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# Wydawnictwa / Czasopisma IMPAN / Studia Mathematica / Wszystkie zeszyty

## An $L^q(L^2)$-theory of the generalized Stokes resolvent system in infinite cylinders

### Tom 178 / 2007

Studia Mathematica 178 (2007), 197-216 MSC: 35Q30, 76D07, 42A45, 46E40. DOI: 10.4064/sm178-3-1

#### Streszczenie

Estimates of the generalized Stokes resolvent system, i.e. with prescribed divergence, in an infinite cylinder ${\mit\Omega}={\mit\Sigma}\times\mathbb R$ with ${\mit\Sigma}\subset \mathbb R^{n-1}$, a bounded domain of class $C^{1,1}$, are obtained in the space $L^q(\mathbb R;L^2({\mit\Sigma}))$, $q\in (1,\infty)$. As a preparation, spectral decompositions of vector-valued homogeneous Sobolev spaces are studied. The main theorem is proved using the techniques of Schauder decompositions, operator-valued multiplier functions and $R$-boundedness of operator families.

#### Autorzy

• Reinhard FarwigDepartment of Mathematics