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Integral operators on the halfspace in generalized Lebesgue spaces \(L^{p(\cdot)}\). I, II. (English) Zbl 1128.47044
To study existence and regularity results for generalized Newtonian fluids and electrorheological fluids, the authors consider the generalized Sobolev space \(W^{k,p(x)}\) with variable exponent, which is a natural energy space for the system of electrorheological fluids. This Sobolev space is built upon the corresponding variable-exponent Lebesgue space \(L^{p(x)}\). The usually applied techniques in this direction often use optimal estimates for solutions to linear elliptic equations and systems, namely the Laplace equation, the Stokes system and the divergence equation. While such estimates are quite well-known in classical Lebesgue spaces, in the spaces \(L^{p(x)}\) with variable exponent only little is known. In their earlier work [J. Reine Angew. Math. 563, 197–220 (2003; Zbl 1072.76071)], the authors treated the divergence equation and extended the classical results of Calderón and Zygmund on principal value integrals and Calderón–Zygmund operators to the spaces \(L^{p(x)}\).
In the paper under review (Parts I and II), the corresponding results for a half-space are established. Such results are indispensable for the study of regularity near the boundary. In Part I, the Calderón–Zygmund theorem on principal value integrals is generalized to kernel operators which do not satisfy standard estimates on \(\mathbb R^{d+1}\).
In Part II, the results from Part I are used to obtain a generalization of the classical theorem of S. Agmon, A. Douglis and L. Nirenberg [Commun. Pure Appl. Math. 12, 623–727 (1959; Zbl 0093.10401)] to the spaces \(L^{p(x)}\). In particular, this result provides the desired estimates in the variable-exponent Sobolev space \(W^{2,p(x)}\) on the half-space for the Laplace equation and the Stokes system.

MSC:
47G10 Integral operators
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
76D05 Navier-Stokes equations for incompressible viscous fluids
76W05 Magnetohydrodynamics and electrohydrodynamics
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