On the Gevrey regularity for Sums of Squares of vector fields, study of some models

TL;DR

This paper investigates the Gevrey regularity properties of sums of squares of vector fields with real analytic coefficients, providing detailed micro-local analysis and some partial regularity results.

Contribution

It offers new insights into the micro-local Gevrey regularity of sums of squares of vector fields, including partial regularity findings.

Findings

Detailed micro-local Gevrey regularity analysis

Partial regularity results for sums of squares

Insights into real analytic coefficient cases

Abstract

The micro-local Gevrey regularity of a class of "sums of squares" with real analytic coefficients is studied in detail. Some partial regularity result is also given.

Equations

$$D_1^2+x_1^{2(p-1)}{D}_2^2+x_1^{2(q-1)}{D}_3^2+x_1^{2(r-1)}{x}_2^{2k}{D}_4^2+x_1^{2(r+\ell -1)}{D}_4^2,$$

$$D_1^2+x_1^{2(p-1)}{D}_2^2+x_1^{2(q-1)}{D}_3^2+x_1^{2(r-1)}{x}_3^{2k}{D}_4^2+x_1^{2(r+\ell -1)}{D}_4^2,$$

$$D_1^2+x_1^{2(p-1)}{D}_2^2+x_1^{2(q-1)}{D}_3^2+x_1^{2(r-1)}{x}_2^{2k}{D}_4^2+x_1^{2(f-1)}{x}_3^{2\ell }{D}_4^2+x_1^{2(f+e-1)}{D}_4^2$$

$$P(x,D)=D_1^2+\sum _{j=2}^{n-1}{x}_1^{2(r_j-1)}{D}_j^2+\left(x_1^{2(r_n-1)}{x}_i^{2k}+x_1^{2(r_n+\ell -1)}\right)D_n^2,2\le i\le n-1,$$

$$Char(P_1)=\{(x;\xi )\in T^{\ast }{\mathbb{R}}^4\setminus \{0\}:x_1=0,{\xi }_1=0\}.$$

$$Char(P_1)={\Sigma }_{p,\pm }\cup {\Sigma }_{q,\pm }\cup {\Sigma }_{r,\pm }\cup {\Sigma }_{r+kp,\pm },$$

$$\begin{array}[]{rcl}\Sigma_{p,\pm}&=&\{(x;\xi)\in T^{*}\mathbb{R}^{4}\setminus\{0\}:\,x_{1}=0,\,\xi_{1}=0,\,\xi_{2}\gtrless 0\},\\[5.0pt] \Sigma_{q,\pm}&=&\{(x;\xi)\in T^{*}\mathbb{R}^{4}\setminus\{0\}:\,x_{1}=0,\,\xi_{1}=0,\,\xi_{2}=0,\,\xi_{3}\gtrless 0\},\\[5.0pt] \Sigma_{r,\pm}&=&\{(x;\xi)\in T^{*}\mathbb{R}^{4}\setminus\{0\}:\,x_{1}=0,\,\xi_{1}=0,\,\xi_{2}=0,\,\xi_{3}=0,\,\xi_{4}x_{2}\neq 0\}\\[5.0pt] \Sigma_{r+pk,\pm}&=&\{(x;\xi)\in T^{*}\mathbb{R}^{4}\setminus\{0\}:\,x_{1}=0,\,\xi_{1}=0,\,\xi_{2}=0,\,\xi_{3}=0,\,x_{2}=0,\,\xi_{4}\gtrless 0\}.\end{array}$$

$$\|u\|^{2}_{\frac{1}{r+kp}}+\sum_{j=1}^{5}\|X_{j}u\|^{2}\leq C\big{(}|\langle P_{1}u,u\rangle|+\|u\|^{2}\big{)}.$$

$$\|u\|^{2}_{\frac{1}{r}}+\sum_{j=1}^{5}\|X_{j}u\|^{2}\leq C\big{(}|\langle Pu,u\rangle|+\|u\|^{2}\big{)},$$

$$\Vert {\phi }_N{D}_4^{N}u{\Vert }_{\frac{1}{r+kp}}^2+\sum _{j=1}^5\Vert X_j{\phi }_N{D}_4^{N}u{\Vert }^2\le C\left(|⟨P_1{\phi }_N{D}_4^{N}u,{\phi }_N{D}_4^{N}u⟩|+\Vert {\phi }_N{D}_4^{N}u{\Vert }^2\right).$$

$$\begin{array}{rl}& ⟨{\phi }_N{D}_4^N{P}_{1}u,{\phi }_N{D}_4^{N}u⟩+\sum _{j=1}^5⟨[X_j^2,{\phi }_N{D}_4^N]u,{\phi }_N{D}_4^{N}u⟩\\ & =2\sum _{j=1}^5⟨[X_j,{\phi }_N{D}_4^N]u,X_j{\phi }_N{D}_4^{N}u⟩+\sum _{j=1}^5⟨[[X_j,{\phi }_N{D}_4^N],X_j]u,{\phi }_N{D}_4^{N}u⟩\\ & +⟨{\phi }_N{D}_4^N{P}_{1}u,{\phi }_N{D}_4^{N}u⟩.\end{array}$$

$$\begin{array}{rl}& |⟨[X_4,{\phi }_N{D}_4^N]u,X_4{\phi }_N{D}_4^{N}u⟩|+|⟨[[X_4,{\phi }_N{D}_4^N],X_4]u,{\phi }_N{D}_4^{N}u⟩|\\ & =2|⟨x_1^{r-1}{x}_2^k{\phi }_N^{(1)}{D}_4^{N}u,X_4{\phi }_N{D}_4^{N}u⟩|+|⟨x_1^{2(r-1)}{x}_2^{2k}{\phi }_N^{(2)}{D}_4^{N}u,{\phi }_N{D}_4^{N}u⟩|.\end{array}$$

$$\begin{array}{rl}|⟨x_1^{r-1}{x}_2^k{\phi }_N^{(1)}{D}_4^{N}u,X_4{\phi }_N{D}_4^{N}u⟩|& \le \sum _{j=1}^N{C}_j\Vert X_4{\phi }_N^{(j)}{D}_4^{N-j}u{\Vert }^2\\ & +\sum _{j=1}^{N+1}\frac{1}{C_j}\Vert X_4{\phi }_N{D}_4^{N}u{\Vert }^2+C_{N+1}\Vert {\phi }_N^{(N+1)}u{\Vert }^2,\end{array}$$

$$\begin{array}{rl}|⟨x_1^{2(r-1)}{x}_2^{2k}{\phi }_N^{(2)}{D}_4^{N}u,{\phi }_N{D}_4^{N}u⟩|& \le \frac{1}{2N^2}\Vert X_4{\phi }_N^{(2)}{D}_4^{N-1}u{\Vert }^2+\frac{N^2}{2}\Vert X_4{\phi }_N{D}_4^{N-1}u{\Vert }^2\\ & +|⟨x_1^{r-1}{x}_2^k{\phi }_N^{(2)}{D}_4^{N-1}u,X_4{\phi }_N^{(1)}{D}_4^{N-1}u⟩|\\ & +|⟨\frac{1}{N}{x}_1^{r-1}{x}_2^k{\phi }_N^{(3)}{D}_4^{N-1}u,N{X}_4{\phi }_N{D}_4^{N-1}u⟩|\\ & +|⟨x_1^{2(r-1)}{x}_2^{2k}{\phi }_N^{(3)}{D}_4^{N-1}u,{\phi }_N^{(1)}{D}_4^{N-1}u⟩|.\end{array}$$

$$|⟨[X_3,{\phi }_N{D}_4^N]u,X_3{\phi }_N{D}_4^{N}u⟩|\le C\Vert x_1^{q-1}{\phi }_N^{(1)}{D}_4^{N}u\Vert +\frac{1}{C}\Vert X_3{\phi }_N{D}_4^{N}u\Vert .$$

$$\Vert x_1^{q-1}{\phi }_N^{(1)}{D}_4^{N}u\Vert \le \Vert x_1^{q-1}{\phi }_N^{(1)}\left(1-{\chi }_N(N^{-1}{D}_4)\right)D_4^{N}u\Vert +\Vert x_1^{q-1}{\phi }_N^{(1)}{\chi }_N(N^{-1}{D}_4)D_4^{N}u\Vert .$$

$$\Vert x_1^{q-1}{\phi }_N^{(1)}\left(1-{\chi }_N(N^{-1}{D}_4)\right)D_4^{N}u\Vert \le C^{N+1}{N}^N,$$

$$\left[{\omega }_N\left(N^{-1}D\right)D^{\theta },{\phi }_N(x)\right]{\chi }_N\left(N^{-1}D\right)D^{N-\theta }=\sum _{j=1}^N{a}_{N,j}(x,D){\chi }_N\left(N^{-1}D\right)D^N,$$

$$|{\partial }_{\xi }^{\alpha }{a}_{N,k}(x,\xi )|\le C_a^{j+1}{N}^{j+\alpha }{\xi }^{-k-\alpha },1\le j\le N,\alpha \le N.$$

$$a_{N,k}(x,D){\chi }_N\left(N^{-1}D\right)D^N=\frac{\theta (\theta -1)\cdots (\theta -j+1)}{j!}{D}_x^j{\phi }_N(x){\chi }_N\left(N^{-1}D\right)D^{N-j}.$$

