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 Trudy Mat. Inst. Steklova, 2001, Volume 232, Pages 58–71 (Mi tm204)

Norm-Generating Pseudodifferential Operators in the Spaces $W_p^s(\mathbb R^n)$

K. O. Besov

Abstract: A class of operators $A$ is analyzed in the Sobolev–Slobodetskii spaces $W_p^s$ on $\mathbb R^n$, $s\in\mathbb R_+$, such that the corresponding equation $Au=f$ is uniquely solvable for any right-hand side. The operators constituting this class—the so-called norm-generating operators—are analogues of known operators of the $p$-Laplacian type in the Sobolev spaces $W_p^s$, $s\in \mathbb N$. In the case of a Hilbert space $W_2^s$, the operators considered are ordinary linear pseudodifferential operators. In the general case when $p\ne 2$ and $s\notin\mathbb N$, the operators are nonlinear and nonlocal and define a one-to-one mapping of the space $W_p^s$ onto the adjoint space $W_{p'}^{-s}$. In addition to the analysis of the properties of these operators, examples of norm-generating operators in $W_p^s$ are presented that specify a more complicated structure of the mapping (that is not one-to-one).

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English version:
Proceedings of the Steklov Institute of Mathematics, 2001, 232, 52–65

Bibliographic databases:
UDC: 513.881+517.43+517.944

Citation: K. O. Besov, “Norm-Generating Pseudodifferential Operators in the Spaces $W_p^s(\mathbb R^n)$”, Function spaces, harmonic analysis, and differential equations, Collected papers. Dedicated to the 95th anniversary of academician Sergei Mikhailovich Nikol'skii, Trudy Mat. Inst. Steklova, 232, Nauka, MAIK «Nauka/Inteperiodika», M., 2001, 58–71; Proc. Steklov Inst. Math., 232 (2001), 52–65

Citation in format AMSBIB
\Bibitem{Bes01} \by K.~O.~Besov \paper Norm-Generating Pseudodifferential Operators in the Spaces $W_p^s(\mathbb R^n)$ \inbook Function spaces, harmonic analysis, and differential equations \bookinfo Collected papers. Dedicated to the 95th anniversary of academician Sergei Mikhailovich Nikol'skii \serial Trudy Mat. Inst. Steklova \yr 2001 \vol 232 \pages 58--71 \publ Nauka, MAIK «Nauka/Inteperiodika» \publaddr M. \mathnet{http://mi.mathnet.ru/tm204} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1851440} \zmath{https://zbmath.org/?q=an:1006.46025} \transl \jour Proc. Steklov Inst. Math. \yr 2001 \vol 232 \pages 52--65 

• http://mi.mathnet.ru/eng/tm204
• http://mi.mathnet.ru/eng/tm/v232/p58

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This publication is cited in the following articles:
1. Besov K.O., “Eigenfunctions of some nonlinear nonlocal operators”, Differential Equations, 38:4 (2002), 510–519
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