Global analytic and Gevrey surjectivity of the Mizohata operator D 2 + i ⁢ x 2 2 ⁢ k ⁢ D 1

Lamberto Cattabriga; Luisa Zanghirati

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1990)

  • Volume: 1, Issue: 1, page 37-39
  • ISSN: 1120-6330

Abstract

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The surjectivity of the operator D 2 + i ⁢ x 2 2 ⁢ k ⁢ D 1 from the Gevrey space E s ⁢ R 2 , s ≥ 1 , onto itself and its non-surjectivity from E s ⁢ R 3 to E s ⁢ R 3 is proved.

How to cite

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Cattabriga, Lamberto, and Zanghirati, Luisa. "Global analytic and Gevrey surjectivity of the Mizohata operator \( D_2 + i x^{2k}_{2} D_{1} \)." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 1.1 (1990): 37-39. <http://eudml.org/doc/244233>.

@article{Cattabriga1990,
abstract = {The surjectivity of the operator \( D\_2 + i x^\{2k\}\_\{2\} D\_\{1\} \) from the Gevrey space \( \mathcal\{E\}^\{\\{s\\}\}(\mathbb\{R\}^\{2\}) \), \( s \geq 1 \), onto itself and its non-surjectivity from \( \mathcal\{E\}^\{\\{s\\}\}(\mathbb\{R\}^\{3\}) \) to \( \mathcal\{E\}^\{\\{s\\}\}(\mathbb\{R\}^\{3\}) \) is proved.},
author = {Cattabriga, Lamberto, Zanghirati, Luisa},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Linear partial differential equations; Surjectivity; Gevrey spaces; Mizohata operator; surjectivity; Gevrey space; non-surjectivity},
language = {eng},
month = {2},
number = {1},
pages = {37-39},
publisher = {Accademia Nazionale dei Lincei},
title = {Global analytic and Gevrey surjectivity of the Mizohata operator \( D\_2 + i x^\{2k\}\_\{2\} D\_\{1\} \)},
url = {http://eudml.org/doc/244233},
volume = {1},
year = {1990},
}

TY - JOUR
AU - Cattabriga, Lamberto
AU - Zanghirati, Luisa
TI - Global analytic and Gevrey surjectivity of the Mizohata operator \( D_2 + i x^{2k}_{2} D_{1} \)
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1990/2//
PB - Accademia Nazionale dei Lincei
VL - 1
IS - 1
SP - 37
EP - 39
AB - The surjectivity of the operator \( D_2 + i x^{2k}_{2} D_{1} \) from the Gevrey space \( \mathcal{E}^{\{s\}}(\mathbb{R}^{2}) \), \( s \geq 1 \), onto itself and its non-surjectivity from \( \mathcal{E}^{\{s\}}(\mathbb{R}^{3}) \) to \( \mathcal{E}^{\{s\}}(\mathbb{R}^{3}) \) is proved.
LA - eng
KW - Linear partial differential equations; Surjectivity; Gevrey spaces; Mizohata operator; surjectivity; Gevrey space; non-surjectivity
UR - http://eudml.org/doc/244233
ER -

References

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  2. CATTABRIGA, L., Solutions in Gevrey spaces of partial differential equations with constant coefficients. Astérisque, 89-90, 1981, 129-151. Zbl0496.35018MR666406
  3. CATTABRIGA, L., On the surjectivity of differential polynomials on Gevrey spaces. Rend. Sem. Mat. Univ. Politec. Torino, Fasc. speciale, Convegno Linear partial and pseudodifferential operators, Torino, 30 Sett. - 2 Ott. 1982, 41, 1983, 81-89. Zbl0561.35008MR745976
  4. DE GIORGI, E., Solutions analytiques des équations aux dérivées partielles à coefficients constants. Séminaire Goulauic-Schwartz 1971-72, Equations aux dérivées partielles et analyse fonctionelle, Exp. N. 29, Ecole Polytech. Centre de Math., Paris1972. Zbl0244.35017MR397137
  5. DE GIORGI, E. - CATTABRIGA, L., Una dimostrazione diretta dell'esistenza di soluzioni analitiche nel piano reale di equazioni a derivate parziali a coefficienti costanti. Boll. Un. Mat. Ital. (4), 4, 1971, 1015-1027. Zbl0224.35018MR382820
  6. EHRENPREIS, L., Lewy's operator and its ramifications. J. Funct. Anal, 68, 1986, 329-265. Zbl0638.35002MR859139DOI10.1016/0022-1236(86)90102-3
  7. FRIBERG, J., Estimates for partially hypoelliptic differential operators. Medd. Lund Univ. Mat. Sem., 17, 1963, 1-93. Zbl0139.28403MR157109
  8. HÖRMANDER, L., On the existence of real analytic solutions of partial equations with constant coefficients. Inventiones Math., 21, 1973, 151-182. Zbl0282.35015MR336041
  9. HÖRMANDER, L., The analysis of linear partial differential operators. Springer Verlag, vol. IV, 1983. Zbl0521.35002
  10. KAWAI, T., On the global existence of real analytic solutions of linear differential equations I and II. J. Math. Soc. Japan, 24, 1972, 481-517; 25, 1973, 644-647. Zbl0259.35062MR310412
  11. KAWAI, T., On the global existence of real analytic solutions and hyperfunction solutions of linear differential equations. Publ. RIMS Kyoto University, 555, 1986. 
  12. KOMATSU, H., An analogue of the Cauchy-Kowalevsky theorem for ultradifferentiable functions and a division theorem of ultradistributions as its dual. J. Fac. Sci. Univ. Tokyo, Sec. IA, 26, 1979, 239-254. Zbl0424.46032MR550685
  13. MIZOHATA, S., Solutions nulles et solutions non analytiques. J. Math. Kyoto Univ., 1, 1962, 271-302. Zbl0106.29601MR142873
  14. RODINO, L., On linear partial differential operators with multiple characteristics. Proceedings Conf. on Partial Differential Equations (Holzhau, GDR, 1988), Teubner Text zur Mathematik. Zbl0681.35093MR1105815
  15. ZAMPIERI, G., An application of the fundamental principle of Ehrenpreis to the existence of global Gevrey solutions of linear differential equations. Boll. Un. Mat. It. (6), V-B, 1986, 361-392. Zbl0624.35011MR860634

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