# First kind Volterra integral equation regularity

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.

• 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 $g$. – Hans Engler Dec 15 '18 at 15:40
• 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. – Gustave Dec 16 '18 at 10:26