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In 1958 L. M. Kelly and W, O, J. Moser showed that apart from a pencil, any configuration of n lines in the real projective plane has at least 3n/7 ordinary or simple points of intersection, with equality in the Kelly-Moser example (a complete quadrilateral with its three diagonal lines). In 1981 S. Hansen claimed to have improved this to n/2 (apart from(More)
A!"#$%&#. In dimension n ≥ 3, we deÞne a generalization of the classical two dimensional partial Legendre transform, that reduces interior regularity of the generalized Monge-Ampère equation detD2u = k (x, u,Du) to regularity of a divergence form quasilinear system of special form. This is then used to obtain smoothness of C2,1 solutions, having n − 1(More)
Let T be a positive linear operator defined for nonnegative functions on a rj-finite measure space {X,m,fi). Given 1 < p < oo and a nonnegative weight function w on X , it is shown that there exists a nonnegative weight function v , finite /¿-almost everywhere on X , such that (1) I \Tf)*wdfi< j fvd/i, for all/>0, J x J x tere exists <j> posi ( h if and(More)
Let E ⊂ C be a compact set, g : C → C be a K-quasiconformal map, and let 0 < t < 2. Let H denote t-dimensional Hausdorff measure. Then H(E) = 0 =⇒ H ′ (gE) = 0 , t′ = 2Kt 2 + (K − 1)t . This is a refinement of a set of inequalities on the distortion of Hausdorff dimensions by quasiconformal maps proved by K. Astala [2] and answers in the positive a(More)
where to fix ideas, the operator T is either the Hardy-Littlewood maximal operator or any Calderón-Zygmund Operator. Versions of these type of inequalities were studied by Sawyer in [Sa] motivated by the work of Muckenhoupt and Wheeden [MW] (see also the works [AM] and [MOS]). E. Sawyer proved that inequality (1.1) holds in R when T = M is the(More)
We discuss the mth order Bergman metric and the mth order Carathéodory-Reiffen metric of Burbea, and a new higher order metric arised in the study of Berezin’s operator calculus on bounded domains in C. Some comparison results among them and the corresponding classical intrinsic metrics are established on certain domains. Inequalities for eigenvalues of(More)