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Let be a convex curve in the plane and let be the arc-length measure of Let us rotate by an angle and let be the corresponding measure. Let . Then This is optimal for an arbitrary . Depending on the curvature of , this estimate can be improved by introducing mixed-norm estimates of the form where and are conjugate exponents. 相似文献
In this paper the authors give a criterion on the weighted boundedness of the multilinear oscillatory singular integral operators with rough kernels.
A theorem of M. F. Driscoll says that, under certain restrictions, the probability that a given Gaussian process has its sample paths almost surely in a given reproducing kernel Hilbert space (RKHS) is either or . Driscoll also found a necessary and sufficient condition for that probability to be .
Doing away with Driscoll's restrictions, R. Fortet generalized his condition and named it nuclear dominance. He stated a theorem claiming nuclear dominance to be necessary and sufficient for the existence of a process (not necessarily Gaussian) having its sample paths in a given RKHS. This theorem - specifically the necessity of the condition - turns out to be incorrect, as we will show via counterexamples. On the other hand, a weaker sufficient condition is available.
Using Fortet's tools along with some new ones, we correct Fortet's theorem and then find the generalization of Driscoll's result. The key idea is that of a random element in a RKHS whose values are sample paths of a stochastic process. As in Fortet's work, we make almost no assumptions about the reproducing kernels we use, and we demonstrate the extent to which one may dispense with the Gaussian assumption.
Let , be a sequence of bounded pseudoconvex domains that converges, in the sense of Boas, to a bounded domain . We show that if can be described locally as the graph of a continuous function in suitable coordinates for , then the Bergman kernel of converges to the Bergman kernel of uniformly on compact subsets of . 相似文献