SINGULAR-INTEGRALS WITH EXPONENTIAL WEIGHTS

We study the operators (V) over bar f(t) = 1/w(t)V(f(r)w(r))(t) where V is the Hardy-Littlewood maximal function, the Hilbert transform or C arleson operator. Under suitable conditions on the weight w(t) of expo nential type, we prove boundedness of (V) over bar from L(p) spaces, d efined on [1, +infinity) with respect to the measure w(2)(t)dt to L(p) + L(2), 1 < p less than or equal to 2, with the same density measure. These operators, that arise in questions of harmonic analysis or nonc ompact symmetric spaces, are bounded from L(p) to L(p), 1 < p < infini ty, if and only if p = 2.