Young's inequality for convolutions

In mathematics, Young's convolution inequality is a mathematical inequality about the convolution of two functions, named after William Henry Young.

In real analysis, the following result is called Young's convolution inequality:

Suppose f is in L^p(R^d) and g is in L^q(R^d) and

with 1 ≤ p, q, r ≤ ∞. Then

Here the star denotes convolution, L^p is Lebesgue space, and

denotes the usual L^p norm.

Equivalently, if ${\displaystyle p,q,r\geq 1}$ and ${\displaystyle \textstyle {\frac {1}{p}}+{\frac {1}{q}}+{\frac {1}{r}}=2}$ then

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