On Carleson’s inequality

We present a new proof of Hardy’s inequality by giving an Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L^{p}$\end{document} version of Carleson’s inequality.


Introduction
The classical Hardy inequality reads where f is a nonnegative measurable function on (0, ∞) and p > 1. A weighted modification of (1.1) was proved also by Hardy [1]: where f is a nonnegative measurable function on (0, ∞) and p > 1, α < p -1. The constant p p -1α p is the best possible.
The importance and the usefulness of these inequalities could never have been overestimated. Connected with Hardy's inequality, the following two inequalities frequently appeared in the literature. See [2,3], and [4].
for any measurable f ≥ 0.

Theorem 1.2 (Carleson's inequality)
Let F(x) be a convex function for x ≥ 0, satisfying F(0) = 0. If -1 < α < ∞, the following inequality holds true: As we shall see in Sect. 3.2, the following can be derived from (1.4) for any measurable f ≥ 0: The discrete version of inequality (1.3) is known as Carleman's inequality [5]. Equation (1.3) is a special case of (1.5), and (1.5) can be regarded as a special case of (1.4).
The goal of this note is to complete the following diagram by finding "(2.1)", which gives (1.2) in the same manner as (1.4) gives (1.5) as a special case.
This will be done in Sect. 2.1.

An inequality of Carleson type
There are a lot of different ideas in the literature for the proof of Hardy's inequality. With the special property of the mean 1 x x 0 f (t) dt, the proof of (1.1) or (1.2) should depend on the convexity of the function x p , p > 1. See [2] and the references therein.

L p version inequality
The following theorem is a straightforward consequence of Minkowski's inequality which comes from the convexity. It verifies Hardy's inequality (1.2) as we shall see in Sect. 3.1.

Remarks
(1) The point of Theorem 2.1 is In fact, we have by (2.4) and (2.6). (2) The conditions of Theorem 2.1 seem to be rather complicated so that one may suspect the existence of such a F. We give a simple example: Let α, β be chosen to be αp + 1 < 0, 0 < β ≤ p, and αp + β + 1 > 0. Take F(x) = x β/p for 0 ≤ x ≤ 1 and Then F is nonnegative increasing concave and F(0) = 0. A simple calculation shows whence we have (2.1) (and (2.7) with A = ∞).
On the other hand, the example F(x) = 0 for 0 ≤ x ≤ 1 and F(x) = 1 for 1 < x < ∞ shows that (2.1) fails if the concavity hypothesis is omitted.
(3) Note that a concave function on an interval is absolutely continuous on each closed subinterval. While absolute continuity (on a closed interval) is equivalent to be an indefinite integral. So, the concavity of F with F(0) = 0 in Theorem 2.1 implies which implies the existence of F (x) almost everywhere and f = F decreasing.

Hardy's inequality (1.2) follows from (2.1)
It is simple to see that Hardy's inequality (1.2) follows immediately from our Theorem 2.1. First assume α = 0 and show (1.1). We may assume f ∈ L p (0, ∞). Let f be the decreasing rearrangement of f : Then f is decreasing and Hence it is sufficient to show (1.1) under the hypothesis that f is decreasing. Let f be decreasing and F(x) = Since F = f is nonnegative decreasing, F is increasing and concave. Thus, applying Theorem 2.1 with α = 0, we obtain (1.1).
Next, for general f and α, (1.2) follows from (1.1) by a simple change of variable: setting it is straightforward to see

Conclusion
From the observation that the Polya-Knopp inequality is a limit version of Hardy's inequality and Carleson's inequality verifies Polya-Knopp inequality as a special case, an inequality verifying Hardy's inequality by the same pattern was called for. In Theorem 2.1, the main result of this article, we give such an inequality: inequality (2.1). A simple application of Minkowski's inequality proved Theorem 2.1. In view of Section 3.3 Remarks (2), it seems that there should have been an easy proof of Hardy's inequality in terms of Minkowski's inequality early in the literature.