Pointwise estimates for a class of strongly degenerate elliptic operators : a geometrical approach
B. Franchi, R. Serapioni (1987)
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
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B. Franchi, R. Serapioni (1987)
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
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Guozhen Lu (1996)
Publicacions Matemàtiques
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Stephen M. Buckley, Pekka Koskela, Guozhen Lu (1995)
Publicacions Matemàtiques
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We obtain (weighted) Poincaré type inequalities for vector fields satisfying the Hörmander condition for p < 1 under some assumptions on the subelliptic gradient of the function. Such inequalities hold on Boman domains associated with the underlying Carnot- Carathéodory metric. In particular, they remain true for solutions to certain classes of subelliptic equations. Our results complement the earlier results in these directions for p ≥ 1.
Piotr Hajlasz, Juha Kinnunen (1998)
Revista Matemática Iberoamericana
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We prove that every Sobolev function defined on a metric space coincides with a Hölder continuous function outside a set of small Hausdorff content or capacity. Moreover, the Hölder continuous function can be chosen so that it approximates the given function in the Sobolev norm. This is a generalization of a result of Malý [Ma1] to the Sobolev spaces on metric spaces [H1].
Guozhen Lu (1992)
Revista Matemática Iberoamericana
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In this paper we mainly prove weighted Poincaré inequalities for vector fields satisfying Hörmander's condition. A crucial part here is that we are able to get a pointwise estimate for any function over any metric ball controlled by a fractional integral of certain maximal function. The Sobolev type inequalities are also derived. As applications of these weighted inequalities, we will show the local regularity of weak solutions for certain classes of strongly degenerate differential...
José C. Fernandes (1991)
Revista Matemática Iberoamericana
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In this paper we study the behavior of degenerate parabolic equations of the form v(x) ut(x,t) = Σn i,j=1 Dxi (aij(x,t) Dxi u(x,t)), where the coefficients are measurable functions.
Bruno Franchi, Guozhen Lu, Richard L. Wheeden (1995)
Annales de l'institut Fourier
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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 versions of Poincaré’s inequality, we obtain relative isoperimetric inequalities.
Paul MacManus (2002)
Publicacions Matemàtiques
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Our understanding of the interplay between Poincaré inequalities, Sobolev inequalities and the geometry of the underlying space has changed considerably in recent years. These changes have simultaneously provided new insights into the classical theory and allowed much of that theory to be extended to a wide variety of different settings. This paper reviews some of these new results and techniques and concludes with an example on the preservation of Sobolev spaces by the maximal function. ...