Multilineární singulární integrální operátory a jejich omezenost na prostorech funkcí
| Thesis title in Czech: | Multilineární singulární integrální operátory a jejich omezenost na prostorech funkcí |
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| Thesis title in English: | Multilinear singular integral operators and their boundedness properties |
| Academic year of topic announcement: | 2024/2025 |
| Thesis type: | dissertation |
| Thesis language: | |
| Department: | Department of Mathematical Analysis (32-KMA) |
| Supervisor: | RNDr. Lenka Slavíková, Ph.D. |
| Author: |
| Guidelines |
| The student will first familiarize himself with recent results involving boundedness properties of multilinear singular integral operators and multilinear Fourier multiplier operators. Afterwards, he will work on extending the existing results to new frameworks. This goal may be approached from different perspectives, e.g. by generalizing a bilinear result to the multilinear setting, by considering new types of singular integral operators or new function spaces. The conditions guaranteeing boundedness of the operators will often be expressed in terms of various spaces of functions and emphasis will be put on their optimality. Another possible research direction involves investigating applications of these results, for instance in ergodic theory.
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| References |
| G. Dosidis, B. Park and L. Slavíková: "Boundedness criteria for bilinear Fourier multipliers via shifted square function estimates", preprint 2024, arXiv:2402.15785.
J. Duoandikoetxea and J. L. Rubio de Francia: "Maximal and singular integral operators via Fourier transform estimates", Invent. Math. 84 (1986), 541 - 561. P. Durcik, V. Kovač, K. A. Škreb, and C. Thiele: "Norm variation of ergodic averages with respect to two commuting transformations", Ergodic Theory Dynam. Systems, 39(3), 658 - 688, 2019. L. Grafakos: "Modern Fourier analysis", third edition. Graduate Texts in Mathematics, 250. Springer, New York, 2014. L. Grafakos, D. He, and P. Honzík: "Rough bilinear singular integrals", Adv. Math. 326 (2018) 54 - 78. V. Kovač: "Boundedness of the twisted paraproduct", Rev. Mat. Iberoam. 28 (2012), no. 4, 1143 - 1164. C. Muscalu, W. Schlag: "Classical and Multilinear Harmonic Analysis", Vol. II., Cambridge University Press, Cambridge, 2013. L. Pick, A. Kufner, O. John and S. Fučík: "Function Spaces", Vol. 1., De Gruyter, Berlin, 2013. and other papers and monographs as needed |