(2-s)p'}}\, d\xi \right) ^{1/p'}\\&\le K \delta ^{\rho +\widetilde{s}-s} \Vert \varphi \Vert _{W^{2-s}_{p'}}\left( 1+\Vert \chi _j^\varepsilon \Vert _{W^{3-s}_{p'}}+\Vert f\Vert _{W^{s}_p}\right) +\frac{\nu }{8}\left\| \chi _j^\varepsilon (\varphi -\varphi _\delta )\right\| _{W^{2-s}_{p'}}. \end{aligned} \end{aligned}$$

(80)

Invoking (4) and Lemma 4.3 (i), the estimates (75) and (80) lead us to

$$\begin{aligned} \Vert T_{j,\varepsilon }^1[\varphi ]\Vert _{W^{2-s}_{p'}}&\le \frac{\nu }{2}\Vert \varphi \Vert _{W^{2-s}_{p'}}+K\delta ^{\rho +\widetilde{s}-s}\Vert \varphi \Vert _{W^{2-s}_{p'}}. \end{aligned}$$

We now chose $\delta =\delta (\varepsilon )\in (0,1]$ sufficiently small to ensure that $K\delta ^{\rho +\widetilde{s}-s}\le \nu /2.$ This choice together with (74) shows that the estimates in (73) hold true and the proof is complete.

The next lemma is the second main result of this section and describes how to freeze the kernels at infinity.

Lemma 4.5

Let $n,\, m \in {\mathbb N}$, $p\in (1,2],$ $s\in (1+1/p,2)$, $\rho \in (0,(s-1-1/p)/2)$, ${\nu \in (0,\infty )}$, and $f\in W^s_p({\mathbb R})$ be given. For sufficiently small $\varepsilon \in (0,1)$ there exists a constant K that depends on $\varepsilon ,$ n,  m,  s,  p,  $\rho ,$ and $\Vert f\Vert _{W^s_p}$ such that

$$\begin{aligned} \left\| \pi _N^\varepsilon B_{0,m}^0(f)[ \overline{\omega }]-B_{0,0}\left[ \pi _N^\varepsilon \overline{\omega }\right] \right\| _{W^{ s-2}_p}\le \nu \left\| \pi _N^\varepsilon \overline{\omega }\right\| _{W^{ s-2}_p}+K\Vert \overline{\omega }\Vert _{W^{ s-2-\rho }_p} \end{aligned}$$

(81)

and

$$\begin{aligned} \left\| \pi _N^\varepsilon B_{n,m}^0(f)[ \overline{\omega }]\right\| _{W^{ s-2}_p}\le \nu \left\| \pi _N^\varepsilon \overline{\omega }\right\| _{W^{ s-2}_p}+K\Vert \overline{\omega }\Vert _{W^{s-2-\rho }_p},\quad n\ge 1, \end{aligned}$$

(82)

for all $\overline{\omega }\in W^{s-2}_p({\mathbb R})$.

Proof

Similarly as in the proof of Lemma 4.2 we write

$$\begin{aligned} \pi _N^\varepsilon B_{0,m}^0(f)[\overline{\omega }]-B_{0,0}\left[ \pi _N^\varepsilon \overline{\omega }\right] =T_{a}+ T_{b}, \end{aligned}$$

where

$$\begin{aligned} T_{a}&:=\left( \chi _N^\varepsilon B_{0,0}\left[ \pi _N^\varepsilon \overline{\omega }\right] - B_{0,0}\left[ \chi _N^\varepsilon (\pi _N^\varepsilon \overline{\omega })\right] \right) -\chi _N^\varepsilon ( B_{0,m}^0 (f)[\pi _N^\varepsilon \overline{\omega }]- \pi _N^\varepsilon B_{0,m}^0 (f)[ \overline{\omega }]),\\ T_{b}&:= \chi _N^\varepsilon ( B_{0,m}^0 (f)[\pi _N^\varepsilon \overline{\omega }]- B_{0,0}[\pi _N^\varepsilon \overline{\omega }]). \end{aligned}$$

Since $\pi _j^\varepsilon$ is a pointwise multiplier for $W^{s-2-\rho }_p({\mathbb R})$ and $\chi _j^\varepsilon$ is a pointwise multiplier for $W^{s-2}_p({\mathbb R})$, we get in view of Lemma 4.1 that

$$\begin{aligned} \Vert T_{a}\Vert _{W^{ s-2}_p}\le K\Vert \overline{\omega }\Vert _{W^{ s-2-\rho }_p}. \end{aligned}$$

