CONVOLUTION INEQUALITIES IN WEIGHTED LORENTZ SPACES: CASE 0 < q < 1

Let g be a fixed nonnegative radially decreasing kernel g. In this paper, boundedness of the convolution operator T(g)f := f*g between the weighted Lorentz spaces Gamma(q)(w) and Lambda(p)(v) is characterized in the case 0 < q < 1. The conditions are sufficient if the kernel g is just a general measurable function. Furthermore, the largest rearrangement-invariant (quasi-)space Y is found such that the Young-type inequality parallel to f*g parallel to(Gamma q(w)) <= C parallel to f parallel to (Lambda p(v))parallel to g parallel to Y holds for all f is an element of Lambda(p)(v) and g is an element of Y.