  We provide a new algorithm for the treatment of the deconvolution problem on
the sphere which combines the traditional SVD inversion with an appropriate
thresholding technique in a well chosen new basis. We establish upper bounds
for the behavior of our procedure for any $\mathbb {L}_p$ loss. It is important
to emphasize the adaptation properties of our procedures with respect to the
regularity (sparsity) of the object to recover as well as to inhomogeneous
smoothness. We also perform a numerical study which proves that the procedure
shows very promising properties in practice as well.