(83)

With respect to $T_b$ we infer from (14) that

$$\begin{aligned} T_{b}&=- \chi _N^\varepsilon \sum _{i=1}^{m} B_{2,i}^0(f)\left[ \pi _N^\varepsilon \overline{\omega }\right] . \end{aligned}$$

If $\varepsilon$ sufficiently small, it follows from Lemma 4.6 that

$$\begin{aligned} \Vert T_b\Vert _{W^{s-2}_p}\le \nu \Vert \pi _N^\varepsilon \overline{\omega }\Vert _{W^{ s-2}_p}+K\Vert \overline{\omega }\Vert _{W^{ s-2-\rho }_p}. \end{aligned}$$

(84)

The claim (81) follows now from (83) and (84). Finally, the assertion (82) is obtained by arguing as in the proof of (81).

Lemma 4.6 below provides the crucial estimate in the proof of Lemma 4.5.

Lemma 4.6

Let $n,\, m \in {\mathbb N}$, $n\ge 1$, $p\in (1,2],$ $1+1/p<\widetilde{s}<s<2$, $\nu \in (0,\infty )$, and ${f\in W^s_p({\mathbb R})}$ be given. For sufficiently small $\varepsilon \in (0,1)$, there exists a positive constant $K=K(\varepsilon ,\, n,\, m,\, s,\,\widetilde{s},\,\Vert f\Vert _{W^s_p})$ such that

$$\begin{aligned} \left\| \chi _N^\varepsilon B_{n,m}^0(f)\left[ \pi _N^\varepsilon \overline{\omega }\right] \right\| _{W^{ s-2}_p}\le \nu \left\| \pi _N^\varepsilon \overline{\omega }\right\| _{W^{ s-2}_p}+K\Vert \overline{\omega }\Vert _{W^{ \widetilde{s}-2}_p} \end{aligned}$$

(85)

for all $\overline{\omega }\in W^{s-2}_p({\mathbb R})$.

Proof

Given $\varphi ,\, \overline{\omega }\in \mathrm{C}^\infty _0({\mathbb R})$, it follows by arguing as in the proof of Lemma 4.4 that

$$\begin{aligned} \left\langle \chi _N^\varepsilon B_{n,m}^0(f)\left[ \pi _N^\varepsilon \overline{\omega }\right] \Big |\varphi \right\rangle _{W^{s-2}_p({\mathbb R})\times W^{2-s}_{p'}({\mathbb R})} =- \left\langle \pi _N^\varepsilon \overline{\omega }|\chi _N^\varepsilon B_{n,m}^0(f)\left[ \chi _N^\varepsilon \varphi \right] \right\rangle _{W^{s-2}_p({\mathbb R})\times W^{2-s}_{p'}({\mathbb R})}. \end{aligned}$$

We write $\chi _N^\varepsilon B_{n,m}^0(f)[\chi _N^\varepsilon \varphi ]=T_{\varepsilon }^1[\varphi ]+T^2_{\varepsilon }[\varphi ]$ where, for sufficiently small $\varepsilon$, we have

$$\begin{aligned} \left\| T_{\varepsilon }^1[\varphi ]\right\| _{W^{2-s}_{p'}}\le \nu \Vert \varphi \Vert _{W^{2-s}_{p'}}\qquad \text {and}\qquad \left\| T_{\varepsilon }^2[\varphi ]\right\| _{W^{2-\widetilde{s}}_{p'}}\le K \Vert \varphi \Vert _{W^{2-s}_{p'}} \end{aligned}$$

(86)

for all $\varphi \in W^{2-s}_{p'}({\mathbb R}).$ The estimates in (86) together with the previous identity imply

$$\begin{aligned}& \left| \left\langle \chi _N^\varepsilon B_{n,m}^0(f)\left[ \pi _N^\varepsilon \overline{\omega }\right] \Big |\varphi \right\rangle _{W^{s-2}_p({\mathbb R})\times W^{2-s}_{p'}({\mathbb R})}\right| \\&\le \left\| \pi _N^\varepsilon \overline{\omega }\right\| _{W^{s-2}_p}\left\| T_{\varepsilon }^1[\varphi ]\right\| _{W^{2-s}_{p'}} +\left\| \pi _N^\varepsilon \overline{\omega }\right\| _{W^{\widetilde{s}-2}_p}\left\| T_{\varepsilon }^2[\varphi ]\right\| _{W^{2-\widetilde{s}}_{p'}}\\&\le \left( \nu \Vert \pi _N^\varepsilon \overline{\omega }\Vert _{W^{s-2}_p}+K\Vert \overline{\omega }\Vert _{W^{\widetilde{s}-2}_{p}}\right) \Vert \varphi \Vert _{W^{2-s}_{p'}}, \end{aligned}$$

and (85) follows.

