Complex powers of degenerating differential operators connected with the Klein--Gordon--Fock operator

D. V. Vozhzhov ; V. A. Nogin

Izvestiâ vysših učebnyh zavedenij. Matematika, no. 9 (2009), pp. 3-12.

Résumé

We develop a theory of complex powers of the generalized Klein–Gordon–Fock operator $$ m^2-\square-i\lambda\frac{\partial^2}{\partial x^2_1},\qquad\lambda>0. $$ The negative powers of this operator are realized as potential-type integrals with nonstandard metrics, while positive powers inverse to negative ones are realized as approximative inverse operators.

Détail

Keywords: potential-type operator, inverse operator, approximative inverse operator.
Mots-clés : symbol

Bibliographie

[1] Nogin V. A., Sukhinin E. V., Obraschenie i opisanie giperbolicheskikh besselevykh potentsialov s $L_p$-plotnostyami, Dep. v VINITI 07.12.93, No 3027–V93, Rost. gos. un-t, Rostov n/D., 1993, 88 pp. | Zbl

[2] Samko S. G., Hypersingular integrals and their applications, Analytical methods and special functions, 5, Taylor Frances, London–New York, 2002, 358 pp. | Zbl

[3] Nogin V. A., Samko S. G., “Method of approximating inverse operators and its applications to the inversion of potential-type integral transforms”, Int. Transforms Special Funct., 6:2 (1999), 89–104 | DOI | MR

[4] Nogin V. A., Samko S. G., “Some applications of potentials and approximative inverse operators in multi-dimensional fractional calculus”, Fract. Calc. Appl. Anal., 2:2 (1999), 205–228 | MR | Zbl

[5] Samko S. G., “Inversion theorems for potential-type integral transforms in $\mathbb R^n$ and on $S^{n-1}$”, Int. Transforms Special Funct., 1:2 (1993), 145–163 | DOI | MR | Zbl

[6] Karasev D. N., Nogin V. A., “$L_p$–$L_q$ otsenki dlya akusticheskikh potentsialov i ikh prilozheniya”, Izv. vuzov. Sev.-Kavk. region. Estestvennye nauki, 2006, no. 5, 3–7

[7] Chegolin A. P., Nogin V. A., “Integral transforms related to complex powers of the generalized Schrödinger operator”, Int. Transforms Special Funct., 17:6 (2006), 409–420 | DOI | MR | Zbl

[8] Karapetyants A. N., Nogin V. A., “Complex powers of the second order non-homogeneous elliptic differential operators with degenerating symbols in the spaces $L_p(\mathbb R^n)$”, Bol. Soc. Mat. Mexicana, 7 (2001), 193–209 | MR | Zbl

[9] Karapetyants A. N., Karasev D. N., Nogin V. A., “$L_p$–$L_q$-estimates for the fractional acoustic potentials and some related operators”, Fract. Calc. Appl. Anal., 8:2 (2005), 155–172 | MR | Zbl

[10] Karasev D. N., Nogin V. A., “Estimates for the acoustic potentials and their application”, Proc. A. Razmadze Math. Inst., 129 (2002), 29–51 | MR | Zbl

[11] Nogin V. A., Sukhinin E. V., “Function spaces connected with Klein–Gordon integral transforms”, Int. Transforms Special Funct., 7:3–4 (1998), 265–278 | MR

[12] Rubin B., “Hypersingular integrals of Marchand type and the inversion problem”, Math. Nachr., 165 (1994), 256–258 | DOI

[13] Lizorkin P. I., “Obobschennoe liuvillevskoe differentsirovanie i metod multiplikatorov v teorii vlozhenii klassov differentsiruemykh funktsii”, Tr. MIAN, 105, 1969, 89–167 | MR | Zbl

[14] Vozhzhov D. V., Nogin V. A., “Obraschenie nekotorykh operatorov tipa potentsiala s simvolami, vyrozhdayuschimisya na giperboloidakh ili paraboloidakh”, Matem. zametki, 80:6 (2006), 814–824 | MR | Zbl

[15] Nogin V. A., Chegolin A. P., “Kompleksnye stepeni telegrafnogo i blizkikh k nemu operatorov v $L_p$-prostranstvakh”, Differents. uravneniya, 39:3 (2003), 402–409 | MR | Zbl