The Weighted $\boldsymbol{L}^{\boldsymbol{p}}$ estimates for the fractional Hardy operator and a class of integral operators on the Heisenberg group

Abstract: In the setting of a Heisenberg group, we first studied the sharp weak estimate for the $n$-dimensional fractional Hardy operator from $L<sup>p$ to $L<sup>{q,\infty}$. Next, we studied the sharp bounds for the $m$-linear $n$-dimensional integral operator with a kernel on weighted Lebesgue spaces. As an application, the sharp bounds for Hardy, Hardy-Littlewood-Pólya, and Hilbert operators on weighted Lebesgue spaces were obtained. Finally, according to the previous steps, we also found the estimate for the Hausdorff operator on weighted $L<sup>p$ spaces.

• The paper introduces sharp weighted Lp and weak Lq norm estimates for the fractional Hardy operator on the Heisenberg group, proving best constants using scaling invariance and duality.
• It extends the framework to multilinear integral operators with radial kernels, including classical Hardy, Hilbert, and Hausdorff operators, by adapting Euclidean techniques to noncommutative geometry.
• The results provide robust analytical tools for advanced harmonic analysis on the Heisenberg group, impacting studies in sub-Riemannian geometry and partial differential equations.

Weighted $L^p$ Estimates for Fractional Hardy and Related Integral Operators on the Heisenberg Group

Introduction and Context

This paper develops the theory of sharp weighted $L^p$ and weak $L^{q, \infty}$ estimates for the fractional Hardy operator, multilinear integral operators with radial kernels, and notable operator subclasses (Hardy, Hardy-Littlewood-Pólya, Hilbert, and Hausdorff operators) on the Heisenberg group $\mathbb{H}^n$. The results fill a critical gap in noncommutative harmonic analysis, transferring and extending endpoint and sharp constant theory from Euclidean and commutative settings to the highly nontrivial geometry of the Heisenberg group.

Harmonic analysis on $\mathbb{H}^n$ is essential for understanding sub-Riemannian geometry, PDEs, and several areas of mathematical analysis, due to the group's stratified, non-abelian, and nilpotent structure. The calculation of operator norms and identification of best possible (sharp) constants on weighted Lebesgue spaces underpins various embedding theorems and advances in analysis on graded Lie groups.

Main Results

Sharp Weak Estimates for the Fractional Hardy Operator

Let $Q = 2n + 2$ denote the homogeneous dimension of the Heisenberg group. The $n$-dimensional fractional Hardy operator on $\mathbb{H}^n$ is defined by

$$\mathcal{H}_\alpha f(x) = \frac{1}{|x|_h^{Q - \alpha}} \int_{|y|_h < |x|_h} f(y) \, dy, \quad 0 < \alpha < Q.$$

The central achievement is the derivation of the sharp* weak* type $(p, q)$ norm for $L^p$0: $L^p$1 for $L^p$2, $L^p$3, $L^p$4, $L^p$5, $L^p$6, and $L^p$7. The sharp constant is attained on characteristic functions of balls (or power weights), and the proof synthesizes Hölder’s inequality, dilation invariance, and careful testing against optimal functions.

For $L^p$8, the optimal weak endpoint estimate is

$L^p$9

Optimal Weighted $L^{q, \infty}$0 Bounds for Multilinear Integral Operators

For a general $L^{q, \infty}$1-linear, $L^{q, \infty}$2-dimensional operator with a measurable radial kernel $L^{q, \infty}$3,

$L^{q, \infty}$4

the operator norm on the product of weighted $L^{q, \infty}$5 spaces to $L^{q, \infty}$6 with weight $L^{q, \infty}$7 and exponents $L^{q, \infty}$8 is given by

$L^{q, \infty}$9

The proof employs symmetrization, optimal testing via scaling and radial reductions, and duality via explicit function constructions.

Explicit Formulas for Classical Operators

The Hardy, Hardy-Littlewood-Pólya, and Hilbert operators emerge as special cases of the above setting:

• $\mathbb{H}^n$0-Linear Hardy Operator: Kernel $\mathbb{H}^n$1. The sharp norm is expressed using gamma functions and is nontrivial in $\mathbb{H}^n$2.
• $\mathbb{H}^n$3-Linear Hardy-Littlewood-Pólya: Kernel is the reciprocal of the maximum power, and the norm involves explicit rational expressions in $\mathbb{H}^n$4.
• $\mathbb{H}^n$5-Linear Hilbert: Kernel is the reciprocal of a sum of the form $\mathbb{H}^n$6, leading to a formula in terms of gamma functions.

Sharp Bound for the Hausdorff Operator

The most general result covers the weighted $\mathbb{H}^n$7-boundedness for the $\mathbb{H}^n$8-linear $\mathbb{H}^n$9-dimensional Hausdorff operator $\mathbb{H}^n$0 under integral constraints on the Hausdorff symbol $\mathbb{H}^n$1, providing the exact formula for the operator norm via an explicit radial and angular integral: $\mathbb{H}^n$2

Methodology and Technical Innovations

The proofs combine homogeneity, reduction to radial functions, sharp testing via scaling invariant families, and detailed integration in polar coordinates on $\mathbb{H}^n$3, employing the structure of noncommutative dilations, the explicit form of the homogeneous norm $\mathbb{H}^n$4, and the interplay with weights.

A crucial aspect is the systematic translation of Euclidean sharp estimate techniques—based on best constants, precise calculation, and duality arguments—to the Heisenberg group context, which is notably more intricate due to its noncommutative geometry.

Implications and Future Work

The results have significant consequences for functional analysis on stratified Lie groups. The precise operator norms and endpoint bounds established here pave the way for:

• Refined embeddings and interpolation results for function spaces on $\mathbb{H}^n$5.
• Development of real-variable harmonic analysis, particularly for weighted norm inequalities.
• Applications to sharp estimates in PDEs posed on the Heisenberg group, including subelliptic operators and fractional Sobolev-type inequalities.

Future work may focus on removing technical restrictions on weight exponents, further generalizing to other homogeneous groups, or extending to strongly singular and commutator operators, where the geometric aspect of $\mathbb{H}^n$6 plays an even more dominant role.

Conclusion

This paper establishes sharp weighted $\mathbb{H}^n$7 and weak-$\mathbb{H}^n$8 estimates for the fractional Hardy operator and a broad class of integral operators on the Heisenberg group, including explicit treatment of classical subcases and the Hausdorff operator. The combination of explicit norm calculations and reduction to radial symmetry provides a robust analytical toolkit for further study of sharp inequalities in noncommutative harmonic analysis.