$$\begin{array}{rl}\Vert x_1^{q-1}{\phi }_N^{(1)}{\chi }_N(N^{-1}{D}_4)D_4^{N}u\Vert \le & \Vert x_1^{q-1}{\phi }_N^{(1)}{\chi }_N(N^{-1}{D}_4)D_4^{N-\frac{1}{r+kp}}u{\Vert }_{\frac{1}{r+kp}}\\ & +\sum _{j=1}^{N-1}{c}_j\Vert x_1^{q-1}{\phi }_N^{(j+1)}{\chi }_N(N^{-1}{D}_4)D_4^{N-j}u\Vert \\ & +\Vert x_1^{q-1}{a}_{N,N}(x,D){\chi }_N(N^{-1}{D}_4)D_4^{N}u\Vert .\end{array}$$

$$c_j\Vert x_1^{q-1}{\phi }^{(j+1)}{\chi }_N(N^{-1}{\xi }_4)D_4^{N-j-1/(r+kp)}u{\Vert }_{\frac{1}{r+kp}},$$

$$\begin{array}{rl}|⟨[[X_3,{\phi }_N{D}_4^N],X_3]u,{\phi }_N{D}_4^{N}u⟩|& =|⟨x_1^{2(q-1)}{\phi }_N^{(2)}{D}_4^{N}u,{\phi }_N{D}_4^{N}u⟩\\ & \le \frac{1}{2N^2}\Vert x_1^{q-1}{\phi }_N^{(2)}{D}_4^{N}u{\Vert }^2+\frac{N^2}{2}\Vert x_1^{q-1}{\phi }_N{D}_4^{N}u{\Vert }^2.\end{array}$$

$$\begin{array}{rl}& \Vert x_1^{q-1}{\phi }_N^{(1)}{\chi }_N(N^{-1}{\xi }_4)D_4^{N-\frac{1}{r+kp}}u{\Vert }_{\frac{1}{r+kp}}^2+\sum _{j=1}^5\Vert X_j{x}_1^{q-1}{\phi }_N^{(1)}{\chi }_N(N^{-1}{\xi }_4)D_4^{N-\frac{1}{r+kp}}u{\Vert }^2\\ & \le \Vert x_1^{q-1}{\phi }_N^{(1)}{\chi }_N(N^{-1}{\xi }_4)D_4^{N-\frac{1}{r+kp}}Pu{\Vert }^2+\Vert x_1^{q-1}{\phi }_N^{(1)}{\chi }_N(N^{-1}{\xi }_4)D_4^{N-\frac{1}{r+kp}}u{\Vert }^2\\ & +2\sum _{j=1}^5|⟨X_j[X_j,x_1^{q-1}{\phi }_N^{(1)}]{\chi }_N(N^{-1}{\xi }_4)D_4^{N-\frac{1}{r+kp}}u,x_1^{q-1}{\phi }_N^{(1)}{\chi }_N(N^{-1}{\xi }_4)D_4^{N-\frac{1}{r+kp}}u⟩|\\ & +\sum _{j=1}^5|⟨[X_j,[X_j,x_1^{q-1}{\phi }_N^{(1)}]]{\chi }_N(N^{-1}{\xi }_4)D_4^{N-\frac{1}{r+kp}}u,x_1^{q-1}{\phi }_N^{(1)}{\chi }_N(N^{-1}{\xi }_4)D_4^{N-\frac{1}{r+kp}}u⟩|.\end{array}$$

$$\begin{array}{rl}& C(q-1{)}^2\Vert x_1^{q-2}{\phi }_N^{(1)}{\chi }_N(N^{-1}{\xi }_4)D_4^{N-\frac{1}{r+kp}}u{\Vert }^2+2C\Vert x_1^{2(q-1)}{\phi }_N^{(2)}{\chi }_N(N^{-1}{\xi }_4)D_4^{N-\frac{1}{r+kp}}u{\Vert }^2\\ & +\frac{1}{N^2}\Vert x_1^{2(q-1)}{\phi }_N^{(3)}{\chi }_N(N^{-1}{\xi }_4)D_4^{N-\frac{1}{r+kp}}u{\Vert }^2+N^2\Vert x_1^{2(q-1)}{\phi }_N^{(1)}{\chi }_N(N^{-1}{\xi }_4)D_4^{N-\frac{1}{r+kp}}u{\Vert }^2\\ & +2C\Vert x_1^{r+q-2}{x}_2^k{\phi }_N^{(2)}{\chi }_N(N^{-1}{\xi }_4)D_4^{N-\frac{1}{r+kp}}u{\Vert }^2\\ & +2C\Vert x_1^{r+\ell +q-2}{\phi }_N^{(2)}{\chi }_N(N^{-1}{\xi }_4)D_4^{N-\frac{1}{r+kp}}u{\Vert }^2\\ & +|⟨x_1^{2(r-1)+q-1}{x}_2^{2k}{\phi }_N^{(3)}{\chi }_N(N^{-1}{\xi }_4)D_4^{N-\frac{1}{r+kp}}u,x_1^{q-1}{\phi }_N^{(1)}{\chi }_N(N^{-1}{\xi }_4)D_4^{N-\frac{1}{r+kp}}u⟩\\ & +|⟨x_1^{2(r+\ell -1)+q-1}{\phi }_N^{(3)}{\chi }_N(N^{-1}{\xi }_4)D_4^{N-\frac{1}{r+kp}}u,x_1^{q-1}{\phi }_N^{(1)}{\chi }_N(N^{-1}{\xi }_4)D_4^{N-\frac{1}{r+kp}}u⟩\end{array}$$

$$\begin{array}{rl}\Vert x_1^{q-2}{\phi }_N^{(1)}{\chi }_N(N^{-1}{D}_4)D_4^{N-\frac{1}{r+kp}}u\Vert \le & \Vert x_1^{q-2}{\phi }_N^{(1)}{\chi }_N(N^{-1}{D}_4)D_4^{N-\frac{2}{r+kp}}u{\Vert }_{\frac{1}{r+kp}}\\ & +\sum _{j=1}^{N-1}{c}_j\Vert x_1^{q-2}{\phi }_N^{(j+1)}{\chi }_N(N^{-1}{D}_4)D_4^{N-j-\frac{1}{r+kp}}u\Vert \\ & +\Vert x_1^{q-2}{a}_{N,N}(x,D){\chi }_N(N^{-1}{D}_4)D_4^{N-\frac{1}{r+kp}}u\Vert .\end{array}$$

$$c_j\Vert x_1^{q-1}{\phi }^{(j+1)}{\chi }_N(N^{-1}{D}_4)D_4^{N-j-\frac{2}{(r+kp)}}u{\Vert }_{\frac{1}{r+kp}},$$

$$\Vert x_1^{q-j-1}{\phi }_N^{(1)}{\chi }_N(N^{-1}{\xi }_4)D_4^{N-\frac{j+1}{r+kp}}u{\Vert }_{\frac{1}{r+kp}}^2.$$

$$C^s\Vert {\phi }_N^{(s)}{\chi }_N(N^{-1}{\xi }_4)D_4^{N-s\frac{q}{r+kp}}u{\Vert }_{\frac{1}{r+kp}}^2.$$

$$\Vert x_1^{h(q-1)}{\phi }_N^{(h)}{\chi }_N(N^{-1}{\xi }_4)D_4^{N-\frac{h-1}{r+kp}}u{\Vert }^2.$$

Full text

1. I Introduction

2. II Proof of Theorem I.1

3. II.1 Gevrey Regularity for $P_{1}(x,D)$.

4. II.2 Gevrey Regularity for $P_{2}(x;D)$

5. III Proof of Theorem I.2

6. IV On the partial regularity of the operators $\textit{{P}}_{\!\text{{1}}}$ and $\textit{{P}}_{\!\text{{2}}}$

7. V Additional material: the $\textit{{n}}-$dimensional case

Abstract. The local and micro-local Gevrey hypoellipticity of a class of “sums of squares” with real analytic coefficients is studied in detail. Some partial regularity result is also given.

I Introduction

The purpose of this paper is to discuss the Gevrey hypoellipticity properties of three model operators that are sums of squares of vector fields in four dimensions. The operators have analytic coefficients and verify the Hörmander condition: the Lie algebra generated by the vector fields as well as by their commutators has, in every point, dimension equal to the dimension of the ambient space. Hence in view of the celebrated Hörmander theorem, H67 , the operators are $C^{\infty}$-hypoelliptic.

Let $P(x;D)=\sum_{1}^{k}X_{j}^{2}(x,D)$, $X_{j}(x,D)$ vector fields with real analytic coefficients on $\Omega$ open subset $\mathbb{R}^{n}$. We say that $P$ is $C^{\infty}\left(G^{r}\right)$-hypoelliptic, $r\geq 1$, in $\Omega$ if for every $U$ open subset of $\Omega$ and every $u\in\mathscr{D}^{\prime}\left(U\right)$, $Pu\in C^{\infty}(U)$ $\left(G^{r}(U)\right)$ implies $u\in C^{\infty}(U)$ $\left(G^{r}(U)\right)$. When $r=1$ we say that $P$ is analytic hypoelliptic. We recall that $G^{r}(U)$ denotes the $r-$Gevrey class of function on $U$: a $C^{\infty}-$function, $f$, on $U$ belongs to $G^{r}(U)$, $1\leq r\leq\infty$, if for every $K$ compact subset of $U$ there is a constant $C_{K}$ such that $\displaystyle|\partial^{\alpha}f(x)|\leq C_{K}^{|\alpha|+1}(\alpha!)^{r}$ for every $\alpha\in\mathbb{N}^{n}$ and $x\in K$.

