# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] Agmon, S.; Douglis, A.; Nirenberg, L., Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I, Comm. pure appl. math., 12, 623-727, (1959) · Zbl 0093.10401 [2] Alvarez, J.; Pérez, C., Estimates with $$A_\infty$$ weights for various singular integral operators, Boll. un. mat. ital. A (7), 8, 123-133, (1994) · Zbl 0807.42013 [3] Calderón, A.P.; Zygmund, A., On singular integrals, Amer. J. math., 78, 289-309, (1956) · Zbl 0072.11501 [4] D. Cruz-Uribe, A. Fiorenza, C.J. Neugebauer, The maximal function on variable $$L^p$$ spaces, preprint (2002) · Zbl 1064.42500 [5] Diening, L.; Růžička, M., Calderón – zygmund operators on generalized Lebesgue spaces $$L^{p(\cdot)}$$ and problems related to fluid dynamics, J. reine angew. math., 563, 197-220, (2003) · Zbl 1072.76071 [6] Diening, L.; Růžička, M., Integral operators on the halfspace in generalized Lebesgue spaces $$L^{p(\cdot)}$$, part II, J. math. anal. appl., 298, 572-588, (2004), this issue · Zbl 1128.47044 [7] Diening, L., Maximal function on generalized Lebesgue spaces $$L^{p(\cdot)}$$, Math. inequalities appl., (2002), in press · Zbl 1071.42014 [8] L. Diening, Riesz potential and Sobolev embeddings of generalized Lebesgue and Sobolev spaces $$L^{p(\cdot)}$$ and $$W^{k, p(\cdot)}$$, preprint, University of Freiburg (2002) [9] L. Diening, Theoretical and numerical results for electrorheological fluids, Ph.D. thesis, Mathematische Fakultät, Universität Freiburg im Breisgau, 2002, p. 156 · Zbl 1022.76001 [10] Edmunds, D.E.; Lang, J.; Nekvinda, A., On $$L^{p(x)}$$ norms, Proc. roy. soc. London ser. A, 455, 219-225, (1999) · Zbl 0953.46018 [11] Edmunds, D.E.; Rákosni´k, J., Sobolev embeddings with variable exponent, Studia math., 143, 267-293, (2000) · Zbl 0974.46040 [12] Fan, X.; Shen, J.; Zhao, D., Sobolev embedding theorems for spaces $$W^{k, p(x)}(\Omega)$$, J. math. anal. appl., 262, 749-760, (2001) · Zbl 0995.46023 [13] Frehse, J.; Málek, J.; Steinhauer, M., An existence result for fluids with shear dependent viscosity—steady flows, (), Part 5, pp. 3041-3049 · Zbl 0902.35089 [14] Hudzik, H., The problems of separability, duality, reflexivity and of comparison for generalized orlicz – sobolev spaces $$W_M^k(\Omega)$$, Comment. math. prace mat., 21, 315-324, (1980) [15] Kováčik, O.; Rákosni´k, J., On spaces $$L^{p(x)}$$ and $$W^{k, p(x)}$$, Czechoslovak math. J., 41, 592-618, (1991) · Zbl 0784.46029 [16] Málek, J.; Nečas, J.; Rokyta, M.; Růžička, M., Weak and measure-valued solutions to evolutionary pdes, (1996), Chapman & Hall London · Zbl 0851.35002 [17] Málek, J.; Rajagopal, K.R.; Růžička, M., Existence and regularity of solutions and the stability of the rest state for fluids with shear dependent viscosity, Math. models methods appl. sci., 5, 789-812, (1995) · Zbl 0838.76005 [18] Nekvinda, A., Hardy – littlewood maximal operator on $$L^{p(x)}(\mathbb{R}^n)$$, Math. inequalities appl., (2002), in press [19] Růžička, M., Electrorheological fluids: modeling and mathematical theory, Lecture notes in mathematics, vol. 1748, (2000), Springer-Verlag Berlin · Zbl 0968.76531 [20] Samko, S.G., Density $$C_0^\infty(\mathbf{R}^n)$$ in the generalized Sobolev spaces $$W^{m, p(x)}(\mathbf{R}^n)$$, Dokl. akad. nauk, 369, 451-454, (1999) · Zbl 1052.46028 [21] Stein, E.M., Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, (1993), Princeton Univ. Press Princeton, NJ, with the assistance of T.S. Murphy, Monographs in Harmonic Analysis, III · Zbl 0821.42001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.