The Schauder estimate for kinetic integral equations

TL;DR

This paper proves interior Schauder estimates for kinetic equations involving integro-differential diffusion operators, providing a priori regularity bounds under ellipticity and continuity conditions.

Contribution

It establishes the first Schauder estimates for kinetic equations with integro-differential diffusion, extending regularity theory to this class of equations.

Findings

Derived a priori Schauder estimates for kinetic integro-differential equations.

Established regularity results under ellipticity and H"older continuity assumptions.

Extended classical Schauder theory to kinetic equations with nonlocal diffusion.

Abstract

We establish interior Schauder estimates for kinetic equations with integro-differential diffusion. We study equations of the form $f_t + v \cdot \nabla_x f = \mathcal L_v f + c$, where $\mathcal L_v$ is an integro-differential diffusion operator of order $2s$ acting in the $v$-variable. Under suitable ellipticity and H\"older continuity conditions on the kernel of $\mathcal L_v$, we obtain an a priori estimate for $f$ in a properly scaled H\"older space.

Equations

$$f_t+v\cdot {\nabla }_{x}f={\int }_{{\mathbb{R}}^d}(f^{\prime }-f)K(t,x,v,v^{\prime })\mathrm{d}{v}^{\prime }+c_0(t,x,v).$$

$$K_{(t,x,v)}(w):=K(t,x,v,v+w)$$

$${\int }_{B_r}|w{|}^{2}K(w)\mathrm{d}w\le \Lambda r^{2-2s}.$$

$${\iint }_{B_R\times B_R}|\phi (v)-\phi (v^{\prime }){|}^{2}K(v^{\prime }-v)\mathrm{d}{v}^{\prime }\mathrm{d}v\ge \lambda {\iint }_{B_{R/2}\times B_{R/2}}|\phi (v)-\phi (v^{\prime }){|}^2|v^{\prime }-v{|}^{-d-2s}\mathrm{d}{v}^{\prime }\mathrm{d}v.$$

$$\underset{|e|=1}{\inf }{\int }_{B_r}(w\cdot e{)}_{+}^{2}K(w)\mathrm{d}w\ge \lambda r^{2-2s}.$$

$${\int }_{B_r}|K_{z_1}(w)-K_{z_2}(w)||w{|}^2\mathrm{d}w\le A_0{r}^{2-2s}{d}_{\ell }(z_1,z_2{)}^{\alpha },$$

$$\Vert f{\Vert }_{C_{\ell }^{2s+\alpha }(Q_{1/2})}\le C\left(\Vert f{\Vert }_{C_{\ell }^{\gamma }([-1,0]\times B_1\times {\mathbb{R}}^d)}+\Vert c{\Vert }_{C_{\ell }^{\alpha }(Q_1)}\right).$$

$$f_t+(-\Delta {)}_v^{s}f=0\text{in }{Q}_1,$$

$$\Vert f{\Vert }_{C_{\ell }^{2s+\alpha }(Q_{1/2})}\le C\Vert f{\Vert }_{L^{\infty }([-1,0]\times B_1\times {\mathbb{R}}^d)}.$$

$$f_t+v\cdot {\nabla }_{x}f=a_{ij}(t,x,v){\partial }_{v_i{v}_j}f+c(t,x,v).$$

$$[f{]}_{C_{\ell }^{2+\alpha }(Q_{1/2})}\lesssim \Vert f{\Vert }_{L^{\infty }(Q_1)}+\Vert f_t+v\cdot {\nabla }_{x}f-{\Delta }_{v}f{\Vert }_{C^{\alpha }(Q_1)}\le \Vert f{\Vert }_{H^{\alpha }(Q_1)}.$$

$$(t_1,x_1,v_1)\circ (t_2,x_2,v_2)=(t_1+t_2,x_1+x_2+t_2{v}_1,v_1+v_2).$$

$$d_{\ell }(z_1,z_2):=\underset{w\in {\mathbb{R}}^d}{\min }\left\{\max \left(|t_1-t_2{|}^{\frac{1}{2s}},|x_1-x_2-(t_1-t_2)w{|}^{\frac{1}{1+2s}},|v_1-w|,|v_2-w|\right)\right\}.$$

$$\Vert (t,x,v)\Vert =\max \left\{|t{|}^{1/(2s)},|x{|}^{1/(1+2s)},|v|\right\}.$$

$$d_{\ell }(z_1,z_2)$$

$$\approx \Vert z_1^{-1}\circ z_2\Vert ,$$

$$\approx \underset{w\in {\mathbb{R}}^d}{\inf }|t_2-t_1{|}^{1/(2s)}+|x_1-x_2+(t_1-t_2)w{|}^{1/(1+2s)}+|v_1-w|+|v_2-w|.$$

$$Q_r=\{(t,x,v):t\le 0\text{ and }{d}_{\ell }(0,(t,x,v))<r\}.$$

$$Q_r(z_0)=\{z=(t,x,v):t\le t_0\text{ and }{d}_{\ell }(z_0,z)<r\}.$$

$$d_{\ell }(z_1,0)=\max \left(|t_1{|}^{\frac{1}{2s}},|x_1+t_1{w}_1{|}^{\frac{1}{1+2s}},|v_1-w_1|,|w_1|\right),$$

$$d_{\ell }(z_2,0)=\max \left(|t_2{|}^{\frac{1}{2s}},|x_2+t_2{w}_2{|}^{\frac{1}{1+2s}},|v_2-w_2|,|w_2|\right).$$

$$d_{\ell }(z_1,z_2)$$

$$|t_1-t_2{|}^{\frac{1}{2s}}\le |t_1{|}^{\frac{1}{2s}}+|t_2{|}^{\frac{1}{2s}}\le d_{\ell }(z_1,0)+d_{\ell }(z_2,0).$$

$$|v_1-w_1-w_2|\le |v_1-w_1|+|w_2|\le d_{\ell }(z_1,0)+d_{\ell }(z_2,0).$$

$$|v_2-w_1-w_2|\le |v_2-w_2|+|w_1|\le d_{\ell }(z_2,0)+d_{\ell }(z_1,0).$$

$$|x_1-x_2-(t_1-t_2)(w_1+w_2){|}^{\frac{1}{1+2s}}$$

$$\le |d_{\ell }(z_1,0{)}^{1+2s}+d_{\ell }(z_2,0{)}^{1+2s}$$

$$+d_{\ell }(z_1,0{)}^{2s}{d}_{\ell }(z_2,0)+d_{\ell }(z_2,0{)}^{2s}{d}_{\ell }(z_1,0){|}^{\frac{1}{1+2s}},$$

$$\le d_{\ell }(z_1,0)+d_{\ell }(z_2,0).$$

$$a^{1+p}+b^{1+p}+a^{p}b+a{b}^p\le (a+b{)}^{1+p}.$$

$$|f(z)-p(z)|\le C{d}_{\ell }(z,z_0{)}^{\alpha }.$$

$$|f(z_0\circ z)-p_0(z)|\le C\Vert z{\Vert }^{\alpha }.$$

$$p_0(t,x,v)=a_0+a_{1}t+a_2\cdot x+a_3\cdot v+\dots ,$$

Full text

The authors would like to thank C. Mouhot for fruitful discussions. LS is supported in part by NSF grants DMS-1254332 and DMS-1764285.

1. Introduction

We study kinetic equations with integral diffusion of the form

[TABLE]

Here, we used the notation from kinetic equations $f=f(t,x,v)$ and $f^{\prime}=f(t,x,v^{\prime})$. The main result in this article is a Schauder estimate for equations of the form (1.1) whose kernels $K_{(t,x,v)}$ are elliptic and Hölder continuous.

Note that the integral diffusion term on the right hand side of (1.1) acts on the velocity variable only. The regularization effect on the $x$ variable is a consequence of the interaction between the integral diffusion and the transport term. The equation (1.1) should be understood as a Kolmogorov-type hypoelliptic integro-differential equation. The diffusion is of fractional order. We work with Hölder-like spaces (given in Definition 2.3) that are adapted to the particular scaling of this equation in each direction.

Our methods allow us to consider a very general class of kernels $K$. This is essential for the eventual applications of our result to the Boltzmann equation. We start by specifying our notion of ellipticity and Hölder continuity for the kernel $K$. In (1.1), $K(t,x,v,v^{\prime})$ denotes a function that maps the variables $(t,x,v)$ into a nonnegative Radon measure $K_{(t,x,v)}$ in $\mathbb{R}^{d}\setminus\{0\}$:

[TABLE]

such that for all $(t,x,v)$, $K_{(t,x,v)}$ belongs to the following ellipticity class of kernels.

Definition 1.1** (The ellipticity class).**

Given the order $2s\in(0,2)$ and ellipticity constants $0<\lambda<\Lambda$, we say that a nonnegative Radon measure $K$ in $\mathbb{R}^{d}\setminus\{0\}$ belongs to the ellipticity class $\mathcal{K}$ when the following conditions are met.

• •

(Symmetry)* $K(w)=K(-w)$.*

• •

(Upper bound)* for all $r>0$*

[TABLE]

• •

(Coercivity estimate)* for any $R>0$ and $\varphi\in C^{2}(B_{R})$,*

[TABLE]

In case $s<(0,1/2)$, we add the following non-degeneracy assumption to the kernel.

[TABLE]

Remark 1.2*.*

Strictly speaking, (1.2), (1.3) and (1.4) should be written with integrals on balls minus the origin. It is customary to extend the measure $K\;\mathrm{d}w$ to have zero point mass at the origin. Other choices do not make any difference since all our integrands equal zero at $w=0$.

Remark 1.3*.*

We write $K$ to denote a nonnegative measure on $\mathbb{R}^{d}$. Even though we use the notation $K(v^{\prime})\;\mathrm{d}v^{\prime}$ as if this measure was absolutely continuous, it does not need to be. We abuse notation in this way because it makes some formulas look simpler. For example, we write $K(v^{\prime}-v)\;\mathrm{d}v^{\prime}$ to denote the measure in terms of the variables $v^{\prime}$ and translated by $v$. Otherwise, for a measure $\mu=K(w)\;\mathrm{d}w$, it would typically be written $\;\mathrm{d}(\tau_{v}\mu(v^{\prime}))$ which is arguably more confusing.

Remark 1.4*.*

When $s<1/2$ we complement the coercivity estimate (1.3) with the non-degeneracy assumption (1.4). These two assumptions may be redundant. Indeed, for stable-like kernels of the form $K(w)=|w|^{-d-2s}a(w/|w|)$, they are equivalent. This follows easily by computing the Fourier symbol of the operator associated with $K$ (see for example [17], and also [16]). For non-stable-like processes the situation is less clear. We do not know any example of a kernel $K$ satisfying the upper bound (1.2) that satisfies one of the assumptions in the coercivity estimate but not the other. The non-degeneracy assumption (1.4) is typically much easier to check than the first coercivity estimate (1.3).

