Hausdorff–Young inequality


The Hausdorff−Young inequality is a foundational result in the mathematical field of Fourier analysis. As a statement about Fourier series, it was discovered by and extended by. It is now typically understood as a rather direct corollary of the Plancherel theorem, found in 1910, in combination with the Riesz-Thorin theorem, originally discovered by Marcel Riesz in 1927. With this machinery, it readily admits several generalizations, including to multidimensional Fourier series and to the Fourier transform on the real line, Euclidean spaces, as well as more general spaces. With these extensions, it is one of the best-known results of Fourier analysis, appearing in nearly every introductory graduate-level textbook on the subject.
The nature of the Hausdorff-Young inequality can be understood with only Riemann integration and infinite series as prerequisite. Given a continuous function, define its "Fourier coefficients" by
for each integer. The Hausdorff-Young inequality says that
Loosely speaking, this can be interpreted as saying that the "size" of the function, as represented by the right-hand side of the above inequality, controls the "size" of its sequence of Fourier coefficients, as represented by the left-hand side.
However, this is only a very specific case of the general theorem. The usual formulations of the theorem are given below, with use of the machinery of spaces and Lebesgue integration.

The conjugate exponent

Given a nonzero real number, define the real number by the equation
If is equal to one, this equation has no solution, but it is interpreted to mean that is infinite, as an element of the extended real number line. Likewise, if is infinite, as an element of the extended real number line, then this is interpreted to mean that is equal to one.
The commonly understood features of the conjugate exponent are simple:

Fourier series

Given a function one defines its "Fourier coefficients" as a function by
although for an arbitrary function, these integrals may not exist. Hölder's inequality shows that if is in for some number ∈, then each Fourier coefficient is well-defined.
The Hausdorff-Young inequality says that, for any number p in the interval (1,2], one has
for all in. Conversely, still supposing ∈(1,2], if is a mapping for which
then there exists whose Fourier coefficients are c and with
References. Section XII.2 in volume II of Zygmund's book

Multidimensional Fourier series

The case of Fourier series generalizes to the multidimensional case. Given a function define its Fourier coefficients by
As in the case of Fourier series, the assumption that f is in Lp for some value of p in ensures, via the Hölder inequality, the existence of the Fourier coefficients. Now, the Hausdorff-Young inequality says that if p is in the range , then
for any in.
References. Page 248 of Folland's book

The Fourier transform

One defines the multidimensional Fourier transform by
The Hausdorff-Young inequality, in this setting, says that if p is a number in the interval , then one has
for any f in Lp.
References. page 114 of Grafakos' book, page 165 of Hörmander's book, page 11 of Reed and Simon's book, or section 5.1 of Stein and Weiss' book. Hörmander and Reed-Simon's books use conventions for the definition of the Fourier transform which are different from those of this article.

The language of normed vector spaces

The above results can be rephrased succinctly as:
Here we use the language of normed vector spaces and bounded linear maps, as is convenient for application of the Riesz-Thorin theorem. There are two ingredients in the proof:
The operator norm of either linear maps is less than or equal to one, as one can directly verify. One can then apply the Riesz–Thorin theorem.

Beckner's sharp Hausdorff-Young inequality

Equality is achieved in the Hausdorff-Young inequality for Fourier series by taking
for any particular choice of integers In the above terminology of "normed vector spaces", this asserts that the operator norm of the corresponding bounded linear map is exactly equal to one.
Since the Fourier transform is closely analogous to the Fourier series, and the above Hausdorff-Young inequality for the Fourier transform is proved by exactly the same means as the Hausdorff-Young inequality for Fourier series, it may be surprising that equality is not achieved for the above Hausdorff-Young inequality for the Fourier transform, aside from the special case for which the Plancherel theorem asserts that the Hausdorff-Young inequality is an exact equality.
In fact,, following a special case appearing in, showed that if is a number in the interval, then
for any in. This is an improvement of the standard Hausdorff-Young inequality, as the context and ensures that the number appearing on the right-hand side of this "Babenko–Beckner inequality" is less than or equal to 1. Moreover, this number cannot be replaced by a smaller one, since equality is achieved in the case of Gaussian functions. In this sense, Beckner's paper gives an optimal version of the Hausdorff-Young inequality. In the language of normed vector spaces, it says that the operator norm of the bounded linear map, as defined by the Fourier transform, is exactly equal to

The condition on the exponent

The condition is essential. If, then the fact that a function belongs to, does not give any additional information on the order of growth of its Fourier series beyond the fact that it is in.

Research articles