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In the field of mathematical analysis, an interpolation inequality is an inequality of the form
where for 0 ≤ k ≤ n {\displaystyle 0\leq k\leq n} , u k {\displaystyle u_{k}} is an element of some particular vector space X k {\displaystyle X_{k}} equipped with norm ‖ ⋅ ‖ k {\displaystyle \|\cdot \|_{k}} and α k {\displaystyle \alpha _{k}} is some real exponent, and C {\displaystyle C} is some constant independent of u 0 , . . , u n {\displaystyle u_{0},..,u_{n}}. The vector spaces concerned are usually function spaces, and many interpolation inequalities assume u 0 = u 1 = ⋯ = u n {\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 u 0 , . . , u n {\displaystyle u_{0},..,u_{n}} , including Hölder's Inequality and Young's inequality for convolutions which are also presented below.