In
mathematics, Young's convolution inequality is a
mathematical inequality about the
convolution
In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution'' ...
of two functions, named after
William Henry Young
William Henry Young FRS (London, 20 October 1863 – Lausanne, 7 July 1942) was an English mathematician. Young was educated at City of London School and Peterhouse, Cambridge. He worked on measure theory, Fourier series, differential ca ...
.
Statement
Euclidean Space
In
real analysis
In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include conv ...
, the following result is called Young's convolution inequality:
Suppose
is in the
Lebesgue space and
is in
and
with
Then
Here the star denotes
convolution
In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution'' ...
,
is
Lebesgue space, and
denotes the usual
norm.
Equivalently, if
and
then
Generalizations
Young's convolution inequality has a natural generalization in which we replace
by a
unimodular group If we let
be a bi-invariant
Haar measure on
and we let
or
be integrable functions, then we define
by
Then in this case, Young's inequality states that for
and
and