Agmon's inequality

Agmon's inequality bounds a function's supremum norm by interpolating between lower-order and higher-order Sobolev norms. It is a dimension-dependent estimate used to control pointwise values from integral regularity.

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Sobolev embedding theorem: A family of results embedding Sobolev spaces into spaces with stronger integrability or differentiability properties. Agmon's estimate combines Sobolev embeddings to obtain a supremum bound.
Navier–Stokes equations: Nonlinear partial differential equations governing the motion of viscous fluids. Agmon-type bounds help control nonlinear terms involving velocity fields.
Weak derivative: A derivative defined through integration against smooth test functions rather than pointwise differentiation. Sobolev norms measure weak derivatives, including for nonsmooth functions.
Morrey's inequality: An embedding estimate that bounds Hölder continuity using sufficiently many integrable derivatives. Morrey's inequality yields a continuity modulus, while Agmon's estimate targets a supremum bound.
Gagliardo–Nirenberg interpolation inequality: An inequality that bounds one norm of a function using interpolated norms of its derivatives. Agmon's inequality is a specialized interpolation estimate in this family.
Nonlinear Schrödinger equation: A nonlinear partial differential equation describing wave envelopes in dispersive media and quantum systems. Uniform estimates can control nonlinearities that depend on solution amplitude.
Lebesgue space: A function space defined by integrability of a function's absolute value to a specified power. Sobolev norms are built from Lebesgue norms of functions and derivatives.
Ladyzhenskaya inequality: A dimension-specific interpolation inequality relating Lebesgue norms to a function and its derivatives. It gives a different norm interpolation commonly used in fluid equations.
Sobolev space: A function space whose norm controls a function and some of its weak derivatives in an integrable sense. The inequality's input and output are norms on Sobolev spaces.
Elliptic regularity: The principle that solutions of elliptic equations gain smoothness from the equation and its data. Agmon's inequality converts derivative estimates into pointwise bounds for elliptic solutions.