For local equations, a Schauder estimate refers to an estimate in a Hölder space when coefficients of the equations are Hölder continuous. For non-local equations, the regularity of the coefficients is replaced with the Hölder continuous dependence of the kernel with respect to the variable $z=(t,x,v)$.

Assumption 1.5** (Hölder continuity of coefficients).**

Given $z_{1}=(t_{1},x_{1},v_{1})$ and $z_{2}=(t_{2},x_{2},v_{2})$, for any $r>0$ we have

[TABLE]

where $d_{\ell}(z_{0},z_{1})$ stands for the kinetic distance, see Definition 2.1 below.

We can now state the Schauder estimate for the equations we consider.

Theorem 1.6** (The Schauder estimate).**

Let $0<\gamma<\min(1,2s)$ and $\alpha=\frac{2s}{1+2s}\gamma$. Let $K(t,x,v,v^{\prime})$ be a kernel such that the two following conditions hold true.

• •

(Ellipticity)* For each $z=(t,x,v)\in Q_{1}$, the kernel $K_{z}(w)=K(t,x,v,v+w)$ belongs to the class $\mathcal{K}$ described in Definition 1.1.*

• •

(Hölder continuity)* Assumption 1.5 holds.*

If $f\in C_{\ell}^{\gamma}([-1,0]\times B_{1}\times\mathbb{R}^{d})$ solves (1.1) in $Q_{1}$, then the following estimate holds

[TABLE]

The constant $C$ depends on $d$, $s$, $\lambda$, $\Lambda$, and $A_{0}$.

We use the same notation as in [10]: $Q_{r}$ denotes the kinetic cylinder $Q_{r}:=(-r^{2s},0]\times B_{r^{1+2s}}\times B_{r}$.

Remark 1.7*.*

The Hölder norms $C_{\ell}^{2s+\alpha}$ and $C_{\ell}^{\alpha}$ must be appropriately understood. They refer to the usual notion of $C^{2s+\alpha}$ and $C^{\alpha}$ regularity with respect to the $v$ variable. The order of regularity in the other directions is adjusted in terms of the invariant structure of the class of equations. On the one hand, the fact that the diffusion is of order $2s$ yields an invariant scaling. On the other hand the equation enjoys Galilean invariance, yielding a Lie group structure. We discuss other choices of distances and their differences in Section 2.6. Hölder spaces are introduced in Definition 2.3 below. The subindex “$\ell$” refers to the fact that the Hölder norm is taken with respect to a distance that is left-invariant by the Lie group structure.

Remark 1.8*.*

Note that our theorem holds for $\alpha=\frac{2s}{1+2s}\gamma<\gamma$. The distinction between these two Hölder exponents $\alpha$ and $\gamma$ comes from technical reasons related to the fact that the class of equations is left invariant, but not right invariant with respect to the Lie group structure.

Remark 1.9*.*

Note that the global $C_{\ell}^{\gamma}$ norm of $f$ cannot be replaced by its global $L^{\infty}$ norm even in the case of the space-homogeneous parabolic fractional heat equation. A solution $f(t,v)$ to

[TABLE]

does not satisfy the estimate

[TABLE]

This is because $f$ will not be better than Lipschitz in time, even though it will have more regularity in space (See [1, section 2.4.1]). The Hölder space $C_{\ell}^{2s+\alpha}$ would impose $C^{1}$ regularity in time for any $\alpha>0$.

Remark 1.10*.*

Note that the equation (1.1) does not have a structure compatible with the notion of weak solutions in the sense of distributions. It is not an equation in divergence form. Our result in this paper is an a priori estimate provided that all quantities involved make sense classically. It is possible to define a weaker notion of solution of (1.1) in the viscosity sense, and presumably our result in Theorem 1.6 applies to that case as well. However, we do not pursue that direction in this paper since it would add some technical difficulties obfuscating the proofs. The result as currently stated is what we need for our intended applications to the Boltzmann equation.

Remark 1.11*.*

If we want our estimates to hold uniformly as $s\to 1$, we would have to replace the constant $\lambda$ in (1.3) by $(1-s)\lambda$. The results in this article hold in the local case $s=1$ as well, with considerably simplified proofs. It would apply to an equation of the form

[TABLE]

Schauder estimates for these and more general equations have been studied before (see the next subsection). Our approach is quite different to earlier works, starting from the fact that we use a different definition of the Hölder norm. In the case $s=1$, some of the difficulties in the proofs presented here disappear. The ellipticity class in Definition 1.1 would be replaced by the usual uniform ellipticity condition of the coefficients $a_{ij}$. The Assumption 1.5 would translate as the Hölder continuity assumption of these coefficients. The section about weak limits of kernels would be unnecessary since it would be replaced by the simple convergence of the matrix of coefficients. The majorant function defined in (3.3), that plays an important role later in the proof of the Liouville theorem, would be irrelevant since the equation is local. Consequently, the final estimate would be in terms of $\|f\|_{L^{\infty}(Q_{1})}$ instead of $\|f\|_{C^{\gamma}((-1,0]\times B_{1}\times\mathbb{R}^{d})}$. Moreover, the equation (4.1) in the Liouville theorem could be written in terms of $f$ directly, instead of introducing the function $g$. The final result is an interior estimate for the $C^{2+\alpha}_{\ell}$ norm with $s=1$. This norm is comparable but contains more explicit information than the norms used previously in the literature (see the norm $H^{\alpha}$ in [9] for example, or equivalently the norm $\|\cdot\|_{2+\alpha,\Omega}$ in [18]). Indeed, the inequality $\|f\|_{H^{\alpha}}\lesssim\|f\|_{C^{2+\alpha}_{\ell}}$ follows easily from Lemma 2.7. The inequality in the opposite direction is much less obvious. A posteriori, one can get it (away from the boundary) as a consequence of our Schauder estimates. Indeed, if $f\in H^{\alpha}$, then it is clear that $f_{t}+v\cdot\nabla_{x}f-\Delta_{v}f\in C_{\ell}^{\alpha}$. Our Schauder estimate for the case $s=1$ tells us that

[TABLE]

For $s<1$, it is not in general possible to redefine the space $C^{2s+\alpha}_{\ell}$ in terms of Hölder norms of derivatives as in the classical definition of $H^{\alpha}$.

1.1. Schauder estimates for kinetic and non-local equations

Linear kinetic equations of second order are a particular instance of the more general theory of ultraparabolic equations of Kolmogorov type. Results involving regularity estimates in Hölder spaces for these equations appeared especially in the late 1990’s. See [22, 13, 12, 4, 2, 15], and the survey article [18]. More recently in [7, 9], Schauder estimates were applied to bootstrap higher regularity estimates for second order models in kinetic theory, including the Landau equation with moderately soft potentials. The Boltzmann equation can be written in the form (1.1) for a kernel $K$ depending on the solution itself (see [21]). It is our intention to use the result of Theorem 1.6 to derive higher order regularity estimates for the non-cutoff Boltzmann equation in (shortly) forthcoming work.

Schauder estimates for integro-differential equations have been obtained in recent years, see [14, 11, 19, 20, 8, 3]. They have the well known difficulty that the smoothness of the tails of the integrals outside of the domain of the equation are difficult to control. There are two common workarounds that have been used in the literature. One workaround that works well in the elliptic case is to impose some regularity in the kernels $K$ with respect to the variable of integration $w$. Another approach, arguably more delicate, imposes extra regularity on the values of the function $f$ outside of the domain. That is the case in [19], [20],[3] and it is also the approach we take here. In our kinetic setting, this restriction goes a bit further by requiring the function $f\in C_{\ell}^{\gamma}$ outside of $Q_{1}$, with $\gamma>\alpha$.

In this paper, we use some key ideas that originated in [19] and simplify enormously the general procedure to prove the Schauder estimates in the nonlocal setting. The key of the proof of Theorem 1.6 is a combination of a blow-up technique (see Proposition 5.1) with a Liouville theorem (see Theorem 4.1).

1.2. Possible extensions and outstanding questions

It is most probably possible to extend Theorem 1.6 to higher values of $\gamma$ such that $\gamma\notin\mathbb{N}+2s\mathbb{N}$. It would require the extension of Assumption 1.5 for $\alpha$ large, or a version of (4.1) involving higher order incremental quotients. It would not be obvious how to imply the result of Theorem 1.6 for higher values of $\gamma$ by simply taking derivatives. Knowing $f\in C_{\ell}^{\gamma}$ gives us a clean estimate for $f_{t}+v\cdot\nabla_{x}f$ in $C_{\ell}^{\gamma-2s}$ and $\partial_{v_{i}}f$ in $C_{\ell}^{\gamma-1}$. These two derivatives do not solve an equation like (1.1). One might attempt to apply hypoelliptic estimates to derive corresponding Hölder spaces for $f_{t}$ and $\partial_{v_{i}}f+v_{i}\partial_{x_{i}}f$. But then we would loose a fraction of the Hölder exponent that goes beyond the order of differentiation. This is somehow reflected in the final relation between $\alpha$ and $\gamma$ in the statement of Theorem 1.6.

In the statement of Theorem 1.6, we make the assumption $\alpha\leq\frac{2s}{1+2s}\gamma$. We know that the theorem would not hold with $\alpha>\gamma$. The range $\frac{2s}{1+2s}\gamma<\alpha\leq\gamma$ is currently unclear.

1.3. Organization of the article

The article is organized as follows. In Section 2, Hölder spaces adapted to the study of kinetic equations are introduced. In particular, a kinetic degree of polynomials and differential operators and a kinetic distance are defined. Section 3 is devoted to the study of integral operators associated with the class $\mathcal{K}$ of elliptic kernels from Definition 1.1. We then state and prove a Liouville type theorem in Section 4. The final section 5 is devoted to the proof of the main theorem. It is done by contradiction through a blowup argument.

2. Kinetic Hölder spaces

In this preliminary section, we mainly introduce the Hölder spaces we need to derive the Schauder estimate for kinetic integro-differential equations. We first define a kinetic distance, then the kinetic degrees of polynomials and differential operators. The definition of Hölder spaces is then given and an interpolation inequality is proved.

2.1. The kinetic distance

The following Lie group structure of $\mathbb{R}^{1+2d}$ plays a key role in all our computations. The product is defined as

[TABLE]

Note that this product is not commutative. The class equations we will be working with (as in (3.19)) are left-invariant, in the sense that if $f(z)$ is a solution of (1.1), then $f_{0}(z):=f(z_{0}\circ z)$ is also a solution of a similar equation with a translated right hand side and a translated kernel in the same ellipticity class.

There is also invariance by scaling. We define $S_{R}(t,x,v)=(R^{2s}t,R^{1+2s}x,Rv)$. If $f(z)$ solves an equation like (1.1), then $f(S_{R}z)$ solves a similar equation with a scaled right hand side and a scaled kernel in the same ellipticity class.

Because of this property, it is good to work with a notion of distance, Hölder norms, degree and differential operators, that are homogeneous respect to this kinetic scaling, and are left invariant by the action of the Lie group.

