Suppose k is a locally-integrable function in \(M_+({{\mathbb {R}}}^n),\) the class of nonnegative Lebesgue-measurable functions on \({{\mathbb {R}}}^n.\) We define the convolution operator \(T_k\) at suitable \(f\in M_+({{\mathbb {R}}}^n)\) by \(\begin{aligned} (T_kf)(x)=\int _{{{\mathbb {R}}}^n}k(x-y)f(y)\, dy, x\in {{\mathbb {R}}}^n. \end{aligned}\) Our interest is in inequalities of the form \(\begin{aligned} \rho _1(T_kf)\le C\rho _2(f), \end{aligned}\) in which \(\rho _1\) and \(\rho _2\) are functionals on functions \(f\in M_+({{\mathbb {R}}}^n).\) Specifically, \(\rho _1\) and \(\rho _2\) are so-called Orlicz–Lorentz functionals \(\lambda _{\Phi ,u}\) given at \(f\in M_+({{\mathbb {R}}}^n)\) by \(\begin{aligned} \lambda _{\Phi ,u}(f)=\rho _{\Phi ,u}(f^*), \end{aligned}\) with \(\rho _{\Phi ,u}\) the weighted Luxemburg functional \(\begin{aligned} \rho _{\Phi ,u}(g)=\inf \left\{ \lambda >0: \int _{{{\mathbb {R}}}_+}\Phi \left( \frac{g(t)}{\lambda }\right) u(t)\, dt\le 1\right\} , \end{aligned}\) \(g\in M_+({{\mathbb {R}}}_+),\) \({{\mathbb {R}}}_+=(0, \infty ).\) The function \(f^*,\) called the nonincreasing rearrangement of f on \({{\mathbb {R}}}_+,\) is given by \(f^*(t)=\mu _f^{-1}(t),\) \(t\in {{\mathbb {R}}}_+,\) where \(\begin{aligned} \mu _f(\lambda )=|\{t \in {{\mathbb {R}}}_+: f(t)> \lambda \}|, \quad \lambda \in {{\mathbb {R}}}_+. \end{aligned}\)
