An Extension of Young's Inequality

Young's inequality is extended to the context of absolutely continuous measures. Several applications are included.


Introduction
Young's inequality [18] asserts that every strictly increasing continuous function See [4], [8], [9] and [14] for details and significant applications. Several questions arise naturally in connection with this classical result. (Q1): Is the restriction on strict monotonicity (or on continuity) really necessary? (Q2): Is there any weighted analogue of Young's inequality? (Q3): Can Young's inequality be improved?
F. Cunningham Jr. and N. Grossman [2] noticed that the question (Q1) has a positive answer (correcting the prevalent belief that Young's inequality is the business of strictly increasing continuous functions). The aim of the present paper is to extend the entire discussion to the framework of locally absolutely continuous measures and to prove several improvements.
As well known, Young's inequality is an illustration of the Legendre duality. Precisely, the functions It turns out that each of these formulas produces a convex function (possibly on a different interval). Some details are in order.
By definition, the conjugate of a convex function F defined on a nondegenerate interval I is the function with domain I * = {y ∈ R : F * (y) < ∞}. Necessarily I * is an non-empty interval and F * is a convex function whose level sets {y : F * (y) ≤ λ} are closed subsets of R for each λ ∈ R (usually such functions are called closed convex functions).
A convex function may not be differentiable, but it admits a good substitute for differentiability.
The subdifferential of a real function F defined on an interval I is a multivalued function ∂F : I → P(R) defined by Geometrically, the subdifferential gives us the slopes of the supporting lines for the graph of F . The subdifferential at a point is always a convex set, possibly empty, but the convex functions F : I → R have the remarkable property that ∂F (x) = ∅ at all interior points. It is worth noticing that ∂F (x) = {F (x)} at each point where F is differentiable (so this formula works for all points of I except for a countable subset). See [9], page 30. Lemma 1. (Legendre duality, [9], page 41). Let F : I → R be a closed convex function. Then its conjugate F * : I * → R is also convex and closed and: i) xy ≤ F (x) + F * (y) for all x ∈ I, y ∈ I * ; ii) xy = F (x) + F * (y) if, and only if, y ∈ ∂F (x); iii) ∂F * = (∂F ) −1 (as graphs); iv) F * * = F.
Recall that the inverse of a graph Γ is the set Γ −1 = {(y, x) : (x, y) ∈ Γ} . How far is Young's inequality from the Legendre duality? Surprisingly, they are pretty closed in the sense that in most cases the Legendre duality can be converted into a Young like inequality. Indeed, every continuous convex function admits an integral representation.
Lemma 2. (See [9], page 37). Let F be a continuous convex function defined on an interval I and let ϕ : I → R be a function such that ϕ(x) ∈ ∂F (x) for every x ∈ I. Then for every a < b in I we have As a consequence, the heuristic meaning of the formula i) in Lemma 1 is the following Young like inequality, where ϕ and ψ are selection functions for ∂F and respectively (∂F ) −1 . Now it becomes clear that Young's inequality should work outside strict monotonicity (as well as outside continuity). The details are presented in Section 2. Our approach (based on the geometric meaning of integrals as areas) allows us to extend the framework of integrability to all positive measures ρ which are locally absolutely continuous with respect to the planar Lebesgue measure dxdy. See Theorem 1 below.
A special case of Young's inequality is which works for all x, y ≥ 0, and p, q > 1 with 1/p + 1/q = 1. Theorem 1 yields the following companion to this inequality in the case of Gaussian measure is the Gauss error function (or the erf function). The precision of our generalization of Young's inequality makes the objective of Section 3.
In Section 4 we discuss yet another extension of Young's inequality, based on recent work done by J. Jakšetić and J. E. Pečarić [13].
The paper ends by noticing the connection of our result to the theory of cconvexity (that is, of convexity associated to a cost density function).
Last but not the least, all results in this paper can be extended verbatim to the framework of nondecreasing functions f : In other words, the interval [0, ∞) plays no special role in Young's inequality.
Besides, there is a straightforward companion of Young's inequality for nonincreasing functions, but this is outside the scope of the present paper.

Young's inequality for weighted measures
In what follows f : [0, ∞) −→ [0, ∞) will denote a nondecreasing function such that f (0) = 0 and lim x→∞ f (x) = ∞. Since f is not necessarily injective we will attach to f a pseudo-inverse by the following formula: } ; here f (x−) and f (x+) represent the lateral limits at x. When f is also continuous, Remark 1. (F. Cunningham Jr. and N. Grossman [2]). Since pseudo-inverses will be used as integrands, it is convenient to enlarge the concept of pseudo-inverse by referring to any function g such that Necessarily, g is nondecreasing and any two pseudo-inverses agree except on a countable set (so their integrals will be the same).
Given 0 ≤ a < b, we define the epigraph and the hypograph of f | [a,b] respectively by Notice that our definitions of epigraph and hypograph are not the standard ones, but agree with them in the context of monotone functions.
We will next consider a measure ρ on [0, ∞) × [0, ∞) , which is locally absolutely continuous with respect to the Lebesgue measure dxdy, that is, ρ is of the form Clearly, Moreover, K (x, y) dy dx. and The discussion above can be summarized as follows: Then for every Lebesgue locally integrable function K : K (x, y) dydx.
We can now state the main result of this section: Theorem 1. (Young's inequality for nondecreasing functions). Under the assumptions of Lemma 3, for every pair of nonnegative numbers a < b, and every number If in addition K is strictly positive almost everywhere, then the equality occurs if Proof. We start with the case where f (a) ≤ c ≤ f (b−). See Figure 1.

