Revistes Catalanes amb Accés Obert (RACO)

Composition of maximal operators

Menita Carozza, Antonia Passarelli di Napoli

Resum


Consider the Hardy-Littlewood maximal operator
$$
Mf(x)=\sup_{Q\owns x}\frac{1}{|Q|}\int_Q |f(y)|\,dy.
$$
It is known that $M$ applied to $f$ twice is pointwise comparable to the maximal operator $M_{L\log L}f$, defined by replacing the mean value of $|f|$ over the cube $Q$ by the $L\log L$-mean, namely
$$
M_{L\log L}f(x)=\sup_{x\in Q} \frac{1}{|Q|}\int_Q|f(y)| \log\left(e+\frac{|f|}{|f|_Q}\right)(y)\,dy,
$$
where $|f|_Q=\frac{1}{|Q|}\int_Q|f|$ (see \cite{L}, \cite{LN}, \cite{P}).

In this paper we prove that, more generally, if $\Phi(t)$ and $\Psi(t)$ are two Young functions, there exists a third function $\Theta(t)$, whose explicit form is given as a function of $\Phi(t)$ and $\Psi(t)$, such that the composition $M_\Psi\circ M_\Phi$ is pointwise comparable to $M_{\Theta}$. Through the paper, given an Orlicz function $A(t)$, by $M_A f$ we mean
$$
M_{A}f(x)=\sup_{Q\owns x}||f||_{A, Q}
$$
where $||f||_{A, Q}=\inf \left\{\lambda > 0:\frac{1}{|Q|}\int_{Q} A\left(\frac{|f|}{\lambda}\right)(x)\, dx\le 1\right\}$.

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