On Some Properties of Lorentz-Sobolev Spaces with Variable Exponent

In recent years there has been an increasing interest in the study of various mathematical problems with variable exponent Lebesgue spaces. There are also a lot of published papers in these spaces. Spaces of weakly differentiable functions, so called Sobolev spaces, play an important role in modern Analysis. The theory of variable exponent Sobolev spaces is useful theoretical tool to study the variable exponent problems, such as solutions of elliptic and parabolic partial differentiable equations, calculus of variations, nonlinear analysis, capacity theory and compact embeddings. Moreover, several authors studied some continuous embeddings from Sobolev spaces to Lorentz spaces. These kinds of embedding results are very interesting and valuable in analysis, and there are many applications of them in various fields. In this paper we define variable exponent Lorentz-Sobolev spaces and prove the boundedness of maximal function in these spaces. Also we will show that there is a continuous embedding between variable exponent Lorentz-Sobolev spaces and variable exponent Lorentz spaces under some conditions.