Summary: 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 \(L^p\) version of the BlockShrink estimator developed by T. T. Cai [Ann. Stat. 27, No. 3, 898–924 (1999; Zbl 0954.62047); Stat. Sin. 12, No. 4, 1241–1273 (2002; Zbl 1004.62036)]. First we show that it is (near) optimal for numerous statistical models, including certain inverse problems. In this statistical context, it achieves better rates of convergence than the hard thresholding estimator introduced by D. L. Donoho and I. M. Johnstone [J. Am. Stat. Assoc. 90, No. 432, 1200–1224 (1995; Zbl 0869.62024)]. We apply this general result to a deconvolution problem.