Let be a metric space endowed with a regular Borel measure such that , . is called a doubling metric measure space if . A positive function on is called admissible if, for some , , . For an admissible function and , a function is said to be in if
Similarly, a real valued function is said to be in if, in the definition of , is replaced by . In the case and , coincides with the Goldberg’s local BMO space bmo. If is an admissible function, .
Now let be an admissible function and be a family of operators bounded on with integral kernels satisfying that there exist , , , such that , and . Using this family of operators, the authors define the Littlewood-Paley -function , Lusin’s area function and the function by
Their main result is the following: Let be a doubling metric measure space satisfying one more condition, “the -annular decay property”. Let be an admissible function. Assume that the Littlewood-Paley -function is bounded on . Then .
If , the same result holds for the function without assuming “the -annular decay property”.
Relating to their boundedness results, the authors give a nonnegative which is not in .