Analyticity of Logarithmic Integrals

Question

Assume $f\in L^2[0,1]$ and let $g(x)=\int_0^1f(y)\ln|x-y|dy$. Is it true that $g\in C^\infty(0,1)$? Is it true that $g$ is analytic in $(0,1)$? Can you refer me to a right reference to look up such integrals and their properties.

Answer 1

The rough approach is to differentiate $g$ formally, and then look at whether the integral has a reason to converge. If it does, you should have the derivative of that order. If it does not , then you shouldn't expect to have that derivative.

So, you shouldn't expect even one derivative here. Indeed, taking $f = \chi_{[0,1/2]}$ you will see that $g'(1/2)=\infty$. Just calculate $f$ explicitly: say, for $y>1/2$ it is $$ g(y) = y\ln (2y)+(1/2-y)\ln(2y-1)-(1/2)\ln(2) -\frac12 $$ where the second term has unbounded derivative at $1/2$.