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Weighted estimates for nonhomogeneous quasilinear equations with discontinuous coefficients. (English) Zbl 1228.35260
The paper deals with local and global estimates for gradient of solutions to a nonhomogeneous quasilinear equation on certain Lebesgue spaces. The equation under consideration is of weak type and has measurable coefficients $$A_{ij}$$. The matrix $$A=A_{ij}(x)$$ is symmetric and satisfies the ellipticity condition.
The author obtains a weighted regularity estimate assuming that, in the bounded domain under study, the equation is of class $$C^1$$ and that each of its components belongs to the Sarason class VMO. The approach could be easily modified to cover the case where the domain $$\Omega$$ where the functions under consideration belong, is Lipschitz with small Lipschitz constant and $$A$$ has small bounded mean oscillation coefficients (i.e., they are in the John-Nirenberg class BMO).
The approach makes use of the Fefferman-Stein sharp maximal function and $$C^{1,\alpha}$$ regularity estimates obtained earlier for homogeneous $$p$$-Laplace type equations. In particular, a weighted version of Fefferman-Stein inequality for a local dyadic sharp maximal function is obtained, which enables the author to apply some available results to the weighted case.

##### MSC:
 35R05 PDEs with low regular coefficients and/or low regular data 35J15 Second-order elliptic equations 42B25 Maximal functions, Littlewood-Paley theory 35J92 Quasilinear elliptic equations with $$p$$-Laplacian 42B37 Harmonic analysis and PDEs
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