[Paper Review] Harmonic Analysis Operators Associated with Multidimensional Bessel Operators

Jorge J. Betancor, Alejandro J. Castro+1 more|arXiv (Cornell University)|Mar 1, 2010

Advanced Harmonic Analysis Research29 references3 citations

TL;DR

This paper establishes the weak type (1,1) boundedness of the maximal operator and Littlewood-Paley g-function associated with the heat semigroup of multidimensional Bessel operators, and proves the strong type (p,p) and weak type (1,1) boundedness of Riesz transforms in the multidimensional Bessel setting for all $1 < p < ∞$. The results extend classical harmonic analysis to higher-dimensional Bessel contexts using weighted measure spaces and dyadic decomposition techniques.

ABSTRACT

In this paper we establish that the maximal operator and the Littlewood-Paley g-function associated with the heat semigroup defined by multidimensional Bessel operators are of weak type (1,1). Also, we prove that Riesz transforms in the multidimensional Bessel setting are of strong type (p,p), for every $1

Motivation & Objective

• To extend harmonic analysis operators—maximal operators, Littlewood-Paley g-functions, and Riesz transforms—to the multidimensional Bessel setting.
• To establish weak type (1,1) and strong type (p,p) boundedness for these operators in weighted $L^p$ spaces with respect to the measure $\prod x_j^{2\lambda_j} dx$.
• To generalize one-dimensional Bessel harmonic analysis results to higher dimensions ($n \geq 2$) using the heat semigroup and Riesz transform theory.
• To provide a complete $L^p$-boundedness theory for key harmonic operators in the multidimensional Bessel setting, including a.e. convergence of the heat semigroup.

Proposed method

• Adapted techniques from Nowak and Sjögren (2010) to prove weak type (1,1) boundedness of the maximal operator via dyadic decomposition and weak-type estimates.
• Used the heat semigroup $\{W_t^{\lambda_1,\dots,\lambda_n}\}_{t>0}$ generated by the multidimensional Bessel operator $\Delta_{\lambda_1,\dots,\lambda_n}$, which is a symmetric diffusion semigroup with respect to the measure $m_{\lambda_1,\dots,\lambda_n}$.
• Defined the Littlewood-Paley g-function as $g^{\lambda_1,\dots,\lambda_n}(f)(x) = \left(\int_0^\infty \left|t \frac{\partial}{\partial t} W_t^{\lambda_1,\dots,\lambda_n}(f)(x)\right|^2 \frac{dt}{t}\right)^{1/2}$ and proved its weak type (1,1) boundedness.
• Introduced the Riesz transforms $R_i^{\lambda_1,\dots,\lambda_n}$ via the factorization $\Delta_{\lambda_i} = \left(\frac{d}{dx_i}\right)^* \frac{d}{dx_i}$ and proved their $L^p$-boundedness using maximal function estimates and dyadic decomposition.
• Employed dyadic cubes $\prod_{j=1}^n [2^{m_j}, 2^{m_j+1})$ to localize the operators and control weak-type norms via maximal function estimates.
• Combined $L^p$-boundedness of the maximal operator and the g-function with $L^1$-weak estimates to deduce $L^p$-boundedness of Riesz transforms for $1 < p < \infty$ and weak type (1,1) via interpolation and standard limiting arguments.

Experimental results

Research questions

• RQ1 Is the maximal operator associated with the multidimensional Bessel heat semigroup of weak type (1,1) with respect to the measure $\prod x_j^{2\lambda_j} dx$?
• RQ2 Does the Littlewood-Paley g-function associated with the multidimensional Bessel heat semigroup map $L^1$ into weak $L^1$?
• RQ3 Are the Riesz transforms in the multidimensional Bessel setting bounded on $L^p$ for all $1 < p < \infty$?
• RQ4 Is the Riesz transform of weak type (1,1) in the multidimensional Bessel setting?
• RQ5 Does the heat semigroup associated with the multidimensional Bessel operator converge a.e. for $L^p$ functions?

Key findings

• The maximal operator $W_*^{\lambda_1,\dots,\lambda_n}$ is of weak type (1,1) with respect to the measure $m_{\lambda_1,\dots,\lambda_n}$, i.e., $\|W_*^{\lambda_1,\dots,\lambda_n}(f)\|_{L^{1,\infty}} \leq C \|f\|_{L^1}$.
• The Littlewood-Paley g-function $g^{\lambda_1,\dots,\lambda_n}$ is bounded from $L^1((0,\infty)^n, \prod x_j^{2\lambda_j} dx)$ into weak $L^1((0,\infty)^n, \prod x_j^{2\lambda_j} dx)$.
• For every $1 < p < \infty$, the Riesz transforms $R_i^{\lambda_1,\dots,\lambda_n}$ are bounded on $L^p((0,\infty)^n, \prod x_j^{2\lambda_j} dx)$.
• The Riesz transforms $R_i^{\lambda_1,\dots,\lambda_n}$ are of weak type (1,1), meaning $\|R_i^{\lambda_1,\dots,\lambda_n}(f)\|_{L^{1,\infty}} \leq C \|f\|_{L^1}$.
• For every $f \in L^p((0,\infty)^n, \prod x_j^{2\lambda_j} dx)$ with $1 \leq p < \infty$, the heat semigroup satisfies $\lim_{t \to 0^+} W_t^{\lambda_1,\dots,\lambda_n}(f)(x) = f(x)$ a.e. $x \in (0,\infty)^n$.
• The Riesz transforms are well-defined a.e. as principal value integrals: $R_i^{\lambda_1,\dots,\lambda_n}(f)(x) = \lim_{\varepsilon \to 0^+} \int_{|x-y| > \varepsilon} R_i^{\lambda_1,\dots,\lambda_n}(x,y) f(y) dy$ a.e. $x \in (0,\infty)^n$.