If the function ( f ) to be estimated were Lipschitz continuous or differentiable, the results regarding the minimax estimation error would likely change in terms of the decay rates of the minimax risks ( R_2 ) and ( R_\\infty ). For Lipschitz continuous functions, the smoothness condition implies a stronger constraint on the function's variation compared to ( \\beta )-Hölder continuous functions. Specifically, Lipschitz continuity (where ( \\beta = 1 )) would lead to a potentially faster decay rate in the minimax risk, as the estimation error could be bounded more tightly due to the limited oscillation of the function.
In the case of differentiable functions, the analysis could further refine the estimation process, as differentiability provides additional information about the function's behavior. The minimax rates might improve, reflecting the increased regularity of the function. The theoretical framework established in the paper could be adapted to account for these different smoothness properties, leading to new bounds and insights into the estimation error in the presence of speckle and additive noise.