On embeddings between spaces of functions of generalized bounded variation

Abstract

In this note, we aim to establish a number of embeddings between various function spaces that are frequently considered in the theory of Fourier series. More specifically, we give sufficient conditions for the embeddings $\Phi V[h]\subseteq \Lambda\text{BV}^{(p_n\uparrow p)}$, $\Lambda V[h_1]^{(p)}\subseteq\Gamma V[h_2]^{(q)}$ and $\Lambda\text{BV}^{(p_n\uparrow p)}\subseteq\Gamma\text{BV}^{(q_n\uparrow q)}$. Our results are new even for the well-known spaces that have been studied in the literature. In particular, a number of results due to M. Avdispahi\'{c}, that describe relationships between the classes $\Lambda\text{BV}$ and $V[h]$, are derived as special cases.