Operator estimates for Fredholm modules

We study estimates of the type parallel to phi(D) - phi(D-o)parallel to(E(M,tau)) less than or equal to C. parallel to D - D(o)parallel to(alpha), alpha = 1/2, 1 where phi(t) = t(1 + t(2))(-1/2), D-o = D-o* is an unbounded linear operato r affiliated with a semifinite von Neumann algebra M, D - D-o is a bounded self-adjoint linear operator from M and (1 + D-0(2))(-1/2) is an element of E(M, tau), where E(M, tau) is a symmetric operator space associated with M . In particular, we prove that phi(D) - phi(D-0) belongs to the non-commuta tive L-p-space for some p is an element of (1,infinity), provided (1 + D-0( 2))(-1/2) belongs to the noncommutative wear: L-r-space for some r is an el ement of [1, p). In the case M = B(H) and 1 less than or equal to p less th an or equal to 2, we show that this result continues to hold under the weak er assumption (1 + D-0(2))(-1/2) is an element of C-p. This may be regarded as an odd counterpart of A. Connes' result For the case of even Fredholm m odules.