Hardy-littlewood maximal operators in Sobolev spaces

The aim of this thesis is to investigate properties of maximal operators in Sobolev spaces. In chapter one, we first introduce the importance of the Sobolev spaces and maximal function. We then focus our attention on the boundedness of maximal operator in the Sobolev space. In chapter two, we briefly summarize the main points made in many articles and books which contained in references. In chapter three, we present various concepts, definitions, and elementary results from functional analysis and measure theory that will be required. We then give a detailed analysis of Sobolev spaces and the Hardy-Littlewood maximal function. In chapter four, we discuss and share the results of the research articles that made on the boundedness of Hardy-Littlewood maximal operator on Sobolev space 〖 W〗^(1,p) (Ω). In particular, we analyze weak derivative of maximal function in Sobolev space. Keywords: Sobolev space, Maximal operator, Bounded operator, Weak derivative, Weak compactness