Abstract
We consider a special class of quasielliptic matrix operators and establish isomorphic properties of these operators in special scales of weighted Sobolev spaces. We give an example of application of these results to systems of differential equations that are not solved with respect to the derivative.
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Demidenko G. V. and Uspenskiî, S. V., Equations and Systems That Are Not Solved with Respect to the Higher Derivative [in Russian], Nauchnaya Kniga, Novosibirsk (1998).
Demidenko G. V., “On weighted Sobolev spaces and integral operators determined by quasielliptic equations,” Dokl. Akad. Nauk, 334, No. 4, 420-423 (1994).
Demidenko G. V., “On quasielliptic operators in Rn,” Sibirsk. Mat. Zh., 39, No. 5, 1028-1037 (1998).
Kudryavtsev L. D., “Embedding theorems for functions defined on the whole space or half-space,” I: Mat. Sb., 69, No. 4, 616-639 (1966); II: Mat. Sb., 70, No. 1, 3-35 (1966).
Kudryavtsev L. D. and Nikol'skiî, S. M., “Spaces of di_erentiable functions in several variables and embedding theorems,” in: Contemporary Problems of Mathematics. Fundamental Trends [in Russian], VINITI, Moscow, 1988, 26, pp. 5-157 (Itogi Nauki i Tekhniki).
Sobolev S. L., “On a new problem of mathematical physics,” Izv. Akad. Nauk SSSR Ser. Mat., 18, No. 1, 3-50 (1954).
Maslennikova V. N., “Estimates in L p and asymptotic behavior of a solution to the Cauchy problem for the Sobolev system as t → ∞,” Trudy Mat. Inst. Steklov, 103, 117-141 (1968).
Cantor M., “Elliptic operators and the decomposition of tensor fields,” Bull. Amer. Math. Soc., 5, No. 3, 235-262 (1981).
Choquet-Bruhat Y. and Christodoulou D., “Elliptic systems in H s, σ spaces on manifolds which are Euclidean at infinity,” Acta Math., 146, No. 1/2, 129-150 (1981).
Lockhart R. B. and McOwen R. C., “Elliptic differential operators on noncompact manifolds,” Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 12, No. 3, 409-447 (1985).
Amrouche C., Girault V., and Giroire J., “Espaces de Sobolev avec poids et equation de Laplace dans R n (Partie I),” C. R. Acad. Sci. Paris Ser. I, 315, 269-274 (1992).
Hile G. N. and Mawata C. P., “The behavior of the heat operator on weighted Sobolev spaces,” Trans. Amer. Math. Soc., 350, No. 4, 1407-1428 (1998).
Demidenko G. V., “Isomorphic properties of one class of differential operators,” in: Abstracts: The Fourth Siberian Congress on Applied and Industrial Mathematics Dedicated to the Memory of M. A. Lavrent'ev. Part 1, Inst. Mat. (Novosibirsk), Novosibirsk, 2000, pp. 123-124.
Uspenskiî, S. V., “On representation of functions determined by a certain class of hypoelliptic operators,” Trudy Mat. Inst. Steklov, 117, 292-299 (1972).
Lizorkin P. I., “Generalized Liouville di_erentiation and the multiplier method in the theory of embeddings of classes of di_erentiable functions,” Trudy Mat. Inst. Steklov, 105, 89-167 (1969).
Hardy G. H., Littlewood J. E., and Pólya G., Inequalities [Russian translation], Izdat. Inostr. Lit., Moscow (1948).
Daletskiî, Yu. L. and Kreîn, M. G., Stability of Solutions to Differential Equations in Banach Space [in Russian], Nauka, Moscow (1970).
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Demidenko, G.V. Isomorphic Properties of One Class of Differential Operators and Their Applications. Siberian Mathematical Journal 42, 865–883 (2001). https://doi.org/10.1023/A:1011907425498
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DOI: https://doi.org/10.1023/A:1011907425498