[Paper Review] A sharp integral rearrangement inequality for the dyadic maximal operator and applications

Eleftherios N. Nikolidakis, Antonios D. Melas|arXiv (Cornell University)|May 11, 2013

Advanced Harmonic Analysis Research7 references4 citations

TL;DR

This paper presents a new proof for the Bellman function of three variables associated with the dyadic maximal operator using a generalized symmetrization principle. By establishing a sharp integral rearrangement inequality for the dyadic maximal operator, the authors re-derive the exact formula for $ B_p(f,F,L) $, confirming its sharpness and offering an alternative to prior methods based on PDEs or case analysis.

ABSTRACT

We prove a sharp integral inequality for the dyadic maximal operator and give as an application another proof for the computation of its Bellman function of three variables.

Motivation & Objective

• To provide an alternative, simplified proof for the computation of the Bellman function $ B_p(f,F,L) $ of the dyadic maximal operator.
• To establish a sharp integral rearrangement inequality for the dyadic maximal operator using a generalized symmetrization principle.
• To demonstrate the sharpness of the known formula for $ B_p(f,F,L) $ through a constructive sequence of test functions.
• To unify and extend existing symmetrization techniques in the context of dyadic maximal operators and Bellman functions.

Proposed method

• Generalizing Theorem A from [10], the authors prove a new symmetrization identity for integrals involving the dyadic maximal operator and two increasing functions $ G_1 $, $ G_2 $.
• The key identity is $ \int_K G_1(\mathcal{M}_\mathcal{T}\phi) G_2(\phi)\,d\mu = \int_0^k G_1\left(\frac{1}{t}\int_0^t g\right) G_2(g(t))\,dt $, valid for non-increasing $ g $ and increasing $ G_i $.
• The method applies this identity to the $ L^p $-norm of $ \max(\mathcal{M}_\mathcal{T}\phi, L) $, reducing the problem to optimizing a functional on decreasing functions.
• The authors construct a sequence of decreasing functions $ g_n $ with controlled $ L^1 $ and $ L^p $ norms to verify the sharpness of the upper bound in the case $ L \geq \frac{p}{p-1}f $.
• They analyze the asymptotic behavior of the maximal function and use integral estimates involving level sets to show convergence to the target expression.
• The proof relies on the inverse function $ \omega_p = H_p^{-1} $, where $ H_p(z) = -(p-1)z^p + pz^{p-1} $, to express the Bellman function in closed form.

Experimental results

Research questions

• RQ1 Can the Bellman function $ B_p(f,F,L) $ for the dyadic maximal operator be re-derived using a generalized symmetrization principle?
• RQ2 What is the sharp integral inequality governing the dyadic maximal operator in terms of decreasing rearrangements?
• RQ3 How does the behavior of $ \mathcal{M}_\mathcal{T}\phi $ change when the essential supremum $ L $ exceeds $ \frac{p}{p-1}f $?
• RQ4 Is the known formula for $ B_p(f,F,L) $ optimal, and can its sharpness be proven via constructive test functions?
• RQ5 Can the symmetrization principle be extended to handle truncated maximal functions $ \max(\mathcal{M}_\mathcal{T}\phi, L)^p $?

Key findings

• The paper establishes a sharp integral rearrangement inequality: $ \int_K G_1(\mathcal{M}_\mathcal{T}\phi) G_2(\phi)\,d\mu = \int_0^k G_1\left(\frac{1}{t}\int_0^t g\right) G_2(g(t))\,dt $, generalizing known symmetrization results.
• The Bellman function $ B_p(f,F,L) $ is proven to be equal to $ F \omega_p\left(\frac{pL^{p-1}f - (p-1)L^p}{F}\right)^p $ when $ L < \frac{p}{p-1}f $, and $ L^p + \left(\frac{p}{p-1}\right)^p (F - f^p) $ when $ L \geq \frac{p}{p-1}f $.
• The sharpness of the upper bound is confirmed by constructing a sequence of functions $ g_n $ such that the $ L^p $-norm of $ \max(\mathcal{M}_\mathcal{T}\phi_n, L) $ converges to the stated expression.
• The limit of the maximal function's $ L^p $-norm under the constructed sequence satisfies $ \lim_n \int_0^1 \max\left(\frac{1}{t}\int_0^t g_n, L\right)^p dt = L^p + \left(\frac{p}{p-1}\right)^p (F - f^p) $ for $ L \geq \frac{p}{p-1}f $.
• The authors show that the error term from the level set integral vanishes as $ n \to \infty $, confirming convergence to the sharp bound.
• The result provides an alternative to PDE-based methods for computing Bellman functions, using only real analysis and symmetrization techniques.