Definition 2.1** (A left-invariant distance).**

Given two points $z_{1}=(t_{1},x_{1},v_{1})$ and $z_{2}=(t_{2},x_{2},v_{2})$ in $\mathbb{R}^{1+2d}$, we define the following distance function

[TABLE]

The subindex “$\ell$” stands for **“l”**eft invariant.

It is convenient to also have a notion of “norm” with the right scale invariance. We define:

[TABLE]

Note that it is not an actual norm in the strictly mathematical sense of the term.

Here are some observations.

• •

The distance $d_{\ell}$ is left invariant by the Lie group action in the sense that $d_{\ell}(z\circ z_{1},z\circ z_{2})=d_{\ell}(z_{1},z_{2})$ for any $z,z_{1},z_{2}\in\mathbb{R}^{1+2d}$.

• •

It is homogeneous with respect to scaling: $d_{\ell}(S_{R}z_{1},S_{R}z_{2})=Rd_{\ell}(z_{1},z_{2})$.

• •

We will see below in Proposition 2.2 that $d_{\ell}$ is indeed a distance when $s\geq 1/2$ in the sense that it satisfies the triangle inequality. When $s<1/2$, the function $d_{\ell}^{2s}$ is a distance. We will still work with $d_{\ell}$ (as opposed to $d_{\ell}^{2s}$) when $s<1/2$ so that we keep a consistent scaling formula (as in the previous bullet point) throughout the paper.

• •

There are other equivalent formulas to measure how far apart $z_{1}$ and $z_{2}$ are. We observe that

[TABLE]

None of the three formulas on the right hand side are proper distances. However, since they give us a good estimate for $d_{\ell}(z_{1},z_{2})$ we will use them whenever it is convenient.

• •

Note that the distance $d_{\ell}$ can be reformulated in the following way: $d_{\ell}(z_{1},z_{2})$ is the infimum value of $r>0$ so that both $z_{1}$ and $z_{2}$ belong to a cylinder $Q_{r}(z)$ for some $z\in\mathbb{R}^{1+2d}$.

• •

Our usual definition for the cylinder $Q_{r}$ would not be affected significantly if we changed it for

[TABLE]

Moreover, because of the Lie group invariance, we could also have for $z_{0}=(t_{0},x_{0},v_{0})$,

[TABLE]

Proposition 2.2**.**

The function $d_{\ell}:\mathbb{R}^{1+2d}\times\mathbb{R}^{1+2d}\to[0,\infty)$ is a distance when $s\geq 1/2$. For $s<1/2$, the function $d_{\ell}^{2s}$ is a distance.

Proof.

We start with the case $s\geq 1/2$. Because of the invariance by the Lie group action, we only need to prove the triangle inequality when one of the three points is the origin. That is, given any $z_{1},z_{2}\in\mathbb{R}^{1+2d}$, we must show that $d_{\ell}(z_{1},0)+d_{\ell}(0,z_{2})\geq d_{\ell}(z_{1},z_{2})$.

Let $w_{1},w_{2}\in\mathbb{R}^{d}$ be the points where the minimum in Definition 2.1 is achieved for $z_{1}$ and $z_{2}$ respectively. That is

[TABLE]

By definition, $d_{\ell}(z_{1},z_{2})$ is the minimum over all choices of $w\in\mathbb{R}^{d}$, so it is less or equal to the value we get by setting $w=w_{1}+w_{2}$.

[TABLE]

We now analyze every one of the four expressions inside the $\max$.

Clearly, since $1/(2s)\leq 1$, we have

[TABLE]

Also, simply by the triangle inequality in $\mathbb{R}^{d}$,

[TABLE]

Likewise,

[TABLE]

The second argument in the max is the only one that requires a nontrivial analysis. We evaluate

[TABLE]

The last inequality follows from the following elementary calculus fact. For any $a,b\geq 0$ and $p\geq 1$, the following inequality holds:

[TABLE]

Clearly, in the last inequality, we applied (2.2) with $a=d_{\ell}(z_{1},0)$, $b=d_{\ell}(z_{2},0)$ and $p=2s$.

The proof for the case $s<1/2$ goes along the same lines, and we conclude the last inequality also applying (2.2) with $p=1/(2s)$. ∎

2.2. Kinetic degree of polynomials

We start by defining a modified notion of degree for a polynomial $p\in\mathbb{R}[t,x,v]$. This special degree, which we will call kinetic degree, matches the scaling of the equation.

In order to compute $\deg_{k}p$, every exponent of the variable $t$ should count times $2s$, every exponent of the variables $x_{i}$ counts times $(1+2s)$ and the exponents of the variables $v_{i}$ count normally. More precisely, if $m\in\mathbb{R}[t,x,v]$ is a monomial, we define its kinetic degree $\deg_{k}m$ is the number $\kappa$ so that $m(S_{R}z)=R^{\kappa}m(z)$.

A polynomial $p$ is always a finite sum of monomials. In general, we define the kinetic degree of a polynomial $p=\sum m_{j}$ as the maximum of $\deg_{k}m_{j}$ for all its monomial terms $m_{j}$.

Note that the degree of a polynomial $p\in\mathbb{R}[t,x,v]$ can be any number in the discrete set $\mathbb{N}+2s\mathbb{N}$.

2.3. Hölder spaces

We now define a properly scaled version of Hölder spaces.

Definition 2.3** (Hölder spaces).**

For any $\alpha\in(0,\infty)$, we say that a function $f:D\to\mathbb{R}$ is $C_{\ell}^{\alpha}$ at a point $z_{0}\in\mathbb{R}^{1+2d}$ if there exists a polynomial $p\in\mathbb{R}[t,x,v]$ such that $\deg_{k}p<\alpha$ and for any $z\in D$

[TABLE]

When this property holds at every point $z_{0}$ in the domain $D$, with a uniform constant $C$, we say $f\in C_{\ell}^{\alpha}(D)$. The semi-norm $[f]_{C_{\ell}^{\alpha}(D)}$ is the smallest value of the constant $C$ so that the inequality above holds for all $z_{0},z\in D$. The norm $\|f\|_{C_{\ell}^{\alpha}(D)}$ is $[f]_{C_{\ell}^{\alpha}(D)}+[f]_{L^{\infty}(D)}$.

Remark 2.4*.*

Note that $d_{\ell}$, $\deg_{k}$, and therefore also $C^{\alpha}_{\ell}$, depend on $s$ implicitly.

Remark 2.5*.*

With the above definition, when $\alpha\in\mathbb{N}+2s\mathbb{N}$, the $C_{\ell}^{\alpha}$ space corresponds to a Lipschitz-type space instead of the classical $C^{1}$ space.

Using the invariance by left translations, we can rephrase the Hölder regularity of $f$ at $z_{0}$ in the following way. There exists a polynomial $p_{0}$ such that $\deg_{k}p_{0}<\alpha$ and for all $z\in\mathbb{R}^{1+2d}$ such that $z_{0}z\in D$, we have

[TABLE]

In this case $p_{0}(z)=p(z_{0}\circ z)$ where $p$ is the polynomial of Definition 2.3. If the polynomial $p_{0}$ is given by

[TABLE]

it is easy to verify that $a_{0}=f(z_{0})$, $a_{1}=\partial_{t}f+v_{0}\cdot\nabla_{x}f$, $a_{2}=\nabla_{x}f$ and $a_{3}=\nabla_{v}f$.

As it is standard for some proofs of the Schauder estimates (see for example Section 4 in [5]), we define the adimensional Hölder spaces.

Definition 2.6** (Adimensional Hölder spaces).**

Given a kinetic cylinder $Q\subset\mathbb{R}^{1+2d}$ and any $\alpha>0$, we define

[TABLE]

where

[TABLE]

Naturally, we also define $\|f\|_{C_{\ell,\ast}^{\alpha}(Q)}:=\|f\|_{C_{\ell}^{0}(Q)}+[f]_{C_{\ell,\ast}^{\alpha}(Q)}$.

Here, we used the notation $\partial_{p}Q_{r}(t_{0},x_{0},v_{0}):=\partial Q_{r}(t_{0},x_{0},v_{0})\setminus\{t=t_{0}\}$.

2.4. Hölder norms and differential operators

The differential operators $(\partial_{t}+v\cdot\nabla_{x})$, $\partial_{x_{i}}$ and $\partial_{v_{i}}$ commute with left translations. They do not commute with each other, and they do not keep the equation (3.19) invariant (with $c=0$). The operators that commute with the equation (3.19) are $\partial_{t}$, $\partial_{x_{i}}$ and $\partial_{v_{i}}+v_{i}\partial_{x_{i}}$, which are the ones that commute with right translations (instead of left translations).

It may be convenient to define the kinetic degree of a differential operator. We say that the kinetic degree of $\partial_{t}+v\cdot\nabla_{x}$ is $2s$, the kinetic degree of $\partial_{x_{i}}$ is $(1+2s)$ and the kinetic degree of $\partial_{v_{i}}$ is $1$.

It is convenient to relate the definition of the Hölder spaces $C_{\ell}^{\alpha}$ with these operators. The following (deceivingly simple) lemma will be used repeatedly.

Lemma 2.7** (Derivatives and kinetic Hölder spaces).**

Let $D=\partial_{t}+v\cdot\nabla_{x}$, $D=\partial_{x_{i}}$ or $D=\partial_{v_{i}}$. Let $f$ be a $C_{\ell}^{\alpha}$ function in a cylinder $Q$ and let $\deg_{k}D=\kappa$ with $\kappa<\alpha$. Then $Df\in C_{\ell}^{\alpha-\kappa}$ and

[TABLE]

Before proving Lemma 2.7, we prove the following auxiliary lemma about polynomials.

Lemma 2.8**.**

Let $p(z)$ be a polynomial in $t$, $x$ and $v$ of kinetic degree $k$. Let us write $p$ as

[TABLE]

where

[TABLE]

Assume that

[TABLE]

Then, for each $j$, we have

[TABLE]

where the constant $C$ depends on $\deg_{k}p$ and dimension only.

Proof.

We observe that once we establish this lemma for $r=1$, the other values of $r$ follow by scaling.

The space of polynomials of kinetic degree $k$ in $\mathbb{R}^{1+2d}$ is finite dimensional. Recall that all norms are equivalent in spaces of finite dimension. The result for $r=1$ follows easily by comparing the two norms given by

[TABLE]

This concludes the proof of the technical lemma. ∎

Proof of Lemma 2.7.

