[Paper Review] Some new moment rearrangement invariant spaces; theory and applications

E. Ostrovsky, L. Sirota|ArXiv.org|May 29, 2006

Advanced Harmonic Analysis Research3 citations

TL;DR

This paper introduces and analyzes a new class of Banach spaces—moment rearrangement invariant spaces—defined via norms that control $ L^p $-norms of functions relative to a weight function $ \psi(p) $. The authors establish foundational properties such as separability and reflexivity, and demonstrate applications to Fourier series, singular integral operators, and martingale theory, showing that these spaces capture sharp integrability and boundedness behavior in harmonic analysis and probability.

ABSTRACT

In this article we introduce and investigate some new Banach spaces, so - called moment spaces, and consider applications to the Fourier series, singular integral operators, theory of martingales.

Motivation & Objective

• To define and study a new class of Banach spaces—moment spaces—based on $ L^p $-norms weighted by a function $ \psi(p) $, generalizing classical Lorentz and Orlicz spaces.
• To establish necessary and sufficient conditions for a function $ \psi(p) $ to arise as the $ L^p $-norm of a measurable function, using convexity and Bernstein-type representation theorems.
• To investigate structural properties of the resulting moment spaces, including separability, reflexivity, and embedding theorems.
• To apply the theory to classical problems in harmonic analysis, such as convergence of Fourier series and boundedness of singular integral operators.
• To extend the framework to martingale theory, demonstrating the relevance of moment spaces in stochastic analysis.

Proposed method

• Define the moment space $ G(\psi) $ via the norm $ \|f\|_{G(\psi)} = \sup_{p \in (a,b)} |f|_p / \psi(p) $, where $ \psi \in U\Psi $, the set of positive continuous functions on $ (a,b) $.
• Use the convexity of $ p \mapsto p \log \psi(p) $ and Bernstein’s theorem to characterize functions $ \psi(p) $ that arise as $ L^p $-norms of measurable functions.
• Introduce the class $ \Psi(a,b) $ of functions $ \psi $ for which $ \psi^p(p) $ admits a representation as a sum of absolutely and relatively monotonic functions.
• Construct specific examples of $ \psi(p) $, such as $ \zeta(p) = \min\{(p-a)^\alpha, (b-p)^\beta\} $, to define concrete moment spaces $ G(a,b;\alpha,\beta) $.
• Apply the theory to analyze the behavior of Fourier transforms and Hilbert transforms in these spaces, showing sharp growth rates in terms of $ p $-norms.
• Use the theory of Orlicz and Lorentz spaces as a foundation, extending them via moment-based norms to capture intermediate or logarithmic growth.

Experimental results

Research questions

• RQ1 Which functions $ \psi(p) $ on $ (a,b) $ can arise as the $ L^p $-norm of a measurable function $ f $, and what are the necessary and sufficient conditions for such $ \psi $?
• RQ2 How do the structural properties (separability, reflexivity, embedding) of the moment spaces $ G(\psi) $ depend on the growth behavior of $ \psi(p) $?
• RQ3 What is the role of moment spaces in characterizing the integrability and boundedness of singular integral operators, such as the Hilbert transform?
• RQ4 How do moment spaces refine the classical theory of Fourier series and transforms, particularly in capturing logarithmic or power-type growth in $ p $-norms?
• RQ5 In what way do moment spaces provide a natural framework for studying martingale transforms and their $ L^p $-boundedness?

Key findings

• A function $ \psi(p) $ belongs to $ \Psi(a,b) $ if and only if $ \psi^p(p) $ can be decomposed into the sum of an absolutely monotonic and a relatively monotonic function on $ (a,b) $, via Bernstein’s theorem.
• The space $ G(\psi) $ is non-trivial: all bounded functions with finite measure support belong to $ G(\psi) $ for any $ \psi \in U\Psi $.
• For $ \psi(p) \asymp \min\{(p-a)^\alpha, (b-p)^\beta\} $, the space $ G(a,b;\alpha,\beta) $ captures sharp $ p $-norm growth rates, such as $ |f|_p \asymp p^d $ for functions with logarithmic singularities.
• The Hilbert transform $ H[f] $ of a function $ f $ with $ |f|_p \asymp p^d $ satisfies $ |H[f]|_p \asymp p^{d+1} $, showing that $ H[f] \in G(a,b;\alpha,\beta) $ with $ \alpha = d+1, \beta = 0 $, while $ f \in G(a,b;\alpha,\beta) $ with $ \alpha = d, \beta = 0 $.
• For $ f(x) = \sum_{n=2}^\infty n^{-1} \log^d n \, \cos(nx) $, the transform $ H[f] $ satisfies $ |H[f]|_p \asymp p^{d+1} $, confirming that $ H[f] \in G(1,\infty;1,d) $, and $ f \in G(1,\infty;1,d) $, but $ H[f] \notin G^o(1,\infty;1,d) $, indicating sharpness of the norm.
• The theory applies to the real-line Hilbert transform: for $ f(x) = \int_3^\infty t^{d-1} \sin(tx) \, dt $, $ |f(x)| \asymp |H[f](x)| \asymp f_{1/d,1}(x) $, and $ f, H[f] \in G \setminus G^o(1,1/d;1,d) $, showing the space captures critical growth.