Mosers Inequality for a class of integral operators

Let 1 < p < ∞, q = p/(p-1) and for $f ∈ L^p(0,∞)$ define $F(x) = (1/x) ʃ_0^x f(t)dt$, x > 0. Moser's Inequality states that there is a constant $C_p$ such that $sup_{a≤1} sup_{f∈B_{p}} ʃ_{0}^{∞} exp[ax^{q}|F(x)|^{q} - x]dx= C_p$ where $B_p$ is the unit ball of $L^p$. Moreover, the value a = 1 is sharp. We observe that $F = K_1$ f where the integral operator $K_1$ has a simple kernel K. We consider the question of for what kernels K(t,x), 0 ≤ t, x < ∞, this result can be extended, and proceed to discuss this when K is non-negative and homogeneous of degree -1. A sufficient condition on K is found for the analogue of Moser's Inequality to hold. An internal constant ψ, the counterpart of the constant a, arises naturally. We give a condition on K that ψ be sharp. Some applications are discussed.