Derridj showed in (D-71) that for $P$ as above, the Hörmander condition is necessary for the analytic hypoellipticity but it is not sufficient. An example of operator sum of squares of real analytic vector fields satisfying the Hörmander condition but not analytic hypoelliptic was given by Baouendi and Goulaouic in (BG-72). At the present, there aren’t general analytic hypoellipticity results. Some results, in this direction, were obtained by Treves, Tr-78, Tartakoff, Tart-80, and Albano and Bove, (AB-13). For completeness, we recall that, with regard to the Gevrey regularity, if no additional assumption is made on the operator $P$, the (local) optimal characterization was obtained by Derridj and Zuily, DZ . In 1999 Treves formulated a conjecture which related the analytic hypoellipticity with geometrical properties of the characteristic variety of $P$, see Tr-99 and Tr-2006 .

In recent papers Albano, Bove and Mughetti, ABM-TrC1-2016 , and Bove and Mughetti, Bove_Mughetti-TrC2-2016 , showed that the sufficient part of the Treves’ conjecture does not hold neither locally nor microlocally. More precisely in ABM-TrC1-2016 and Bove_Mughetti-TrC2-2016 the authors produced and studied the first models which are not consistent with the Treves conjecture, Tr-2006 . However, contrary to the cases of ABM-TrC1-2016 and Bove_Mughetti-TrC2-2016 , the operators studied here have no exceptional strata because the symbols do not depend on the tangent variables of the “inner most” stratum.

Our results can be stated as follows:

Theorem I.1**.**

Let $P_{1}(x,D)$ the sum of squares given by

[TABLE]

where $p,q,r,k$ and $\ell$ are positive integers such that $p<q<r$ and $pk<\ell$ and $P_{2}(x,D)$ the sum of squares given by

[TABLE]

where $p,q,r,k$ and $\ell$ are positive integers such that $p<q<r$ and $qk<\ell$. We have:

• i)

$P_{1}(x,D)$* is $G^{s}$-hypoelliptic with s=\sup\Big{\{}\frac{r+kp}{q},\frac{r}{p}\Big{\}}.*

• ii)

$P_{2}(x,D)$* is $G^{s}\!$-hypoelliptic with $s\!=\frac{r+kq}{p}$.*

The strategy used to obtain the above results shows, without particular technical trouble, that:

Remark I.1**.**

If $pk\geq\ell$ then $P_{1}$ is $G^{s}$-hypoelliptic with s=\sup\Big{\{}\frac{r+\ell}{q},\frac{r}{p}\Big{\}} and $P_{2}$ is $G^{s}$-hypoelliptic with $s=\frac{r+\ell}{p}$.

We recall that by the result of Derridj and Zuily, DZ , $P_{1}$ is $(r+kp)$-Gevrey hypoelliptic and $P_{2}$ is $(r+kq)$-Gevrey hypoelliptic when $kq<\ell$ and they are both $(r+\ell)$-Gevrey hypoelliptic when $kp\geq\ell$.

Theorem I.2**.**

Let the operator $P_{3}(x,D)$ be given by

[TABLE]

in $\Omega$, open neighborhood of the origin in $\mathbb{R}^{4}$, where $p,q,r,k,f,\ell$ and $e$ are positive integers such that $p<q<r<f$ and $e>\sup\{pk,q\ell\}$. We have: $P_{3}$ is $G^{s}$-hypoelliptic, with s=\displaystyle\sup\Big{\{}\frac{r+kp}{q},\frac{r}{p}\Big{\}} if $f>r+kp$ and s=\displaystyle\sup\Big{\{}\frac{f}{q},\frac{r}{p}\Big{\}} if $f\leq r+kp$.

We point out that, in accordance with the results in DZ , the operator $P_{3}$ is $(r+kp)$-Gevrey hypoelliptic if $f>r+kp$ and $\inf\{r+kp,f+q\ell\}$-Gevrey hypoelliptic if $f<r+kp$.

The strategy used to proof the Theorem I.2 shows, without particular technical trouble, that:

Remark I.2**.**

If $e<\sup\{pk,q\ell\}$ we can distinguish two cases: $f+e<r+kp$, the operator $P_{3}$ is a generalization of the Oleĭnik-Radkevič operator and it is $G^{\frac{f+e}{p}}$-hypoelliptic and $kp<e<lq$, i. e. $r+kp<f+e$, $P_{3}$ is a generalization of the operator $P_{2}$, it is $G^{s}$-hypoelliptic with $s=\sup\{\frac{r+kp}{q},\frac{r}{p}\}$.

Remark I.3**.**

If $p<q<f<r$ and $e>\sup\{r+pk,f+q\ell\}$ we can distinguish two cases: $f+q\ell>r$, $P_{3}$ is $G^{\frac{r}{p}}$-hypoelliptic, and $f+q\ell<r$, $P_{3}$ is $G^{\frac{f+q\ell}{p}}$-hypoelliptic.

Even if, at the present, the proof of the optimality of the operators $P_{1}$, $P_{2}$ and $P_{3}$ is an open problem, we think that the Gevrey regularities obtained are optimal.

Remark I.4**.**

The results stated above can be extended to the operators

[TABLE]

defined in $\Omega$, open neighborhood of the origin in $\mathbb{R}^{n}$, where $r_{j}$, $j=1,\dots,n$, $k$ and $\ell$ are positive integers such that $r_{1}<r_{2}<\dots<r_{n}$. We have: if $kr_{i}<\ell$, $P(x,D)$ is $G^{s}$-hypoelliptic with s=\displaystyle\sup\Big{\{}\frac{r_{n}+kr_{2}}{r_{3}},\frac{r_{n}}{r_{2}}\Big{\}} if $i=2$ and $s=\displaystyle\frac{r_{n}+kr_{i}}{r_{2}}$ if $i\neq 2$; if $kr_{i}\geq\ell$, $P(x,D)$ is $G^{s}$-hypoelliptic with s=\displaystyle\sup\Big{\{}\frac{r_{n}+\ell}{r_{3}},\frac{r_{n}}{r_{2}}\Big{\}} if $i=2$ and $s=\displaystyle\frac{r_{n}+\ell}{r_{2}}$ if $i\neq 2$. This situation does not present additional difficulties compared to that we are going to handle.

In the last section we analyze the partial regularity, that is, following the ideas in BT , we study the non-isotropic Gevrey regularity of $P_{1}$ and $P_{2}$. In addiction to give a more precise characterization of the local regularity of the operators in a neighborhood of the origin, the purpose is to make in evidence further differences with the Oleĭnik-Radkevič operator: as shown the operator $P_{1}$, both in the case $pk<\ell$ that in the case $pk\geq\ell$, does not have directions with analytic growth, thing which, on the contrary, occurs both in $P_{2}$ and in the Oleĭnik-Radkevič operator.

Acknowledgements:

The author was partially supported by a postdoctoral fellowship from FAPESP, Grant 2013/08238-6. The author is also happy to thank the “Instituto de Matemática e Estatística” of the University of São Paulo for its generous and kind hospitality and in particular the research group on Partial Differential Equations and Complex Analysis for the excellent working conditions.

II Proof of Theorem I.1

II.1 Gevrey Regularity for $P_{1}(x,D)$.

The characteristic variety of $P_{1}$ is:

[TABLE]

Following the ideas in Tr-2006 and BoTr it can be seen as the disjoint union of analytic submanifolds, strata:

[TABLE]

where

[TABLE]

By the results of Derridj and Zuily, DZ , and Rothschild and Stein, RS , the operator $P_{1}$ has the following sub-elliptic estimate with loss of $2(1-1/(r+kp))$ derivatives:

[TABLE]

Here $X_{1}=D_{1}$, $X_{2}=x_{1}^{p-1}D_{2}$, $X_{3}=x_{1}^{q-1}D_{3}$, $X_{4}=x_{1}^{r-1}x_{2}^{k}D_{4}$, $X_{5}=x_{1}^{r+\ell-1}D_{4}$, $\|\cdot\|_{s}$ denotes the $H^{s}$ Sobolev norm and $\|\cdot\|=\|\cdot\|_{0}$ denotes the $L^{2}$ norm on the fixed open set $\Omega$.

To study the regularity of the solutions we estimate the high order derivatives of the solutions in $L^{2}$ norm. As a matter of fact we estimate a suitable localization of a high derivative using the above estimate. For $x_{1}\neq 0$ the operator $P_{1}$ is elliptic and we shall not examine this region, elliptic operators are Gevrey hypoelliptic in any class $G^{s}$ for $s\geq 1$.

Let $\varphi_{N}(x)$ be a cutoff function of Ehrenpreis-Hörmander type: $\varphi_{N}\in C^{\infty}_{0}\left(\Omega\right)$ non negative such that $\varphi_{N}\equiv 1$ on $\Omega_{0}$, $\Omega_{0}$ neighborhood of the origin compactly contained in $\Omega$, and exists a constant $C$ such that for every $|\alpha|\leq 2(r+pk)N$, $\alpha\in\mathbb{N}^{4}$, we have $|D^{\alpha}\varphi_{N}|\leq C^{|\alpha|+1}N^{|\alpha|}$.

We may assume that $\varphi_{N}$ is independent of the $x_{1}$-variable since every $x_{1}$-derivative landing on $\varphi_{N}$ would leave a cut off function supported where $x_{1}$ is bounded away from zero, where the operator is elliptic. Moreover we may assume that $\varphi_{N}$ is independent of the $x_{2}$-variable since every $x_{2}$-derivative landing on $\varphi_{N}$ would leave a cut off function supported where $x_{2}$ is bounded away from zero, in this region the operator satisfies the Hörmander-Lie algebra condition at the step $r$. The operator $P_{1}$, in this region, has the following estimate with loss of $2(1-1/r)$ derivatives:

[TABLE]

where $u\in C^{\infty}_{0}(K)$ with $K\cap\{x_{2}=0\}=\emptyset$. In this region the operator is a generalization of the Oleĭnik-Radkevič operator then $P_{1}$ is $G^{r/p}$-hypoelliptic and not better, for more details see C and BT . Then, we can conclude that if $v$ solves the equation $P_{1}v=f$ and $f$ is analytic then the points $\displaystyle\rho_{{}_{1}}\in\Sigma_{r}$ does not belong to $WF_{\frac{r}{p}}(v)$.

