Melin inequality for paradifferential operators and applications.

*(English)*Zbl 1028.35176The main goal of this paper is to study the interior regularity for the equation
\[
F(x,u,\dots, \partial^\beta u,\dots)_{|\beta|\leq m}= 0
\]
assuming that the linearized symbol associated to this equation is subelliptic. The function \(F\) is assumed to be \(C^\infty\) and \(x\in\mathbb{R}^n\).

In order to state the results proved in this paper, it is necessary first to introduce some notation. Given \(m,\rho\in\mathbb{R}\) the author considers real functions of the form \[ p(x,\xi)= p_m(x,\xi)+ p_{m-1}(x,\xi)+\cdots+ p_{m-[\rho]}(x,\xi), \] where \(p_{m-k}(x,\xi)\) is homogeneous of degree \(m-k\) and of class \(C^\infty\) in the variable \(\xi\) and it belongs to the Zygmund class \(C^{\rho-k}_{*\text{loc}}\) in \(x\). Functions \(p(x,\xi)\) satisfying these conditions are called paradifferential symbols and the family of paradifferential symbols is denoted \(\Sigma^m_\rho\).

A properly supported operator \(T: D'\to D'\) is called a paradifferential operator of order \(m\) if there exists a function \(p\in\Sigma^m_\rho\), called the symbol of \(T\), such that for every compact subset \(K\subset\mathbb{R}^n\) and for every \(\chi\in C^\infty_0\) the mapping \(P-\chi\widetilde{p\chi}(x,D)\) is continuous from the Sobolev space \(H^s_{\text{comp}}(K)\) to the Sobolev space \(H^{s-m+\rho}\). The class of paradifferential operators of order \(m\) is denoted \(OP(\Sigma^m_\rho)\). We must add that \(\widetilde{p\chi}(x,D)\) indicates the classical pseudo-differential operator with symbol \(\widetilde{p\chi}\).

The main result in this paper is as follows. For \(\rho> 3\), let \(P\) be a selfadjoint paradifferential operator with symbol \(p= p_m+ p_{m-1}+\cdots+ p_{m-[\rho]}\). It is assumed that \(p_m\geq 0\) and \(p_{m-1}\) satisfies certain positivity condition. Then for every compact subset \(K\subset\mathbb{R}^n\) and for every \(C^\infty\) function \(u\) supported in \(K\) there exist \(c_K\), \(C_K> 0\) so that the operator \(P\) satisfies the following lower bound \[ (Pu, u)\geq c_K\|u\|^2_{(m- 1)/2}- C_K\|u\|^2_{(m/2)- 1}. \] That is to say, this result is an extension to paradifferential operators of the lower bounds obtained by A. Melin [Ark. Mat. 9, 117-140 (1971; Zbl 0211.17102)] for pseudodifferential operators.

The author applies this lower bound to prove the hypoellipticity of a second-order nonlinear equation \[ F(x,u,\partial u,\partial^2 u)= 0, \] for which the linearized symbol satisfies the positivity conditions mentioned above.

The author includes a section with preliminary definitions and results on microlocal analysis, as well as two appendices where the author recalls several auxiliary results.

In order to state the results proved in this paper, it is necessary first to introduce some notation. Given \(m,\rho\in\mathbb{R}\) the author considers real functions of the form \[ p(x,\xi)= p_m(x,\xi)+ p_{m-1}(x,\xi)+\cdots+ p_{m-[\rho]}(x,\xi), \] where \(p_{m-k}(x,\xi)\) is homogeneous of degree \(m-k\) and of class \(C^\infty\) in the variable \(\xi\) and it belongs to the Zygmund class \(C^{\rho-k}_{*\text{loc}}\) in \(x\). Functions \(p(x,\xi)\) satisfying these conditions are called paradifferential symbols and the family of paradifferential symbols is denoted \(\Sigma^m_\rho\).

A properly supported operator \(T: D'\to D'\) is called a paradifferential operator of order \(m\) if there exists a function \(p\in\Sigma^m_\rho\), called the symbol of \(T\), such that for every compact subset \(K\subset\mathbb{R}^n\) and for every \(\chi\in C^\infty_0\) the mapping \(P-\chi\widetilde{p\chi}(x,D)\) is continuous from the Sobolev space \(H^s_{\text{comp}}(K)\) to the Sobolev space \(H^{s-m+\rho}\). The class of paradifferential operators of order \(m\) is denoted \(OP(\Sigma^m_\rho)\). We must add that \(\widetilde{p\chi}(x,D)\) indicates the classical pseudo-differential operator with symbol \(\widetilde{p\chi}\).

The main result in this paper is as follows. For \(\rho> 3\), let \(P\) be a selfadjoint paradifferential operator with symbol \(p= p_m+ p_{m-1}+\cdots+ p_{m-[\rho]}\). It is assumed that \(p_m\geq 0\) and \(p_{m-1}\) satisfies certain positivity condition. Then for every compact subset \(K\subset\mathbb{R}^n\) and for every \(C^\infty\) function \(u\) supported in \(K\) there exist \(c_K\), \(C_K> 0\) so that the operator \(P\) satisfies the following lower bound \[ (Pu, u)\geq c_K\|u\|^2_{(m- 1)/2}- C_K\|u\|^2_{(m/2)- 1}. \] That is to say, this result is an extension to paradifferential operators of the lower bounds obtained by A. Melin [Ark. Mat. 9, 117-140 (1971; Zbl 0211.17102)] for pseudodifferential operators.

The author applies this lower bound to prove the hypoellipticity of a second-order nonlinear equation \[ F(x,u,\partial u,\partial^2 u)= 0, \] for which the linearized symbol satisfies the positivity conditions mentioned above.

The author includes a section with preliminary definitions and results on microlocal analysis, as well as two appendices where the author recalls several auxiliary results.

Reviewer: Josefina Alvarez (Las Cruces)

##### MSC:

35S50 | Paradifferential operators as generalizations of partial differential operators in context of PDEs |

35A27 | Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs |

35H10 | Hypoelliptic equations |

##### Keywords:

interior regularity; linearized symbol; subelliptic; selfadjoint paradifferential operator; symbol; lower bound; hypoellipticity; microlocal analysis
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\textit{F. Hérau}, Commun. Partial Differ. Equations 27, No. 7--8, 1659--1680 (2002; Zbl 1028.35176)

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