Uniform Boundedness for Approximations of the Identity with Nondoubling Measures

Let be a nonnegative Radon measure on which satisfies the growth condition that there exist constants and such that for all and , , where is the open ball centered at and having radius . In this paper, the authors establish the uniform boundedness for approximations of the identity introduced by Tolsa in the Hardy space and the BLO-type space RBLO . Moreover, the authors also introduce maximal operators (homogeneous) and (inhomogeneous) associated with a given approximation of the identity , and prove that is bounded from to and is bounded from the local atomic Hardy space to . These results are proved to play key roles in establishing relations between and , BMO-type spaces RBMO and rbmo as well as RBLO and rblo , and also in characterizing rbmo and rblo .