Now, we are interested to the microlocal regularity in $\displaystyle\rho_{{}_{0}}\in\Sigma_{r+kp}$. To obtain this it is sufficient to study the microlocal regularity of $P_{1}$ in $(0;e_{4})$. Indeed the microlocal regularity in a generic point $\displaystyle\rho_{{}_{0}}$ can be obtained following the same strategy below with the only difference that the cut-off function $\varphi_{N}(x)$ will be identically equal to 1 in $\Omega_{0}$ neighborhood of $\pi_{x}(\rho_{{}_{0}})=(0,0,x_{3}^{0},x_{4}^{0})$, where $\pi_{x}$ is the projection in the space variables. Thus since we are interested to the microlocal regularity of $P_{1}$ in $(0;e_{4})$ we take $\varphi_{N}(x)=\varphi_{N}(x_{3},x_{4})$.

We replace $u$ by $\varphi_{N}(x)D_{4}^{N}u$ in ( II.4). We have

[TABLE]

The scalar product in the right hand side leads to

[TABLE]

The last term is trivial to estimate since $P_{1}u$ is analytic; we may assume without loss of generality, that is zero. Since $\varphi_{N}$ depends only by $x_{3}$ and $x_{4}$ we must analyze the commutators with $X_{3}$, $X_{4}$ and $X_{5}$. Before to give the general form of the terms which appear inside of the iterating process we begin to analyze some particular situations.

Case $X_{4}$. We have

[TABLE]

The first term, we have

[TABLE]

the constants $C_{j}$ are arbitrary, we make the choice $C_{j}=\varepsilon^{-1}2^{j}$, $\varepsilon$ suitable small positive constant. The terms of the form $C_{j}^{-1}\|X_{4}\varphi_{N}D_{4}^{N}u\|^{2}$ can be absorbed on the right hand side of (II.5). We have $\|\varphi^{(N+1)}_{N}u\|\leq C^{N+1}\alpha!,$ the analytic growth. Finally we observe that the terms in the first sum have the same form as $\|X_{4}\varphi_{N}D_{4}^{N}u\|^{2}$ where one or more $x_{4}$-derivatives have been shifted from $u$ to $\varphi_{N}$; on these terms we can take maximal advantage from the sub-elliptic estimate restarting the process.

With regard to the second term on the right hand side of (II.7) we have

[TABLE]

The last term is the same of the left hand side in which one $x_{4}$-derivative has been shifted from $u$ to $\varphi_{N}$ on both side, we can restart the above process. On the first two terms we can take maximal advantage from the sub-elliptic estimate restarting the process. We point out that the “ weight” $N$ introduced above helps to balance the number of $x_{4}$-derivatives on $u$ with the number of derivatives on $\varphi_{N}$, we take the factor $N$ as a derivative on $\varphi_{N}$ and $N^{-1}\varphi^{(2)}_{N}$ as $\varphi^{(1)}_{N}$. The other two terms have the same form of the term on the left hand side of (II.8), the second one with the help of the weight $N$, we can handled both in the same way.

The same strategy can be used to handle the case involving the field $X_{5}$.

The case $X_{3}$. We have

[TABLE]

The second term can be absorbed on the left hand side of (II.5), if $C^{-1}$ is chosen small enough. Since the first term does not have sufficient power of $x_{1}$ to take maximal advantage from the sub-elliptic estimate, we will use the sub-ellipticity. To do this we will pull back $D_{4}^{1/(r+kp)}$. Let $\chi_{N}(\xi_{4})$ be an Ehrenpreis-Hörmander cutoff function such that $\chi_{N}$ is $C^{\infty}(\mathbb{R})$ non negative function such that $\chi_{N}=0$ for $\xi_{4}<3$ and $\chi_{N}=1$ for $\xi_{4}>4$. We have

[TABLE]

Since $1-\chi_{N}(N^{-1}D_{4})$ has support for $\xi_{4}<4N$ we can estimate the first therm of the above inequality with

[TABLE]

where $C$ is a positive constant independent by $N$, but depending on $u$. As already mentioned, to handle the second term of the above inequality we pull back $D_{4}^{1/(r+kp)}$. This is well defined since $\xi_{4}>1$, but is a pseudodifferential operator, and its commutator with $\varphi_{N}$ needs to some care. We use Lemma B.1 and Corollary B.1 in ABM-TrC1-2016 . For completeness we recall them. Let $\omega_{N}\in C^{\infty}(\mathbb{R})$ be an Ehrenpreis type cutoff such that $\omega_{N}=1$ for $x>2$ and $\omega_{N}=0$ for $x<1$, $\omega_{N}$ non negative and such that $\omega_{N}\chi_{N}=\chi_{N}$. Then we have

Lemma II.1** (ABM-TrC1-2016 ).**

Let $0<\theta<1$. Then

[TABLE]

where $a_{N,j}$ is a pseudo-differential operator of order $-k$ such that

[TABLE]

Corollary II.1** (ABM-TrC1-2016 ).**

For $1\leq j\leq N-1$ in (II.9) we have that

[TABLE]

Applying these results we find that

[TABLE]

The last term has analytic growth. To handle the first term on the right hand side we will apply the sub-elliptic estimate. Concerning the the terms in the summation, we need, as done previously, to pull back $D^{1/(r+kp)}$ once more in order to use the sub-elliptic estimate, this will produce either terms with analytic growth or terms of the form

[TABLE]

which can be handled as the first term.

Before to analyze the first term on the right hand side of the above inequality we remark that

[TABLE]

As above we use the “weight ” $N$ to balance the number of $x_{4}-$ derivatives on $u$ with the number of derivatives on $\varphi_{N}$. The two terms on the right hand side have the same form as $\|x_{1}^{q-1}\varphi^{(1)}_{N}D_{4}^{N}u\|$, we can use the same strategy to analyze these two terms.

Then the only term that we have to handle is the term $\|x_{1}^{q-1}\varphi^{(1)}_{N}\chi_{N}(N^{-1}\xi_{4})D_{4}^{N-\frac{1}{r+pk}}u\|_{1/r+pk}$. Once again, to estimate this term we use the sub-elliptic estimate (II.4) replacing $u$ with $x_{1}^{q-1}\varphi^{(1)}_{N}\chi_{N}(N^{-1}\xi_{4})D_{4}^{N-\frac{1}{r+pk}}u$. We have

[TABLE]

The right hand side of the above equation can be estimate by

[TABLE]

modulo terms which can be absorbed on the left hand side or which give analytic growth. We remark that on the last four terms we can take maximal advantage from the sub-elliptic estimate restarting the processes; moreover in view of the role of the weight $N$ the third and the fourth term have the same form of the second one. Before to give the general form of the terms which appear inside of the iterating process we analyze the particular situations. To handle the first term on the right hand side of (II.12) we must use the sub-ellipticity, i.e. we pull back $D_{4}^{1/(r+kp)}$. Using the Lemma II.1 and the Corollary II.1 we have

[TABLE]

The last term has analytic growth. To handle the first term on the right hand side we will apply the sub-elliptic estimate. Concerning the the terms in the summation, we need, as done previously, to pull back $D^{1/(r+kp)}$ once more in order to use the sub-elliptic estimate, this will produce either terms with analytic growth or terms of the form

[TABLE]

which can be handled as the first term.

Iterating the above strategy at the $j$-th step we obtain a term of the form

[TABLE]

When $j=q-1$ we have $\|\varphi^{(1)}_{N}\chi_{N}(N^{-1}\xi_{4})D_{4}^{N-\frac{q}{r+kp}}u\|^{2}_{1/r+kp}$. Iterating this cycle $s$-times we obtain a term of the form

[TABLE]

Using up all $x_{4}$-derivatives we estimate this term, hence the right hand side of (II.5), with $C^{2(N+1)}N^{2N(r+kp)/q}$. We have a growth corresponding to $G^{(r+kp)/q}$.

The second term on the right hand side of (II.12), $\|x_{1}^{2(q-1)}\varphi^{(2)}_{N}\chi_{N}(N^{-1}\xi_{4})D_{4}^{N-\frac{1}{r+kp}}u\|^{2}$, once again we must use the sub-ellipticity, that is using the Lemma II.1 and the Corollary II.1 we pull back $D_{4}^{1/(r+kp)}$ restarting the process.

Iterating this strategy at the $h$-th step we obtain a term of the form

[TABLE]

Let $0<\beta<1$ a parameter that will be chosen later. Using the Lemma II.1 and the Corollary II.1 we pull back $D_{4}^{\beta}$; we can estimate the above quantity with

[TABLE]

modulo terms of the form $c_{j}\|x_{1}^{h(q-1)}D_{4}\varphi^{(h+j)}_{N}\chi_{N}(N^{-1}\xi_{4})D_{4}^{N-\frac{h-1}{r+kp}-j}u\|^{2}$, $j=1,\dots,N-1$, and $\|x_{1}^{h(q-1)}a_{N,N}(x,D)\chi_{N}(N^{-1}\xi_{4})D_{4}^{N-\frac{h-1}{r+kp}}u\|^{2}$. The last one gives analytic growth, the others can be estimated restarting the process, i.e. pulling back $D_{4}^{\beta}$ and using the same process to estimate (II.13), that we will show below. The term (II.13) can be estimated by

[TABLE]

where $C_{2}$ is a small suitable constant. The last term can be absorbed on the left hand side. Choosing $m=k$, $\beta=(k+1)^{-1}$ and $h=(r-1+k(p-1))/((k+1)(q-1))$ we obtain

[TABLE]

Restarting the process, taking maximum advantage from the sub-elliptic estimate we obtain after $s$ step

[TABLE]

Iterating until all the $x_{4}$-derivatives are used up, that is until $N-s((h-1)(r+kp)^{-1}+\beta)\sim 0$, we have the growth corresponding to $G^{(r+kp)/q}$.

