Hypoellipticity of Fediĭ’s type operators under Morimoto’s logarithmic condition

Abstract

We prove hypoellipticity of second order linear operators on $\mathbb {R}^{n+m}$ of the form $L(x,y,D_x,D_y) = L_1(x,D_x) + g(x) L_2(y,D_y)$, where $L_j$, $j=1,2$, satisfy Morimoto’s super-logarithmic estimates $||\log \!\left<\xi \right>^2 \hat{u}(\xi )||^2 \le \varepsilon (L_j u,u) + C_{\varepsilon ,K} ||u||^2$, and g is smooth, nonnegative, and vanishes only at the origin in $\mathbb {R}^n$ to any arbitrary order. We also show examples in which our hypotheses are necessary for hypoellipticity.

Appendix

1.1 Notation

The following multipliers will be frequently used, where as usual multipliers are defined via the Fourier transform $\mathcal {F} \left( m(D)f\right) (\xi ) := m(\xi ) \hat{f}(\xi )$. In particular for any $s\in {\mathbb {R}}$

$$\begin{aligned} \!\left<\xi \right>^s :=(e^2+\xi ^2)^\frac{s}{2} \end{aligned}$$

Then $\left<(\xi ,\eta )\right>^s=\!\left<D\right>^s \in S^s$.

For cut off functions we will use the following notation,

$$\begin{aligned} \phi \Subset \psi \in C^\infty _0(K) \end{aligned}$$

(83)

to mean that $\psi \equiv 1$ on the support of $\phi$, and $\phi \equiv 1$ on a typically predefined compact set.

Norms without subscript will be $L^2$ norms, i.e. $||f|| = ||f||_{L^2}$ and subscripts would mean $H^s$ norms, i.e. $||f||_{s}:=||f||_{H^s} := \left( \int \!\left<\xi \right>^{2s} |\hat{f}(\xi )|^2 d\xi \right) ^{\frac{1}{2}}$.

1.2 Positivity of elliptic operator

First we establish positivity of a second order elliptic operator, which is used throughout the paper.

Lemma 43

Let L be degenerately elliptic, i.e. of the form (2). Then for any compact set $K\Subset {\mathbb {R}}^n$ there is a constant $C_K$, such that for all $u\in C^\infty _0(K)$

$$\begin{aligned} Re(Lu,u) \ge - C_K||u||^2 \end{aligned}$$

(84)

Proof

We consider terms one by one from the second order down to the second.

Let $I_2$ denote all second order terms and integrate by parts:

$$\begin{aligned} I_2:=&-\sum _{j,k} \int a_{jk}(x)\partial _{x_j}\partial _{x_k} u\cdot u dx\\ =&\sum _{j,k} \int a_{jk}(x)\partial _{x_j} u \cdot \partial _{x_k} u dx+\sum _{j,k} \int \partial _{x_k} a_{jk}(x)\partial _{x_j} u \cdot u dx \end{aligned}$$

Observe that $\partial _{x_k} u \cdot u =\frac{1}{2}\partial _{x_k} (u^2)$. With this knowledge in mind integrate the second term by parts

$$\begin{aligned} I_2 = \sum _{j,k} \int a_{jk}(x)\partial _{x_j} u \cdot \partial _{x_k} u dx - \frac{1}{2}\sum _{j,k}\int \partial _{x_j} \partial _{x_k} a_{jk}(x) u \cdot u dx \end{aligned}$$

The non-negative definite property of $a_{jk}\ge 0$ implies the first term above is non-negative. While for the second term we apply the Hölder inequality:

$$\begin{aligned} I_2 \ge 0 -\frac{1}{2}\sum _{j,k}||\partial _{x_j}\partial _{x_k} a_{jk}(x)||_{L^\infty _x}\cdot ||u||^2 \end{aligned}$$

By Holder inequality

$$\begin{aligned} |(a_0 u,u)| \le C_K||u||^2 \end{aligned}$$

(85)

The first and zero-th terms are treated similarly to $\int \partial _{x_k} a_{jk}(x)\partial _{x_j} u \cdot u dx$ and $\int \partial _{x_k} a_{jk}(x)\partial _{x_j} u \cdot u dx$ respectively. $\square$

