Ezequiel Rela

A new quantitative two weight theorem for the Hardy-Littlewood maximal operator

A quantitative two weight theorem for the Hardy-Little- wood maximal operator is proved improving the known ones. As a consequence a new proof of the main results in (HP) and (HPR12) is obtained which avoids the use of the sharp quantitative reverse Holder inequality for A1 proved in those papers. Our results are valid within the context of spaces of homogeneous type without imposing the non- empty annuli condition.

Refined Size Estimates for Furstenberg Sets via Hausdorff Measures: A Survey of Some Recent Results

In this survey we collect and discuss some recent results on the so-called “Furstenberg set problem”, which in its classical form concerns the estimates of the Hausdorff dimension (dimH) of the sets in the Fα-class: for a given α ∈ (0, 1], a set E ⊆ ℝ2 is in the Fα-class if for each e ∈ 𝕊 there exists a unit line segment l e in the direction of e such that dimH(l ∩ E) ≥ α. For α=1, this problem is essentially equivalent to the “Kakeya needle problem”. Define γ(α)= inf{dimH(E):E ∈F α}. The best-known results on γ(α) are the following inequalities:

Improving Bounds for Singular Operators via Sharp Reverse Höolder Inequality for \( A_\infty \)

In this expository article we collect and discuss some recent results on different consequences of a Sharp Reverse Holder Inequality for \( A\infty \) weights. For two given operators T and S, we study \( L^{p}(w) \) bounds of Coifman– Fefferman type:

Optimal exponents in weighted estimates without examples

\parallel T\;f \parallel_{L^p}(w)\;\leq \; c_{n,w,p}\parallel S\;f \parallel _{L^p}(w),

Furstenberg sets for a fractal set of directions

that can be understood as a way to control T by S.