Combining and iterating the above processes more time, removing powers of $x_{1}$ and $x_{2}$ with $D_{4}$ and taking profit from the sub-ellipticity we may estimate the left hand side of (II.5) with terms of the form

[TABLE]

where $h$ is as above, $(h-1)(r+kp)^{-1}-\beta=qh(r+kp)^{-1}$ and $m_{4}(q-1)-m_{5}\leq(q-1)$. Iterating until all $x_{4}$-derivatives are used up, that is $N-(m_{1}+m_{2}h)q(r+pk)^{-1}-m_{3}\sim 0$ and $N-m_{1}+(m_{3}+m_{2}h)q(r+kp)^{1}-(m_{4}+m_{5})(r+kp)^{1}-m_{6}+m_{6}(r+kp)^{-1}\sim 0$ we have that $m_{1}+hm_{2}+m_{3}$ and $m_{1}+hm_{2}+m_{3}+m_{4}$ , since $m_{3}\geq 1$ and $m_{6}\geq 1$, are small or equal to $(r+kp)Nq^{-1}$. We can conclude

[TABLE]

where $C$ is independent by $N$ but depends on $u$. This conclude the proof.

Remark II.5**.**

In particular we have that if $u$ solves the equation $P_{1}u=f$ and $f$ is analytic, if $\rho_{{}_{0}}\in\Sigma_{r+kp}$ then $\rho_{{}_{0}}\notin WF_{\frac{r+kp}{q}}(u)$ and if $\rho_{{}_{1}}\in\Sigma_{r}$ then $\rho_{{}_{1}}\notin WF_{\frac{r}{p}}(u)$.

II.2 Gevrey Regularity for $P_{2}(x;D)$

The characteristic variety of $P_{2}$ is:

[TABLE]

Following the ideas in Tr-2006 and BoTr it can be seen as the disjoint union of analytic submanifolds:

[TABLE]

where

[TABLE]

Once more by the results in DZ and RS the operator $P_{2}$ has the following sub- elliptic estimate with loss of $2(1-1/(r+kq))$ derivatives:

[TABLE]

Here $X_{1}=D_{1}$, $X_{2}=x_{1}^{p-1}D_{2}$, $X_{3}=x_{1}^{q-1}D_{3}$, $X_{4}=x_{1}^{r-1}x_{3}^{k}D_{4}$, $X_{5}=x_{1}^{r+\ell-1}D_{4}$, $\|\cdot\|_{s}$ denotes the $H^{s}$ Sobolev norm and $\|\cdot\|=\|\cdot\|_{0}$ denotes the $L^{2}$ norm on the fixed open set $\Omega$.

To study the regularity of the solutions we estimate the high order derivatives of the solutions in $L^{2}$ norm, as in the case of $P_{1}$. For $x_{1}\neq 0$ the operator $P_{2}$ is elliptic and we shall not examine this region, elliptic operators are Gevrey hypoelliptic in any class $G^{s}$ for $s\geq 1$.

Let $\varphi_{N}(x)$ be a cutoff function of Ehrenpreis-Hörmander type with the same properties described in the beginning of the previous paragraph.

We assume that $\varphi_{N}$ is independent of the $x_{1}$-variable for the same reason described in the proof of the regularity of $P_{1}$. Moreover we may assume that $\varphi_{N}$ is independent of the $x_{3}$-variable since every $x_{3}$-derivative landing on $\varphi_{N}$ would leave a cut off function supported where $x_{3}$ is bounded away from zero, in this region the operator satisfies the Hörmander-Lie algebra condition at the step $r$. The operator $P_{2}$ is sub-elliptic with loss of $2(1-1/r)$ derivatives. In this region the operator is a generalization of the Oleĭnik-Radkevič operator then $P_{2}$ is $G^{r/p}$-hypoelliptic and not better, for more details see C and BT . Thus we can conclude that if $v$ solves the equation $P_{1}v=f$ and $f$ is analytic then the points $\displaystyle\rho_{{}_{1}}=\in\Sigma_{r}$, does not belong to $WF_{\frac{r}{p}}(v)$.

Now, we are interested to the microlocal regularity in $\displaystyle\rho_{{}_{0}}\in\Sigma_{r+kp}$, to obtain this it is sufficient to study the microlocal regularity of $P_{2}$ in $(0;e_{4})$. The microlocal regularity in a generic point $\displaystyle\rho_{{}_{0}}$ can be obtained following the same strategy below with the only difference that the cut-off function $\varphi_{N}(x)$ will be identically equal to 1 in $\Omega_{0}$ neighborhood of $\pi_{x}(\rho_{{}_{0}})=(0,x_{2}^{0},0,x_{4}^{0})$. Thus since we are interested to the microlocal regularity of $P_{2}$ in $(0;e_{4})$ we take $\varphi_{N}(x)=\varphi_{N}(x_{2},x_{4})$.

We replace $u$ by $\varphi_{N}(x)D_{4}^{N}u$ in ( II.14). We have

[TABLE]

As in the case of the operator $P_{1}$ we want to estimate terms of the form:

[TABLE]

Since $\varphi_{N}$ depends only by $x_{2}$ and $x_{4}$ we must analyze the commutators with $X_{2}$, $X_{4}$ and $X_{5}$. The cases $X_{4}$ and $X_{5}$ give analytic growth, they can handled in same way as done in the study of $P_{1}$; in these cases we can take maximal advantage from the sub-elliptic estimate. The case $X_{2}$. In this case we have to estimate the term

[TABLE]

Since it does not have sufficient power of $x_{1}$ to take maximal advantage from the sub-elliptic estimate, we will use the sub-ellipticity. To do this we will pull back $D_{4}^{1/(r+kq)}$. Using the same strategy employed to study the case of the vector field $X_{3}$ in the study of the regularity of $P_{1}$, here we have $x_{1}^{p-1}$ instead of $x_{1}^{q-1}$. Following the same strategy used to deduce the regularity of $P_{1}$, we conclude that

[TABLE]

where $C$ is independent by $N$ but depends on $u$. We have that the point $(0;e_{4})$ and more in general that the points $\rho_{0}\in\Sigma_{r+kq}$ do not belong to $WF_{\frac{r+kq}{p}}(u)$. This conclude the proof of the theorem.

Remark II.6**.**

In particular we have that if $u$ solves the equation $P_{1}u=f$ and $f$ is analytic, if $\rho_{{}_{0}}\in\Sigma_{r+kp}$ then $\rho_{{}_{0}}\notin WF_{\frac{r+kp}{p}}(u)$ and if $\rho_{{}_{1}}\in\Sigma_{r}$ then $\rho_{{}_{1}}\notin WF_{\frac{r}{p}}(u)$.

III Proof of Theorem I.2

The characteristic variety of $P_{3}$ is:

[TABLE]

Following the ideas in Tr-2006 and BoTr it can be seen as the disjoint union of analytic submanifolds:

[TABLE]

if $f>pk+r$ there is only one more stratum of depth $r+kp$:

[TABLE]

otherwise if $f<pk+r$ there are other two strata of depth $f$,

[TABLE]

and of depth $r+kp$ if $r+kp<f+q\ell$,

[TABLE]

or of depth $f+q\ell$ if $r+kp>f+q\ell$,

[TABLE]

Case $\textit{f}>\textit{r}+\textit{kp}$: In this case the Hörmander condition is satisfied at the step $r+kp$. Once more by the results in DZ and RS $P_{3}$ satisfies the following sub-elliptic estimate with loss of $2(1-1/(r+kp))$ derivatives:

[TABLE]

Here $X_{1}=\!D_{1}$, $X_{2}=\!x_{1}^{p-1}D_{2}$, $X_{3}=\!x_{1}^{q-1}D_{3}$, $X_{4}=\!x_{1}^{r-1}x_{2}^{k}D_{4}$, $X_{5}=\!x_{1}^{f-1}x_{3}^{\ell}D_{4}$ and $X_{6}=\!x_{1}^{f+e-1}D_{4}$.

The result can be archived following the some strategy used to characterize the regularity of the operator $P_{1}(x,D)$, Theorem I.1–i. In fact the presence of the additional vector field $X_{5}=\!x_{1}^{f-1}x_{3}^{\ell}D_{4}$ gives, in the algorithm developed to handle the operator $P_{1}$, only a negligible contribution, i.e. analytic growth: to estimate the terms $|\langle x_{1}^{f-1}x_{3}^{\ell}\varphi^{(1)}_{N}D_{4}^{N}u,X_{5}\varphi_{N}D_{4}^{N}u\rangle|$ and $|\langle x_{1}^{2(f-1)}x_{3}^{2\ell}\varphi^{(2)}_{N}D_{4}^{N}u,\varphi_{N}D_{4}^{N}u\rangle|$ can take maximal advantage from the sub-elliptic estimate.

