Boundedness of Differential Transforms for Heat Semigroups Generated by Schrodinger Operators

In this paper we analyze the convergence of the following type of series T-N(L) f(x) = Sigma(N2)(j=N1) v(j) (e(j+1)(-a)() pound f(x) - e(-aj ) pound f(x)), x is an element of R-n, where {e(-t )}(t pound > 0) is the heat semigroup of the operator = pound -Delta + V with Delta being the dassical laplacian, the nonnegative potential V belonging to the reverse Holder class RHq with q > n/2 and n >= 3, N = (N-1 , N-2) is an element of Z(2) with N-1 < N-2, {v(j)}(j is an element of Z) is a bounded real sequences, and {a(j)}(j is an element of Z) is an increasing real sequence. Our analysis will consist in the boundedness, in L-p(R-n) and in BMO(R-n), of the operators TNLand its maximal operator T* f (x) = sup(N) T-N() pound f (x). It is also shown that the local size of the maximal differential transform operators (with V = 0) is the same with the order of a singular integral for functions f having local support. Moreover, if {v(j)}(j is an element of Z) is an element of l(p)(Z), we get an intermediate size between the local size of singular integrals and Hardy-Littlewood maximal operator.