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

K. O. Besov

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function spaces, harmonic analysis, and differential equations, Tome 232 (2001), pp. 58-71

Résumé

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).