Let again $\{\eta _\delta \}_{\delta >0}$ be a mollifier and set

$$\begin{aligned} T_{\varepsilon }^1[\varphi ]=\chi _N^\varepsilon B_{n,m}^0(f)\left[ \chi _N^\varepsilon (\varphi -\varphi _\delta )\right] \qquad \text {and}\qquad T_{\varepsilon }^2[\varphi ]= \chi _N^\varepsilon B_{n,m}^0(f)\left[ \chi _N^\varepsilon \varphi _\delta \right] , \end{aligned}$$

where $\varphi _\delta :=\varphi *\eta _\delta$ and with $\delta =\delta (\varepsilon )\in (0,1]$ which we fix later on. Taking advantage of Lemma 2.4 (with ${r=2-\widetilde{s}\in (0,1-1/p)}$) and of the fact that $\chi _N^\varepsilon \varphi _\delta \in W^{2-\widetilde{s}}_{p'}({\mathbb R})$, we conclude that $T_{\varepsilon }^2[\varphi ]\in W^{2-\widetilde{s}}_{p'}({\mathbb R})$ and together with (69) we obtain the estimate

$$\begin{aligned} \left\| T_{\varepsilon }^2[\varphi ]\right\| _{W^{2-\widetilde{s}}_{p'}}\le K \left\| \chi _N^\varepsilon \varphi _\delta \right\| _{W^{2-\widetilde{s}}_{p'}}\le K\delta ^{\widetilde{s} -s}\Vert \varphi \Vert _{W^{2-s}_{p'}}. \end{aligned}$$

(87)

We now estimate $T_{\varepsilon }^1[\varphi ].$ Combining Lemma 2.1 and (70), we have

$$\begin{aligned} \left\| T_{\varepsilon }^1[\varphi ]\right\| _{p'}\le C \left\| \chi _N^\varepsilon (\varphi _\delta -\varphi )\right\| _{ p' }\le C\Vert \varphi -\varphi _\delta \Vert _{p'}\le C\delta ^{2-s}\Vert \varphi \Vert _{W^{2-s}_{p'}}, \end{aligned}$$

(88)

and it remains to consider the seminorm $[T_{\varepsilon }^1[\varphi ]]_{W^{2-s}_{p'}}$. Using (14), we get

$$\begin{aligned} T_{\varepsilon }^1[\varphi ]-\tau _\xi T_{\varepsilon }^1[\varphi ]=T_{1}(\xi )+T_{2}(\xi )+ T_{3}(\xi ),\qquad \xi \in {\mathbb R}, \end{aligned}$$

where

$$\begin{aligned} T_{1}(\xi )&:=(\chi _N^\varepsilon -\tau _\xi \chi _N^\varepsilon )\tau _\xi \left( B_{n,m}^0(f)[\chi _N^\varepsilon (\varphi -\varphi _\delta )]\right) ,\\ T_{2}(\xi )&:= \chi _N^\varepsilon B_{n,m}^0(f)\left[ \chi _N^\varepsilon (\varphi -\varphi _\delta ) -\tau _\xi (\chi _N^\varepsilon (\varphi -\varphi _\delta ))\right] ,\\ T_{3}(\xi )&:= \chi _N^\varepsilon \sum _{j=1}^{n}B_{n,m}(f,\ldots ,f)\left[ \underset{(j-1)-\mathrm{times}}{\underbrace{\tau _\xi f,\ldots ,\tau _\xi f}},f-\tau _\xi f,f,\ldots , f,\tau _\xi (\chi _N^\varepsilon (\varphi -\varphi _\delta ))\right] \\&+\chi _N^\varepsilon \sum _{j=1}^m B_{n+2,m+1}^j \left[ \tau _\xi f,\ldots ,\tau _\xi f,\tau _\xi f+f,\tau _\xi f-f,\tau _\xi (\chi _N^\varepsilon (\varphi -\varphi _\delta ))\right] , \end{aligned}$$

with $B_{n+2,m+1}^j$ as defined in the proof of Lemma 4.4. Arguing as in Lemma 4.4, we obtain

$$\begin{aligned} \Vert T_{1}(\xi )\Vert _{p'}\le C\delta ^{2-s} \left\| \chi _N^\varepsilon -\tau _\xi \chi _N^\varepsilon \right\| _{W^1_{p'}}\Vert \varphi \Vert _{W^{2-s}_{p'}} \end{aligned}$$

