Moser's Inequality for a class of integral operators

Let 1 < p < ∞, q = p/(p-1) and for ${\displaystyle f\in L^p(0,\infty )}$ define ${\displaystyle F\left(x\right)=\left(\frac{1}{x}\right){ʃ}_0^{x}f\left(t\right)dt}$, x > 0. Moser's Inequality states that there is a constant ${\displaystyle C_p}$ such that ${\displaystyle {\supset }_{a\le 1}{\supset }_{f\in B_p}{ʃ}_0^{\infty }\exp [a{x}^q{\left|F\left(x\right)\right|}^q-x]dx=C_p}$ where ${\displaystyle B_p}$ is the unit ball of ${\displaystyle L^p}$. Moreover, the value a = 1 is sharp. We observe that ${\displaystyle F=K_1}$ f where the integral operator ${\displaystyle K_1}$ 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.