Testing weighted estimates without test functions

EZEQUIEL RELA

Sevilla

Conferencia; Conferencia IMUS; 2016

Insituto de Matemáticas de la Universidad de Sevilla (IMUS)

In general, once a certain norm estimate for a given operator is proved, the next question is about the sharpness of that result. In the theory of weighted inequalities for $A_p$ weights, the expected inequality for an operator $T$ is something like:\[\|T\|_{L^{p}(w)}\le c\, [w]^{\beta}_{A_p} \qquad w \in A_{ p}.\]These type of inequalities were proved for a large class of classical operators, including maximal operator and singular integrals. To test the sharpness of the exponent $\beta$, each situation demands a specific example. Usually, a one parameter family of functions and weights asymptotically verifying that the same inequality cannot hold for a smaller $\beta$.In this talk I will present a recent discovery of a close connection between the best possible $\beta$ for the above inequality and the asymptotic behaviour of the unweighted $L^p$ norm $\|T\|_{L^p}$ as $p$ goes to $1$ and $+\infty$. Using this method, we verify the sharpness of know bounds and provide a lower bound for the exponent $\beta$ for those cases when it is still not know if this best possible value is attained. One of the benefit of such approach is that now we are able to consider in a unified way maximal operators defined over very general Muckenhoupt bases.