Two-weighted inequalities for the fractional integral associated to the Schrödinger operator

Abstract

In this article we prove that the fractional integral operator associated to the Schrödinger second order differential operator L −α/2 = (−Δ + V ) −α/2 maps with continuity weak Lebesgue space L p,∞ (v) into weighted Campanato-Hölder type spaces BMO β L (w) , thus improving regularity under appropriate conditions on the pair of weights (v,w) and the parameters p , α and β . We also prove the continuous mapping from BMO β L (v) to BMO γ L (w) for adequate pair of weights. Our results improve those known for the same weight in both sides of the inequality and they also enlarge the families of weights known for the classical fractional integral associated to the Laplacian operator L = −Δ . Mathematics subject classification (2010): 42B35, 35J10.