Relation between maximal operator and the differentiation properties of a basis

Abstract

The main purpose of this thesis is to investigate the weak type inequalities for the Hardy-Littlewood maximal operators and some differentiation properties of a basis in R^(n). We show that the Hardy-Littlewood maximal operator M:f∈L_(1oc)(R)→Mf∈m(R) is of weak type (p,p) in case when p=1. We demonstrate how to find weak type inequalities for the maximal operators with respect to interval basis in R^(2) and open bounded interval basis in R^(n). Moreover, we establish a converse inequality for the maximal operator open M_(1) associated to the Busemann-Feller basis B_(1) of open cubic intervals in R^(n). Finally, we show that for classes of functions, of L^(p)(R^(n)) where 1≤p<∞, we can reduce some differentiation properties from weak type inequalities for the maximal operator associated to the differentiation basis, and conversely. ;본 논문의 주된 목적은 Hardy-Littlewood maximal operator와 R^(ｎ)에서 basis를 갖는 몇가지 differentiation property 들에 대한 weak type inequality 들을 연구하는 것이다. 우리는 Hardy-Littlewood maximal operator M: fεL_(loc) (R)→ｍ(R)가, p 가 1 인 경우일때, weak type (p,p) 임을 보인다. 그리고 R^(2) 에서의 interval basis와 R^(ｎ)에서의 open bounded interval basis와 관련있는 maximal operator에 대한 weak type inequality들이 어떻게 성립되는가를 증명한다. 또한, R^(ｎ)에서의 open cubic interval의 Busemann- Feller basis β_(1)과 관련있는 maximal operator M_(1)에 대한 converse inequality가 성립된다. 마지막으로, Classes of functions, L^(p) (R^(ｎ) ,｜≤ p < ∞ 에 .대해서 differentiation basis 와 관련있는 maximal operator 에 대한 weak type inequality 들로 부터 몇가지 differentiation property들을 되게 할 수 있고, 역도 성립됨을 보인다.