On Some Properties of a Vector-Valued Function Space

The study of various mathematical problems (such as elasticity, non-Newtonian fluids and electrorheological fluids) with variable exponent growth condition has been received considerable attention in recent years. These problems are interesting in applications and raise many difficult mathematical problems. There are also a lot of published papers in these spaces. Vector-valued Lebesgue spaces are widely used in analysis, abstract evolution equations and in the theory of integral operators. In this paper we recall the weighted vector-valued classical and variable exponent Lebesgue spaces. We define a intersection space of vector-valued weighted classical Lebesgue and variable exponent Lebesgue spaces. We discuss some basic properties, such as, Banach space, dense subspaces and Hölder type inequalities of these spaces. We will also show that every elements of vector-valued these spaces are locally integrable. Moreover, we investigate several embeddings and continuous embeddings properties of these spaces under some conditions with respect to exponents and two weight functions.