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 $$(K_{\alpha }f)(x)={\int }_0^1(1-xy{)}^{\alpha -1}f(y)\mathrm{d}\text{\hspace{0.17em}⁣}y,0<\alpha <1.$$ 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 $\kappa$.