Interpolation inequality

In the field of mathematical analysis, an interpolation inequality is an inequality of the form

${\displaystyle \|u_{0}\|_{0}\leq C\|u_{1}\|_{1}^{\alpha _{1}}\|u_{2}\|_{2}^{\alpha _{2}}\dots \|u_{n}\|_{n}^{\alpha _{n}},\quad n\geq 2,}$

where for ${\displaystyle 0\leq k\leq n}$, ${\displaystyle u_{k}}$ is an element of some particular vector space ${\displaystyle X_{k}}$ equipped with norm ${\displaystyle \|\cdot \|_{k}}$ and ${\displaystyle \alpha _{k}}$ is some real exponent, and ${\displaystyle C}$ is some constant independent of ${\displaystyle u_{0},..,u_{n}}$. The vector spaces concerned are usually function spaces, and many interpolation inequalities assume ${\displaystyle u_{0}=u_{1}=\cdots =u_{n}}$ and so bound the norm of an element in one space with a combination norms in other spaces, such as Ladyzhenskaya's inequality and the Gagliardo-Nirenberg interpolation inequality, both given below. Nonetheless, some important interpolation inequalities involve distinct elements ${\displaystyle u_{0},..,u_{n}}$, including Hölder's Inequality and Young's inequality for convolutions which are also presented below.