Hp spaces for generalized Schrodinger operators and applications

In this paper, we study the Hardy type space H-L(p)(R-n) by means of local maximal functions associated with the heat semigroup e(-tL) generated by -L , where = -Delta +mu pis the generalized Schrodinger operator in R-n(n >= 3) and mu not equivalent to 0 is a nonnegative Radon measure satisfying certain scale-invariant Kato conditions and doubling conditions. Via the equivalence of the norms between various local maximal functions, we show that the norms parallel to f parallel to(p)(n)( )(HLp)((R)())and parallel to f parallel to H(mp,q)((R)())(p)(n)are equivalent for n/n+delta' < p <= 1 <= q <= infinity (p not equal q) with some delta' > 0. As applications, we prove that Calderon-Zygrnund operators related to the auxiliary function m(x, mu) are bounded from H-L((R))(p)n into L-p (n)((R)()) for n/n+gamma(1) < p <= 1 with gamma(1)> 0. In particular, we show that the Riesz transform del(-Delta + mu)(-1/2), which is a special example of the above Calderon-Zygrnund operators, is bounded from H-L(p) (R-n) into H-p (R-n ) for n/n+gamma(1) < p <= 1 with 0 < gamma(1) < 1.