Norm convolution inequalities in Lebesgue spaces (Q1645296)

The convolution operator \(A\) with kernel \(K\) has the form \[ A(f)(x)= \int_R K(x-t)f(t)\,dt \] and it follows according to Young's inequality that if \(K\in L^q(R)\), where \(q\geq 1\), and if \(1/r= (1/p)+(1/q)-1\), \(\leq p\leq r\) then \(A: L^p\to L^r\), and \(\| A(f)\|_r\leq\| K\|_q\| f\|_p\). In some extensions of Young's inequality, including the Hardy-Littlewood-Sobolev theorems, \(K\) is not in \(L^q\), and \(\| K\|_q\) is replaced by a finite constant \(A_{p\to r}\). Using estimates involving rearrangements of functions, the authors derive conclusions of the form \[ \begin{aligned} A_{p\to r}\leq C(p,r)D,\,\text{where}\,D&=\sup\{m^0(E)^{(1/r)-(1/p)}\int_E |K(x)|\,dx,\, E\in{\mathcal U}\} \text{ a family sets,} \tag{1}\\m^0(E-E)^{-1/p)} m^0(E)^{1/r}|&\int_E K(x)\,dx|\leq A_{p\to r}\tag{2}\end{aligned} \]