[Paper Review] Commutators of singular integrals on generalized $L^p$ spaces with variable exponent

Alexei Yu. Karlovich, Andrei K. Lerner|ArXiv.org|Jan 26, 2004

Advanced Harmonic Analysis Research14 references4 citations

TL;DR

This paper extends the classical Coifman-Rochberg-Weiss theorem on commutators of Calderón-Zygmund singular integrals to generalized Lebesgue spaces with variable exponent $L^{p(ullet)}(\mathbb{R}^n)$. It proves that a function $b$ belongs to $BMO(\mathbb{R}^n)$ if and only if the commutator $[b,T]$ is bounded on $L^{p(\cdot)}(\mathbb{R}^n)$, provided $p$ and its conjugate exponent $p'$ satisfy the Muckenhaupt-type condition for the boundedness of the Hardy-Littlewood maximal operator.

ABSTRACT

A classical theorem of Coifman, Rochberg, and Weiss on commutators of singular integrals is extended to the case of generalized $L^p$ spaces with variable exponent.

Motivation & Objective

• To generalize the classical Coifman-Rochberg-Weiss theorem on commutators of singular integrals to the setting of generalized $L^p$ spaces with variable exponent.
• To establish a characterization of $BMO(\mathbb{R}^n)$ via the boundedness of commutators $[b,T]$ on variable exponent Lebesgue spaces $L^{p(\cdot)}(\mathbb{R}^n)$.
• To address the challenge that $L^{p(\cdot)}(\mathbb{R}^n)$ spaces are not translation-invariant or rearrangement-invariant, requiring new techniques beyond classical harmonic analysis.
• To extend the duality and sharp maximal function methods to variable exponent spaces, particularly using local sharp maximal functions and duality inequalities.

Proposed method

• Adapts a localized version of the Fefferman-Stein sharp maximal function inequality for $L^{p(\cdot)}(\mathbb{R}^n)$ spaces, relying on recent results by Diening and Růžička.
• Employs a duality inequality due to Lerner [15, Theorem 1] to relate the norm of the commutator to the BMO seminorm of $b$.
• Uses a sharp function inequality for commutators from Strömberg [11] and Pérez [22, Lemma 3.1] in the variable exponent setting.
• Establishes boundedness of $[b,T]$ on $L^{p(\cdot)}(\mathbb{R}^n)$ via approximation by compactly supported bounded functions and norm continuity extension.
• Proves that if $[b,T]$ is bounded on $L^{p(\cdot)}(\mathbb{R}^n)$ and $\Omega$ is odd, then $[b,T]$ is also bounded on $L^{p'(\cdot)}(\mathbb{R}^n)$, enabling duality arguments.
• Applies Janson’s theorem on $L^2(\mathbb{R}^n)$ boundedness to deduce $b \in BMO(\mathbb{R}^n)$, leveraging the boundedness of $[b,T]$ on $L^2$ via interpolation and norm estimates.

Experimental results

Research questions

• RQ1 Does the classical characterization of $BMO(\mathbb{R}^n)$ via commutator boundedness on $L^p(\mathbb{R}^n)$ extend to variable exponent Lebesgue spaces $L^{p(\cdot)}(\mathbb{R}^n)$?
• RQ2 Can the boundedness of the commutator $[b,T]$ on $L^{p(\cdot)}(\mathbb{R}^n)$ imply $b \in BMO(\mathbb{R}^n)$ even when $L^{p(\cdot)}(\mathbb{R}^n)$ lacks translation invariance?
• RQ3 What conditions on the variable exponent $p(\cdot)$ ensure that the Hardy-Littlewood maximal operator is bounded on $L^{p(\cdot)}(\mathbb{R}^n)$, enabling the extension of classical results?
• RQ4 How can duality and sharp maximal function techniques be adapted to variable exponent spaces where standard rearrangement and good-$\lambda$ methods fail?
• RQ5 Is the boundedness of $[b,T]$ on $L^{p(\cdot)}(\mathbb{R}^n)$ sufficient to imply boundedness on $L^{p'(\cdot)}(\mathbb{R}^n)$, and what role does the oddness of the kernel $\Omega$ play?

Key findings

• If $b \in BMO(\mathbb{R}^n)$ and $p, p' \in \mathcal{M}(\mathbb{R}^n)$, then the commutator $[b,T]$ is bounded on $L^{p(\cdot)}(\mathbb{R}^n)$ with operator norm satisfying $\|[b,T]\|_{\mathcal{B}(L^{p(\cdot)})} \leq C_p \|b\|_*$.
• Conversely, if $\Omega$ is odd, $b \in L\log L(Q)$ for every cube $Q$, and $[b,T]$ is bounded on $L^{p(\cdot)}(\mathbb{R}^n)$, then $b \in BMO(\mathbb{R}^n)$ with $\|b\|_* \leq C_p' \|[b,T]\|_{\mathcal{B}(L^{p(\cdot)})}$.
• The boundedness of $[b,T]$ on $L^{p(\cdot)}(\mathbb{R}^n)$ implies its boundedness on $L^{p'(\cdot)}(\mathbb{R}^n)$ under the oddness condition on $\Omega$, with $\|[b,T]\|_{\mathcal{B}(L^{p'(\cdot)})} \leq r_p \|[b,T]\|_{\mathcal{B}(L^{p(\cdot)})}$.
• The commutator $[b,T]$ is bounded on $L^2(\mathbb{R}^n)$, and from this, $b \in BMO(\mathbb{R}^n)$ follows via Janson’s theorem, with $\|b\|_* \leq c_2(K) \|[b,T]\|_{\mathcal{B}(L^2)}$.
• The constant $C_p'$ in the reverse inequality is explicitly bounded by $2r_p c_2(K)$, where $r_p$ depends on $p$ and $c_2(K)$ is a kernel-dependent constant.
• The condition $p' \in \mathcal{M}(\mathbb{R}^n)$ can be removed due to recent results showing $p \in \mathcal{M}(\mathbb{R}^n)$ implies $p' \in \mathcal{M}(\mathbb{R}^n)$, making the result more widely applicable.