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Sharp estimates for pseudo-differential operators with symbols of limited smoothness and commutators. (English) Zbl 1099.35191

The author studies continuity properties and commutator estimates for a class of pseudo-differential operators with nonclassical, nonsmooth (in \(x\)) symbols \(\sigma(x,\xi)\). This study is motivated by problems from the theory of nonlinear water waves where symbols of form \(\sqrt{(1+|\nabla a(x)|^2)|\xi|^2 - (\nabla a(x)\cdot\xi)^2}\) appear in a natural way. To accommodate such symbols the author introduces the symbol class \(\Gamma^m_s\), \(m\in\mathbb R\), \(s>d/2\), consisting of functions \(\sigma : \mathbb R^d\times\mathbb R^d \to \mathbb R\) satisfying \(\sigma|\mathbb R^d\times \overline{B_1(0)} \in L^\infty(\overline{B_1(0)}, H^s(\mathbb R^d))\) —here \(H^s\) denotes the usual \(L^2\) Sobolev space— and for all \(\beta\in\mathbb N_0^d\) the inequality \[ \sup_{|\xi|>1/4} (1+|\xi|)^{|\beta|-m} \|\partial_\xi^\beta \sigma(\cdot,\xi)\|_{H^s}<\infty \] holds. The first main result are inequalities of the type \[ \|\sigma(x,D)u\|_{H^s} \leq c(\sigma,m,s,d)\|u\|_{H^{m+t_0}} + C(\sigma,m,d)\|u\|_{H^{m+s}} \] where \(s\in [t_0,s_0]\), \(\sigma\in \Gamma^m_{s_0}\), \(m\in\mathbb R\) and \(d/2<t_0\leq s_0\). The dependence of the constants \(c,C\) on \(\sigma\) is explicitly expressed in terms of seminorms on the symbol space \(\Gamma^m_s\). The proof relies on a decomposition of the symbol \(\sigma\) into four components (one of them is a paradifferential symbol) which are then separately estimated. The second main result is on various commutator estimates, e.g.
(i) of Kato-Ponce type: \[ \begin{split} \|[\sigma^1(D),\sigma^2(\cdot,D)]u - \{\sigma^1,\sigma^2\}_n(\cdot,D)u\|_{H^s} \\ \leq C(\sigma^1,\sigma^2)\|u\|_{H^{s+m_1+m_2-n-1}} + \|\sigma^2\|_{H^{s+\min(m_1,s)}}\|u\|_{H^{m_1+m_2+t_0-\max(m_1,n)}} \end{split} \] where \(\{\cdot,\cdot\}\) is the Poisson bracket, \(d/2<t_0\leq s_0\), \(\sigma^1\) is a certain Fourier-multiplier of order \(m_1\) and \(\sigma^2\in \Gamma^{m_2}_{s_0+1+\max(m_1,n)}\). Again the dependence of the constant \(C\) on \(\sigma^1, \sigma^2\) is explicitly given. Various variants of the above estimates are given if the symbols have additional properties;
(ii) of Calderon-Coifman-Meyer type: under the same assumptions as stated in (i) one has \[ \|[\sigma^1(D),\sigma^2(\cdot,D)]u - \{\sigma^1,\sigma^2\}_n(\cdot,D)u\|_{H^s} \leq c'(\sigma^1)\|\nabla^{n+1}_x\sigma^2\|_{H^{t_0}}\|u\|_{H^{s+m_1+m_2-n-1}} \] with explicitly given dependence of the constants.
Both results are finally extended to the case where \(\sigma^1\) is not only a Fourier multiplier but even a symbol from the class \(\Gamma^m_s\). It is stated (without proof) that an extension of the above estimates to \(L^p\)-Sobolev spaces is indeed possible; for the sake of clarity, however, the exposition covers only \(L^2\)-Sobolev spaces.

MSC:

35S05 Pseudodifferential operators as generalizations of partial differential operators
35S50 Paradifferential operators as generalizations of partial differential operators in context of PDEs
47G30 Pseudodifferential operators
47A30 Norms (inequalities, more than one norm, etc.) of linear operators
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