 The dissertation of Carleman under Erik Albert Holmgren, as well as his work in the early 1920s, was devoted to singular integral equations.He developed the spectral theory of integral operators with "Carleman kernels", that is, kernels "K"("x", "y") such that "K"("y", "x") = "K"("x", "y") for almost every ("x", "y"), and 
for almost every "x".In the mid-1920s, Carleman developed the theory of quasi-analytic functions.