An upper bound on the Kolmogorov widths of a certain family of integral operators

We consider the family of integral operators $(K_{alpha}f)(x)$ from $L^p[0,1]$ to $L^q[0,1]$ given by $$(K_{alpha}f)(x)=int_0^1(1-xy)^{alpha -1},f(y),operatorname{d}!y, qquad 0<alpha<1.$$ The main objective is to find upper bounds for the Kolmogorov widths, where the $n$th Kolmogorov width is the infimum of the deviation of $(K_{alpha}f)$ from an $n$-dimensional subspaces of $L^p[0,1]$ (with the infimum taken over all $n$-dimensional subspaces), and is therefore a measure of how well $K_{alpha}$ can be approximated. We find upper bounds for the Kolmogorov widths in question that decrease faster than $exp(-kappa sqrt{n})$ for some positive constant $kappa$.