Case $\textit{f}<\textit{r}+\textit{kp}$: In this case we distinguish two different situations: $r+kp<f+\ell q$ and $f+\ell q<r+kp$. Since the only difference between the two cases is the subelliptic index, that is in the first case the Hörmander condition is satisfied at the step $r+kp$ and in the other at the step $f+\ell q$ we will analyze only the first one.

Case $\textit{r}+\textit{kp}<\textit{f}+\mathbf{\ell}\textit{q}\,$: The operator $P_{3}$ is sub-elliptic with loss of $2(1-1/(r+kp))$ derivatives, as above the sub-elliptic a priori estimate (III.17) holds.

Let $\varphi_{N}(x)$ be a localizing cutoff function of Ehrenpreis-Hörmander type. We may assume that $\varphi_{N}$ is independent of the $x_{1}$-variable since every $x_{1}$-derivative landing on $\varphi_{N}$ would leave a cutoff function supported where $x_{1}$ is bounded away from zero, where the operator is elliptic. We can also assume that $\varphi_{N}$ is independent of the $x_{2}$-variable. If $x_{2}\neq 0$ the operator $P_{3}$ is an operator of Oleĭnik-Radkevič type, OR1973 , in view of the result obtained in (C, ), in this region, the operator is $G^{r/p}$-hypoelliptic. We can conclude that if $u$ solves the equation $P_{3}u=g$ and $g$ is analytic then the points $\rho_{{}_{6}}\in\Sigma_{r}$, does not belong to $WF_{r/p}(u)$.

Moreover we may assume that $\varphi_{N}$ is independent of the $x_{3}$-variable. Every $x_{3}$-derivative landing on $\varphi_{N}$ would leave a cut off function supported where $x_{3}$ is bounded away from zero, in this region the Hörmander condition is satisfied at the step $f$. The operator $P_{3}$ has the same form of the operator $P_{1}$, (I.1), in the Theorem I.1, with $pk>\ell$. We can conclude that if $u$ solves the equation $P_{3}u=g$ and $g$ is analytic then the points $\rho_{{}_{5}}\in\Sigma_{f}$, do not belong to $WF_{f/p}(u)$.

We assume that $\varphi_{N}(x)=\varphi_{N}(x_{4})$. We replay $u$ by $\varphi_{N}(x_{4})D_{4}^{N}u$ in (III.17). We have

[TABLE]

We have to estimate terms of the form:

[TABLE]

Since $\varphi_{N}$ depends only by $x_{4}$, $X_{1}$, $X_{2}$ and $X_{3}$ commute with $\varphi_{N}$. We must only analyze the commutators with $X_{4}$, $X_{5}$ and $X_{6}$. These cases give analytic growth, we can take maximal advantage from the sub-elliptic estimate. They can be handled as the field $X_{4}$, (II.7), in the proof of the Theorem I.1. We conclude that the point $(0;e_{4})$ and more in general the points $\rho_{{}_{4}}\in\Sigma_{r+kp}$ do not belong to $WF_{a}(u)$.

Remark III.7**.**

In particular we have that if $u$ solves the equation $P_{3}u=f$ and $f$ is analytic, if $f>r+kp$ then if $\displaystyle\rho_{{}_{2}}\in\displaystyle\Sigma_{r+kp}$ then $\displaystyle\rho_{{}_{2}}\notin WF_{\frac{r+kp}{q}}(u)$ and if $\displaystyle\rho_{{}_{3}}\in\Sigma_{r}$ then $\displaystyle\rho_{{}_{3}}\notin WF_{\frac{r}{p}}(u)$; if $f\leq r+kp$ then if $\displaystyle\rho_{{}_{4}}\in\displaystyle\Sigma_{r+kp}$ then $\displaystyle\rho_{{}_{4}}\notin WF_{a}(u)$, if $\displaystyle\rho_{{}_{5}}\in\Sigma_{f}$ then $\displaystyle\rho_{{}_{5}}\notin WF_{\frac{f}{p}}(u)$ and if $\displaystyle\rho_{{}_{6}}\in\Sigma_{r}$ then $\displaystyle\rho_{{}_{6}}\notin WF_{\frac{r}{p}}(u)$.

IV On the partial regularity of the operators $\textit{{P}}_{\!\text{{1}}}$ and $\textit{{P}}_{\!\text{{2}}}$

In this section, following the ideas in BT , we analyze the partial regularity in a neighborhood of the origin of the operators $P_{1}$, (I.1), and $P_{2}$, (I.2). We recall the definition of the non-isotropic Gevrey classes:

Definition IV.1**.**

A smooth function $f(x_{0},x_{1},\dots,x_{n})$ belongs to the Gevrey space $G^{(\alpha_{0},\alpha_{1},\dots,\alpha_{n})}$ at the point $x_{0}$ provided that there exists a neighborhood, $U$, of $x_{0}$ and a constant $C_{f}$ such that for all multi-indices $\beta$

[TABLE]

where $\beta!^{\alpha}=\beta_{0}!^{\alpha_{0}}\beta_{1}!^{\alpha_{1}}\dots\beta_{n}!^{\alpha_{n}}$.

Our result can be stated as follows:

Proposition IV.1**.**

Let $P_{1}$ be as in the Theorem I.1, where $pk<\ell$. If $u$ solves the problem $P_{1}u=f$ and $f$ is analytic then $u\in G^{(s_{1},s_{2},s_{3},s_{4})}$ where $s_{4}\geq\sup\{\frac{r+kp}{q},\frac{r}{p}\}$, $s_{2}\geq\frac{k}{k+1}+\frac{1}{k+1}\frac{r+kp}{q}$, $s_{3}\geq\frac{r(q-1)}{r(p-1)+q-p}$ and s_{1}\geq 1+\sup\Big{\{}\frac{1}{p(k+1)}\left(\frac{r+kp}{q}-1\right),\frac{1}{r}\left(\frac{r+kp}{q}-1\right),\frac{1}{r}\left(\frac{r}{p}-1\right),\frac{(r-1)(q-p)}{q(r(p-1)+q-p)}\Big{\}}.

The same strategy used in the proof of the above Proposition shows that:

Remark IV.8**.**

If $pk>\ell$ then $u\in G^{(s_{1},s_{2},s_{3},s_{4})}$ where $s_{4}\geq\sup\{\frac{r+\ell}{q},\frac{r}{p}\}$, $s_{2}\geq\frac{k}{k+1}+\frac{1}{k+1}\frac{r+\ell}{q}$, $s_{3}\geq\frac{r(q-1)}{r(p-1)+q-p}$ and s_{1}\geq 1+\sup\Big{\{}\frac{1}{p(k+1)}\left(\frac{r+\ell}{q}-1\right),\frac{1}{r}\left(\frac{r+\ell}{q}-1\right),\frac{1}{r}\left(\frac{r}{p}-1\right),\frac{(r-1)(q-p)}{q(r(p-1)+q-p)}\Big{\}}.

Remark IV.9**.**

Let $P_{2}(x;D)$ as in the Theorem I.1. If $qk<\ell$ and $u$ solves the problem $P_{2}u=f$, $f$ analytic, then $u\in G^{(s_{1},s_{2},s_{3},s_{4})}$ where $s_{4}\geq\frac{r+kq}{p}$, $s_{2}\geq 1$, $s_{3}\geq\frac{k}{k+1}+\frac{1}{k+1}\frac{r+kq}{p}$ and s_{1}\geq\sup\Big{\{}1+\frac{1}{p(k+1)}\left(\frac{r+kq}{p}-1\right),1+\frac{1}{r}\left(\frac{r+kq}{p}-1\right)\Big{\}}. Otherwise if $qk\geq\ell$ then $u\in G^{(s_{1},s_{2},s_{3},s_{4})}$ where $s_{4}\geq\frac{r+\ell}{p}$, $s_{2}\geq 1$, $s_{3}\geq\frac{k}{k+1}+\frac{1}{k+1}\frac{r+\ell}{p}$ and s_{1}\geq\sup\Big{\{}1+\frac{1}{p(k+1)}\left(\frac{r+\ell}{p}-1\right),1+\frac{1}{r}\left(\frac{r+\ell}{p}-1\right)\Big{\}}.

Proof Proposition IV.1.

Since the regularity in the direction $D_{4}$ has been obtained in the Theorem I.1 we have only to analyze the direction $D_{1}$, $D_{2}$ and $D_{3}$. The primary tool will be once again the subelliptic estimate (II.4). Roughly speaking the strategy will be to transform the derivatives in the directions $D_{2}$ and $D_{1}$ in powers of the derivative in the direction $D_{4}$, this will allow us to use the result in the Theorem I.1. Concerning the direction $D_{3}$ we will obtain the result directly.