(89)

and, for some fixed $\rho \in (s-\widetilde{s},\min \{2-\widetilde{s},2(s-\widetilde{s})\})$,

$$\begin{aligned} \Vert T_{3}(\xi )\Vert _{p'}\le K\delta ^{\rho +\widetilde{s}-s}\Vert f-\tau _\xi f\Vert _{W_{p'}^{2s-1-2/p}}\Vert \varphi \Vert _{W^{2-s}_{p'}}. \end{aligned}$$

(90)

In order to estimate $T_{2}(\xi ),$ let $F_\varepsilon$ denote the Lipschitz continuous function equal to f on the set $\{|x|\ge 1/\varepsilon -2\varepsilon \}$ and which is linear in the interval $\{|x|\le 1/\varepsilon -2\varepsilon \}$. If $|\xi |\ge \varepsilon ,$ Lemma 2.1 and (70) yield

$$\begin{aligned} \Vert T_{2}(\xi )\Vert _{p'}\le C\Vert \varphi -\varphi _\delta \Vert _{p'}\le C\delta ^{2-s}\Vert \varphi \Vert _{W^{2-s}_{p'}} . \end{aligned}$$

(91)

If $|\xi |<\varepsilon$, we note that $\xi + \mathop \mathrm{supp}\nolimits \chi _N^\varepsilon \subset \{|x|\ge 1/\varepsilon -2\varepsilon \}$. Lemma 2.1, the definition of $F_\varepsilon$, and the observation that $\Vert F_\varepsilon '\Vert _{\infty }\le 2\varepsilon \Vert f\Vert _\infty +\Vert f'\Vert _{L_\infty (\{|x|\ge 1/\varepsilon -2\varepsilon \})}$ (which holds if $\varepsilon$ is sufficiently small), then lead to

$$\begin{aligned} \begin{aligned} \Vert T_{2}(\xi )\Vert _{p'}&= \left\| \chi _N^\varepsilon B_{n,m}(f,\ldots ,f)\left[ f,\ldots ,f, F_\varepsilon ,\chi _N^\varepsilon (\varphi -\varphi _\delta ) -\tau _\xi (\chi _N^\varepsilon (\varphi -\varphi _\delta ))\right] \right\| _{p'}\\&\le C\Vert F_\varepsilon ' \Vert _{\infty }\left\| \chi _N^\varepsilon (\varphi -\varphi _\delta ) -\tau _\xi (\chi _N^\varepsilon (\varphi -\varphi _\delta ))\right\| _{p'}\\&\le \frac{\nu }{24 }\left\| \chi _N^\varepsilon (\varphi -\varphi _\delta ) -\tau _\xi (\chi _N^\varepsilon (\varphi -\varphi _\delta ))\right\| _{p'}, \end{aligned} \end{aligned}$$

(92)

provided that $\varepsilon$ is sufficiently small. To obtain the last inequality we have taken advantage of the fact that $f'\in W^{s-1}_p({\mathbb R})$ vanishes at infinity.

From (89) to (92), we conclude, for $\varepsilon$ sufficiently small, that

$$\begin{aligned} {\left[ T_{\varepsilon }^1[\varphi ]\right] _{W^{2-s}_{p'}}}\le K \delta ^{\rho +\widetilde{s}-s} \Vert \varphi \Vert _{W^{2-s}_{p'}}\left( 1+\Vert 1-\chi _N^\varepsilon \Vert _{W^{3-s}_{p'}}+\Vert f\Vert _{W^{s}_p}\right) +\frac{\nu }{8}\Vert \chi _N^\varepsilon (\varphi -\varphi _\delta )\Vert _{W^{2-s}_{p'}}. \end{aligned}$$

(93)

Combining (4), Lemma 4.3 (i), (88), and (93) we get

$$\begin{aligned} \Vert T_{\varepsilon }^1[\varphi ]\Vert _{W^{2-s}_{p'}}&\le \frac{\nu }{2}\Vert \varphi \Vert _{W^{2-s}_{p'}}+K\delta ^{\rho +\widetilde{s}-s}\Vert \varphi \Vert _{W^{2-s}_{p'}}. \end{aligned}$$

Choosing now $\delta =\delta (\varepsilon )$ sufficiently small to ensure that $K\delta ^{\rho +\widetilde{s}-s}\le \nu /2,$ we obtain, together with (87), the desired estimates (86) and the proof is complete.

