ﻻ يوجد ملخص باللغة العربية
In this survey we collect and discuss some recent results on the so called Furstenberg set problem, which in its classical form concerns the estimates of the Hausdorff dimension of planar sets containing, for any direction, a subset of an interval poitning in that direction of some prescribed dimension. This problem is closely related to the Kakeya needle problem. In this work we approach this problem from a more general point of view, in terms of generalized Hausdorff measures associated to dimension functions. We generalize the known results in terms of logarithmic gaps and obtain analogues to the classical estimates. Moreover, these analogues allow us to extend our results to the zero dimensional endpoint. We also obtain results about the dimension of a variation of Furstenberg sets defined for a fractal set of directions. We prove analogous inequalities reflecting the interplay between the size of the set of directions and the size of each fiber. This problem is also studied in the general scenario of Hausdorff measures.
We establish $L^ptimes L^q$ to $L^r$ estimates for some paraproducts, which arise in the study of the bilinear Hilbert transform along curves.
The purpose of this paper is to establish some one-sided estimates for oscillatory singular integrals. The boundedness of certain oscillatory singular integral on weighted Hardy spaces $H^{1}_{+}(w)$ is proved. It is here also show that the $H^{1}_{+
In this paper, we are interested in the following bilinear fractional integral operator $Bmathcal{I}_alpha$ defined by [ Bmathcal{I}_{alpha}({f,g})(x)=int_{% %TCIMACRO{U{211d} }% %BeginExpansion mathbb{R} %EndExpansion ^{n}}frac{f(x-y)g(x+y)}{|y|^{
In this note, we obtain a full characterization of radial Carleson measures for the Hilbert-Hardy space on tube domains over symmetric cones. For large derivatives, we also obtain a full characterization of the measures for which the corresponding em
We prove a number of textit{a priori} estimates for weak solutions of elliptic equations or systems with vertically independent coefficients in the upper-half space. These estimates are designed towards applications to boundary value problems of Diri