# Optimal exponents in weighted estimates

We are interested in proving the optimality of weighted inequalities of the form: \begin{align}\label{optimal} \|Tf\| \lesssim{n,p,T}\, [w]\|f\|{Lp(w)}, \end{align} \begin{align} \|Tf\|(w)} \lesssim{n,p,T}\, [w]\|f\|{Lp(w)}. \end{align} for a certain operator $T$ and $w$ an $A_p$ weight. In the first part of the seminar, we show that whenever \eqref{optimal} is true, then necessarily $\beta$ satisfies a lower bound which is a function of the asymptotic behaviour of the unweighted $Lp$ norm $\|T\|_{Lp(\mathbb{R}^n)}$ as $p$ goes to $1$ and $+\infty$. By combining these results with known weighted inequalities, we derive the sharpness of this exponent $\beta$, without building any specific example, for maximal, Calder\’on—Zygmund and fractional integral operators. The main underlying idea of this result comes from extrapolation theory and the Rubio de Francia algorithm.

In the second part, we focus on the case where $T$ is the strong maximal function and $w$ is a \emph{strong}-$A_p$ weight. Although for this operator no such quantitative estimates are currently known, we describe some partial results we have obtained.

This talk is part of the Analysis Seminar series.