Let $z_{0}$ and $z_{1}$ be two points in $Q$. Since $f\in C_{\ell}^{\alpha}(Q)$, there exist polynomials $q_{0}$ and $q_{1}$ of degree less than $\alpha$ so that for all $z$ so that $z_{0}\circ z\in Q$ and $z_{1}\circ z\in Q$,

[TABLE]

where $C=[f]_{C_{\ell}^{\alpha}}$. We used the fact that $d_{\ell}$ is left invariant and $d_{\ell}(z,0)\leq\|z\|$. Let $r=d_{\ell}(z_{0},z_{1})\approx\|z_{0}^{-1}\circ z_{1}\|$ and let us pick any $z$ so that $\|z\|<r$ 111It is slightly problematic when $z_{0}\circ z$ or $z_{1}\circ z$ fall outside the domain $Q$. It can be handled like in the proof of Proposition 2.10 by using the equivalent norm in the space of polynomials of degree $<\alpha$ that considers only the points that fall inside the domain.. From the triangle inequality (modified with the power $2s$ when $s<1/2$), we have $\|z_{0}^{-1}\circ z_{1}\circ z\|\approx d_{\ell}(z_{0},z_{1}\circ z)\leq d_{\ell}(z_{0},z_{1})+d_{\ell}(z_{1},z_{1}\circ z)\lesssim r$. We apply the two inequalities above for $z_{0}^{-1}\circ z_{1}\circ z$ and $z$ respectively to obtain

[TABLE]

Therefore, for all $\|z\|\leq r$,

[TABLE]

Let us write the coefficients of both polynomials $q_{1}(z)$ and $q_{0}(z_{0}^{-1}\circ z_{1}\circ z)$.

[TABLE]

The coefficients $a_{j}(z_{1})$ can be computed in terms of the coefficients of $q_{0}$ and the value of $z_{0}^{-1}\circ z_{1}$. It is not hard to see that each coefficient is a polynomial in $z_{0}^{-1}\circ z_{1}$ and thus of $z_{1}$ whose degree is not larger than $\deg_{k}q_{0}$ minus the degree of the corresponding monomial. Applying Lemma 2.8, we see that

[TABLE]

Here $k_{1}=\deg_{k}t=\deg_{k}(\partial_{t}+v\cdot\nabla_{x})=2s$, $k_{2}=\deg_{k}x=\deg_{k}\partial_{x_{i}}=1+2s$ and $k_{3}=\deg_{k}v=\deg_{k}\partial_{v_{i}}=1$, i.e. $k_{j}=\kappa$ in short. Since $b_{1}=(\partial_{t}+v_{1}\nabla_{x})f(z_{1})$, $b_{2}=\nabla_{x}f(z_{1})$ and $b_{3}=\nabla_{v}f(z_{1})$, i.e. $b_{j}=Df(z_{1})$ in short, we have

[TABLE]

This finishes the proof. ∎

Remark 2.9*.*

Note we can also apply Lemma 2.8 to the other coefficients of the polynomial $q_{1}(z)$. If $p_{z}$ is the polynomial expansion of $f$ at $z$ of kinetic degree less than $\alpha$ and it has the form

[TABLE]

then, by a direct computation, the coefficients $a_{j}(z)$ correspond to

[TABLE]

Note that $(\partial_{t}+v\cdot\nabla_{x})$ and $\partial_{v}$ do not commute.

2.5. Interpolation inequalities

The usual interpolation estimates for Hölder spaces will hold.

Proposition 2.10** (Interpolation).**

Given $\alpha_{1}<\alpha_{2}<\alpha_{3}$ so that $\alpha_{2}=\theta\alpha_{1}+(1-\theta)\alpha_{3}$, then for any function $f\in C_{\ell}^{\alpha_{3}}(Q_{1})$,

[TABLE]

Also, for any function $f\in C_{\ell,\ast}^{\alpha_{3}}(Q_{1})$,

[TABLE]

Remark 2.11*.*

We classically get from the previous estimates that for all $\varepsilon>0$,

[TABLE]

Proof.

We prove the first interpolation inequality. The second one follows as a consequence by scaling. The statement says precisely that the function $\alpha\mapsto\log[f]_{C_{\ell}^{\alpha}}$ is convex. This is a local property, so we only need to prove it for $\alpha_{3}$ sufficiently close to $\alpha_{1}$. Because of this, it is enough to prove the interpolation inequality assuming that $(\mathbb{N}+2s\mathbb{N})\cap[\alpha_{1},\alpha_{3})$ contains at most one element.

Let $q^{1}_{z}$, $q^{2}_{z}$ and $q^{3}_{z}$ be the polynomial expansions of $f$ at $z$ of kinetic degrees less than $\alpha_{1}$, $\alpha_{2}$ and $\alpha_{3}$ respectively such that for all $z\circ\xi\in Q_{1}$,

[TABLE]

The polynomials $q^{1}_{z}$, $q^{2}_{z}$ and $q^{3}_{z}$ are increasingly higher order expansions at the same point $z$. Therefore, $q^{2}_{z}$ contains all the terms in $q^{1}_{z}$ plus perhaps higher order ones. In the same way, $q^{3}_{z}$ contains all the terms of $q^{2}_{z}$ plus perhaps higher order ones. Because of our assumption that $(\mathbb{N}+2s\mathbb{N})\cap[\alpha_{1},\alpha_{3})$ has at most one element, which we call $\bar{\alpha}$, there can be at most one degree of homogeneity in the difference between the polynomials $q_{z}^{1}$ and $q_{z}^{3}$. The polynomial $q^{2}_{z}$ coincides with either $q^{1}_{z}$ or $q^{3}_{z}$ depending on whether $\bar{\alpha}\geq\alpha_{2}$ or $\bar{\alpha}<\alpha_{2}$. When $(\mathbb{N}+2s\mathbb{N})\cap[\alpha_{1},\alpha_{3})=\emptyset$, we have $q_{z}^{1}=q^{2}_{z}=q^{3}_{z}$ that is easier to analyze. Let us consider the case in which there is an $\bar{\alpha}$ and the polynomials are not equal.

Like in Lemma 2.8, we write

[TABLE]

where each $m_{j}$ is a monomial.

Substracting (2.6) for $i=1,2$, whenever $z\circ\xi\in Q_{1}$, we have

[TABLE]

From this inequality, we want to infer an estimate for $\|q_{z}^{3}-q_{z}^{1}\|$. Let us first make some remarks about the norm of a polynomial. The space of polynomials of kinetic degree less than $\alpha_{3}$ is finite dimensional. So, all norms that we can write are equivalent. A natural choice is perhaps

[TABLE]

If we change that radius $1$ for any other universal constant, we would obtain an equivalent norm. Note that translations of polynomials are also polynomials of the same degree. Therefore, for any two universal constants $c$ and $C$, $z_{0}\in\mathbb{R}^{1+2d}$, and $\deg_{k}q<\alpha_{3}$, we deduce that

[TABLE]

The factors in $\approx$ depend naturally on $c$ and $C$.

Coming back to (2.7), for any $N\in(0,1]$ and $z\in Q_{1}$, let us pick some point $\xi_{1}\in Q_{1}$ such that

• •

$\|\xi_{1}\|<N$,

• •

Whenever $d_{\ell}(\xi_{1},\xi)<cN$, then $\|\xi\|\leq N$ and $z\circ\xi\in Q_{1}$. Here $c$ is a universal constant.

It is not hard to see that for any $z\in Q_{1}$, such $\xi_{1}$ exists (in fact plenty).

From (2.7) we get,

[TABLE]

Since $q_{z}^{3}-q_{z}^{1}$ is homogeneous of degree $\bar{\alpha}$,

[TABLE]

Since $\|\xi_{1}\|\leq N$, then $\|S_{1/N}\xi_{1}\|\leq 1$. From the triangle inequality (modified with power $2s$ when $s<1/2$), there is a universal constant $C$ so that whenever $\|\xi\|\leq 1$, then $\xi\in d_{\ell}(S_{1/N}\xi_{1},\xi)\leq C$.

According to the discussion above, the fact that all norms are equivalent in the space of polynomials implies that

[TABLE]

We now optimize for $N\in[0,1]$ and obtain

[TABLE]

where $\bar{\alpha}=\bar{\theta}\alpha_{1}+(1-\bar{\theta})\alpha_{2}$.

Now we estimate $f(z\circ\xi)-q_{z}^{2}(\xi)$ using both $[f]_{C_{\ell}^{\alpha_{1}}}$ and $[f]_{C_{\ell}^{\alpha_{3}}}$. There are two cases depending on whether $\alpha_{2}>\bar{\alpha}$ or $\alpha_{2}\leq\bar{\alpha}$. The proofs are very similar, so let us do only the later. In this case $q^{2}_{z}=q^{1}_{z}$. We have

[TABLE]

One can easily verify that the right hand side is less than $[f]_{C_{\ell}^{\alpha_{1}}}^{\theta}[f]_{C_{\ell}^{\alpha_{3}}}^{1-\theta}\|\xi\|^{\alpha_{2}}$ for any value of $\|\xi\|$. Indeed, if $\|\xi\|\geq([f]_{C_{\ell}^{\alpha_{3}}}/[f]_{C_{\ell}^{\alpha_{2}}})^{1/(\alpha_{3}-\alpha_{1})}$, we have

[TABLE]

Otherwise, we have

[TABLE]

If $\alpha_{2}>\bar{\alpha}$, we would have $q^{2}_{z}=q^{3}_{z}$ and the term $|q^{3}_{z}-q^{1}(z)|\leq\left([f]_{C_{\ell}^{\alpha_{1}}}^{\bar{\theta}}[f]_{C_{\ell}^{\alpha_{3}}}^{1-\bar{\theta}}+[f]_{C_{\ell}^{\alpha_{1}}}\right)\|\xi\|^{\bar{\alpha}}$ would appear on the first line of the inequality. If $(\mathbb{N}+2s\mathbb{N})\cap[\alpha_{1},\alpha_{3})=\emptyset$, we have $q_{z}^{1}=q^{2}_{z}=q^{3}_{z}$ and the extra term involving $\bar{\alpha}$ would not be there. ∎

2.6. Discussion about choices of distance

We use the distance $d_{\ell}$ which is invariant by left translations and by the scaling $S_{R}$. Let us analyze the consequences of this choice and compare with the other possible choices.

We could define a distance $d_{r}$ that is invariant by right translations of the Lie group. It would be given by:

[TABLE]

Moreover, it would be comparable to the following expressions.

[TABLE]

Alternatively, we could ignore the Lie group structure and define a distance $d_{s}$ that only takes scaling into account.

[TABLE]

Here $\|.\|$ stands for the scaled norm as in (2.1).

The most brutal choice would be to ignore both the Lie group action and scaling and use the plain Euclidean distance in $\mathbb{R}^{1+2d}$.

[TABLE]

The definition of Hölder spaces (Definition 2.3) depends on the choice of distance. We can thus consider the four possible candidates $C_{\ell}^{\alpha}$, $C_{r}^{\alpha}$, $C_{s}^{\alpha}$ and $C_{e}^{\alpha}$. The distances are not equivalent, and these four spaces are all different. Their only equivalence appears when measuring distances from the origin $d_{\ell}(0,z)\approx d_{r}(0,z)\approx d_{e}(0,z)$. Thus $C^{\alpha}_{\ell}(0)=C^{\alpha}_{r}(0)=C^{\alpha}_{e}(0)$ (by $C^{\alpha}_{\ast}(0)$ we mean the functions that are $C^{\alpha}_{\ast}$ at the point [math]).

