[Paper Review] On a Hardy-Littlewood theorem

Elijah Liflyand, Ulrich Stadtmüller|arXiv (Cornell University)|Mar 7, 2013

Mathematics and Applications17 references3 citations

TL;DR

This paper extends the classical Hardy-Littlewood theorem on absolute convergence of Fourier series to functions on the real line, proving that if a function and its conjugate (via the modified Hilbert transform) are both of bounded variation and vanish at infinity, then their Fourier transforms are integrable. The result is derived using Hardy-type inequalities and the real Hardy space $ H^1(\mathbb{R}) $, with the original periodic theorem shown to be a special case when the function has compact support.

ABSTRACT

A known Hardy-Littlewood theorem asserts that if both the function and its conjugate are of bounded variation, then their Fourier series are absolutely convergent. It is proved in the paper that the same result holds true for functions on the whole axis and their Fourier transforms with certain adjustments. The proof of the original Hardy-Littlewood theorem is derived from the obtained assertion. It turned out that the former is a partial case of the latter when the function is supposed to be of compact support. A similar result as the obtained one but for radial functions is derived from the one-dimensional case.

Motivation & Objective

• To generalize the classical Hardy-Littlewood theorem on absolute convergence of Fourier series to functions on the real line.
• To establish conditions under which the Fourier transform of a function on $ \mathbb{R} $ is integrable, replacing Fourier series convergence with $ L^1 $-integrability of the Fourier transform.
• To show that the original Hardy-Littlewood theorem is a special case of the new result when the function has compact support.
• To extend the one-dimensional result to radial functions in higher dimensions using Leray's formula and the one-dimensional theory.

Proposed method

• Use the modified Hilbert transform $ \widetilde{f}(x) $, defined with a correction term $ \frac{t}{1+t^2} $, to ensure existence for bounded functions.
• Apply the identity $ \frac{d}{dx}\widetilde{f}(x) = \mathcal{H}f'(x) $ almost everywhere at Lebesgue points, relying on integration by parts and local absolute continuity.
• Establish that $ f' \in H^1(\mathbb{R}) $ by verifying $ \|f'\|_{L^1} + \|\mathcal{H}f'\|_{L^1} < \infty $, using the bounded variation of $ f $ and $ \widetilde{f} $.
• Employ a Hardy-type inequality: $ \int_{\mathbb{R}} \frac{|\widehat{f'}(x)|}{|x|} dx \leq \|f'\|_{H^1} $, linking the $ L^1 $-norm of the Fourier transform to the $ H^1 $-norm.
• Use the relation $ \widehat{\mathcal{H}f'}(x) = i \operatorname{sign}(x) \widehat{f'}(x) $ to connect the Fourier transform of $ \widetilde{f} $ to $ \widehat{f'} $, enabling $ L^1 $-integrability of $ \widehat{\widetilde{f}} $.
• Apply Leray's formula for radial functions to reduce the multidimensional Fourier transform to a one-dimensional oscillatory integral, then apply the main theorem to the resulting kernel function.

Experimental results

Research questions

• RQ1 Under what conditions on a function $ f $ on $ \mathbb{R} $ is its Fourier transform $ \widehat{f} $ integrable?
• RQ2 Can the classical Hardy-Littlewood theorem on absolute convergence of Fourier series be derived as a special case of a broader result on the real line?
• RQ3 How can the conjugate function concept be adapted for non-periodic, bounded variation functions on $ \mathbb{R} $, where the standard Hilbert transform may not exist?
• RQ4 What conditions ensure that the Fourier transform of a radial function in $ \mathbb{R}^n $ is Lebesgue integrable, based on one-dimensional properties of its radial profile?
• RQ5 To what extent does the $ H^1(\mathbb{R}) $ space and Hardy-type inequalities facilitate the extension of classical Fourier analysis results to the non-periodic setting?

Key findings

• If $ f $ is of bounded variation on $ \mathbb{R} $, vanishes at infinity, and its conjugate $ \widetilde{f} $ is also of bounded variation, then both $ \widehat{f} $ and $ \widehat{\widetilde{f}} $ are integrable on $ \mathbb{R} $.
• The derivative of the conjugate function satisfies $ \frac{d}{dx}\widetilde{f}(x) = \mathcal{H}f'(x) $ almost everywhere, which is essential for linking to the $ H^1(\mathbb{R}) $ space.
• The function $ f' $ belongs to the real Hardy space $ H^1(\mathbb{R}) $, as both $ \|f'\|_{L^1} $ and $ \|\mathcal{H}f'\|_{L^1} $ are finite under the given assumptions.
• The Hardy-type inequality $ \int_{\mathbb{R}} \frac{|\widehat{f'}(x)|}{|x|} dx \leq \|f'\|_{H^1} $ implies that $ \|\widehat{f}\|_{L^1} \leq \|f'\|_{H^1} $, establishing the integrability of $ \widehat{f} $.
• The original Hardy-Littlewood theorem is recovered as a special case when $ f $ has compact support, since then the modified Hilbert transform reduces to the standard one and the Fourier transform of the compactly supported function is integrable.
• For radial functions in $ \mathbb{R}^n $, if the $ (n-1) $-th derivative of the associated kernel $ I(t) $ satisfies the conditions of the one-dimensional theorem, then the multidimensional Fourier transform of $ f $ is integrable.