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Izv. Akad. Nauk SSSR Ser. Mat., 1973, Volume 37, Issue 2, Pages 356–385 (Mi izv2252)  

This article is cited in 14 scientific papers (total in 14 papers)

On $(p,l)$-capacity, inbedding theorems, and the spectrum of a selfadjoint elliptic operator

V. G. Maz'ya

Abstract: Necessary and sufficient conditions are found for continuity, compactness, and closability of imbedding operators of some function spaces into the space $L_p$. These results (for $p=2$) give criteria for positive definiteness and discreteness of the spectrum of the Dirichlet problem for a selfadjoint elliptic operator of arbitrary order. Some integral inequalities are considered for differentiable functions on a cube.

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English version:
Mathematics of the USSR-Izvestiya, 1973, 7:2, 357–387

Bibliographic databases:

UDC: 513.88
MSC: Primary 46E35, 47F05; Secondary 35J40
Received: 20.03.1972

Citation: V. G. Maz'ya, “On $(p,l)$-capacity, inbedding theorems, and the spectrum of a selfadjoint elliptic operator”, Izv. Akad. Nauk SSSR Ser. Mat., 37:2 (1973), 356–385; Math. USSR-Izv., 7:2 (1973), 357–387

Citation in format AMSBIB
\by V.~G.~Maz'ya
\paper On $(p,l)$-capacity, inbedding theorems, and the spectrum of a~selfadjoint elliptic operator
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1973
\vol 37
\issue 2
\pages 356--385
\jour Math. USSR-Izv.
\yr 1973
\vol 7
\issue 2
\pages 357--387

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    This publication is cited in the following articles:
    1. P. I. Lizorkin, M. Otelbaev, “Imbedding theorems and compactness for spaces of Sobolev type with weights”, Math. USSR-Sb., 36:3 (1980), 331–349  mathnet  crossref  mathscinet  zmath  isi
    2. O. D. Apyshev, M. Otelbaev, “On the spectrum of a class of differential operators and some imbedding theorems”, Math. USSR-Izv., 15:1 (1980), 1–24  mathnet  crossref  mathscinet  zmath  isi
    3. I. D. Chueshov, “A remark on the Schrödinger operator with a highly singular potential”, Funct. Anal. Appl., 15:4 (1981), 310–311  mathnet  crossref  mathscinet  zmath  isi
    4. V. G. Maz'ya, T. O. Shaposhnikova, “Theory of multipliers in spaces of differentiable functions”, Russian Math. Surveys, 38:3 (1983), 23–95  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    5. A. Fraguela Collar, “On perturbation of a polyharmonic operator by delta-like potentials”, Math. USSR-Sb., 58:2 (1987), 389–396  mathnet  crossref  mathscinet  zmath
    6. Regina Kleine, “Warped products with discrete spectra”, Results. Math, 15:1-2 (1989), 81  crossref
    7. M. van den Berg, “On the spectral counting function for the Dirichlet Laplacian”, Journal of Functional Analysis, 107:2 (1992), 352  crossref
    8. N. N. Tarkhanov, “Approximation on compact sets by solutions of systems with surjective symbol”, Russian Math. Surveys, 48:5 (1993), 103–145  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    9. Guo Zhen Lu, “Potential Analysis on Carnot Groups, Part II: Relationship between Hausdorff Measures and Capacities”, Acta Math Sinica, 20:1 (2004), 25  crossref  mathscinet  zmath
    10. Vladimir Kondratiev, Vladimir Maz'ya, Mikhail Shubin, “Discreteness of Spectrum and Strict Positivity Criteria for Magnetic Schrödinger Operators”, Communications in Partial Differential Equations, 29:3-4 (2005), 489  crossref
    11. Vladimir Kondratiev, Vladimir Maz'ya, Mikhail Shubin, “Gauge Optimization and Spectral Properties of Magnetic Schrödinger Operators”, Communications in Partial Differential Equations, 34:10 (2009), 1127  crossref  elib
    12. L. E. Fraenkel, “A lower bound for electrostatic capacity in the plane”, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 88:3-4 (2011), 267  crossref
    13. K. N. Ospanov, “Discreteness and estimates of spectrum of a first order difference operator”, Eurasian Math. J., 9:2 (2018), 89–94  mathnet
    14. L. M. Mustafina, V. V. Zhurov, N. F. Abaeva, K. M. Akhmetov, “Raznostnye vesovye teoremy vlozheniya v odnom vyrozhdennom sluchae”, Mezhdunar. nauch.-issled. zhurn., 2018, no. 5(71), 18–24  mathnet  crossref
  • Известия Академии наук СССР. Серия математическая Izvestiya: Mathematics
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