The class of equations (1.1) is invariant by left translations. Because of that, the norm $d_{\ell}$ is the most appropriate to work with. For example, if we proved an estimate for solutions of (1.1) of the sort $[f]_{C_{\ell}^{\alpha}(0)}\lesssim\|f\|_{L^{\infty}(Q_{1})}$, it implies by simple translations that $[f]_{C_{\ell}^{\alpha}(Q_{1/2})}\lesssim\|f\|_{L^{\infty}(Q_{1})}$. This implication does not hold true for $C_{r}^{\alpha}(Q_{1/2})$, $C_{s}^{\alpha}(Q_{1/2})$ or $C_{e}^{\alpha}(Q_{1/2})$ (at least not true keeping the same exponent $\alpha$).

In previous works, people have taken more or less attention to these distinctions. The results in [6] and [10] are oblivious of the choice of distance. That is because these results are about an estimate in Hölder spaces for an undetermined exponent $\alpha>0$. For any pair of points $z_{1}$ and $z_{2}$ in $Q_{1}$, the following inequality holds

[TABLE]

where $d_{1}$ and $d_{2}$ are any two choices among $d_{\ell}$, $d_{r}$, $d_{s}$ and $d_{e}$. Thus, the main theorems in [6] and [10] hold for the $C_{\ell}^{\alpha}$ norm defined in terms of any of these distances, modulo an adjustments of the constants and Hölder exponent $\alpha$.

For Schauder estimates, the distinction between different distances plays a crucial role. In this case we want to obtain an estimate with the precise exponent $C^{2s+\alpha}$ when the right hand side is $C^{\alpha}$. It seems that such an estimate can only be true with the distance $d_{\ell}$.

For right-invariant Hölder spaces $C^{\alpha}_{r}$ in terms of $d_{r}$, the corresponding statement of Lemma 2.7 would be in terms of the operators $\partial_{t}$, $\partial_{x_{i}}$ and $\partial_{v_{i}}+v_{i}\partial_{x_{i}}$. These differential operators have the advantage that they commute with the equation (3.19). For regular Hölder spaces $C^{\alpha}_{s}$ or $C^{\alpha}_{e}$, Lemma 2.7 would of course hold with pure derivatives $\partial_{t}$, $\partial_{x_{i}}$ and $\partial_{v_{i}}$.

Our Liouville theorem 4.1 holds for any choice of distance $d_{\ell}$, $d_{r}$ or $d_{s}$. This is because in the step 1 of the proof we establish that the function is constant in $x$. After that, we ignore the $x$ coordinate and the three distances are the same.

In the proof of Lemma 5.2, we select a sequence of functions $\tilde{f}_{j}$ that are scaled left-translations of a sequence of solutions $f_{j}$. If we used a different choice of distance that is not invariant by left translations, we would not be able to conclude anything about their $C^{\beta}$ norms.

One needs to be careful throughout this paper to make sure we do not implicitly use the exact triangle inequality for $s<1/2$, we do not commute group operation $\circ$, and that we do not accidentally apply $d_{r}$ or $d_{s}$ instead of $d_{\ell}$.

3. Integral operators

This section is devoted to the integral coperators

[TABLE]

associated with fixed kernels $K$ from the elliptic class $\mathcal{K}$ given in Definition 1.1. We first explain when these integral operators can be evaluated pointwise. We then turn to limits of kernels and integral operators. We conclude this section by proving Hölder estimates that will be used in the proof of the Schauder estimate.

3.1. Evaluating operators pointwise

In this subsection, we discuss how to evaluate pointwise operators associated with kernels in the elliptic class $\mathcal{K}$. More precisely, we want to explain the conditions that a function $f:\mathbb{R}^{d}\to\mathbb{R}$ must meet in order for the integral in (3.1) to be well defined at the point $v_{0}$. On one hand, it must be sufficiently regular so that the integral does not diverge in a neighborhood of $v_{0}$. On the other hand, it must also satisfy some growth conditions so that the integral does not diverge at infinity. Let us split the domain of integration accordingly and analyze conditions for convergence of each part.

[TABLE]

When $s\geq 1/2$, the first term must be understood in the principal value sense, even when $f$ is smooth. Using the symmetry condition $K(w)=K(-w)$, we can symmetrize the integral and remove the principal value.

[TABLE]

Because of (1.2), this integral is classically computable when $f\in C^{2s+\varepsilon}(B_{1}(v_{0}))$ for some $\varepsilon>0$. Indeed,

[TABLE]

In order to analyze the tail of the integral, we introduce the following function

[TABLE]

We observe that, because of (1.2), the function $\omega_{v_{0}}$ can be used to bound the tail of the integral. We state the estimate in a lemma for later use.

Lemma 3.1**.**

Let $R>0$.

[TABLE]

Proof.

Using the definition of $\omega_{v_{0}}(r)$, we can write

[TABLE]

Using (1.2) yields the result. ∎

Summarizing, we have the estimate

[TABLE]

Moreover, the integral expression in (3.1) is classically computable whenever the right hand side of the inequality is finite.

3.2. Weak limits of kernels

We now discuss how to pass to the limit in kernels. We first define the notion of weak-$\ast$ convergence and we then prove that the set $\mathcal{K}$ is compact for the corresponding topology.

Definition 3.2** **(Weak-$\ast$ convergence of

kernels).

We say that a sequence $K_{j}$ of Radon measures in $\mathbb{R}^{d}\setminus\{0\}$ converges weakly-$\ast$ to the Radon measure $K_{\infty}$ if for any continuous function $\varphi:\mathbb{R}^{d}\to\mathbb{R}$, compactly supported, whose support does not include the origin, we have

[TABLE]

Lemma 3.3** **(Closedness of $\mathcal{K}$ under weak-$\ast$

limit).

If the kernels $K_{j}$ belong to the class $\mathcal{K}$ of Definition 1.1 and $K_{j}$ converges weakly-$\ast$ to $K_{\infty}$, then $K_{\infty}$ also belongs to the class $\mathcal{K}$.

Proof.

The fact that $K_{\infty}$ is a non-negative Radon measure is classical.

As far as the upper bound is concerned, it is enough to consider a cut-off function $\varphi_{r}$ valued in $[0,1]$ with $\varphi_{r}\equiv 1$ in $B_{r}\setminus B_{\eta}$ and whose compact support is contained in $B_{r+\varepsilon}\setminus B_{\eta/2}$ for some $\varepsilon,\eta>0$. Then we write

[TABLE]

Passing to the limit as $j\to+\infty$, we get

[TABLE]

Since $\varepsilon$ and $\eta$ are arbitrary, $K_{\infty}$ satisfies the upper bound.

As far as the coercivity estimate is concerned, let $R>0$ and $\varphi\in C^{2}(B_{R})$. Since $K_{j}\in\mathcal{K}$, we have

[TABLE]

For all $r>0$, consider a cut-off function $\Psi_{r}$ valued in $[0,1]$, $\Psi_{r}\equiv 1$ in $B_{r/2}$ and $\Psi_{r}\equiv 0$ outside $B_{r}$. Thanks to the uniform upper bound, we have

[TABLE]

Combining the two previous estimates, we get

[TABLE]

We can now pass to the limit as $j\to\infty$ and obtain

[TABLE]

Letting $r\to 0^{+}$ yields the result. ∎

Lemma 3.4** **(Compactness of $\mathcal{K}$ for weak-$\ast$

topology).

If $K_{j}$ is a sequence of kernels in the class $\mathcal{K}$ of Definition 1.1, then it has a weak-$\ast$ convergent subsequence.

Proof.

We split $\mathbb{R}^{d}\setminus\{0\}$ into $\bigcup_{k\in\mathbb{Z}}C_{k}$ with $C_{k}=B_{2^{k-1}}\setminus B_{2^{k-1}}$. The sequence of Radon measures $\{K_{j}|_{C_{k}}\}_{j}$ in $C_{k}$ are compact because of Banach-Alaoglu theorem. Thanks to a diagonal argument, we can thus extract a sequence $\{K_{l(j)}\}_{j}$ converging towards $K_{\infty}$ on each ring $C_{k}$. In particular, this sequence weak-$\ast$ converges to $K_{\infty}$ in the sense of Definition 3.2. ∎

3.3. Limits of operators

Lemma 3.5** (Limits of operators).**

Let $\Omega$ be a open bounded set of $(-\infty,0]\times\mathbb{R}^{d}\times\mathbb{R}^{d}$ and $K_{j}$ and $f_{j}:(-\infty,0]\times\mathbb{R}^{d}\times\mathbb{R}^{d}\to\mathbb{R}$ be a sequence of kernels and functions respectively so that the following conditions hold.

1. (1)

Each $K_{j}$ belongs to the class $\mathcal{K}$. 2. (2)

The sequence $K_{j}\rightharpoonup K_{\infty}$ weakly-$\ast$* as $j\to\infty$.* 3. (3)

The sequence $f_{j}\to f$ locally uniformly in $(-\infty,0]\times\mathbb{R}^{d}\times\mathbb{R}^{d}$ as $j\to\infty$. 4. (4)

The sequence $f_{j}$ is uniformly bounded in $C_{\ell}^{2s+\eta}(\Omega)$ for some $\eta>0$. 5. (5)

There is a function $\omega:[1,\infty)\to\mathbb{R}$ so that $\int_{1}^{\infty}\omega(r)r^{-1-2s}\;\mathrm{d}r<+\infty$ and for every $j\in\mathbb{N}$,

[TABLE]

Then we have

[TABLE]

where $\mathcal{L}_{j}$ is the integral operator corresponding to $K_{j}$, see (3.1).

Proof.

Let $\varepsilon>0$ be arbitrary.

We use the assumption (4) to bound the part of the integrals in $\mathcal{L}_{j}f_{j}$ and $\mathcal{L}f$ around the origin. Thanks to the symmetry assumption of the kernels,

[TABLE]

We use the assumption (5) to bound the tails of the integrals. Note that for any $v\in\Omega$ and $r>\operatorname{diam}(\Omega)$, we can obtain a common majorant function $\omega_{v}(r)$ for all functions $f_{j}$, as in (3.3), by the formula

[TABLE]

Using Lemma 3.1, for $R$ sufficiently large,

[TABLE]

Using that $K_{j}\in\mathcal{K}$ and $f_{j}\to f$ locally uniformly, then for $j$ sufficiently large

[TABLE]

Finally, since $K_{j}\to K$ weak-$\ast$, then for $j$ large

[TABLE]

Note that because $f$ is a continuous function on $\mathbb{R}^{1+2d}$, the choice of $j$ can be made uniform with respect to the point $z=(t,x,v)\in\Omega$ by the argument that led to (3.7).

