Before revision:  We also give lower bounds of the L^p ( 0, 1 ^d \le \le  ) approximation error for Sobolev spaces, which show that our construction  is asymptotically optimal up to a logarithm factor.
After revision:  We also give lower bounds of the L^p ( 1 ^d \le \le ) approximation error for Sobolev spaces, which show that our construction is asymptotically optimal up to a logarithm factor.