Endpoint regularity of maximal functions in higher dimensions

It is well known that the Hardy-Littlewood maximal operator is bounded on Lebesgue spaces if the exponent is strictly larger than one, and that this bound fails when the Lebesgue exponent is equal to one. Similarly, the gradient of the Hardy-Littlewood maximal function is bounded by the gradient of the function when the Lebesgue exponent is strictly larger than one, but it has been an open question whether this also holds when the Lebesgue exponent equals one. This endpoint regularity bound has been conjectured to hold, but only proven fully in one imension, using a simple formula for the variation of semi-continuous functions on the real line. In higher-dimensional Euclidean spaces the bound has been proven for the maximal function of radial functions, where again one-dimensional considerations suffice. The only fully known endpoint regularity bounds in higher dimensions concern some fractional maximal operators, which however are not of the same form as in the aforementioned conjecture. In this thesis we present the first proof of the endpoint boundedness of the gradient of a maximal operator in all dimensions. In the first two papers we prove the endpoint regularity ofthe uncentered Hardy-Littlewood maximal function of characteristic functions and of the dyadic maximal function of any function. We then generalize and combine the insights which we gained in order to prove further endpoint regularity bounds: We prove the corresponding endpoint bound for the gradient of the centered and of the uncentered fractional Hardy-Littlewood maximal function, and we eventually also prove their endpoint continuity. We conclude this thesis by showing a proof for the endpoint regularity bound for the cube maximal function, answering the long-standing endpoint regularity question for an uncentered maximal operator when averaging over cubes instead of balls. Our results also hold for the local versions of the above maximal operators, excluding fractional maximal operators. The starting point in our proofs is to view the variation of a function in terms of the coarea formula. We then prove and apply higher-dimensional geometric tools which involve the interplay between volume and perimeter such as the relative isoperimetric inequality, covering lemmas that concern the boundary of a set, dyadic decompositions of functions, and approximation arguments in Sobolev spaces.