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.



Analyticity of Logarithmic Integrals
No, if $f$ is the characteristic function of $[0,\frac12]$, $g$ will not be smooth at $\frac12$ by a simple explicit calculation. 


We have $g=f\*h$, where $h(x)=\ln|x|$ is in $L^1$ and has singular support ${0}$. Thus $g$ is $C^\infty$ whereever $f$ is. See e.g. Hörmander, The Analysis of Partial Differential Operators II, chap 16 (which perhaps should be considered overkill for this).


