Singular integrals with exponential weights

Abstract

Abstract. We study the operators V f(t) = 1 w(t) V (f(r)w(r))(t)where V is the Hardy-Littlewood maximal function, the Hilbert transform or Carleson operator. Under suitable conditions on the weight w(t) of exponential type, we prove boundedness of V from L p spaces, defined on [1, +∞) with respect to the measure w 2 (t)dt, to L p + L 2 , 1 < p ≤ 2, with the same density measure. These operators, that arise in questions of harmonic analysis on noncompact symmetric spaces, are bounded from L p to L p , 1 < p < ∞, if and only if p = 2.The study of convergence properties of inverse spherical transforms of radial functions on noncompact symmetric spaces [1], [6], [7] requires estimates on singular integrals with exponential weights, as defined below, due to the exponential growth at infinity of the radial part D(t)dt of the measure, wherewith p and q suitable nonnegative integers that depend upon the geometry of the symmetric space.In what follows we define a class of functions w(t), that include exponentials, and prove boundedness of the operators