Conditions are found under which solutions of the Volterra integral equation $x'(t) + \int _0^t a_\lambda (t - s)x(s)ds = k$ are bounded on $\{ 0 \leqq t < \infty \} $, uniformly in $\lambda $, when each $a_\lambda $ is nonnegative, nonincreasing and convexx in t.The results generalize earlier work of the author which did not admit certain piecewise linear kernels. The main proof uses a method of Shea and Wainger involving transforms of $H^1 $ functions. Applications to equations in Hilbert space are indicated.