Let \(\varphi: [0,1]\to \mathbb{R}\) be a nonnegative nondeecreasing function on \([0,1]\), and let the operator \({\mathcal V}\) be defined on classes of measurable functions on \([0,1]\) by \[ {\mathcal V}(f)(x)= \int^*_0 \varphi(x-y) f(y) dy,\quad 0\leq x\leq 1. \] If \({\mathcal X}\) is a rearrangement invariant space of functions on \([0,1]\), then \([{\mathcal V},{\mathcal X}]\) denotes \(\{f:{\mathcal V}(|f|\in{\mathcal X}\}\), with \({\mathcal X}\) identified as \((\mathbb{L}^1,\mathbb{L}^\infty)_\rho\), where \((X_0,X_1)_\rho\) denotes the interpolation space between Banach spaces \(X_0\), \(X_1\), and \(\rho\) is a monotone Riesz-Fischer norm. In the first main theorem of this paper it is shown that if \(\varphi\) satisfies ‘quadrature property’ involving inf. over \(0\leq b\leq 1\) of expressions of the form \[ \Biggl(\Biggl( \int^b_a \varphi(y) dy\Biggr)\Biggl/(b- a)\varphi(b)\Biggr), \] then \([{\mathcal V},{\mathcal X}]\) is identifiable with \((L^1(I,w_\varphi), L^1(I,u_\varphi))_\rho\), where \(L^1(I, w)\) is the space of integrable functions on \([0,1]\) with weight function \(w\), \(w_\varphi(y)= \int^{1-y}_0 \varphi(s) ds\), \(u_\varphi(y)= \varphi(1- y)\).
In a second theorem which represents a converse of the first theorem it is shown that if \((L^1(I, w_\varphi), L^1(I, u_\varphi))_\rho\), \([{\mathcal V},{\mathcal X}]\) are isomorphic, then \(\varphi\) satisfies the quadrature property. Other results related to the non-compactness of \({\mathcal V}:[{\mathcal V},{\mathcal X}]\to{\mathcal X}\), when \({\mathcal X}\) is a rearrangement invariant space or \(L^1(0, 1)\) or \(L^\infty(0,1)\).