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Norm inequalities for off-centered maximal operators.

Richard L. Wheeden — 1993

Publicacions Matemàtiques

Sufficient conditions are derived in order that there exist strong-type weighted norm inequalities for some off-centered maximal functions. The maximal functions are of Hardy-Littlewood and fractional types taken over starlike sets in R. The sufficient conditions are close to necessary and extend some previously known weak-type results.

Representation formulas and weighted Poincaré inequalities for Hörmander vector fields

Bruno FranchiGuozhen LuRichard L. Wheeden — 1995

Annales de l'institut Fourier

We derive weighted Poincaré inequalities for vector fields which satisfy the Hörmander condition, including new results in the unweighted case. We also derive a new integral representation formula for a function in terms of the vector fields applied to the function. As a corollary of the L 1 versions of Poincaré’s inequality, we obtain relative isoperimetric inequalities.

Two-weight Sobolev-Poincaré inequalities and Harnack inequality for a class of degenerate elliptic operators

Bruno FranchiCristian E. GutiérrezRichard L. Wheeden — 1994

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

In this Note we prove a two-weight Sobolev-Poincaré inequality for the function spaces associated with a Grushin type operator. Conditions on the weights are formulated in terms of a strong A weight with respect to the metric associated with the operator. Roughly speaking, the strong A condition provides relationships between line and solid integrals of the weight. Then, this result is applied in order to prove Harnack's inequality for positive weak solutions of some degenerate elliptic equations....

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