[Paper Review] Sharp estimates for potential operators associated with Laguerre and Dunkl-Laguerre expansions

Adam Nowak, Krzysztof Stempak|arXiv (Cornell University)|Feb 11, 2014

Advanced Harmonic Analysis Research7 references4 citations

TL;DR

This paper establishes sharp $L^p$-$L^q$ boundedness estimates for potential operators associated with one-dimensional Laguerre and Dunkl-Laguerre expansions, proving qualitatively sharp pointwise estimates for their potential kernels. The key contribution is a complete characterization of admissible $p$ and $q$ for boundedness, extending the classical Hardy-Littlewood-Sobolev theorem to these settings without parameter restrictions on the Laguerre parameter $\alpha$. The results are optimal in the sense of sharp Lorentz space estimates and interpolation theory.

ABSTRACT

We study potential operators associated with Laguerre function expansions of convolution and Hermite types, and with Dunkl-Laguerre expansions. We prove qualitatively sharp estimates of the corresponding potential kernels. Then we characterize those $1 \le p,q \le \infty$, for which the potential operators are $L^p-L^q$ bounded. These results are sharp analogues of the classical Hardy-Littlewood-Sobolev fractional integration theorem in the Laguerre and Dunkl-Laguerre settings.

Motivation & Objective

• To characterize the full range of $L^p$-$L^q$ boundedness for potential operators in Laguerre and Dunkl-Laguerre expansions, extending classical results to these settings.
• To prove qualitatively sharp pointwise estimates for the potential kernels associated with these operators, removing previous restrictions on the Laguerre parameter $\alpha$.
• To establish optimality of earlier $L^p$-$L^q$ bounds from [9], showing they are sharp in the one-dimensional case.
• To provide a foundation for further study of weak-type estimates, two-weight inequalities, and related fractional integrals in these contexts.

Proposed method

• Derivation of sharp pointwise estimates for the potential kernels $K^{\alpha,\sigma}(x,y)$ in the Laguerre setting using asymptotic analysis and comparison with known special functions.
• Application of the Marcinkiewicz interpolation theorem to extend $L^p$-$L^q$ boundedness from strong-type estimates to endpoint cases, particularly when $\frac{1}{q} = \frac{1}{p} - 2\sigma$.
• Use of duality arguments to transfer boundedness results from $U_1^{\alpha,\sigma}$ to $U_2^{\alpha,\sigma}$, especially for the case $q=1$.
• Extension of kernel estimates to the Dunkl-Laguerre setting via decomposition into symmetric and antisymmetric parts, using $\mathcal{K}^{\alpha,\sigma}_{\pm}(x,y)$ and symmetry properties of the Dunkl kernel.
• Reduction of the $L^p$-$L^q$ boundedness problem on $\mathbb{R}$ to the positive half-line using the decomposition $f = f_+ + f_-$ and measure equivalence $\|f\|_{L^p(dw_\alpha)} \simeq \|f_+\|_{L^p(d\mu_\alpha)} + \|f_- one\|_{L^p(d\mu_\alpha)}$.
• Proof of necessity of the boundedness range by showing that if $I_D^{\alpha,\sigma}$ is $L^p$-$L^q$ bounded, then so is $\mathcal{I}_+^{\alpha,\sigma}$, leveraging extension of functions to $\mathbb{R}$ and norm comparison.

Experimental results

Research questions

• RQ1 What is the complete range of $p$ and $q$ for which the potential operator $U_1^{\alpha,\sigma}$ is bounded from $L^p$ to $L^q$ in the Laguerre setting of convolution type?
• RQ2 How do the sharp pointwise estimates of the potential kernel $K^{\alpha,\sigma}(x,y)$ influence the $L^p$-$L^q$ mapping properties of the associated potential operator?
• RQ3 To what extent are the $L^p$-$L^q$ bounds for potential operators in the Dunkl-Laguerre setting sharp, and how do they relate to the classical Hardy-Littlewood-Sobolev theorem?
• RQ4 Can the $L^p$-$L^q$ boundedness results be extended to the full range of $\alpha \in (-1/2, \infty)$ without restrictions, and is this range optimal?

Key findings

• The potential kernel $K^{\alpha,\sigma}(x,y)$ admits qualitatively sharp pointwise estimates for all $\alpha > -1/2$, with explicit decay and growth behavior depending on $\sigma$ and $x,y$.
• For $\sigma < 1/2$, the kernel behaves like $x^{2\sigma-1}$ near zero, and for $\sigma > 1/2$, like $(1+x)^{1-2\sigma}$, with logarithmic corrections at $\sigma = 1/2$.
• The $L^p$-$L^q$ boundedness of $U_1^{\alpha,\sigma}$ holds if and only if $\frac{1}{q} > \frac{1}{p} - 2\sigma$ for $\sigma < 1/2$, and $\frac{1}{q} \geq \frac{1}{p} - 2\sigma$ for $\sigma \geq 1/2$, with weak-type boundedness at the endpoint.
• The $L^p$-$L^q$ boundedness of the Dunkl-Laguerre potential operator $I_D^{\alpha,\sigma}$ holds precisely when the same conditions on $p$ and $q$ hold as in the standard Laguerre case, due to kernel comparability and symmetry.
• The unweighted $L^p$-$L^q$ bounds from [9] are shown to be sharp in the one-dimensional case, confirming the optimality of earlier results.
• The necessity of the boundedness range is proven by showing that if $I_D^{\alpha,\sigma}$ is $L^p$-$L^q$ bounded, then so is $\mathcal{I}_+^{\alpha,\sigma}$, which inherits the sharp range from Theorem 2.2.