Singular measures and convolution operators

We show that in the study of certain convolution operators, functions can be replaced by measures without changing the size of the constants appearing in weak type (1,1) inequalities. As an application, we prove that the best constants for the centered Hardy-Littlewood maximal operator associated to parallelotopes do not decrease with the dimension.Comment: 8 page

Dimension dependency of the weak type (1,1)(1,1) bounds for maximal functions associated to finite radial measures

We show that the best constants appearing in the weak type (1,1) inequalities satisfied by the centered Hardy-Littlewood maximal function associated to some finite radial measures, such as the standard gaussian measure, grow exponentially fast with the dimension.Comment: 7 pages, to appear in the Bull. London Math. So

The weak type (1,1)(1,1) bounds for the maximal function associated to cubes grow to infinity with the dimension

Let $M_d$ be the centered Hardy-Littlewood maximal function associated to cubes in $\mathbb{R}^d$ with Lebesgue measure, and let $c_d$ denote the lowest constant appearing in the weak type (1,1) inequality satisfied by $M_d$. We show that $c_d \to \infty$ as $d\to \infty$, thus answering, for the case of cubes, a long standing open question of E. M. Stein and J. O. Str\"{o}mberg.Comment: A very similar version to this posting (but with fewer explicit constants, in order to satisfy the requests of an anonymous referee) is to appear in Ann. Math.; additionally, and with respect to the last version, this one contains several simplifications and improvements in the exposition, suggested by the refere

Besicovitch type properties in metric spaces

We explore the relationship in metric spaces between different properties related to the Besicovitch covering theorem, with emphasis on geometrically doubling spaces.Comment: 14 page

Strengthened Cauchy-Schwarz and H\"older inequalities

We present some identities related to the Cauchy-Schwarz inequality in complex inner product spaces. A new proof of the basic result on the subject of Strengthened Cauchy-Schwarz inequalities is derived using these identities. Also, an analogous version of this result is given for Strengthened H\"older inequalities