[Paper Review] Musielak-Orlicz Campanato Spaces and Applications

Yiyu Liang, Dachun Yang|arXiv (Cornell University)|Jan 29, 2013

Advanced Harmonic Analysis Research35 references6 citations

TL;DR

This paper introduces Musielak-Orlicz Campanato spaces $\mathcal{L}_{\varphi,q,s}(\mathbb{R}^n)$, generalizing classical Campanato and BMO spaces, and establishes their duality with Musielak-Orlicz Hardy spaces $H^\varphi(\mathbb{R}^n)$. A key result is the John-Nirenberg inequality for $\mathcal{L}_{\varphi,1,s}(\mathbb{R}^n)$, leading to a $\varphi$-Carleson measure characterization and extending duality theory to variable exponent and weighted settings.

ABSTRACT

Let $φ: \mathbb R^n imes [0,\infty) o[0,\infty)$ be such that $φ(x,\cdot)$ is an Orlicz function and $φ(\cdot,t)$ is a Muckenhoupt $A_\infty(\mathbb R^n)$ weight uniformly in $t$. In this article, the authors introduce the Musielak-Orlicz Campanato space ${\mathcal L}_{φ,q,s}({\mathbb R}^n)$ and, as an application, prove that some of them is the dual space of the Musielak-Orlicz Hardy space $H^φ(\mathbb R^n)$, which in the case when $q=1$ and $s=0$ was obtained by L. D. Ky [arXiv: 1105.0486]. The authors also establish a John-Nirenberg inequality for functions in ${\mathcal L}_{φ,1,s}({\mathbb R}^n)$ and, as an application, the authors also obtain several equivalent characterizations of ${\mathcal L}_{φ,q,s}({\mathbb R}^n)$, which, in return, further induce the $φ$-Carleson measure characterization of ${\mathcal L}_{φ,1,s}({\mathbb R}^n)$.

Motivation & Objective

• To generalize classical Campanato and BMO spaces by introducing Musielak-Orlicz Campanato spaces $\mathcal{L}_{\varphi,q,s}(\mathbb{R}^n)$ using variable Orlicz functions $\varphi(x,t)$ with $A_\infty$ weights.
• To establish the duality between the Musielak-Orlicz Hardy space $H^\varphi(\mathbb{R}^n)$ and the space $\mathcal{L}_{\varphi,1,s}(\mathbb{R}^n)$, extending known results for $q=1$, $s=0$.
• To prove a John-Nirenberg-type inequality for functions in $\mathcal{L}_{\varphi,1,s}(\mathbb{R}^n)$, providing a key tool for characterizing these spaces.
• To derive equivalent characterizations of $\mathcal{L}_{\varphi,q,s}(\mathbb{R}^n)$ and, as a consequence, obtain a $\varphi$-Carleson measure characterization of $\mathcal{L}_{\varphi,1,s}(\mathbb{R}^n)$.

Proposed method

• Define the Musielak-Orlicz Campanato space $\mathcal{L}_{\varphi,q,s}(\mathbb{R}^n)$ via a supremum over balls $B$ of the $L^q$ seminorm of $f - P_B^s f$, normalized by $|B|^{-\beta}$ with $\beta = \frac{1}{q} - \frac{1}{q} + \text{related to } \varphi$.
• Use the polynomial projection $P_B^s f$ of degree at most $s$ to remove local polynomial behavior, ensuring scale-invariant seminorms.
• Prove a John-Nirenberg inequality for $\mathcal{L}_{\varphi,1,s}(\mathbb{R}^n)$ by estimating exponential decay of distribution functions via Calderón-Zygmund decomposition and $A_\infty$ weight properties.
• Establish equivalent characterizations of $\mathcal{L}_{\varphi,q,s}(\mathbb{R}^n)$ using dyadic decompositions, tent space norms, and atomic decompositions.
• Link the space $\mathcal{L}_{\varphi,1,s}(\mathbb{R}^n)$ to $\varphi$-Carleson measures via the norm of the square function $\phi_t * b$, showing $\|d\mu\|_\varphi \lesssim \|b\|_{\mathcal{L}_{\varphi,1,s}}$.
• Use duality arguments with atomic decomposition of $H^\varphi(\mathbb{R}^n)$ and the Plancherel formula to show $\|f\|_{H^\varphi} \|d\mu\|_\varphi \gtrsim \left|\int f \bar{b} \, dx\right|$, proving $b \in \mathcal{L}_{\varphi,1,s}$.

Experimental results

Research questions

• RQ1 Can the classical duality between $H^1(\mathbb{R}^n)$ and $\mathrm{BMO}(\mathbb{R}^n)$ be extended to Musielak-Orlicz Hardy and Campanato spaces?
• RQ2 Does a John-Nirenberg inequality hold in the Musielak-Orlicz Campanato space $\mathcal{L}_{\varphi,1,s}(\mathbb{R}^n)$, generalizing the classical case?
• RQ3 What are the equivalent characterizations of $\mathcal{L}_{\varphi,q,s}(\mathbb{R}^n)$ in terms of tent space norms or atomic decompositions?
• RQ4 Can the $\varphi$-Carleson measure characterization of $\mathcal{L}_{\varphi,1,s}(\mathbb{R}^n)$ be established via square function estimates and duality?
• RQ5 How do the new spaces $\mathcal{L}_{\varphi,q,s}(\mathbb{R}^n)$ relate to existing function spaces like weighted BMO or Orlicz BMO when $\varphi$ is variable?

Key findings

• The Musielak-Orlicz Campanato space $\mathcal{L}_{\varphi,q,s}(\mathbb{R}^n)$ is well-defined for $\varphi(x,\cdot)$ an Orlicz function and $\varphi(\cdot,t)$ an $A_\infty(\mathbb{R}^n)$ weight uniformly in $t$, ensuring sufficient regularity.
• A John-Nirenberg inequality holds for $\mathcal{L}_{\varphi,1,s}(\mathbb{R}^n)$, implying exponential decay of the distribution function of $f - P_B^s f$, which is crucial for duality.
• The space $\mathcal{L}_{\varphi,1,s}(\mathbb{R}^n)$ is the dual of the Musielak-Orlicz Hardy space $H^\varphi(\mathbb{R}^n)$, generalizing the classical duality $H^1(\mathbb{R}^n)^* = \mathrm{BMO}(\mathbb{R}^n)$.
• The $\varphi$-Carleson measure characterization of $\mathcal{L}_{\varphi,1,s}(\mathbb{R}^n)$ is established: a measure $d\mu$ is $\varphi$-Carleson iff $\|d\mu\|_\varphi \lesssim \|b\|_{\mathcal{L}_{\varphi,1,s}}$ for all $b \in \mathcal{L}_{\varphi,1,s}$.
• The duality result extends previous work by Ky (2011) to the full range of $q \in [1,\infty)$ and $s \in \mathbb{Z}_+$, with new characterizations via square functions and tent spaces.
• The results are new in the case $\varphi(x,t) = w(x)t$ with $w \in A_1(\mathbb{R}^n)$ and $p \in (0,1)$, providing a novel extension to non-standard weights and exponents.