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Norm-generating pseudodifferential operators in the spaces \(W_p^s (\mathbb{R}^n)\). (English. Russian original) Zbl 1006.46025
Function spaces, harmonic analysis, and differential equations. Collected papers. Dedicated to the 95th anniversary of academician S. M. Nikol’skii. Transl. from the Russian. Moscow: MAIK Nauka/Interperiodika Publishing, Proc. Steklov Inst. Math. 232, 52-65 (2001); translation from Tr. Mat. Inst. Steklova 232, 58-71 (2001).
Let \(s\geq 0\), \(1<p< \infty\), \(\frac{1}{p} + \frac{1}{p'} =1\). Let \(W^s_p ({\mathbb R}^n)\) be the well-known Sobolev-Slobodeckij spaces. A mapping \(A\) from \(W^s_p ({\mathbb R}^n)\) into its dual \(W^{-s}_{p'} ({\mathbb R}^n)\) is called norm-generating if \[ (Au,u) = \|Au |W^{-s}_{p'} ({\mathbb R}^n) \|\cdot \|u |W^s_p ({\mathbb R}^n) \|, \quad u \in W^s_p ({\mathbb R}^n). \] It is the aim of this paper to study the structure of these operators \(A\).
For the entire collection see [Zbl 0981.00017].
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
47G30 Pseudodifferential operators