Adding up (3.5), (3.6), (3.7) and (3.8), we get that $|\mathcal{L}_{j}f-\mathcal{L}f|<\varepsilon$ uniformly in $\Omega$ for $j$ sufficiently large.

∎

3.4. Consequences of Assumption 1.5

We gather here some consequences of Assumption 1.5 that will be used in the next subsection when deriving Hölder estimates.

[TABLE]

Both inequalities are consequences of the fact that 1.5 implies that for all $r>0$,

[TABLE]

To get (3.10), we use dyadic rings $B_{2^{k+1}}\setminus B_{2^{k}}$ and sum over $k$.

3.5. Hölder estimates

We gather here estimates that will be used when proving the main Schauder estimate, see the terms $A$ and $B$ on page 5.

Let us consider a sign changing kernel $K$ such that $K(w)=K(-w)$ and it satisfies the upper bound for all $r>0$,

[TABLE]

Let us study the corresponding integral operator

[TABLE]

We start with a global estimate.

Lemma 3.6**.**

Assume $\alpha<\min(1,2s)$. For any sign-changing symmetric kernel $K$ satisfying (3.11), and $f\in C_{\ell}^{2s+\alpha}(\mathbb{R}^{1+2d})$, we have the estimate

[TABLE]

Proof.

Let us start by fixing some notation. As usual, we denote by $p_{z}$ the polynomial expansion of $f$ at $z$ so that $\deg_{k}p_{z}<2s+\alpha$ and for $z,\xi\in\mathbb{R}^{1+2d}$,

[TABLE]

Let us also write, for $z,\xi,\zeta\in\mathbb{R}^{1+2d}$,

[TABLE]

We must estimate the following quantity

[TABLE]

Since $\alpha<\min(1,2s)$, proving en estimate for $[\mathcal{L}_{K}f]_{C_{\ell}^{\alpha}}$ amount to finding the right upper bound for $|\Delta_{\xi}\mathcal{L}_{K}f(z)|$.

We split the integral above into two subdomains: $B_{R}$ and $\mathbb{R}^{d}\setminus B_{R}$. We will later choose $R=\|\xi\|$.

Estimating the integral in $B_{R}$, we symmetrize using that $K(w)=K(-w)$ and

[TABLE]

When $2s+\alpha>2$, we cannot cancel out the second order terms in $v$ in the polynomial $p_{z}$. Thus, in that case the same computation leads to

[TABLE]

In the estimates above, the value of $z\in\mathbb{R}^{1+2d}$ is arbitrary. The inequalities hold for $z\circ\xi$ just as well. Therefore, applying $\Delta_{\xi}$ we get

[TABLE]

In the last inequality we used Lemma 2.7 for the case $2s+\alpha>2$. Note that for any value of $2s+\alpha$, when we choose $R=\|\xi\|$ we will get

[TABLE]

Now we move on to estimate the part of the integral in $\mathbb{R}^{d}\setminus B_{R}$. We use the following two inequalities

[TABLE]

The second one naturally requires some further analysis. We observe that

[TABLE]

Therefore

[TABLE]

We will split the integral in $\mathbb{R}^{d}\setminus B_{R}$ as the sum of several terms.

[TABLE]

where

[TABLE]

We bound $I_{1}$ easily using (3.12) and (3.11).

[TABLE]

We bound $I_{2}$ following the procedure, but applying (3.14).

[TABLE]

For the analysis of $I_{3}$, we write $p_{z}$ as a sum of monomials.

[TABLE]

Moreover $a_{0}(z)=f(z)$ and

[TABLE]

From Lemma 2.7, we know that $[a_{j}]_{C_{\ell}^{2s+\alpha-\deg_{k}m_{j}}}\leq[f]_{C_{\ell}^{2s+\alpha}}$. Note that $2s+\alpha-\deg_{k}m_{j}<1$ for any monomial such that $0<\deg_{k}m_{j}<2s+\alpha$. Thus,

[TABLE]

Therefore,

[TABLE]

Regarding $I_{4}$, note that since $2s+\alpha<1+2s$, the polynomial $p_{z}$ cannot have a term that involves its second component ($x$). Since $(0,0,w)^{-1}\circ\xi\circ(0,0,w)$ and $\xi$ differ only on their second component, then actually $I_{4}=0$.

When we choose $R=\|\xi\|$, the estimates of all terms are $\lesssim\Lambda[f]_{C_{\ell}^{2s+\alpha}}\|\xi\|^{\alpha}$. And therefore we conclude the proof. ∎

We next derive a local estimate from the global one.

Lemma 3.7**.**

Let $\alpha=\frac{2s}{1+2s}\gamma$ with $\gamma<\min(1,2s)$. Then $[\mathcal{L}_{K}f]_{C_{\ell}^{\alpha}(Q_{1/2})}\lesssim[f]_{C_{\ell}^{2s+\alpha}(Q_{1})}+[f]_{C_{\ell}^{\gamma}((-1,0]\times B_{1}\times\mathbb{R}^{d})}.$

Proof.

It is enough to write $\mathcal{L}_{K}f(z)=\mathcal{L}_{\tilde{K}}f(z)+C(z)$ where $\tilde{K}(w)=\mathbf{1}_{B_{\rho}}(w)K(w)$ and $C(z)$ is $\mathcal{L}_{K}$ where $K$ is replaced with $(1-\mathbf{1}_{B_{\rho}})K(w)$ and $\rho$ small. From the previous lemma, we have

[TABLE]

Let us prove that

[TABLE]

In order to do so, we write for $z_{1},z_{2}\in Q_{1/2}$,

[TABLE]

and we first prove that

[TABLE]

On the one hand, since $f\in C_{\ell}^{\alpha}(Q_{1/2})$ and $\alpha\leq\gamma$, we have

[TABLE]

On the other hand, the $C_{\ell}^{\gamma}$ regularity of $f$ yields

[TABLE]

We now compute $(0,0,-w)\circ z_{2}^{-1}\circ z_{1}\circ(0,0,w)=z_{2}^{-1}\circ z_{1}-(0,(t_{1}-t_{2})w,0)$, and get

[TABLE]

Combining the three previous estimates yields (3.18). Since $\int_{\mathbb{R}^{d}\setminus B_{\rho}}(1+|w|^{\frac{\gamma}{1+2s}})k(w)\;\mathrm{d}w\lesssim\Lambda C_{\rho}$, thanks to Assumption (1.2) and the fact that $\gamma<2s$, (3.18) implies (3.17). This achieves the proof of the lemma. ∎

3.6. The local Hölder estimate

The symmetry condition $K(t,x,v,v+w)=K(t,x,v,v-w)$ corresponds to equations in non-divergence form, in the sense that the integro-differential operator has a structure similar to that of elliptic equations of non-divergence form (as in $a_{ij}(t,x,v)\partial_{v_{i}v_{j}}f$). It is different of the other symmetry condition that would make the operator self adjoint $K(t,x,v,v^{\prime})=K(t,x,v^{\prime},v)$, and corresponds to equations in divergence form. The weak Harnack inequality, in the style of De Giorgi, obtained in [10] does not apply to (1.1) precisely because of this distinction of symmetry assumptions. Our kernels $K$ do not satisfy the cancellation conditions (1.6) and (1.7) from [10].

The situation is simpler when we take a translation invariant kernel $K(w)$ and consider the equation

[TABLE]

It is an integro-differential analog of an equation with constant coefficients. There is no distinction in this case between divergence and non-divergence form. The kernel $\tilde{K}(t,x,v,v^{\prime})=K(v^{\prime}-v)$ satisfies the symmetry condition (and thus also the cancellation condition) $\tilde{K}(t,x,v,v^{\prime})=\tilde{K}(t,x,v^{\prime},v)$ for any kernel $K$ in $\mathcal{K}$.

The regularity of the solution $f$ to (3.19) is not important. It is straight forward to approximate any (weak/viscosity) solution to (3.19) with $C^{\infty}$ solutions by mollification. Indeed, if $f$ solves (3.19), then for any smooth compactly supported function $\varphi:\mathbb{R}^{1+2d}\to\mathbb{R}$, the function

[TABLE]

also solves (3.19) (perhaps in a slightly smaller domain depending on the support of $\varphi$). Naturally, the function $\varphi\ast_{k}f\in C^{\infty}$ whenever $\varphi\in C^{\infty}$. Taking $\varphi$ to be an approximation of the unit mass at the origin, we approximate any solution of (3.19) by a smooth one. Therefore, we can safely assume, without loss of generality, that every function is $C^{\infty}$ for the purposes of the results in this section.

We apply the main result from [10] to our setting.

Theorem 3.8** (Local Hölder estimate).**

Let $\mathcal{L}$ be an integral operator corresponding to a kernel in the class $\mathcal{K}$ (as in Definition 1.1). $f:(-1,0]\times B_{1}\times\mathbb{R}^{d}\to\mathbb{R}$ be a function that solves the equation (3.19) in $Q_{1}$. Then, the following estimate holds

[TABLE]

Here $\delta>0$ and $C$ are constants depending only on dimension and the parameters $\lambda$ and $\Lambda$ of Definition 1.1.

Remark 3.9*.*

Note that Theorem 3.8 and its corollaries below hold for several different choices of the distance function. See Section 2.6.

Proof.

The coercivity conditions in the definition of the class $\mathcal{K}$ of elliptic kernels slightly differ from the coercivity condition imposed in [10]. Let $f:\mathbb{R}^{d}\to\mathbb{R}$ be supported in some ball $B_{\bar{R}}$.

[TABLE]

with $\bar{\Lambda}\simeq\bar{R}^{-2s}$. The other conditions from [10] are satisfied straightforwardly from our assumptions in Definition 1.1. ∎

Note that the right hand side depends on the $L^{\infty}$ norm of $f$ with respect to all values of $v\in\mathbb{R}^{d}$. This is a common inconvenience with nonlocal equations. The result can be easily improved to allow functions $f$ that are unbounded as $|v|\to\infty$. Let $\omega_{0}$ be the majorant function as in (3.3), centered at the origin. That is

[TABLE]

We derive the following improvement of Theorem 3.8.

Corollary 3.10**.**

Let $\mathcal{L}$ be an integral operator corresponding to a kernel in the class $\mathcal{K}$ (as in Definition 1.1). $f:(-1,0]\times B_{1}\times\mathbb{R}^{d}\to\mathbb{R}$ be a function that solves the equation (3.19) in $Q_{1}$. Then, the following estimate holds

[TABLE]

Here $\delta>0$ and $C$ are constants depending only on dimension and the parameters $\lambda$ and $\Lambda$ of Definition 1.1.

Proof.

