MathSciDoc: An Archive for Mathematician ∫

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On déduit des estimations gaussiennes supérieures du noyau de la chaleur des estimations du même type pour les premières dérivées spatiales. On obtient ainsi des estimations gaussiennes inférieures du noyau de la chaleur. On donne des applications de ces résultats.

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A sharp result on global small solutions to the Cauchy problem $$u_t = \Delta u + f\left( {u,Du,D^2 u,u_t } \right)\left( {t > 0} \right),u\left( 0 \right) = u_0 $$

We give a characterization of Possonian domains in$R$^{$n$}, i.e., those domains for which every bounded harmonic function is the harmonic extension of some function in$L$^{∞}of harmonic measure. We deduce several properties of such domains, including some results of Mountford and Port. In two dimensions we give an additional characterization in terms of the logarithmic capacity of the boundary. We also give a necessary and sufficient condition for the harmonic measures on two disjoint planar domains to be mutually singular.

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Let ℒ($H$) denote the space of operators on a Hilbert space$H$. We show that the extreme points of the unit ball of the space of continuous functions$C(K, ℒ(H))$($K$-compact Hausdorff) are precisely the functions with extremal values. We show also that these extreme points are (a) strongly exposed if and only if dim$H$<∞ and card$K$<∞, (b) exposed if and only if$H$is separable and$K$carries a strictly positive measure.

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