Dyadic systems in doubling metric spaces and applications to positive integral operators
Year: 2013
Type: dissertation
Abstract: This dissertation brings contribution to two interrelated topics. The first contribution concerns the so-called systems of dyadic cubes in the context of metric spaces. Second contribution is applications to one and two weight norm inequalities for linear and sublinear positive integral operators. Both of the topics are important in harmonic analysis and an ongoing area of study. The main novelties of the presented works consist of improving and extending existing results into more general frameworks. The work consists of four research articles and an introductory part. The first two articles, written in collaboration with T. Hytönen, study systems of dyadic cubes in metric spaces. In the Euclidean space, dyadic cubes are well-known and define a convenient structure with useful covering and intersection properties. Such dyadic structures are central especially in the modern trend of harmonic analysis. In the first article extensions of these structures are constructed in general geometrically doubling metric spaces. These consist of a refinement of existing constructions and a completely new construction of finitely many adjacent dyadic systems which behave like "translates" of a fixed system but without requiring a group structure. In this context, "cubes" are not properly cubes but rather more complicated sets that collectively have properties reminiscent of those in the Euclidean case. However, it is natural to ask what type of sets could or should be regarded cubes. In the second paper, we give a complete answer to this question in the general framework of a geometrically doubling metric space making use of the "plumpness" notion already appeared in the geometric measure theory. From another side; the two latter articles study weighted norm inequalities. Via the new construction of adjacent dyadic systems, weighted estimates for positive integral operators are obtained in a general framework. In the third paper, the two-weight problem is investigated for potential-type operators. Both strong and weak type estimates are characterized by "testing type" conditions: to show the full norm inequality it suffices to test the desired estimate on a specific class of simple test functions only. The results improve some previous results in the sense that the considered ambient space is more general (with more general measures and no additional geometric assumptions) and the testing is over a countable collection of test function only (instead of a significantly larger collection appearing in the previous works on the topic). The main technical novelty of the proof is a decomposition of the operator, along dyadic systems giving rise to certain finitely many "dyadic" versions of the original operator. In the fourth article, the focus is on sharp constant estimates for generalized fractional integral operators. A positive answer and its sharpness are given in the context of a space of homogeneous type. The result is reduced to weak-type inequalities using the results in the third paper. The sharpness requires a construction of functions that locally behave similarly to the basic power functions on the Euclidean space. The result extends a recent Euclidean result. Keywords: dyadic cube, adjacent dyadic systems, metric space, space of homogeneous type, potential type operator, testing condition, weighted norm inequality, sharp bound
Source: Työväentutkimus Vuosikirja
Author: Anna Kairema
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W1157599850