Wavelet modifications have been successful in optimally stabilizing the hierarchical basis methods. Such modifications primarily rely on establishing an optimal BPX preconditioner for the meshes under consideration. Existing literature on optimality of both the BPX preconditioner and preconditioners arising from wavelet modified hierarchical basis methods have been mostly restricted to uniformly refined meshes. We extend optimality results of the above methods to locally refined 2 and 3 dimensional meshes by using variants of red-green and red refinement strategies. Our framework provides rigorous relations about the neighboring simplices and eliminates the requirement of continuous partial differential equation coefficients. Numerical experiments are presented for 2 and 3 dimensional problems with various local refinement strategies.