Resumen de Maximal functions and related weight classes
The famous result of Muckenhoupt on the connection between weights $\omega$ in $A_p$-classes and the boundedness of the maximal operator in $L_p(\omega)$ is extended to the case $p=\infty$ by the introduction of the geometrical maximal operator. Estimates of the norm of the maximal operators are given in terms of the $A_p$-constants. The equality of two differently defined $A_{\infty}$-constants is proved. Thereby an answer is given to a question posed by R. Johnson. For non-increasing functions on the positive real line a parallel theory to the $A_p$-theory is established for the connection between weights in $B_p$-classes and maximal functions, thereby extending and developing the recent results of Ario and Muckenhoupt.
