Equivalence of Sobolev norms in Lebesgue spaces for Hardy operators in a half-space

1. Problem Statement

The paper investigates the equivalence of homogeneous Sobolev norms generated by Hardy operators in the half-space $\mathbb{R}^d_+ = \mathbb{R}^{d-1} \times (0, \infty)$.

The operator of interest is the generalized Schrödinger operator:
$$L_\lambda^{(\alpha)} := (-\Delta)_{\mathbb{R}^d_+}^{\alpha/2} + \lambda x_d^{-\alpha}$$
where:

• $(-\Delta)_{\mathbb{R}^d_+}^{\alpha/2}$ is the regional fractional Laplacian (reducing to the Dirichlet Laplacian when $\alpha=2$).
• $\lambda x_d^{-\alpha}$ is a singular potential depending on the distance to the boundary, with coupling constant $\lambda$.
• $\alpha \in (0, 2]$.

The central question is whether the Sobolev norms defined by the operator with the potential, $\|(L_\lambda)^{s/2} u\|_{L^p(\mathbb{R}^d_+)}$, are equivalent to the norms defined by the operator without the potential, $\|(L_0)^{s/2} u\|_{L^p(\mathbb{R}^d_+)}$, for $p \in (1, \infty)$ and $s > 0$.

Previous work (specifically Frank and Merz, 2023) established this equivalence for $p=2$. This paper extends the result to the full range $p \in (1, \infty)$, addressing the significant analytical challenges that arise when $p \neq 2$, particularly regarding the definition of fractional powers of the operator in $L^p$ spaces and the behavior of heat kernels with singular weights.

2. Methodology

The authors employ a combination of heat kernel analysis, square function estimates, and singular integral theory. The proof strategy is structured around three main pillars:

A. Heat Kernel Estimates

The authors rely on precise pointwise bounds for the heat kernel $p_t^\lambda(x, y)$ of $e^{-tL_\lambda}$.

• They utilize known bounds for $\lambda \geq 0$ (and $\lambda \geq -1/4$ for $\alpha=2$) which exhibit decay controlled by a parameter $\sigma$ related to $\lambda$.
• They extend these bounds to complex time $z \in \mathbb{C}$ using a Phragmén–Lindelöf argument, which is crucial for estimating derivatives of the heat semigroup $(tL_\lambda)^k e^{-tL_\lambda}$.
• They derive spatially averaged estimates (off-diagonal decay) for operators involving the heat kernel, which are necessary for applying singular integral theorems.

B. Continuous Square Function Estimates

A major technical hurdle is defining $(L_\lambda)^{s/2}f$ in $L^p$ for $p \neq 2$. The authors bypass the need for a spectral multiplier theorem (which fails due to singular weights when $\lambda < 0$) by using continuous square functions based on the heat semigroup:
$$(S_{L_\lambda, \gamma} f)(x) := \left( \int_0^\infty |(tL_\lambda)^\gamma e^{-tL_\lambda} f|^2 \frac{dt}{t} \right)^{1/2}$$
They prove that $\|S_{L_\lambda, \gamma} f\|_{L^p} \simeq \|f\|_{L^p}$ for all $1 < p < \infty$. This equivalence allows them to characterize the domain of $(L_\lambda)^{s/2}$ and relate it to the standard fractional Laplacian.

C. Reversed and Generalized Hardy Inequalities

To compare the norms of $L_\lambda$ and $L_0$, the authors establish two critical inequalities:

1. Reversed Hardy Inequality: This provides a lower bound on the difference between the square functions of $L_\lambda$ and $L_0$ in terms of the Hardy potential $x_d^{-\alpha s/2}$. It relies on bounds for the difference of kernels $tL_0 e^{-tL_0} - tL_\lambda e^{-tL_\lambda}$.
2. Generalized Hardy Inequality: This provides an upper bound on the Hardy potential in terms of the operator $L_\lambda$. It involves proving the boundedness of an integral operator with a specific kernel derived from the Riesz potential of $L_\lambda$.