Estimates for some pointwise multipliers

In this appendix we present the proofs of the estimates (6)–(7). The estimate (7) is used in the proof of Theorem  3.3 and (6) is an important argument when establishing (7).

Proof of (6)

Since $W^{r}_p({\mathbb R})\hookrightarrow \mathrm{C}({\mathbb R})$, we have

$$\begin{aligned} \Vert gh\Vert _{p'}\le \Vert g\Vert _\infty \Vert h\Vert _{p'}, \end{aligned}$$

(94)

and it remains to estimate the term

$$\begin{aligned}{}[gh]_{W^{1-r}_{p'}}^{p'}=\int _{\mathbb R}\frac{\Vert gh-\tau _\xi (gh)\Vert _{p'}^{p'}}{|\xi |^{1+(1-r)p'}}\, d\xi \le 2^{p'}\left( \Vert g\Vert _\infty ^{p'}\int _{\mathbb R}\frac{\Vert h-\tau _\xi h\Vert _{p'}^{p'}}{|\xi |^{1+(1-r)p'}}\, d\xi + \int _{\mathbb R}\frac{\Vert (g-\tau _\xi g)\tau _\xi h\Vert _{p'}^{p'}}{|\xi |^{1+(1-r)p'}}\, d\xi \right) . \end{aligned}$$

According to [5, Theorem 4.1], the multiplication

$$\begin{aligned}{}[(g,h)\mapsto gh]: W^{2(r-1/p)}_{p'}({\mathbb R})\times W^{1-r-\rho }_{p'}({\mathbb R})\rightarrow L_{p'}({\mathbb R}) \end{aligned}$$

is continuous, hence

$$\begin{aligned}{}[gh]_{W^{1-r}_{p'}}^{p'}\le 2^{p'}\Vert g\Vert _\infty ^{p'}\Vert h\Vert _{W^{1-r}_{p'}}^{p'}+C\Vert h\Vert _{W^{1-r-\rho }_{p'}}^{p'}\int _{\mathbb R}\frac{\Vert g-\tau _\xi g\Vert _{W^{2(r-1/p)}_{p'}}^{p'}}{|\xi |^{1+(1-r)p'}}\, d\xi . \end{aligned}$$

Lemma 2.3 (with $t'=2(r-1/p)$ and $t=r+1-2/p$) yields

$$\begin{aligned} \int _{\mathbb R}\frac{\Vert g-\tau _\xi g\Vert _{W^{2(r-1/p)}_{p'}}^{p'}}{|\xi |^{1+(1-r)p'}}\, d\xi \le \Vert g\Vert _{W^{r+1-2/p}_{p'}}^{p'}\le C \Vert g\Vert _{W^{r}_p}^{p'}. \end{aligned}$$

We thus conclude that

$$\begin{aligned}{}[gh]_{W^{1-r}_{p'}}^{p'}\le 2^{p'}\Vert g\Vert _\infty ^{p'}\Vert h\Vert _{W^{1-r}_{p'}}^{p'}+C \Vert g\Vert _{W^{r}_p}^{p'}\Vert h\Vert _{W^{1-r-\rho }_{p'}}^{p'}, \end{aligned}$$

(95)

and (94) together with (95) lead to the desired claim.

We are now in a position to prove (7).

Proof of (7)

Given $h,\, \psi \in \mathrm{C}^\infty _0({\mathbb R})$, we have

$$\begin{aligned} \langle \varphi h|\psi \rangle _{W^{r-1}_p({\mathbb R})\times W^{1-r}_{p'}({\mathbb R})}=\langle h|\varphi \psi \rangle _{W^{r-1}_p({\mathbb R})\times W^{1-r}_{p'}({\mathbb R})} =\langle h|T_1[\psi ]+T_2[\psi ]\rangle _{W^{r-1}_p({\mathbb R})\times W^{1-r}_{p'}({\mathbb R})}, \end{aligned}$$

where $T_1[\psi ]\in W^{1-r}_{p'}({\mathbb R})$ and $T_2[\psi ]\in W^{1- r+\rho }_{p'}({\mathbb R})$ are defined in (97) and satisfy

$$\begin{aligned} \Vert T_1[\psi ]\Vert _{W^{1-r}_{p'}}\le 5\Vert \varphi \Vert _\infty \Vert \psi \Vert _{W^{1-r}_{p'}}\quad \text {and}\quad \Vert T_2[\psi ]\Vert _{W^{1-r+\rho }_{p'}}\le C\frac{ \Vert \varphi \Vert _{W^{r}_{p}}^{1+2r/\rho }}{\Vert \varphi \Vert _{\infty }^{2r/\rho }}\Vert \psi \Vert _{W^{1-r}_{p'}}. \end{aligned}$$

