Direct and inverse theorems of approximation theory in Lebesgue spaces with Muckenhoupt weights

In this work we establish direct and inverse theorems of approximation theory in Lebesgue spaces $L_{p,w}$ with Muckenhoupt weights $w$ on the axis and on a period. The classical definition of the modulus of continuity can be meaningless in weighted spaces. Therefore, as the modules of continuity, including non-integer order, we use the norms of powers of deviation of Steklov means. The properties of these quantities are derived, some of which are similar to the properties of usual modules of continuity. In addition to the direct and inverse theorems, we obtain equivalence relations between the modules of continuity and the $K$- and $R$-functionals.
The proofs are based on estimates for the norms of convolution operators and they do not employ a maximal function. This allows us to establish the results for all $p\in[1,+\infty)$ not excluding the case $p=1$. Previously used methods that employed the maximal function in one form or another are unsuitable for $p\to1$. In addition, by the convolution-based approach we can obtain results simultaneously in the periodic and non-periodic case. With rare exceptions, constants are not specified explicitly, but their dependence on parameters is always tracked. All constants in the estimates depend on $[w]_p$ (Muckenhoupt characteristics of weight $w$), and there is no other dependence on $w$ and $p$. The norms of convolution operators are estimated explicitly in terms of $[w]_p$. The methods of this work can be applied to prove direct and inverse theorems in more general functional spaces.