Direction ${\bm{D}_{3}}$: Let $\varphi_{N}(x_{3},x_{4})$ be a cut off function of Ehrenpreis-Hörmander type described in the proof of the Theorem I.1-A to analyze the direction $x_{4}$. We replay $u$ by $\varphi_{N}D_{3}^{N}u$ in ( II.4). We have

[TABLE]

The scalar product in the right hand side leads to

[TABLE]

The last term has a trivial estimate since $P_{N}u$ is analytic. Without loss of generality we can assume that it is zero. We focus our attention only on the vector field $X_{3}$, the case $X_{4}$ and $X_{5}$ can be handled in the same way, these vector fields have coefficients with power of $x_{1}$ greater than $q-1$. We have

[TABLE]

The constant $C_{j}$ are arbitrary, we make the choice $C_{j}=\delta^{-1}2^{j}$, for a suitable fixed small $\delta$. We can absorb each term of the form $C^{-1}_{j}\|X_{3}\varphi_{N}D^{\alpha}_{3}u\|^{2}$ on the left hand side of (IV.19). The term $C_{N+1}\|\varphi^{N+1}_{N}u\|^{2}$ is smaller than $C^{2(N+1)}N!^{2}$, that is it gives analytic growth. To estimate the terms $C_{j}\|X_{3}\varphi^{j}_{N}D_{3}^{N-j}u\|^{2}$, we observe that for each of them there has been a shift of one or more $x_{3}$-derivatives from $u$ to $\varphi_{N}$, but they have the same form as $\|X_{3}\varphi_{N}D_{3}^{N}u\|^{2}$. We have to estimate the sum

[TABLE]

We start from the first term in the sum. We use the Rothschild-Stein sub-elliptic estimate replacing $u$ with $\varphi^{(1)}_{N}D_{3}^{N-1}u$, repeating the above procedure we have

[TABLE]

modulo terms which give analytic growth or which have the following form $|\langle[X_{3},[X_{3},\varphi^{(1)}_{N}]]D_{3}^{N-1}u,\varphi^{(1)}_{N}D_{3}^{N-1}u\rangle|$; we remark that for each of them there has been a shift of $x_{3}$-derivatives from $u$ to $\varphi_{N}$, but essentially they have the same form as $|\langle[X_{3},[X_{3},\varphi_{N}]]D_{3}^{N}u,\varphi^{(1)}_{N}D_{3}^{N-1}u\rangle|$ in (IV.20), for the discussion of these terms see in the continuations of the proof. As before we may absorb the second term in the left hand side of the estimate. Repeating the above process $s$ times we have

[TABLE]

modulo terms which can be absorbed on the left hand side or which give analytic growth or which have the form $|\langle[X_{3},[X_{3},\varphi^{(j)}_{N}]]D_{3}^{N-1}u,\varphi^{(j)}_{N}D_{3}^{N-j}u\rangle|$, $1\leq j\leq s-1$. With the same procedure, after $N-1$ iterates, we obtain a term of the form

[TABLE]

This term can be estimate by $C^{2(N+1)}(N!)^{2}$, we have analytic growth.

On the other hand we have

[TABLE]

We study any single term. Term $H_{0}$:

[TABLE]

As done previously the weight $N$ is introduced to balance the number of $x_{3}$-derivatives on $u$ with the number of derivatives on $\varphi_{N}$. The terms on the right hand side have the same form as $\|X_{3}\varphi_{N}D_{3}^{N}u\|^{2}$. We can restart the process.

The term $H_{1}$:

[TABLE]

The above sum can be handled with the same strategy used to estimate the sum (IV.21). The last term give analytic growth.

The term $H_{2}$:

[TABLE]

The last term gives analytic growth. To estimate the terms in the sums, we observe that with the help of the weight $N$ we have essentially, on each of them, shifted one or more $x_{3}$-derivatives from $u$ to $\varphi_{N}$; they have the same form as $\|X_{3}\varphi_{N}D_{3}^{N}u\|^{2}$.

The term $H_{3}$:

[TABLE]

Iterating we obtain

[TABLE]

We observe that the terms in the first sum have the same form as $H_{0}$, the terms in the second sum have the same form as $H_{1}$ and those in the third sum have the same form as $H_{2}$, we can handle each of them as above. Finally, the last term gives analytic growth. Using the estimate (II.4) with $u$ replaced by $N^{i}\varphi^{(j)}_{N}D_{3}^{N-(j+i)}u$ or $N^{-i}\varphi^{(j+i)}_{N}D_{3}^{N-j}u$ and applying recursively the same strategy followed above we are able to shift all free derivatives on $\varphi_{N}$.

As previously observed, to analyze the case $X_{4}$ and $X_{5}$ we can use the same strategy used to study the case $X_{3}$. Indeed since the commutators $[X_{4},\varphi_{N}]$, $[X_{5},\varphi_{N}]$, $[X_{4},[X_{4},\varphi_{N}]]$ and $[X_{5},[X_{5},\varphi_{N}]]$ give terms with powers of $x_{1}$ greater than $q-1$, we can take again maximum advantage from the sub-elliptic estimate. Also in these cases we have analytic growth.

Hence we have

[TABLE]

To obtain the result we need to consider when $x_{2}\neq 0$. To do it since when $x_{2}\neq 0$ the operator $P_{1}$ is an operator of Oleĭnik-Radkevič type, OR1973 , we use the following result in C :

Theorem IV.3** (C ).**

Let $P$ be the operator given by

[TABLE]

We have that $P$ is $G^{r_{n}/r_{1}}$ hypoelliptic and not better. More precisely we have that if $u$ solves the equation $Pu=f$ and $f$ is analytic then if $\rho_{j}\in\Sigma_{r_{j}-1}$ then $\rho_{j}\notin WF_{r_{j}/r_{1}}(u)$ and moreover $u\in G^{(s_{0},s_{1},\dots,s_{n})}$ where

[TABLE]

where $r^{*}=\displaystyle\sup_{j}\left\{1-\frac{1}{r_{j}}+\frac{\beta_{j}}{r_{j}}\right\}$, in particular $s_{2}\geq 1$ and $s_{n}\geq r_{n}/r_{1}$.

We can conclude that we have in the direction $x_{3}$ a growth corresponding to $G^{\frac{r(q-1)}{r(p-1)+q-p}}$.

Direction ${\bm{D}_{2}}$. Once again our primary tool will be the sub-elliptic estimate (II.4). As in the study of the direction $x_{3}$, we replace $u$ by $\varphi_{N}D_{2}^{N}u$ in ( II.4). We recall that $\varphi_{N}$ does not depend on $x_{1}$ and $x_{2}$. We have

[TABLE]

We consider the scalar product in the right hand side of the above inequality. We have to study terms of the type

[TABLE]

Since $X_{3}=x^{q-1}_{1}D_{3}$, $X_{5}=x^{r+\ell-1}_{1}D_{4}$ and $q$ and $r$ are strictly greater than $p$, as seen in the study of the direction $x_{3}$, we can take maximum advantage from the sub-elliptic estimate shifting one derivative from $u$ to $\varphi_{N}$. If we focus our attention only on these terms and we iterate the process we will obtain analytic growth.

The case $X_{4}=x_{1}^{r-1}x_{2}^{k}D_{4}$. We have

[TABLE]

Without loss of generality we analyze one of the terms; a similar method can be used to handle the other terms. We consider the first one: $Nki^{-1}x_{1}^{r-1}D_{2}\varphi_{N}x_{2}^{k-1}D_{2}^{N-2}D_{4}u$. We have to estimate $Nk\|X_{2}\varphi_{N}x_{2}^{k-1}D_{2}^{N-2}D_{4}u\|$. We apply the sub-elliptic estimate with $u$ replayed by $Nk\varphi_{N}x_{2}^{k-1}D_{2}^{N-2}D_{4}u$, arguing as above, we study the first term coming from the commutator with $X_{4}$. We obtain the term $k^{2}N(N-2)x_{1}^{r-1}x_{2}^{2(k-1)}\varphi_{N}D_{2}^{N-3}D_{4}^{2}u$. We have to estimate $k^{2}N(N-2)\|X_{4}x_{2}^{k-2}\varphi_{N}D_{2}^{N-3}D_{4}u\|$. Hence after two steps we have

[TABLE]

Repeating the process $j$-times, we have

[TABLE]

Here the constant $C$ depend by $k$. We stress that $N![(N-1)(N-(j+1))!]^{-1}\sim N^{j}$. In this way after $k$ iterates we have to analyze a term of the form $C_{k}N![(N-1)(N-(k+1))!]^{-1}\varphi_{N}D_{2}^{N-(k+1)}D_{4}u$. Arguing in the same way after $m$ steps we have

[TABLE]

Iterating the cycle $N/(k+1)$-times we use up all free derivatives in $x_{2}$-direction and we are left with

[TABLE]

As well as it was done in the proof of the Theorem I.1 we introduce $\chi_{N}(\xi_{4})$ an Ehrenpreis-Hörmander cutoff function such that $\chi_{N}$ is $C^{\infty}(\mathbb{R})$ non negative function such that $\chi_{N}=0$ for $\xi_{4}<3$ and $\chi_{N}=1$ for $\xi_{4}>4$. We have

[TABLE]

Since $1-\chi_{N}(N^{-1}D_{4})$ has support for $\xi_{4}<4N$ we have

[TABLE]

where $C$ is a positive constant independent by $N$, but depending on $u$ and $k$. To estimate $\|X_{4}\varphi_{N}\chi_{N}(N^{-1}D_{4})D_{4}^{\frac{N}{k+1}}u\|$ we use the same strategy used in the proof of the Theorem I.1. Therefore since in the direction $x_{4}$ we have a growth corresponding to $G^{\frac{r+kp}{q}}$ we can estimate this term with $C^{N+1}(N!)^{\frac{r+k(p+q)}{q(k+1)}}$. We can estimate the left hand side of (IV.23) with this quantity, we have the growth corresponding to $G^{(r+k(p+q))/q(k+1)}$.

More in general applying the sub-elliptic estimate and iterating the above processes more time, we may estimate the left hand side of (IV.23) with terms of the form

[TABLE]

Iterating the procedure until all the $x_{2}$-derivatives are used up we have to apply the sub-elliptic estimate to terms of the form

[TABLE]

To handle these terms we argue as before that is we introduce the cut-off $\chi_{N}$ and we apply the strategy used in the proof of the Theorem I.1 to obtain the Gevrey regularity in the direction $x_{4}$. Since $(r+kp)/q>1$ we can conclude

[TABLE]

To gain the result we need to consider when $x_{2}\neq 0$. To do it since when $x_{2}\neq 0$ the operator $P_{1}$ is an operator of Oleĭnik-Radkevič type, OR1973 , we use Theorem IV.3. We have that when $x_{2}\neq 0$ in the direction $D_{2}$ we have analytic growth. We conclude that in this direction the growth corresponding to $G^{(r+k(p+q))/q(k+1)}$.

