Singular values of compact pseudodifferential operators of variable order with nonsmooth symbol

A. I. Karol

Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 50, Tome 519 (2022), pp. 67-104

Résumé

We consider compact pseudodifferential operators with symbols whose decaying order with respect to the variable $\xi$ depends on the space variable. We obtain the estimates for singular values as well as validity conditions of the Weyl's asymptotics. The results are formulated in terms of the symbol belonging to the classes of multipliers of integral operators. We give applications of the results to the $L_2$ - small ball deviation asymptotics for Gaussian processes with variable Hurst parameter.

Bibliographie

[1] A. Benassi, S. Jaffard, D. Roux, “Gaussian processes and pseudodifferential elliptic operators”, Rev. Math. Iberoamer., 13:1 (1997), 19–81 | DOI | MR

[2] A. Benassi, S. Cohen, J. Istas, “Identifying the multifractional function of a Gaussian process”, Statist. Probab. Lett., 39 (1998), 337–345 | DOI | MR

[3] J. F. Coeurjolly, “Identification of multifractional Brownian motion”, Bernoulli, 11:6 (2005), 987–1008 | DOI | MR

[4] A. I. Karol, A. I. Nazarov, “Spectral Analysis for some Multifractional Gaussian Processes”, Russian J. Math. Phys., 28:4 (2021), 488–500 | DOI | MR

[5] A. Karol, A. Nazarov, Ya. Nikitin, “Small ball probabilities for Gaussian random fields and tensor products of compact operators”, Trans. Amer. Math. Soc., 360:3 (2008), 1443–1474 | DOI | MR

[6] G. Lieberman, “Regularized distance and its applications”, Pacific J. Math., 117:2 (1985), 329–352 | DOI | MR

[7] M. A. Lifshits, Lectures on Gaussian Processes, Springer, New York, 2012 | MR

[8] A. I. Nazarov, “Log-level comparison principle for small ball probabilities”, Stat. Prob. Lett., 79:4 (2009), 481–486 | DOI | MR

[9] R.-F. Peltier, J. Lévy Véhel, Multifractional Brownian motion: definition and preliminary results, Inria research report No 2645, 1995

[10] Theor. Prob. and Math. Stat., 80 (2010), 119–130 | DOI

[11] J. Ryvkina, “Fractional Brownian motion with variable Hurst parameter: definition and properties”, J. Theor. Probab., 28:3 (2015), 866–891 | DOI | MR

[12] E. Seneta, Regularly Varying Functions, Lect. Notes in Math., 508, Springer, Berlin–Heidelberg, 1976 | DOI | MR

[13] M. Sh. Birman, M. Z. Solomyak, “Asimptotika spektra psevdodifferentsialnykh operatorov s anizotropno-odnorodnymi simvolami”, Vestn. LGU, ser. matem., 1977, no. 13, 13–21

[14] M. Sh. Birman, M. Z. Solomyak, “Asimptotika spektra psevdodifferentsialnykh operatorov s anizotropno-odnorodnymi simvolami II”, Vestn. LGU, ser. matem., 1979, no. 13, 5–10

[15] M. Sh. Birman, M. Z. Solomyak, “Otsenki singulyarnykh chisel integralnykh operatorov”, Uspekhi matem. nauk, 32:1 (1977), 17–84 | MR

[16] M. Sh. Birman, M. Z. Solomyak, Spektralnaya teoriya samosopryazhennykh operatorov v gilbertovom prostranstve, Lan, SPb., 2010, 464 pp.

[17] T. Vaidl, “Obschie operatornye idealy slabogo tipa”, Algebra i analiz, 4:3 (1992), 117–144 | MR

[18] A. I. Karol, “Asimptotika spektra kompaktnykh pdo s simvolom, negladkim po prostranstvennym peremennym”, Probl. matem. anal., 89 (2017), 21–39

[19] A. I. Karol, “Asimptotika singulyarnykh chisel kompaktnykh PDO s simvolom, negladkim po prostranstvennym peremennym”, Funkts. analiz i ego pril., 53:4 (2019), 89–92 | MR

[20] A. I. Karol, “Singulyarnye chisla kompaktnykh psevdodifferentsialnykh operatorov s simvolom, negladkim po prostranstvennym peremennym”, Sib. mat. zh., 61:4 (2020), 849–866 | MR

[21] S. M. Nikolskii, Priblizhenie funktsii mnogikh peremennykh i teoremy vlozheniya, Nauka, M., 1977 | MR