zbMATH — the first resource for mathematics

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.

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
Full Text: DOI
[1] Bony J.-M., Ann. Sci. École Norm. Sup. 14 (4) pp 209– (1981) · Zbl 0495.35024
[2] Bony J.-M., Lect. Notes Math. 1495, in: Analyse Microlocale Des Équations Aux Dérivées Partielles Non Linéaires. Microlocal Anal. and Appl. (1991) · Zbl 0747.00025
[3] DOI: 10.1080/03605308808820568 · Zbl 0659.35115
[4] Brummelhuis R., On Melin’s Inequality for Systems (1999) · Zbl 1007.35111
[5] DOI: 10.1007/BF02383640 · Zbl 0211.17102
[6] Hérau F., Melin–Hörmander Inequality in a Wiener Type Pseudo-Differential Algebra · Zbl 1039.35149
[7] DOI: 10.1002/cpa.3160320304 · Zbl 0388.47032
[8] Hörmander L., The Analysis of Linear Partial Differential Operators I,III (1985)
[9] Hörmander L., Lectures on Nonlinear Hyperbolic Differential Equations (1997)
[10] DOI: 10.1080/03605308308820269 · Zbl 0522.35069
[11] DOI: 10.1007/BF02787101 · Zbl 0864.35131
[12] Saint Raymond X., Remarks on Gårding Inequalities for Differential Operators (2000)
[13] Tataru D., On the Fefferman–Phong Inequality and Related Problems (2000) · Zbl 1045.35115
[14] Toft J., Com. Part. Diff. Eq. 25 (7) pp 1201– (2000) · Zbl 0963.35215
[15] Xu C.-J., Notes in Math. Series, in: Nonlinear Microlocal Analysis, Pit. Reas. (1996)
[16] Xu C.-J., Calc. Var. Partial Differential Equations 5 (4) pp 323– (1997) · Zbl 0902.35019
[17] Xu C.J., The Dirichlet Problems for a Class of Semilinear Subelliptic Equations Nonlinear Anal. 37 (8) pp 1039– (1999)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.