Direction ${\bm{D}_{1}}$: As in the study of the other directions, we replace $u$ by $\varphi_{N}(x)D_{1}^{N}u$ in ( II.4). We have

[TABLE]

We consider the scalar product in the right hand side of the above inequality. We have to study terms of the type

[TABLE]

We describe the case $X_{2}$, the other cases can be handled using the same strategy. We have

[TABLE]

Without loss of generality we analyze one of the terms. A similar method can be used to handle the other terms. Consider $N(p-1)D_{1}x_{1}^{p-2}\varphi_{N}D_{1}^{N-2}D_{2}u$ that is we have to estimate a term of the form $N(p-1)\|X_{1}x_{1}^{p-2}\varphi_{N}D_{1}^{N-2}D_{2}u\|$. Applying the sub-elliptic estimate with $u$ replaced by $x_{1}^{p-2}\varphi_{N}D_{1}^{N-2}D_{2}u$ and arguing as above, we study the first term coming from the commutator with $X_{2}$. We obtain the term $N(N-2)(p-1)^{2}x_{1}^{2(p-2)}D_{1}^{N-3}D_{2}^{2}u$. We have to estimate $N(N-2)(p-1)^{2}\|X_{2}x_{1}^{p-3}D_{1}^{N-3}D_{2}u\|$. Hence after two step we have

[TABLE]

Repeating the process $s$-times, we have

[TABLE]

We stress that $N![(N-1)(N-(j+1))!]^{-1}\sim N^{j}$. In this way after $s=p-1$ iterates we have to analyze a term of the form $C_{p}N^{p-1}\|X_{2}\varphi_{N}D_{1}^{N-p}D_{2}u\|$. Arguing in the same way after $m$ steps we have

[TABLE]

Iterating the cycle $N/p$-times we use up all free derivatives in $x_{1}$-direction and we are left with

[TABLE]

Since in the direction $x_{2}$ we have a growth as $G^{\frac{r+k(p+q)}{q(k+1)}}$ we can estimate the above term with

[TABLE]

We have the growth $G^{1+\frac{r+kp-q}{pq(k+1)}}$.

The other cases, that is the terms involving the commutators with $X_{3}$, $X_{4}$ and $X_{5}$, can be handled in the same way achieving analytic growth, $1+(r+kp-q)/rq$-Gevrey growth and $1+(r+kp-q)/(r+\ell)q$-Gevrey growth respectively. We remark that in these three situations, arguing as above, we obtain terms of the form $C_{q}^{N}(N!)^{(q-1)/q}\|X_{3}\varphi_{N}D_{3}^{N/q}u\|$, $C_{r}^{N}(N!)^{(r-1)/r}\|X_{4}\varphi_{N}D_{4}^{N/r}u\|$ and $C_{r+\ell}^{N}(N!)^{(r+\ell-1)/(r+\ell)}\|X_{5}\varphi_{N}D_{4}^{N/(r+\ell)}u\|$. Moreover we point out that also in the general situation we will obtain a Gevrey growth less than or equal to that obtained by analyzing the individual cases. We have obtained a growth corresponding to $G^{s_{1}}$ where $s_{1}=\sup\{1+\frac{r+kp-q}{qp(k+1)},1+\frac{r+kp-q}{rq}\}$.

To obtain the result we need to consider when $x_{2}\neq 0$. To do it since when $x_{2}\neq 0$ the operator $P_{1}$ is an operator of Oleĭnik-Radkevič type, OR1973 , we use Theorem IV.3. We have that when $x_{2}\neq 0$ in the direction $D_{2}$ we have a growth corresponding to $G^{s_{2}}$ where $s_{2}=\sup\{1+\frac{1}{q}\left(\frac{r(q-1)}{r(p-1)+q-p}-1\right),1+\frac{1}{r}\left(\frac{1}{p}-\frac{r}{p}\right)\}$. We conclude that in the direction $x_{2}$ we have a growth corresponding to $G^{s}$ where $s=\sup\{s_{1},s_{2}\}$. We point out that the case $x_{2}\neq 0$ can be directly considered taking the cutoff function $\varphi_{N}$ depending also on the $x_{2}$-variable from the beginning.

∎

V Additional material: the $\textit{{n}}-$dimensional case

Following the some ideas used to archive the Theorems I.1 and I.2 we can extend without particular difficulties such results to the following $n$-dimensional cases, $n\geq 5$. We omit the proofs.

Theorem V.4**.**

Let $P_{i,n}(x;D)$ be the operator given by

[TABLE]

in $\Omega$, open neighborhood of the origin in $\mathbb{R}^{n}$, where $r_{j}$, $j=1,\dots,n$, $k$ and $\ell$ are positive integers such that $r_{1}<r_{2}<\dots<r_{n}$. We have:

• i)

if $kr_{i}<\ell$, $P_{i,n}(x;D)$ is $G^{s}$-hypoelliptic with s=\displaystyle\sup\Big{\{}\frac{r_{n}+kr_{2}}{r_{3}},\frac{r_{n}}{r_{2}}\Big{\}} if $i=2$ and $s=\displaystyle\frac{r_{n}+kr_{i}}{r_{2}}$ if $i\neq 2$. In particular if $u$ solves the equation $P_{i,n}u=f$ and $f$ is analytic then the point $(0,e_{n})$ in $\operatorname{Char}(P_{i,n})$ does not belong to $\operatorname{WF}_{(r_{n}+kr_{2})/r_{3}}(u)$ if $i=2$ and it does not belong to $\operatorname{WF}_{(r_{n}+kr_{i})/r_{2}}(u)$ if $i\neq 2$.

• ii)

if $kr_{i}\geq\ell$, $P_{i,n}(x;D)$ is $G^{s}$-hypoelliptic with s=\displaystyle\sup\Big{\{}\frac{r_{n}+\ell}{r_{3}},\frac{r_{n}}{r_{2}}\Big{\}} if $i=2$ and $s=\displaystyle\frac{r_{n}+\ell}{r_{2}}$ if $i\neq 2$. In particular if $u$ solves the equation $P_{i,n}u=f$ and $f$ is analytic then the point $(0,e_{n})$ in $\operatorname{Char}(P_{i,n})$ does not belong to $\operatorname{WF}_{(r_{n}+\ell)/r_{3}}(u)$ if $i=2$ and it does not belong to $\operatorname{WF}_{(r_{n}+\ell)/r_{2}}(u)$ if $i\neq 2$.

Remark V.10**.**

Let $\widetilde{P}_{i,n}(x;D)$ be the operator given by

[TABLE]

in $\Omega$, open neighborhood of the origin in $\mathbb{R}^{n}$, where $m\geq 3$, $2\leq i\leq m-1$, $r_{j}$, $j=1,\dots,n$, $k$ and $\ell$ positive integers such that $r_{1}<r_{2}<\dots<r_{n}$ and $r_{n}>r_{m}+\sup\{kr_{i},\ell\}$. We have that $\widetilde{P}_{i,n}(x;D)$ is $r_{n}/r_{2}$-Gevrey hypoelliptic. In particular if $u$ solves the equation $\widetilde{P}_{2,n}(x;D)u=f$ and $f$ is analytic then the point $(0,e_{n})\in\operatorname{Char}(\widetilde{P}_{2,n})$ does not belong to $\operatorname{WF}_{r_{n}/r_{3}}(u)$.

Theorem V.5**.**

Let $P_{m,n}(x,D)$ be the operator given by

[TABLE]

*in $\Omega$, open neighborhood of the origin in $\mathbb{R}^{n}$, where $r_{i}$, $i=2,\dots,n$, $k_{i-m}$ and $\ell_{i-m}$, $i=m+2,\dots,n$, are positive integers such that $r_{2}<\cdots<r_{n}$, $k_{2}<\cdots<k_{n-m}$, $\ell_{2}<\cdots<\ell_{n-m}$ and $r_{i}k_{i-m}<\ell_{i-m}$ for every $i$, $i=m+2,\dots,n$, then $P_{m,n}(x,D)$ is $G^{s}$-hypoelliptic with $s=\displaystyle\frac{r_{n}+r_{n-m}k_{n-m}}{r_{2}}$. Moreover if $u$ solves the equation $P_{m,n}u=f$ and $f$ is analytic then the point $(0,e_{n})\in\operatorname{Char}(P_{m,n})$ does not belong to $WF_{\frac{r_{n}+r_{n-m}k_{n-m}}{r_{n-m+1}}}(u)$. *

Remark V.11**.**

Let $\widetilde{P}_{m,n}(x,D)$ be the operator given by

[TABLE]

in $\Omega$, open neighborhood of the origin in $\mathbb{R}^{n}$, where $r_{i}$, $i=2,\dots,n$, $k_{i-m+1}$ and $\ell_{i-m}$, $i=m+3,\dots,n$, are positive integers such that $r_{2}<\cdots<r_{n}$, $k_{3}<\cdots<k_{n-m}$, $\ell_{3}<\cdots<\ell_{n-m}$ and $r_{i}k_{i-m}<\ell_{i-m}$ for every $i$, $i=m+3,\dots,n$, then the point $(0,e_{n})\in\operatorname{Char}(\widetilde{P}_{m,n})$ does not belong to $WF_{\frac{r_{n}+r_{n-m}k_{n-m}}{r_{2}}}(u)$.