Good evening,
I'm so sorry if my question is ridiculous a little bit but it is important for me;
1- I would like to understand what we mean by an interpolation inequality? 2- What are the condition must be satisfied by an inequality to give it the name "interpolation"? 3- The word interpolation comes from where? 4- An interpolation inequality in a Sobolev one?
Thank you very much and sorry again for this question
An interpolation inequality is usually an inequality between normed vector spaces. The inequality relates the norm of a given function in one space to the same function in another space. Since functions can belong to some spaces and not others, they can only hold for functions that can belong to to all the spaces used in the inequality.
For example: the Lebesgue spaces $L^p$ nest: $L^q(X) \subset L^p(X)$ for $1 \leq p < q \leq \infty$, equipped with finite measures (hat-tip: @Ian) so for any function $f \in L^q(X)$ we can attempt to construct an interpolation inequality between $\Vert f \Vert_q$ and $\Vert f \Vert_p$. In particular, these results will hold for fractional values of $p$ and $q$.
The 'condition' the interpolation inequality must satisfy is that it's an inequality between normed vector spaces applied to the same function in each space. There are no constant constraints and the constraints on the function are those that arise from belonging to the spaces in question. Inequalities between Sobolev spaces can be interpolation inequalities, but there exist other types of spaces too (e.g. Besov, Orlicz) for which interpolation inequalities can be defined.
As for the name: interpolation means "adding material between". An interpolation process constructs something between two givens (in this case, two spaces) to give meaning to something. A simple example: linear interpolation between two points means construct a (straight) line between the two points given. Cubic interpolation between two points would construct a cubic curve between the two given points. The interpolation inequality between two normed vector spaces allows us to estimate the value in an intermediate space.