3. Key Contributions

• Extension to $L^p$ Spaces: The primary contribution is extending the Sobolev norm equivalence from $L^2$ to all $L^p$ spaces ($1 < p < \infty$). This is non-trivial because standard spectral multiplier techniques (Mikhlin-Hörmander) are not directly applicable due to the singular nature of the potential and the lack of $L^p$-off-diagonal estimates for negative coupling constants in the fractional case.
• New Square Function Estimates: The paper proves new square function estimates for operators with heat kernels that decay slowly and include singular weights. These estimates are robust enough to handle the specific geometry of the half-space and the inverse-power potential.
• Handling Attractive Potentials ($\lambda < 0$): While the main results are proven for repulsive potentials ($\lambda \geq 0$) in the fractional case ($\alpha < 2$), the authors provide a conditional extension to attractive potentials ($\lambda \in [\lambda_*, 0)$). They show that if the known heat kernel upper bounds for repulsive cases extend to this regime (a conjecture supported by recent work in 1D), then the Sobolev equivalence holds for these attractive potentials as well, provided $p$ is restricted to a specific range.
• Density Results: The authors prove the density of $(L_\lambda)^{s/2} C_c^\infty(\mathbb{R}^d_+)$ in $L^p(\mathbb{R}^d_+)$, which is essential for defining the operator rigorously in $L^p$ and applying the results to a dense set of functions.

4. Main Results

Theorem 1.1 (Equivalence of Sobolev Norms):
Let $p \in (1, \infty)$, $\alpha \in (0, 2]$, and $\lambda$ be in the admissible range ($\lambda \geq 0$ for $\alpha < 2$, $\lambda \geq -1/4$ for $\alpha = 2$). Let $\sigma$ be the parameter defined by the coupling constant $\lambda$ via the relation $C(\sigma) = \lambda$.
For $s \in (0, 2] \cap (0, 2d/\alpha)$:

• If $1/p > \frac{\alpha s}{2} - \sigma$, then $\|(L_0)^{s/2} u\|_p \lesssim \|(L_\lambda)^{s/2} u\|_p$.
• If $1/p > \frac{\alpha s}{2} - (\alpha-1)_+$, then $\|(L_\lambda)^{s/2} u\|_p \lesssim \|(L_0)^{s/2} u\|_p$.
• Conclusion: If $1/p > \max\left\{ \frac{\alpha s}{2} - (\alpha-1)_+, \frac{\alpha s}{2} - \sigma \right\}$, the norms are equivalent:
  $$\|(L_\lambda)^{s/2} u\|_{L^p} \simeq \|(L_0)^{s/2} u\|_{L^p}$$

Corollary 1.2 (Riesz Transforms):
The result implies the $L^p$-boundedness of the Riesz transforms associated with the Hardy operator, specifically the operator $(L_0)^{s/2}(L_\lambda)^{-s/2}$.

5. Significance and Applications

• Mathematical Physics: The results are crucial for the study of many-particle quantum systems with critically strong interactions. Specifically, they facilitate the analysis of relativistic atoms and the Scott conjecture (which describes the asymptotic behavior of the ground state energy of heavy atoms). The equivalence allows researchers to replace the complex Hardy operator with the more tractable fractional Laplacian in estimates.
• Nonlinear PDEs: The equivalence of norms is a powerful tool for studying nonlinear Schrödinger equations (NLS) involving Hardy potentials. It allows for the transfer of well-posedness and scattering results from the standard Laplacian case to the Hardy case.
• Theoretical Advancement: The paper resolves open questions regarding the domain of fractional powers of Hardy operators in $L^p$ spaces. It demonstrates that despite the singularity of the potential, the "scaling" properties of the operator preserve the Sobolev structure, provided the parameters $p, s, \lambda$ satisfy specific constraints.
• Methodological Innovation: The use of continuous square functions and the specific handling of heat kernel differences for singular weights provide a template for analyzing other singular Schrödinger operators where standard spectral theory fails.

In summary, this paper successfully generalizes the equivalence of Sobolev norms for Hardy operators from the Hilbert space setting ($L^2$) to the Banach space setting ($L^p$), overcoming significant analytical barriers related to singular potentials and providing a robust framework for future applications in mathematical physics and PDE theory.