Of Volterra operators in the scale $L_p[0,1]$ $(1\leqslant p\leqslant\infty)$

In this article a method of a priori estimates is used to solve an integro-differential equation and to substantially strengthen previously obtained sufficient conditions for the operator $\mathscr Kf=i\int_0^xk(x,t)f(t)\,dt$ to be similar to the operator $\mathscr Tf=i\int_0^xf(t)\,dt$ in the scale $L_p[0,1]$. Criteria for the similarity of $\mathscr K$ to $\mathscr T$ are found for a wide class of kernels which depend on a difference.
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