Geodesic
On relative boundedness of a class of degenerate differential operators in the lebesgue space
Matematičeskie zametki SVFU, Tome 25 (2018) no. 1, pp. 3-14 Cet article a éte moissonné depuis la source Math-Net.Ru

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In the space $L_p(\Omega)$, where $1 and $\Omega$ is an arbitrary (bounded or unbounded) domain in $R^n$, we investigate relative boundedness for a class of higher order partial differential operators in non-divergent form. These operators have nonpower degeneracy on the whole boundary of $\Omega$ and degeneracy with respect to each of independent variables is characterized by different functions. In the earlier published papers in this direction, as a rule, firstly the operator is defined in $\Omega$ and then functions characterizing degeneracies of the operator's coefficients are defined in this domain. In contrast to that, here we define $\Omega$ and these functions related to each other while fulfilling the “immersion condition” introduced by P. I. Lizorkin in [19]. In addition, differentiability of the functions by which we define degeneracy of the investigated operator is not required. Study of relative boundedness of differential operators is one of the modern directions in such operators theory with results theory of differentiable functions of many variables, the separation theory of differential operators, the spectral theory of differential operators, etc.
Keywords: partial differential operator, non-power degeneracy, relative boundedness of operators, partition of the unit. 
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M. G. Gadoev; F. S. Iskhokov. On relative boundedness of a class of degenerate differential operators in the lebesgue space. Matematičeskie zametki SVFU, Tome 25 (2018) no. 1, pp. 3-14. http://geodesic.mathdoc.fr/item/SVFU_2018_25_1_a0/

[1] Kato T., Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, Heidelberg, New York, 1995 | MR | Zbl

[2] “Separability theorems, weighted spaces and their applications”, Proc. Steklov Inst. Math., 170 (1987), 39–81 | MR | Zbl

[3] Anderson T. G., Hinton D. B., “Relative boundedness and compactness theory for second-order differential operators”, J. Inequal. Appl., 1 (1997), 375–400 | MR | Zbl

[4] Binding P., Hryntiv R., “Relative boundedness and relative compactness for linear operators in Banach space”, Proc. Amer. Math. Soc., 128:8 (2000), 2287–2290 | DOI | MR | Zbl

[5] Otelbaev M., “On separation of elliptic operators”, Dokl. Akad. Nauk SSSR, 234:3 (1977), 540–543 | MR | Zbl

[6] “Coercive estimates and separation theorems of elliptic operators in $R^n$”, Proc. Steklov Inst. Math., 161 (1984), 213–239 | MR | Zbl | Zbl

[7] Muratbekov M., Otelbaev M., “On the existence of a resolvent and separability for a class of singular hyperbolic type differential operators on an unbounded domain”, Euras. Math. J., 7:1 (2016), 50–67 | MR

[8] Boimatov K. Kh., “Separability theorems, weighted spaces and their applications to boundary value problems”, Dokl. Akad. Nauk SSSR, 247:3 (1979), 532–536 | MR

[9] Boimatov K. Kh., “Separation theorems”, Dokl. Akad. Nauk SSSR, 213:5 (1973), 1009–1011 | MR

[10] Zorin V. A., “Method of investigation of essential and discrete spectra of Hamiltonian operator of hydogen-like atom in quantum mechanics of Kuryshkin”, Vestn. Ross. Univ. Druzhby Narodov, Ser. Prikl. Comput. Mat., 3:1 (2004), 121–131

[11] Brown R. C., Hinton D. B., “Relative form boundedness and compactness for a second-order differential operator”, J. Comput. Appl. Math., 171 (2004), 123–140 | DOI | MR | Zbl

[12] Trigub R. M., “Comparison of linear differential operators”, Math. Notes, 82:3 (2007), 380–394 | DOI | DOI | MR | MR | Zbl

[13] Behncke H., Nyamwala F. O., “Spectral analysis of higher order differential operators with unbounded coeffcients”, Math. Nachr., 285:1 (2012), 56–73 | DOI | MR | Zbl

[14] Brown R. C., “Separation and disconjugacy”, J. Inequal. Pure Appl. Math., 4:3 (2003), 56, 1–16 | MR

[15] Ospanov K. N., Akhmetkaliyeva R. D., “Separation and the existence theorem for second order nonlinear differential equation”, Electr. J. Quantat. Theory Diff. Equ., 2012, no. 66, 1–12 | MR

[16] Zayed E. M. E., Omran S. A., “Separation for triple-harmonic differential operator in Hilbert space”, Int. J. Math. Combin., 4 (2010), 13–23 | Zbl

[17] Qi J., Sun H., “Relatively bounded and relatively compact perturbations for limit circle Hamiltonian systems”, Integr. Equ. Oper. Theory, 86:3 (2016), 359–375 | DOI | MR | Zbl

[18] Gadoev M. G., Iskhokov F. S., “On invertibility of a class of degenerate differential operators in the Lebesgue space”, Mat. zamet. SVFU, 23:3 (2016), 3–26 | Zbl

[19] Lizorkin P. I., “Estimate of mixed and intermediate derivatives in weighted $L_p$-norms”, Tr. Mat. Inst. Steklova, 156 (1983), 141–153 | MR | Zbl | Zbl

[20] Gadoev M. G., Iskhokov F. S., “On some functional spaces, which norms are given by differential operators”, Proc. Int. Summer Math. Stechkin School-Conf. Function Theory (Aug. 15-25, 2016), Dushanbe, Tajikistan, 2016, 82–84