The article discusses the convolution and Wiener-Hopf operators in the space of \(BMO^k\) , \(k\in Z_+\) -functions with a limited average oscillation of k-of that order, the specific internal properties of which require a different approach from the case of \(L_p\) -spaces, even in matters of limitation, and the results on limitation, Fredholm, and reversibility are presented. It is interesting to note that when moving to the space \(BMO^k\) , \(k\in {\textbf{N}}\) , the boundedness conditions depend on the order of k, and the Fredholm conditions for convolutions are the same as for \(k=0\) . When studying Fredholm, it becomes necessary to use weighted convolutional rings, with weights having different behavior on \(\pm \infty \) . When considering integral operators, it turned out that estimates of the average values of functions over intervals, depending on the interval itself and its length, play an important role. The boundedness and reversibility of the H convolution operator in the BMO space were shown. Wiener-Hopf operators with a total kernel in the BMO space and in the \(BMO^k\) space were also considered. It turns out that, unlike the spaces \(L_p({\textbf{R}_+^1})\) , \(1\le {p}\le \infty \) , even the boundedness of such operators in \(BMO(\mathbf {R^1_+})\) with analytical symbols (in the upper or lower half-plane) is described by different conditions. When studying the convolution and Wiener-Hopf operators in the \(BMO^k\) space, it turned out that their boundedness depends on k, while the nature of their solvability is the same as in the case of \(k=0\) .
