[Paper Review] Weighted Boundedness of the Maximal, Singular and Potential Operators in Variable Exponent Spaces

Vakhtang Kokilashvili, Stefan Samko|ArXiv.org|May 14, 2008

Advanced Harmonic Analysis ResearchMathematics73 references21 citations

TL;DR

This paper establishes new weighted boundedness criteria for the Hardy-Littlewood maximal operator in variable exponent Lebesgue spaces $L^{p(ullet)}(X, \varrho)$ over metric measure spaces $X$ satisfying the doubling condition. It introduces a Muckenhoupt-type condition and provides sufficient conditions involving Matuszewska-Orlicz-type indices of weights and measure growth, extending known results to variable exponents and general doubling spaces, including bounded and unbounded domains with radial-type weights.

ABSTRACT

We present a brief survey of recent results on boundedness of some classical operators within the frameworks of weighted spaces $L^{p(\cdot)}(\varrho)$ with variable exponent $p(x)$, mainly in the Euclidean setting and dwell on a new result of the boundedness of the Hardy-Littlewood maximal operator in the space $L^{p(\cdot)}(X,\varrho)$ over a metric measure space $X$ satisfying the doubling condition. In the case where $X$ is bounded, the weight function satisfies a certain version of a general Muckenhoupt-type condition For a bounded or unbounded $X$ we also consider a class of weights of the form $\varrho(x)=[1+d(x_0,x)]^{\bt_\infty}\prod_{k=1}^m w_k(d(x,x_k))$, $x_k\in X$, where the functions $w_k(r)$ have finite upper and lower indices $m(w_k)$ and $M(w_k)$. Some of the results are new even in the case of constant $p$.

Motivation & Objective

• To extend weighted boundedness theory of classical harmonic operators to variable exponent Lebesgue spaces with general weights.
• To establish sufficient conditions for the boundedness of the Hardy-Littlewood maximal operator in $L^{p(\cdot)}(X,\varrho)$ over doubling metric measure spaces.
• To generalize Muckenhoupt-type conditions to variable exponent and non-homogeneous settings using Matuszewska-Orlicz indices of measure and weight functions.
• To analyze radial-type oscillating weights of the form $\varrho(x) = [1+d(x_0,x)]^{\beta_\infty} \prod w_k(d(x,x_k))$ in bounded and unbounded domains.
• To unify and extend previous results on variable exponent spaces by incorporating local and global indices of measure and weight growth.

Proposed method

• Introduces a uniform lower Matuszewska-Orlicz index $m(\mu B)$ for the measure $\mu B(x,r)$ to capture local doubling behavior.
• Applies Zygmund-Bary-Stechkin class $\Phi_1^0$ and weighted classes $\widetilde{W}$ to characterize the growth of weight functions $w_k(r)$.
• Derives sufficient conditions for boundedness via inequalities involving $m(w_k)$, $M(w_k)$, $p(x_k)$, and $m(\mu B)$ in Theorems B and C.
• Uses the doubling condition $\mu B(x,2r) \leq C \mu B(x,r)$ to control measure growth and derive index-based estimates.
• Applies Hölder's inequality in variable exponent spaces with $\|f\|_{p(\cdot)}$ and $\|g\|_{p'(\cdot)}$ norms.
• Analyzes both bounded and unbounded metric measure spaces, distinguishing behavior at infinity via $m_\infty(\mu B)$ and $M_\infty(\mu B)$.

Experimental results

Research questions

• RQ1 Under what conditions is the Hardy-Littlewood maximal operator bounded in $L^{p(\cdot)}(X,\varrho)$ for a doubling metric measure space $X$ with variable exponent $p(x)$?
• RQ2 How can Muckenhoupt-type conditions be generalized to variable exponent and non-constant measure growth settings?
• RQ3 What role do the lower and upper Matuszewska-Orlicz indices $m(w_k)$ and $M(w_k)$ play in the boundedness of the maximal operator with radial-type weights?
• RQ4 How do the local and global indices $m(\mu B)$ and $m_\infty(\mu B)$ of the measure influence the boundedness in bounded versus unbounded domains?
• RQ5 What are the precise conditions on power-type weights $\varrho(x) = \prod [d(x,x_k)]^{\beta_k}$ for boundedness in variable exponent spaces?

Key findings

• Theorem B establishes boundedness of the maximal operator in $L^{p(\cdot)}(X,\varrho)$ for bounded $X$ when $-\frac{m(\mu B)}{p(x_k)} < m(w_k) \leq M(w_k) < \frac{m(\mu B)}{p'(x_k)}$ for each $k$, with $m(\mu B)$ defined via the uniform lower index of $\mu B(x,r)$.
• Theorem C extends this to unbounded $X$ with $p(x) \equiv p_\infty$ outside a ball, requiring an additional condition involving $m_\infty(\mu B)$ and $\Delta_{p_\infty} = \frac{M_\infty(\mu B) - m_\infty(\mu B)}{p_\infty}$.
• For power weights $\varrho(x) = (1+d(x_0,x))^{\beta_0} \prod [d(x,x_k)]^{\beta_k}$, the conditions reduce to $-\frac{d}{p(x_k)} < \beta_k < \frac{d}{p'(x_k)}$ and $-\frac{d}{p_\infty} < \sum \beta_k < \frac{d}{p'_\infty}$ in spaces of constant dimension $d$.
• The results are new even in the constant exponent case, particularly for weights in the Zygmund-Bary-Stechkin class and non-constant measure growth.
• The index $m(\mu B)$ captures the uniform lower growth rate of balls and is essential for characterizing the boundedness condition in a way analogous to the classical Muckenhoupt condition.
• The paper shows that the uniform index $m(\mu B)$ is independent of $x$ and can be used to define a generalized Muckenhoupt-type condition for variable exponent spaces on doubling metric measure spaces.