Function spaces of $l_{p(\cdot)}$ $(l_{q(\cdot)})$-type and embedding theorems for spaces with variable smoothness

We define the iterated (quasi)normed function spaces of $L_{p(\cdot)}$ $(L_{q(\cdot)}( \dots))$-type with exponents depending on all variables. Also, we prove an analog of the Minkowski inequality for mixed norms and a multiplicative interpolation type inequality in these spaces and use the relevant theorems for proving an embedding theorem for the function spaces with variable smoothness depending on different directions.