MathSciDoc: An Archive for Mathematician ∫

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Let$F$be a family of meromorphic functions on the unit disc Δ and let$a$and$b$be distinct values. If for every$f$∈$F$,$f$and$f′$share$a$and$b$on Δ, then$F$is normal on Δ.

The main goal of this paper is to present an alternative, real variable proof of the$T$(1)-theorem for the Cauchy integral. We then prove that the estimate from below of analytic capacity in terms of total Menger curvature is a direct consequence of the$T$(1)-theorem. An example shows that the$L$^{∞}-BMO estimate for the Cauchy integral does not follow from$L$^{2}boundedness when the underlying measure is not doubling.

In this paper we show that the pluripolar hull of$E$={($z$, ω)∈C^{2}:ω=$e$^{−1/z},$z$≠0} is equal to$E$. This implies that$E$is plurithin at 0, which answers a question of Sadullaev. The result remains valid if$e$^{−1/z}is replaced by certain other holomorphic functions with an essential singularity at 0.

In his 1983 paper [3], R. F. Gundy introduced a new functional related to the Littlewood-Paley theory, called the$density of the area integral$. In this paper, we prove that this functional (although highly non-linear) can be expressed as the principal value of an explicit singular integral. This result provides us with a new and precise connection between the density of the area integral and the theory of Calderón-Zygmund operators. It does not seem to be a consequence of the standard Calderón-Zygmund-Cotlar theory, because the$sign$of a harmonic function in the half-space fails to have, in some appropriate sense, boundary limits.

This paper is devoted to the study of the Cauchy problem in$C$^{∞}and in the Gevrey classes for some second order degenerate hyperbolic equations with time dependent coefficients and lower order terms satisfying a suitable Levi condition.

Examples of surfaces in$P$^{6}with no trisecant lines are constructed. A partial classification recovering them is given and conjectured to be the complete one.

We establish several conditions, sufficient for a set to be (quasi)conformally removable, a property important in holomorphic dynamics. This is accomplished by proving removability theorems for Sobolev spaces in$R$^{$n$}. The resulting conditions are close to optimal.

We study the ergodic properties of fibered rational maps of the Riemann sphere. In particular we compute the topological entropy of such mappings and construct a measure of maximal relative entropy. The measure is shown to be the unique one with this property. We apply the results to selfmaps of ruled surfaces and to certain holomorphic mapping of the complex projective plane$P$^{2}.

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