Peetre's inequality

Peetre's inequality bounds a polynomial weight of a sum of vectors by a constant times the product of corresponding weights of the individual vectors. It controls polynomial weights in Fourier analysis and function spaces.

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Japanese bracket: The weight ⟨x⟩=(1+|x|²)¹ᐟ², commonly used to express polynomial growth on Euclidean space. Peetre's inequality compares this weight at a sum with weights at its summands.
Schwartz space: The space of smooth functions whose derivatives decay faster than every inverse polynomial. Polynomial weights help express the rapid-decay conditions defining this function space.
Polynomial growth: Growth bounded in magnitude by a fixed power of the variable's size. The inequality is designed to control precisely this growth when arguments are added.
Young's convolution inequality: An inequality bounding the norm of a convolution by norms of its two factors. It controls convolution through integrability, whereas Peetre's inequality controls polynomial weights of summed variables.
Triangle inequality: An inequality stating that the distance between two points does not exceed the sum of their distances from a third point. The Euclidean triangle inequality supplies the basic comparison behind the polynomial weight estimate.
Tempered distribution: A continuous linear functional on Schwartz space, generalizing functions with at most polynomial growth. Peetre's inequality helps control weighted Fourier transforms of these generalized functions.
Euclidean norm: The usual length of a vector, computed as the square root of the sum of squared coordinates. The standard formulation uses vector lengths to define its polynomial weights.
Bernstein's inequality: An estimate relating derivative norms of a frequency-restricted function to its frequency support. Both exploit frequency information, but Bernstein's inequality bounds derivatives rather than weight shifts.
Polynomially weighted estimates: Bounds in which quantities are multiplied by powers of a spatial or frequency variable. Peetre's estimate is a reusable tool for controlling these weights under addition.
Sobolev space: A function space measuring smoothness through integrability of weak derivatives or frequency weights. Frequency weights in Sobolev estimates can be compared across sums using Peetre's bound.