The authors prove that the parametrized area integral and function are bounded from the weighted weak Hardy space to the weighted weak Lebesgue space as satisfies a class of the integral Dini condition, respectively.
"Boundedness for Parametrized Littlewood-Paley Operators with Rough Kernels on Weighted Weak Hardy Spaces." Abstr. Appl. Anal. 2013 1 - 15, 2013. https://doi.org/10.1155/2013/651941