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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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