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. Avdispahić, that describe relationships between the classes $\Lambda \text{BV}$ and $V[h]$, are derived as special cases.

7 pages