Moser's second one-dimensional inequality for a class of integral operators (Q1969091)

The second Moser's inequality can be formulated as follows: Let \(1<p<\infty\). Then \[ \int_{-\infty}^\infty\exp[a|F(x)|^q-|x|] dx\leq C_p \] holds for every \(a\leq 1\) and for every non-negative \(f\in L^p(-\infty,\infty)\) with \(\|f\|_p\leq 1\) and such that \(F\) defined by \(F(x)-F(0)=\int_0^x f(t) dt\) satisfies \(\int_{-\infty}^\infty F(t)\exp(-|t|) dt=0\). The author extends the assertion replacing the definition of \(F\) by a more general integral operator with a non-negative kernel \(K(x,t)\) homogeneous of degree 0 and \(-1\) in \(t\) and \(x\), respectively. For a corresponding extension of the first Moser's inequality see \textit{F. Holland} and \textit{D. Walsh} [Stud. Math. 113, No. 2, 141-168 (1995; Zbl 0843.26008)].