Equality holds if and only if
K (x, y) dx dy, that is, when c = f (b+) (provided that K is strictly positive almost everywhere). See Figure 2.
If c ∈ (f (b−) , f (b+)) , then f −1 sup (c) = b and the inequality in the statement of Theorem 1 is actually an equality. See Figure 3. Remark 2. (The probabilistic companion of Theorem 1). Suppose there is given a nonnegative random variable X : [0, ∞) → [0, ∞) whose cumulative distribution function F X (x) = P (X ≤ x) admits a density, that is, a nonnegative Lebesgueintegrable function ρ X such that The quantile function of the distribution function F X (also known as the increasing rearrangement of the random variable X) is defined by Thus, a quantile function is nothing but a pseudo-inverse of F X . Motivated by Statistics, a number of fast algorithms were developed for computing the quantile functions with high accuracy. See [1]. Without entering the details, we recall here the remarkable formula (due to G. Steinbrecher) for the quantile function of the normal distribution: where c 0 = 1 and for all k ≥ 1.
According to Theorem 1, for every pair of continuous random variables Y, Z : [0, ∞) → [0, ∞) with density ρ Y,Z , and every positive numbers b and c, the following inequality holds: This can be seen as a principle of uncertainty, since it shows that the functions cannot be made simultaneously small. The proof is based on mathematical induction (which is left to the reader). The above inequality cover the n-variable generalization of Young's inequality as obtained by Oppenheim [10] (as well as the main result in [12]).
The following stronger version of Corollary 1 incorporates the Legendre duality.
Proof. According to the Legendre duality, K (x, y) dy, and by integrating both sides from a to b we obtain the inequality K (x, y) dy dx.
In a similar manner, starting with u = 1 and v = f −1 sup (y) a K (x, y) dx, we arrive first at the inequality and then to K (x, y) dydx, and the inequality in the statement of Theorem 2 is now clear.
In the special case where K (x, y) = 1, a = f (a) = 0 and Φ(x) = x p /p (for some p > 1), Theorem 2 yields the following inequality: This remark extends a result due to W. T. Sulaiman [15]. We end this section by noticing the following result that complements Theorem 1.

Proposition 1. Under the assumptions of Lemma
K (x, y) dydx .
Assuming K strictly positive almost everywhere, the equality occurs if and only if The other case, c ≥ f (b), has a similar approach.

The precision in Young's inequality
The main result of this section is as follows: Assuming K strictly positive almost everywhere, the equality occurs if and only if Figure 4. The left-hand side of the inequality in the statement of Theorem 3 represents the measure of the cross-hatched curvilinear trapezium, while right-hand side is the measure of the ABCD rectangle.
The equality holds if f (b) = c or f is an affine function. The inequality sign should be reversed if f has a decreasing derivative on the interval

The connection with c-convexity
Motivated by the mass transportation theory, several people [3], [5] drew a parallel to the classical theory of convex functions by extending the Legendre duality. Technically, given two compact metric spaces X and Y and a cost density function c : X × Y → R (which is supposed to be continuous), we may consider the following generalization of the notion of convex function: We abbreviate (4.1) by writing F = G c . A useful remark is the equality The classical notion of convex function corresponds to the case where X is a compact interval and c(x, y) = xy. The details can be found in [9], pp. 40-42.  In fact, under the hypotheses of this theorem, the functions and verify the relations F c = G and G c = F (due to the equality case as specified in the statement of Theorem 1, so they are both c-convex.
On the other hand, a simple argument shows that F and G are also convex in the usual sense.
Let us call the functions c that admits a representation of the form (4.3) with K ∈ L 1 (R × R), absolutely continuous in the hyperbolic sense. With this terminology, Theorem 1 can be rephrased as follows:  Necessarily, an absolutely continuous function c in the hyperbolic sense, is continuous. It admits partial derivatives of the first order and a mixed derivative ∂ 2 c ∂x∂y almost everywhere. Besides, the functions y → ∂c ∂x (x, y) and x → ∂c ∂y (x, y) are defined everywhere in their interval of definition and represent absolutely continuous functions; they are also nondecreasing provided that ∂ 2 c ∂x∂y ≥ 0 almost everywhere. A special case of Theorem 4 was proved by Zs. Páles [11], [12]