Young's Inequality For Convolutions
In real analysis, the following result is also called Young's inequality:
Suppose f is in Lp(Rd) and g is in Lq(Rd) and
with 1 ≤ p, q, r ≤ ∞. Then
Here the star denotes convolution, Lp is Lebesgue space, and
denotes the usual Lp norm.
In case p, q > 1 the result can be strengthened to a sharp form, viz
where the constant cp,q < 1.
An example application is that Young's inequality can be used to show that the heat semigroup is a contraction semigroup using the L2 norm.
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