Young's Inequality - Young's Inequality For Convolutions

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 c_p,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.