We investigate the asymptotic minimax properties of an adaptive wavelet block thresholding estimator under the ${L}^p$ risk over Besov balls. It can be viewed as a $\mathbb{L}^p$ version of the BlockShrink estimator developed by Cai (1996,1997,2002). Firstly, we show that it is (near) optimal for numerous statistical models, including certain inverse problems. Under this statistical context, it achieves better rates of convergence than the hard thresholding estimator introduced by Donoho and Johnstone (1995). Secondly, we apply this general result to a deconvolution problem.