Operadores maximais fracionários - Maia, Rodrigo Fidélis (2020)
This master's thesis aims to study the centered version of the Fractional Maximal Operator, primarily its regularity in L^p(R^n) spaces along with the Riesz Potential. Next, we will define both operators in measures of R^n in order to prove the B.Muckenhoupt-R.L.Wheeden Theorem. Then we will study two theorems of Juha Kinnunen about the behavior of the operator in Sobolev spaces. We will also present point estimates for the operator's weak gradient, where one of them provides a control of function oscillation. We will analyze the regularity in Sobolev spaces of the local version of the operator M_{alpha,omega} in open ended with finite measure. In addition, we will give a punctual estimate of the weak gradient of the operator, which unlike the non-local case, we will have the addition of an extra term containing the Fractional Local Maximal, then we will mention some examples that will show the optimization of the presented results. Finally, we will study the action of the Fractional Maximal Operator on Campanato spaces L^{p,beta}(X), where X is a measurable metric space equipped with a regular positive Borel measure satisfying the property "Doubling Property Means Condition" .