CONVOLUTION IN WEIGHTED LORENTZ SPACES OF TYPE Γ

We characterize boundedness of the convolution operator between weighted Lorentz spaces Gamma(P) (v) and Gamma(q) (w) for the range of parameters p, q is an element of [1, infinity], or p is an element of (0, 1) and q is an element of {1, infinity} or p = infinity and q is an element of (0, 1). We provide Young-type convolution inequalities of the form parallel to f * g parallel to Gamma(q) (omega) <= C parallel to f parallel to Gamma(p) (v) parallel to g parallel to Y, f is an element of Gamma(p)(v), g is an element of Y, characterizing the optimal rearrangement-invariant space Y for which the inequality is satisfied.