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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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References:

[1] Auscher, P.; Taylor, M., Paradifferential operators and commutator estimates, Comm. Partial Differential Equations, 20, 9-10, 1743-1775 (1995) · Zbl 0844.35149
[2] S. Benzoni-Gavage, R. Danchin, S. Descombes, Multi-dimensional Korteweg model, preprint.; S. Benzoni-Gavage, R. Danchin, S. Descombes, Multi-dimensional Korteweg model, preprint. · Zbl 1114.76058
[3] Bony, J.-M., Calcul symbolique et propagation des singularités pour les équations aux dérivés partielles non linéaires, Ann. Scient. École. Norm. Sup. (4), 14, 2, 209-246 (1981) · Zbl 0495.35024
[4] Bourdaud, G., Une algèbre maximale d’opérateurs pseudo-différentiels, Comm. Partial Differential Equations, 13, 1059-1083 (1988) · Zbl 0659.35115
[5] J.-Y. Chemin, Fluides parfaits incompressibles, Astérisque No. 230, 1995, 177pp.; J.-Y. Chemin, Fluides parfaits incompressibles, Astérisque No. 230, 1995, 177pp. · Zbl 0829.76003
[6] R.R. Coifman, Y. Meyer, Au-delà des opérateurs pseudo-différentiels, Astérisque, vol. 57. Société Mathématique de France, Paris, 1978, i+185pp.; R.R. Coifman, Y. Meyer, Au-delà des opérateurs pseudo-différentiels, Astérisque, vol. 57. Société Mathématique de France, Paris, 1978, i+185pp. · Zbl 0483.35082
[7] Grenier, E., Pseudo-differential energy estimates of singular perturbations, Comm. Pure Appl. Math., 50, 9, 821-865 (1997) · Zbl 0884.35183
[8] L. Hörmander, Lectures on Nonlinear Hyperbolic Differential Equations, Mathématiques & Applications, vol. 26, Springer, Berlin, 1997.; L. Hörmander, Lectures on Nonlinear Hyperbolic Differential Equations, Mathématiques & Applications, vol. 26, Springer, Berlin, 1997.
[9] Hwang, I. L., The \(L^2\)-boundedness of pseudodifferential operators, Trans. Amer. Math. Soc., 302, 1, 55-76 (1987) · Zbl 0651.35089
[10] Kato, T.; Ponce, G., Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41, 7, 891-907 (1988) · Zbl 0671.35066
[11] Lannes, D., Well-posedness of the water-waves equations, J. Amer. Math. Soc., 18, 605-654 (2005) · Zbl 1069.35056
[12] Marschall, J., Pseudodifferential operators with coefficients in Sobolev spaces, Trans. Amer. Math. Soc., 307, 1, 335-361 (1988) · Zbl 0679.35088
[13] G. Métivier, K. Zumbrun, Large viscous boundary layers for noncharacteristic nonlinear hyperbolic problems, American Mathematical Society, Providence, RI, 2005 (Memoirs of the American Mathematical Society 826).; G. Métivier, K. Zumbrun, Large viscous boundary layers for noncharacteristic nonlinear hyperbolic problems, American Mathematical Society, Providence, RI, 2005 (Memoirs of the American Mathematical Society 826).
[14] Y. Meyer, Remarques sur un théorème de J. M. Bony, Supplemento al Rendiconti der Circolo Matematico di Palermo, Serie II, No.1, 1981.; Y. Meyer, Remarques sur un théorème de J. M. Bony, Supplemento al Rendiconti der Circolo Matematico di Palermo, Serie II, No.1, 1981.
[15] Y. Meyer, R.R. Coifman, Ondelettes et opérateurs. III, Opérateurs multilinéaires, Actualités Mathématiques, Hermann, Paris, 1991, pp. i-xii and 383-538.; Y. Meyer, R.R. Coifman, Ondelettes et opérateurs. III, Opérateurs multilinéaires, Actualités Mathématiques, Hermann, Paris, 1991, pp. i-xii and 383-538. · Zbl 0745.42012
[16] Moser, J., A rapidly convergent iteration method and nonlinear partial differential equations, I, Ann. Scuola Norm. Sup. Pisa, 20, 265-315 (1966) · Zbl 0144.18202
[17] M. Taylor, Pseudodifferential Operators and Nonlinear PDE, Progress in Mathematics, vol. 100, Birkhäuser, Boston, Basel, Berlin, 1991.; M. Taylor, Pseudodifferential Operators and Nonlinear PDE, Progress in Mathematics, vol. 100, Birkhäuser, Boston, Basel, Berlin, 1991.
[18] M. Taylor, Partial differential equations. III. Nonlinear equations, Applied Mathematical Sciences, vol. 117, Springer, New York, 1997.; M. Taylor, Partial differential equations. III. Nonlinear equations, Applied Mathematical Sciences, vol. 117, Springer, New York, 1997.
[19] M. Taylor, Tools for PDE. Pseudodifferential operators, paradifferential operators, and layer potentials, Mathematical Surveys and Monographs, vol. 81, American Mathematical Society, Providence, RI, 2000, x+257pp.; M. Taylor, Tools for PDE. Pseudodifferential operators, paradifferential operators, and layer potentials, Mathematical Surveys and Monographs, vol. 81, American Mathematical Society, Providence, RI, 2000, x+257pp. · Zbl 0963.35211
[20] Taylor, M., Commutator estimates, Proc. Amer. Math. Soc., 131, 5, 1501-1507 (2003) · Zbl 1022.35096
[21] Yamazaki, M., A quasi-homogeneous version of paradifferential operators. I. Boundedness on spaces of Besov type, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 33, 131-174 (1986) · Zbl 0608.47058
[22] Yamazaki, M., A quasi-homogeneous version of paradifferential operators, II. A symbol calculus, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 33, 311-345 (1986) · Zbl 0659.47045
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