Interpolation inequality


In the field of mathematical analysis, an interpolation inequality is an inequality of the form
valid for all u0,..., un in some vector spaces X0,..., Xn equipped with norms ‖·‖0, ‖·‖1,..., ‖·‖n, and where C is a constant independent of u0,..., un and α1,..., αn are some real powers. Usually, the elements u0,..., un are all the same element u and only the norms differ, but some interpolation inequalities use different u0,..., un.
The main applications of interpolation inequalities lie in the theory of Sobolev spaces, where spaces of functions that have a non-integer number of derivatives are interpolated from the spaces of functions with integer number of derivatives. The abstract structure of interpolation inequalities is formalized in the notion of an interpolation space.
A simple example of an interpolation inequality — one in which all the uk are the same u, but the norms ‖·‖k are different — is Ladyzhenskaya's inequality for functions u: ℝ2 → ℝ, which states that whenever u is a compactly supported function such that both u and its gradient ∇u are square integrable, it follows that the fourth power of u is integrable and
i.e.
Another simple example of an interpolation inequality — one in which the uk and the norms ‖·‖k are different — is Young's inequality for the convolution of two functions f, g: ℝd → ℝ:
where the exponents p, r and s ≥ 1 are related by

Examples of interpolation inequalities