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\textbf{SPEAKER}:
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Speaker: Carlos P\'erez, University of the Basque Country and Ikerbasque
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{\bf TITLE}: Optimality of the exponents in weighted estimates without examples
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{\bf ABSTRACT:}\\

We present a general approach for proving the optimality of the exponents on weighted estimates. We show that if an operator $T$ 
satisﬁes a bound like
$$
\|T \|_{L^p(w)} \leq  c[w]^\beta_{A_p}, ~~~ w \in A_p,
$$
then the optimal lower bound for $\beta$ is closely related to the asymptotic behaviour of the unweighted $L^p$-norm $\|T \|_{L^p(R^n)}$ as $p$ goes to $1$ and $\infty$.




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We present a general approach for proving the optimality of the exponents on weighted estimates. We show that if an operator T satisﬁes a bound like

kTkLp(w) ≤ c[w]β

Ap w ∈ Ap,

then the optimal lower bound for β is closely related to the asymptotic behaviour of the unweighted Lp norm kTkLp(Rn) as p goes to 1 and +∞.

In most of the classical operators the optimality of the exponents is de-rived by showing some explicit examples.