We consider a $C^{\infty}$ function $\eta:\mathbb{R}^{d}\to[0,1]$ so that $\eta(v)=1$ if $v\in B_{3/2}$ and $\eta(v)=0$ when $v\notin B_{2}$. We apply Theorem 3.8 to the localized function $\tilde{f}(t,x,v)=f(t,x,v)\eta(v)$. We must analyze the equation that $\tilde{f}$ satisfies. We compute directly $\mathcal{L}\tilde{f}$ to get

[TABLE]

where $\tilde{c}=\eta c$ and

[TABLE]

From Theorem 3.8, we have

[TABLE]

It is easy to see that

[TABLE]

It only remains to prove that we have for any $v\in B_{1}$,

[TABLE]

Let us justify this inequality. Arguing as in Lemma 3.1, we have

[TABLE]

Moreover,

[TABLE]

We also have

[TABLE]

This achieves the proof of the corollary. ∎

For convenience, we also state the scaled version of the previous result.

Corollary 3.11**.**

Let $\mathcal{L}$ be an integral operator corresponding to a kernel in the class $\mathcal{K}$ (as in Definition 1.1) and $f:(-R^{2s},0]\times B_{R^{1+2s}}\times\mathbb{R}^{d}$ be a function that solves the equation (3.19) in $Q_{R}$. Then, the following estimate holds

[TABLE]

Here $\delta>0$ and $C$ are constants depending only on dimension and the parameters $\lambda$ and $\Lambda$ of Definition 1.1.

4. Liouville theorem

This section is devoted to the statement and the proof of a theorem of Liouville type.

Theorem 4.1** (Liouville).**

Let $0<\gamma<\min(1,2s)$ and $\alpha=\frac{2s}{1+2s}\gamma$. Assume that $\lfloor 2s+\alpha\rfloor<2s+\alpha^{\prime}<2s+\alpha$ and $\alpha-\alpha^{\prime}<\delta$ where $\delta$ is the constant from Theorem 3.8.

Let $f\in C_{\ell,loc}^{2s+\alpha^{\prime}}((-\infty,0]\times\mathbb{R}^{d}\times\mathbb{R}^{d})$ be a function that satisfies the following conditions.

• (i)

There is a constant $C_{1}>0$ such that for all $R\geq 1$,

[TABLE]

• (ii)

For any $\xi=(h,y,w)\in\mathbb{R}^{1+2d}$, with $h<0$, we define $g(z):=f(\xi\circ z)-f(z)$ or equivalently, $g(t,x,v):=f(t+h,x+y+tw,v+w)-f(t,x,v).$ Then, $g$ solves the equation

[TABLE]

where $\mathcal{L}$ is the operator associated to some kernel $K\in\mathcal{K}$ as defined in (3.1).

Then $f$ is a polynomial of kinetic degree smaller than $2s+\alpha$.

Remark 4.2*.*

Note that the assumption (i) ensures that the tails of $\mathcal{L}g$ are integrable. Indeed, let us take $\beta=\gamma+\varepsilon<\min(1,2s)$ for $\varepsilon$ small. The assumption (i) tells us that

[TABLE]

Note that the condition $\beta<\min(1,2s)$ ensures that the polynomial $q_{z}$ in the definition of $[f]_{C_{\ell}^{\alpha}(z)}$ is the constant $q_{z}(\xi)=f(z)$.

Observe that for $z=(t,x,v)$ and $\xi=(h,y,w)$, we have

[TABLE]

Recalling that $\beta=\gamma+\varepsilon$, we get

[TABLE]

The operator $\mathcal{L}g$ is well defined because this function $\omega$ suffices to bound the expression (3.4). The constant $C_{t,x,\xi}$ depends on $t$, $x$, $\xi$, and the constant in the assumption (i) with $\beta=\gamma+\varepsilon$.

The tails of $\mathcal{L}f$ may not be integrable, and therefore we can only make sense of the equation for $g$, and not for $f$.

Remark 4.3*.*

It is plausible that a version of this Liouville type result holds also for higher values of $\alpha$. In that case, for the equation (4.1) to make sense, we would have to make $g$ a higher order incremental quotient of $f$.

We start with a simpler Liouville type result that is a consequence of the Hölder estimate contained in Theorem 3.8.

Lemma 4.4** (Liouville).**

Let $\delta$ be the constant from Theorem 3.8. Assume $\beta<\delta$ and $f$ is a solution to (3.19) in $(-\infty,0]\times\mathbb{R}^{d}\times\mathbb{R}^{d}$ with $c=0$. Assume further that for all $R\geq 1$,

[TABLE]

then $f$ is constant.

Proof.

We apply Corollary 3.11 in $Q_{R}$ and make $R\to 0$. From our assumption on the growth of $f$, for all $(t,x)\in(-R^{2s},0]\times B_{R^{1+2s}}$ we have that $\omega(t,x,r)\lesssim r^{\beta}$. Thus, we have

[TABLE]

Then, Corollary 3.11 tells us that

[TABLE]

Taking $R\to\infty$, the semi-norm $[f]_{C_{\ell}^{\delta}(Q_{R})}$ converges to zero, and then the function $f$ must be constant. ∎

Proof of Theorem 4.1.

We first claim that it is enough to prove the result assuming that $f\in C^{\infty}$. Indeed, if $f$ is less regular, we can mollify it respecting the Lie group structure then apply the result to the approximate function and pass to the limit.

The remainder of the proof proceeds in several steps.

Step 1: $f$ is constant in $x$. Let $y\in\mathbb{R}^{d}$ and $g(t,x,v):=f(t,x-y,v)-f(t,x,v)$. We apply the assumption (i) with $\beta=2s+\alpha^{\prime}$. Note that $2s+\alpha^{\prime}<1+2s$ by assumption. We get

[TABLE]

Since we assume that $\alpha-\alpha^{\prime}<\delta$, then we can apply Lemma 4.4 and we get that $g$ is constant. Therefore, $f$ must be of the form

[TABLE]

for some constant $a\in\mathbb{R}^{d}$. However, the assumption (i) tells us that for all $R\geq 1$,

[TABLE]

This is only possible if $a=0$ (recall that $a\cdot x$ is a polynomial of order $1+2s>2s+\alpha$). Thus, $f$ is independent of $x$ and from now on we write $f=f(t,v)$.

Step 2: $f_{t}$ is constant. Observe that the kinetic order of $\partial_{t}$ is $2s$. Therefore, $f_{t}$ is well defined since $f\in C_{\ell,loc}^{2s+\alpha^{\prime}}$. Moreover, from the assumption (i) and Lemma 2.7, we deduce that,

[TABLE]

Since $f_{t}(t,v)=\lim_{h\to 0}(f(t+h,v)-f(t,v)))/h$, using (ii) we deduce that

[TABLE]

We omitted the term $v\cdot\nabla_{x}f_{t}$ because it is identically zero.

Using the invariance of the equation by the Lie group action and the fact that $f$ is independent of $x$, we have that for any $(h,w)\in[0,\infty)\times\mathbb{R}^{d}$, the function

[TABLE]

also solves

[TABLE]

Because of (4.6), with $\tilde{\beta}=\alpha^{\prime}$, we get that

[TABLE]

Thus, we obtain that $g$ is constant applying Lemma 4.4. Therefore, $f_{t}$ must be of the form $f_{t}(t,v)=a\cdot v+bt+c$. However, (4.6) implies that the kinetic degree of $f_{t}$ cannot be more than $\alpha$, and therefore $f_{t}$ is a constant.

Since $f$ is independent of $x$ and $f_{t}$ is constant, then $f$ has the form $f=at+\tilde{f}(v)$ for some constant $a$. The function $\tilde{f}$ satisfies

[TABLE]

We are left to prove that $f$ is a polynomial in $v$.

Step 3: $f$ is a polynomial in $v$. The third step is also divided into three cases depending the integer part of $2s+\alpha$. Indeed, the maximum number of terms in the polynomial $\tilde{f}(v)$ will depend of $2s+\alpha$ belonging to the three possible ranges $(0,1)$, $(1,2)$ or $(2,3)$ (recall that $2s+\alpha$ is not an integer).

Let us start by assuming that $2s+\alpha\in(0,1)$. Given any $w\in\mathbb{R}^{d}$, we set $g(v):=\tilde{f}(v+w)-\tilde{f}(v)$. Applying the assumption (i), with $\beta=2s+\alpha^{\prime}$ we get

[TABLE]

Moreover, $g$ solves

[TABLE]

We apply Lemma 4.4 right away. We deduce that $g$ is constant for any $w\in\mathbb{R}^{d}$. Therefore $\tilde{f}$ has the form $\tilde{f}(v)=b\cdot v+c$. However, the assumption (i) with $\beta=0$ implies in this case that $b=0$, so $\tilde{f}$ must be constant. Therefore, in the case $2s+\alpha\in(0,1)$ we conclude that $f(t,x,v)=at+c$ for some constants $a$ and $c$. Thus, $f$ is a polynomial of degree at most $2s$.

In the case $2s+\alpha\in(1,2)$ the function $f$ must be differentiable in $v$ because of the assumption (i) applied with $\beta\in(1,2s+\alpha^{\prime})$. Thus, if we let $f_{j}=\partial_{v_{j}}\tilde{f}$ we get that

• (i’)

There is a $C_{1}>0$ such that for all $\beta\in[0,2s+\alpha^{\prime}-1]$ and $R\geq 1$,

[TABLE]

• (ii’)

For any $w\in\mathbb{R}^{d}$, we define

[TABLE]

Then, $g$ solves

[TABLE]

Therefore, we repeat the proof of the case $2s+\alpha\in(0,1)$ for $f_{j}$ instead of $\tilde{f}$ and get that each partial derivative $f_{j}$ is constant. Therefore, in this case $\tilde{f}$ must be an affine function.

Likewise, in the case $2s+\alpha\in(2,3)$, we apply the argument for $2s+\alpha\in[1,2)$ to each partial derivative $f_{j}=\partial_{v_{j}}\tilde{f}$. In this case we obtain that each $f_{j}$ is affine, and therefore $\tilde{f}$ must be a polynomial in $v$ of degree at most $2$. ∎

5. Blowup argument

In this section, we prove that the Hölder exponent of a solution of a linear equation of the form (1.1) can be improved; moreover, the improvement is quantitative. The result is proved by blowup and compactness. It is first proved for equation with “constant coefficients” (Proposition 5.1) and then proved in the general case (Proposition 5.4).

Proposition 5.1** (Improvement by blow up for “constant coefficients”).**

Let $\alpha$ and $\alpha^{\prime}$ be as in Theorem 4.1. Assume that

[TABLE]

for some function $c_{0}\in C_{\ell}^{\alpha}(Q_{1})$ and some kernel $K\in\mathcal{K}$. Then

[TABLE]

where $C$ only depends on $d$, $s$, $\alpha$, $\alpha^{\prime}$, $\lambda$ and $\Lambda$.

Before proving the proposition, we state a lemma corresponding to [19, Claim 3.2]. Its adaptation to kinetic Hölder spaces is straight forward.

