Optimal Young’s inequality and its converse: a simple proof

We give a new proof of the sharp form of Young’s inequality for convolutions, first proved by Beckner [Be] and Brascamp-Lieb [BL]. The latter also proved a sharp reverse inequality in the case of exponents less than 1. Our proof is simpler and gives Young’s inequality and its converse altogether. The classical convolution inequality of Young asserts that for all functions f ∈ L(R) and g ∈ L(R) we have ‖f ∗ g‖r ≤ ‖f‖p ‖g‖q, where p, q, r are ≥ 1 and 1/p + 1/q = 1 + 1/r. This inequality is sharp only when p or q is one. The best constants in Young’s inequality were found by Beckner [Be], using tensorisation arguments and rearrangements of functions. In [BL], Brascamp and Lieb derived them from a more general inequality, which we will refer to as the Brascamp-Lieb inequality; this Brascamp-Lieb inequality was also successfully applied to several problems in convex geometry by K. Ball (see [B] for one example). The expression of the best constant for Young’s inequality is rather complicated but can be easily memorized via a simple principle: it is obtained when f and g are Gaussian functions on the real line, f(x) = exp(−p′x2) and g(x) = exp(−q′x2), where p′ is the conjugate exponent of p. This principle has been largely developed by Lieb in the more recent paper [Li]; among many other results, this paper contains a new proof of the Brascamp-Lieb inequality (let us also mention [Ba] where we give yet another proof). A reverse form of Young’s inequality was found by Leindler [Le]: for 0 < p, q, r ≤ 1 and f, g non-negative, ‖f ∗ g‖r ≥ ‖f‖p‖g‖q. Again these inequalities are sharp only when p or q is one. The sharp reverse inequalities were obtained by Brascamp and Lieb in the same paper. It is also shown in [BL] that the reverse Young inequalities imply another important inequality, the inequality of Leindler and Prekopa, a close relative of the Brunn-Minkowski inequality ([Le], [Pr]). As far as we know, the proof from [BL] is the only proof available for this sharp reverse Young inequality; in our opinion, it is both rather mysterious and complicated, and uses many 1 ingredients: tensorisation, Schwarz symmetrisation, Brunn-Minkowski and some not so intuitive phenomenon for the measure in high dimension. To the contrary, our argument is elementary and gives a unified treatment of both cases, the Young inequality and the reverse inequality. It is well known that tensorisation arguments allow to deduce the multidimensional case from the one-dimensional (see [Be] for example): if the best constant is C for the real line, it will be C in the case of R . We state now the precise results. For every t > 0, we define t′ by 1/t+ 1/t′ = 1 (notice that t′ is negative when t < 1). Let us introduce for every t > 0 Ct = √ t1/t |t′|1/t′ . The general multi-dimensional result is as follows: Theorem 1. Let p, q, r > 0 satisfy 1/p + 1/q = 1 + 1/r, and let f ∈ L(R ) and g ∈ L(R ) be non-negative functions. If p, q, r ≥ 1 then (1) ‖f ∗ g‖r ≤ (CpCq Cr N ‖f‖p‖g‖q. If p, q, r ≤ 1 then (2) ‖f ∗ g‖r ≥ (CpCq Cr N ‖f‖p‖g‖q. It is easy to check that when N = 1 and p, q 6= 1, there is equality in (1) or (2) for the functions f(x) = exp(−|p′| x) and g(x) = exp(−|q′| x). As was said above, it is enough to prove the inequalities when N = 1. We will prove this case in a modified form (Theorem 2) for which we introduce some notation. The condition 1/p+ 1/q = 1 + 1/r is equivalent to the relation 1/p′ + 1/q′ = 1/r′ for the conjugates, and r′, p′ and q′ have the same sign if p, q, r > 1 or p, q, r < 1. We set c = √ r′/q′ and s = √ r′/p′. Notice that c + s = 1. We also introduce the constant K(p, q, r) = p 1 2p q 1 2q r 1 2r that will appear several times in the rest of this paper. We can now state an equivalent form of Theorem 1. Indeed, a simple change of variables shows that the following Theorem 2 is equivalent to Theorem 1 when N = 1, provided p, q and r are different from 1. 2 Theorem 2. Let p, q, r > 0 satisfy 1/p+1/q = 1+1/r and either p, q, r > 1 or p, q, r < 1. Let c = √ r′/q′, s = √ r′/p′, and let f, g be non-negative functions in L(R). If p, q, r > 1 then