Continuity of Calderon-Zygmund operators on Besov or Triebel-Lizorkin spaces

Calderon-Zygmund operators are playing an important role in real analysis. The continuity of Calderon-Zygmund operators T on L-2 was studied in [2-4] and, here, we are investigating the continuity of T on the Besov spaces B-p(0,q) where 1 <= p, q <= infinity and on the Triebel-Lizorkin spaces F-p(0,q) where 1 <= p < infinity, 1 <= q <= infinity. The exponents measuring the regularity of the distributional kernel K(x, y) of T away from the diagonal are playing a key role in our results. They are smaller than the ones which were assumed by other authors. Moreover, our results are sharp in the case of the Besov spaces B-1(0,q) and of the Triebel-Lizorkin spaces F-1(0,q) when 1 <= q <= infinity. The proof uses a pseudo-annular decomposition of Calderon-Zygmund operators.