On the real line, $\delta$ is in the $L^2$ Sobolev space $H^{-1/2-\epsilon}$ for all $\epsilon>0$ (and is compactly supported). Thus, if you assume a solution $y$ is in $H^{1/2+\epsilon}$ or better, the product is (at least) locally $L^1$, for example, so is again a legitimate distribution. If you assume a solution is locally $H^{1+\epsilon'}$, then the product is locally is $H^{1/2+\epsilon'-\epsilon}$, etc., which by Sobolev imbedding is inside $C^o$, so pointwise values are completely fine.
paul garrett
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