1.3 Properties of pseudodifferential operators

Proposition 44

(Adjoint $\Psi$DO) Let p be the symbol of the $\Psi$DO P. If $P^{*}$, i.e. $(Pu,v)=(u,P^{*}v), \ \forall u,v\in \mathscr {S}$, then the symbol $p^{*}$ has the following asymptotic expansion

$$\begin{aligned} p^{*}(x,\xi )\sim \sum _{\alpha }\frac{i^{|\alpha |}}{\alpha !}\overline{\partial _{\xi }^{\alpha }\partial _{x}^{\alpha }p(x,\xi )}. \end{aligned}$$

Proof

See e.g. [14] Theorem 2.1.7. $\square$

Proposition 45

(Composition of $\Psi$DO) Let p be the symbol of the $\Psi$DO P, and q—the symbol of the $\Psi$DO Q. Then the symbol r of the composition $P\circ Q$ has the following asymptotic expansion

$$\begin{aligned} r(x,\xi )\sim \sum _{\alpha }\frac{(-i)^{|\alpha |}}{\alpha !}\partial _{\xi }^{\alpha }p(x,\xi )\partial _{x}^{\alpha }q(x,\xi ). \end{aligned}$$

Proof

See e.g. [14] Theorem 2.1.7. $\square$

Lemma 46

Let P be a pseudodifferential operator of order m such that $P+P^{*}$ is of order $m-l$. Then for any $v\in H^{m}$ we have

$$\begin{aligned} Re(Pv,v)=\left( \frac{P+P^{*}}{2}v,v\right) \le C||v||^{2}_{\frac{m-l}{2}}. \end{aligned}$$

(86)

Proof

We can write

$$\begin{aligned} Re(Pv,v)= & {} \frac{1}{2}\left( (Pv,v)+\overline{(Pv,v)}\right) \\= & {} \frac{1}{2}\left( (Pv,v)+\overline{(v,P^{*}v)}\right) =\frac{1}{2}\left( (Pv,v)+(P^{*}v,v)\right) , \end{aligned}$$

which establishes the first equality in (86). The inequality in (86) then follows from the boundedness of $\Psi$DO, see [14]. $\square$

1.4 Regularization $S_{\delta _1}$

Lemma 47

Let $S_{\delta _1}$ be defined by the symbol $s_{\delta _1}=\!\left<\delta _1\cdot ( \xi ,\eta )\right>^{-(N+s+3)}$. We then have

• 1. $S_{\delta _1}$ is an operator of order $-N-s-3$ whose seminorms depend on $\delta _1$;

• 2. $S_{\delta _1}$ is an operator of order 0 uniformly in $\delta _1$, i.e. $\limsup _{\delta _1\rightarrow 0}|s_{\delta _1}|<\infty$;

• 3. The operator with the symbol $\frac{\partial _{\xi ,\eta }^{\alpha }s_{\delta _1}}{s_{\delta _1}}$ for $|\alpha |=1$ is uniformly of order $-1$. Inductively, same argument can be repeated for $|\alpha |>1$.

Proof

• 1. This follows immediately from the definition.

• 2. It is easy to check that $|s_{\delta _1}|\le 1\ \forall \xi ,\eta$.

• 3. Differentiating we get

  $$\begin{aligned} \frac{\partial _{\xi _{k}}s_{\delta _1}}{s_{\delta _1}}=\frac{\partial _{\xi _{k}}(1+\delta _1^{2}|\xi |^{2}+\delta _1^{2}|\eta |^{2})^{\frac{-(N+s+3)}{2}}}{(1+\delta _1^{2}|\xi |^{2}+\delta _1^{2}|\eta |^{2})^{\frac{-(N+s+3)}{2}}} =-(N+s+3)\frac{\delta _1^{2}\xi _k}{(1+\delta _1^{2}|\xi |^{2}+\delta _1^{2}|\eta |^{2})} \end{aligned}$$

  which is easily seen to be an operator of order $-1$ uniformly in $\delta _1$.

$\square$