(96)

Having established (96), we get

$$\begin{aligned} |\langle \varphi h|\psi \rangle |_{W^{r-1}_p({\mathbb R})\times W^{1-r}_{p'}({\mathbb R})}&\le \Vert h\Vert _{W^{r-1}_p}\Vert T_1[\psi ]\Vert _{W^{1-r}_{p'}}+\Vert h\Vert _{W^{ r-1-\rho }_p}\Vert T_2[\psi ]\Vert _{W^{1- r+\rho }_{p'}}\\&\le \left( 5\Vert \varphi \Vert _\infty \Vert h\Vert _{W^{r-1}_p}+C\frac{ \Vert \varphi \Vert _{W^{r}_{p}}^{1+2r/\rho }}{\Vert \varphi \Vert _{\infty }^{2r/\rho }}\Vert h\Vert _{W^{r-1-\rho }_p}\right) \Vert \psi \Vert _{W^{1-r}_{p'}}, \end{aligned}$$

and the estimate (7) follows. The functions in (96) are defined by

$$\begin{aligned} T_1[\psi ]:=\varphi (\psi -\psi _\delta )\quad \text {and}\quad T_2[\psi ]:=\varphi \psi _\delta , \end{aligned}$$

(97)

where $\{\eta _\delta \}_{\delta >0}$ is a mollifier, $\psi _\delta =\psi *\eta _\delta$, and $\delta \in (0,1]$ is chosen below. Combining (6), Lemma 4.3 (i), and (70), we get

$$\begin{aligned} \Vert T_1[\psi ]\Vert _{W^{1-r}_{p'}}&\le 2\Vert \varphi \Vert _\infty \Vert \psi -\psi _\delta \Vert _{W^{1-r}_{p'}}+C\Vert \varphi \Vert _{W^{r}_p}\Vert \psi -\psi _\delta \Vert _{W^{1-r-\rho }_{p'}} \\&\le \left( 4\Vert \varphi \Vert _\infty +C\delta ^{\rho }\Vert \varphi \Vert _{W^{r}_p}\right) \Vert \psi \Vert _{W^{1-r}_{p'}}. \end{aligned}$$

After eventually choosing C to be larger than the norm of the embedding $W^r_p({\mathbb R})\hookrightarrow L_\infty ({\mathbb R}),$ we set $\delta = (\Vert \varphi \Vert _\infty /(C\Vert \varphi \Vert _{W^r_p}))^{1/\rho }\in (0,1]$, and obtain that

$$\begin{aligned} \Vert T_1[\psi ]\Vert _{W^{1-r}_{p'}}&\le 5\Vert \varphi \Vert _\infty \Vert \psi \Vert _{W^{1-r}_{p'}}. \end{aligned}$$

With respect to $T_2[\psi ]$, we note that since $\varphi \in W^{r+1-2/p}_{p'}({\mathbb R})$ and $\psi _\delta \in W^{r+1}_{p'}({\mathbb R})$, it holds that $T_2[\psi ]\in W^{r+1-2/p}_{p'}({\mathbb R})\hookrightarrow W^{1-r+\rho }_{p'}({\mathbb R})$, cf. (2), and together with (2) and (69) we get

$$\begin{aligned} \Vert T_2[\psi ]\Vert _{W^{1-r+\rho }_{p'}}\le C\Vert \psi _\delta \Vert _{W^{r+1}_{p'}}\Vert \varphi \Vert _{W^{r+1-2/p}_{p'}}\le C \delta ^{-2r}\Vert \varphi \Vert _{W^{r}_{p}}\Vert \psi \Vert _{W^{1-r}_{p'}} \le C\frac{ \Vert \varphi \Vert _{W^{r}_{p}}^{1+2r/\rho }}{\Vert \varphi \Vert _{\infty }^{2r/\rho }} \Vert \psi \Vert _{W^{1-r}_{p'}}. \end{aligned}$$

Hence, both estimates in (96) hold true and the proof is complete.