  Deep learning has achieved notable success in various fields, including image
and speech recognition. One of the factors in the successful performance of
deep learning is its high feature extraction ability. In this study, we focus
on the adaptivity of deep learning; consequently, we treat the variable
exponent Besov space, which has a different smoothness depending on the input
location $x$. In other words, the difficulty of the estimation is not uniform
within the domain. We analyze the general approximation error of the variable
exponent Besov space and the approximation and estimation errors of deep
learning. We note that the improvement based on adaptivity is remarkable when
the region upon which the target function has less smoothness is small and the
dimension is large. Moreover, the superiority to linear estimators is shown
with respect to the convergence rate of the estimation error.
