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Let $K(x,y) \in {L^2}({(0,1)^2})$ and $g \in {L^2}(0,1)$. We consider the following integral equation $$\int\limits_0^x {K(x,t)f(t)dt = g(x)} $$ My question: what can we say about the regularity of $f$ if it exists? Thanks.

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    $\begingroup$ There is nothing we can say without knowing more about $g$, since any $f \in L^2$ may be the solution of such an equation with a suitable $g$. $\endgroup$ – Hans Engler Dec 15 '18 at 15:40
  • $\begingroup$ Thank you sir. What if wa add $f$ to tge second member which will make the equation of second kind. Can we ensure that $f$ is $L^2$? thank you. $\endgroup$ – Gustave Dec 16 '18 at 10:26

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