Lemma 5.2** (From $C_{\ell}^{\beta^{\prime}}$ to $C_{\ell}^{\beta}$ with $\beta>\beta^{\prime}$).**

Let $0<\beta^{\prime}<\beta$. Let $\nu$ be the maximum number in $\mathbb{N}+2s\mathbb{N}$ such that $\nu<\beta$. Assume that $\nu<\beta^{\prime}<\beta$. Let $f$ be a continuous function in $C_{\ell}^{\beta^{\prime}}((-\infty,0]\times\mathbb{R}^{d}\times\mathbb{R}^{d})$ and let $C_{0}$ be such that

[TABLE]

Then $[f]_{C_{\ell}^{\beta}(Q_{1/2})}\leq C_{0}$. Moreover, if $\beta\notin\mathbb{N}+2s\mathbb{N}$ (and assuming $f$ smooth), the supremum is attained at some $r>0$ and $z\in\overline{Q_{1/2}}$.

We can now turn to the proof of Proposition 5.1.

Proof of Proposition 5.1.

Without loss of generality, we normalize the problem so that

[TABLE]

Under these conditions, we need to prove that $\|f\|_{C_{\ell}^{2s+\alpha}(Q_{1/2})}\lesssim 1$. We proceed by contradiction. Assuming the opposite, there would exist sequences $f_{j}$, $K_{j}\in\mathcal{K}$ and $c_{j}$ such that,

[TABLE]

The last property holds since we cannot apply Lemma 5.2 uniformly. Thanks to Lemma 5.2, there exists $r_{j}>0$ and $z_{j}\in\overline{Q_{1/2}}$ such that

[TABLE]

In particular, $F_{j}\to+\infty$ and $r_{j}\to 0^{+}$.

We define

[TABLE]

where $q_{j}$ denotes the polynomial expansion of $f_{j}$ at $z_{j}$ as in the definition of $C_{\ell}^{2s+\alpha^{\prime}}(Q_{r_{j}}(z_{j}))$. This sequence $\tilde{f}_{j}$ satisfies the following properties.

• •

Since $F_{j}=r_{j}^{\alpha^{\prime}-\alpha}[f_{j}]_{C_{\ell}^{2s+\alpha^{\prime}}(Q_{r_{j}}(z_{j}))}$, we have

[TABLE]

• •

Since $F_{j}\geq r^{\alpha^{\prime}-\alpha}[f_{j}]_{C_{\ell}^{2s+\alpha^{\prime}}(Q_{r}(z_{j}))}$, for all $r>r_{j}$, we have

[TABLE]

• •

Because of the substraction of the polynomial expansion $q_{j}$, we also have

[TABLE]

• •

By interpolation (Proposition 2.10) between the last two items, we deduce for all $\beta\in(0,2s+\alpha^{\prime}]$,

[TABLE]

For each choice of $\xi\in(-\infty,0]\times\mathbb{R}^{d}\times\mathbb{R}^{d}$, we define

[TABLE]

Condition (5.6) allows us to bound the growth of $g_{j}(t,x,v)$ as $|v|\to\infty$ by a single majorant function $\omega(|v|)$ as in (4.4).

Because of (5.3), each function $g_{j}$ satisfies for $j$ large enough (see the choice of $R_{j}$ below) the equation

[TABLE]

where the operator $\tilde{\mathcal{L}}_{j}$ corresponds to a scaled kernel $\tilde{K}_{j}$ and the source term $c_{j}$ is scaled too,

[TABLE]

In particular, we choose the radius $R_{j}\to+\infty$ such that $Q_{2R_{j}r_{j}}(z_{j})\subset Q_{1}$ and $\xi\circ z\in Q_{2R_{j}}(0)$ for $z\in Q_{R_{j}}(0)$.

Because of (5.1), it is straight forward to verify that,

[TABLE]

Therefore, the right hand side of (5.7) converges to zero over any compact set.

We observe that all the kernels $\tilde{K}_{j}$ belong to the class $\mathcal{K}$. Applying Lemma 3.4, they converge weak-$\ast$ to a kernel $K_{\infty}$ up to a subsequence. Because of Lemma 3.3, $K_{\infty}\in\mathcal{K}$.

We pick $\varepsilon>0$ so that $\lfloor 2s+\alpha\rfloor<2s+\varepsilon<2s+\alpha^{\prime}$. Because of Arzela-Ascoli theorem, we take a subsequence so that $\tilde{f}_{j}$ converges to some function $f:(-\infty,0]\times\mathbb{R}^{d}\times\mathbb{R}^{d}$ locally in $C_{\ell}^{2s+\varepsilon}$. This function $f_{\infty}$ also satisfies (5.6).

Since all the $\tilde{f}_{j}$ are controlled by a single majorizer $\omega$, we can apply Lemma 3.5 and we have that the corresponding function $g_{\infty}=\lim\tilde{g}_{j}$ solves the limit equation from (5.7),

[TABLE]

We are then able to apply the Liouville theorem 4.1 and get that $g_{\infty}$ is a polynomial. However, we subtracted the polynomial expansion of $f_{j}$ in the definition of $\tilde{f}_{j}$, forcing $\tilde{f}_{j}$ to have a vanishing polynomial expansion at [math] of order up to $2s+\alpha^{\prime}$. Since $\tilde{f}_{j}\to f_{\infty}$ in $C_{\ell}^{2s+\varepsilon}$ and $2s+\varepsilon>\lfloor 2s+\alpha\rfloor$, then all derivatives of $f_{\infty}$ at the origin must be zero and therefore $f_{\infty}=0$.

This contradicts (5.5) and the proof is complete. ∎

Remark 5.3*.*

We would like to emphasize the importance of the majorant function $\omega$. In order to do so, we point out that it is possible to follow the same outline of the proof by blowup to obtain the result of Lemma 5.2 under more general assumptions, at the expense of a more complicated proof. Again, we should not overlook the importance of the majorant function $\omega$. For example, with a more complicated function $\omega$ it is possible to derive an estimate of the form

[TABLE]

provided that $\alpha\leq 2s\gamma/(1+2s)$. However, the following estimate is not true for any $\gamma<\alpha$, even in the elliptic case

[TABLE]

If we tried to reproduce the proof of Lemma 5.2 for this last inequality, we would still have scaled functions $\tilde{f}_{j}$ that converge locally uniformly to a function satisfying the assumption (i) in Liouville’s theorem 4.1. We can still construct the functions $\tilde{g}_{j}$ that converge uniformly to a function $g_{\infty}$. But unless we have an appropriate control on the tails of the original functions $f_{j}$, we cannot conclude that the limit function $g_{\infty}$ will satisfy any equation.

We now extend the blowup lemma to the case of “variable coefficients”. In order to do so, we are going to use Assumption 1.5 depending on a constant $A_{0}>0$.

Proposition 5.4** **(Improvement by blowup for variable

coefficients).

Let $\alpha,\alpha^{\prime}$ as in Theorem 4.1. Under the assumptions of Theorem 1.6, we have

[TABLE]

where the constant $C$ depends on dimension, $s$, $\alpha$, $\alpha^{\prime}$, $\lambda$ and $\Lambda$.

Proof.

Let $\eta\in C^{\infty}_{c}((-1,0]\times B_{1}\times\mathbb{R}^{d})$. Assume that $\eta=1$ in $Q_{3/4}$ and that $\eta(z)=0$ whenever $z\notin Q_{1}$.

Let $\tilde{f}(z)=\eta(z)f(z)$. Obviously, we have

[TABLE]

We want to estimate the right hand side that would make $\tilde{f}$ satisfy an equation as in (3.19). We freeze coefficients first:

[TABLE]

and let $\mathcal{L}_{0}$ be the corresponding integro-differential operator. A straight-forward computation shows that for all $z\in Q_{1/2}$,

[TABLE]

where

[TABLE]

Lemma 5.5**.**

$[A(z)]_{C_{\ell}^{\alpha}(Q_{1/2})}\lesssim A_{0}\left(\|f\|_{C_{\ell}^{2s+\alpha}(Q_{1})}+\|f\|_{C_{\ell}^{\gamma}((-1,0]\times B_{1}\times\mathbb{R}^{d})}\right).$**

Proof.

We write $A(z_{1})-A(z_{2})=I_{1}+I_{2}$ with

[TABLE]

As far as $I_{1}$ is concerned, we write $I_{1}=I_{1}^{\text{out}}+I_{1}^{\text{in}}$ with

[TABLE]

For the nonsingular part of $I_{1}$, we simply write

[TABLE]

where we used (3.10), which is a consequence of (1.5).

We now turn to $I_{1}^{\text{in}}$. We write

[TABLE]

We used (3.9) which is a consequence of (1.5).

We now estimate $I_{2}$ thanks to Lemma 3.7. This achieves the proof of the lemma. ∎

We now estimate the $C_{\ell}^{\alpha}(Q_{1/2})$ norm of $B$.

Lemma 5.6**.**

$[B]_{C_{\ell}^{\alpha}(Q_{1/2})}\lesssim\|f\|_{C_{\ell}^{\gamma}((-1,0]\times B_{1}\times\mathbb{R}^{d})}.$**

Proof.

For $z_{1},z_{2}\in Q_{1/2}$, we compute,

[TABLE]

with

[TABLE]

We turn to estimate $J_{1}$. Since $\eta$ is smooth, and in particular $C_{\ell}^{\gamma}$, we can apply Lemma 3.7 and get

[TABLE]

As far as $J_{2}$ is concerned, we get

[TABLE]

This achieves the proof of the estimate for $B$. ∎

Thanks to Lemmas 5.5 and 5.6, we conclude the proof of Proposition 5.4 by applying Proposition 5.1 to $\tilde{f}$ (with $Q_{1}$ replaced with $Q_{\frac{1}{2}}$). ∎

Proof of Theorem 1.6.

Without loss of generality, we perform an initial scaling to make sure that the constant $A_{0}$ is small (it is a subcritical parameter).

We will next prove the slightly stronger estimate

[TABLE]

Scaling and translating the estimate from Proposition 5.4, we get that for any cylinder $Q_{\rho}(z_{0})\subset Q_{1}$,

[TABLE]

For any $z_{0}$ in $Q_{1}$, we choose $\rho=d_{\ell}(z_{0},\partial Q_{1})<1$ so that $Q_{\rho}(z_{0})\subset Q_{1}$.

Note that $A_{0}\rho^{\alpha}\leq A_{0}$ and $\rho^{\gamma}[f]_{C_{\ell}^{\gamma}((-1,0]\times B_{1}\times\mathbb{R}^{d})}\leq[f]_{C_{\ell}^{\gamma}((-1,0]\times B_{1}\times\mathbb{R}^{d})}$. We get

[TABLE]

We use the interpolation result of Proposition 2.10 (see Remark 2.11) to get $[f]_{C_{\ell,\ast}^{2s+\alpha^{\prime}}(Q_{1})}\leq\varepsilon[f]_{C_{\ell,\ast}^{2s+\alpha}(Q_{1})}+C_{\varepsilon}\|f\|_{L^{\infty}(Q_{1})}$. This achieves the proof of the main theorem. ∎