Revistes Catalanes amb Accés Obert (RACO)

On the range space of Yano's extrapolation theorem and new extrapolation estimates at infinity

María Jesús Carro Rossell

Resum


Given a sublinear operator $T$ satisfying that $\Vert Tf \Vert_{L^p(\nu)} \le \frac{C}{p-1} \Vert f\Vert_{L^p(\mu)}$, for every $1 < p\le p_0$, with $C$ independent of $f$ and $p$, it was proved in [C] that
$$
\sup_{r>0}\frac{\int_{1/r}^\infty \lambda_{Tf}^\nu(y)\,dy}{1+\log^+ r}\lesssim \int_{\mathcal{M}}|f(x)|(1+\log^+|f(x)|)\,d\mu(x).
$$
This estimate implies that $T\colon L\log L \rightarrow B$, where $B$ is a rearrangement invariant space. The purpose of this note is to give several characterizations of the space $B$ and study its associate space. This last information allows us to formulate an extrapolation result of Zygmund type for linear operators satisfying $\Vert Tf\Vert_{L^p(\nu)}\le C p\Vert f\Vert_{L^p(\mu)}$, for every $p\ge p_0$.

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