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Melin inequality for paradifferential operators and applications. (English) Zbl 1028.35176
The 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.

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
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