Regularity of fractional Schrödinger equations and sub-Laplacian multipliers on the Heisenberg group

Abstract: We define functions of the sub-Laplacian $Δ$ on the Heisenberg group $\mathbb H<sup>d$ as Fourier multipliers. In this setting, we show that the solution $u$ of the free fractional Schrödinger equation $$i\partial_tu + (-Δ)^νu = 0, u|<em>{t=0} = u_0$$, for any $ν &gt; 0$, satisfies the Hardy space estimate that $|u(t,\cdot)|</em>{H^p(\mathbb H<sup>d)}$ is estimated from above by $$(1 + t)^{{Q|1/p-1/2|}|(1-Δ)^{{νQ|1/p-1/2|}u_0|_{H^p(\mathbb}} H<sup>d)}$$, with $Q = 2d + 2$, for all $p \in (0,\infty)$, and the corresponding estimate with $p=\infty$ in $\mathrm{BMO}(\mathbb H<sup>d)$. This is done via a general regularity result for parameter dependent sub-Laplacian Fourier multipliers. We prove also that Bessel potential spaces on the Heisenberg group correspond to Sobolev spaces in the same way as in Euclidean space, also for Hardy spaces.

• The paper extends Euclidean regularity results to the Heisenberg group by deriving sharp estimates for fractional Schrödinger equations in H^p and BMO spaces.
• It introduces a refined multiplier theorem for sub-Laplacian operators, ensuring boundedness and invertibility under precise symbol conditions.
• The work establishes the equivalence of Bessel potential and Sobolev spaces, paving the way for advances in noncommutative harmonic analysis and dispersive PDEs.

Regularity Estimates for Fractional Schrödinger Equations and Sub-Laplacian Multipliers on the Heisenberg Group

Introduction and Motivation

This work establishes sharp regularity estimates for the solution operator to the free fractional Schrödinger equations associated with the sub-Laplacian on the Heisenberg group $\mathbb{H}^d$. The paper also characterizes the mapping properties of general sub-Laplacian Fourier multipliers, including Bessel potential operators, acting on Hardy spaces $H^p(\mathbb{H}^d)$ and $BMO(\mathbb{H}^d)$. This is motivated by advances in the Euclidean setting, specifically Miyachi’s work on $L^p$-Sobolev mappings of the free Schrödinger evolution, and seeks to clarify to what extent these results and methodologies extend or require modification in the context of a non-abelian, stratified nilpotent Lie group.

Main Results

Fractional Schrödinger Regularity in $H^p$ Spaces

The key result states that, for any $\nu > 0$ and $p \in (0, \infty]$, the solution $u(t,\cdot)$ to the Cauchy problem

$$i\,\partial_t u + (-\Delta)^\nu u = 0, \qquad u|_{t=0} = u_0$$

satisfies, for homogeneous dimension $Q = 2d+2$, the optimal estimate

$H^p(\mathbb{H}^d)$0

and analogously for $H^p(\mathbb{H}^d)$1 with the $H^p(\mathbb{H}^d)$2 norm. This extends Miyachi’s Euclidean results to the subelliptic, noncommutative Heisenberg group setting, preserving the quantitative relation between the regularity shift and the group’s homogeneous dimension.

Boundedness and Invertibility of Sub-Laplacian Multipliers

The author provides a refined multiplier theorem for operators of the form $H^p(\mathbb{H}^d)$3 under precise symbol-type conditions:

$H^p(\mathbb{H}^d)$4

for all orders $H^p(\mathbb{H}^d)$5, real $H^p(\mathbb{H}^d)$6, and multi-indices $H^p(\mathbb{H}^d)$7. This general framework covers the solution operators for fractional powers of the sub-Laplacian and Bessel potentials. The mapping property holds:

$H^p(\mathbb{H}^d)$8

Bessel Potential/Sobolev Space Equivalence

An important technical achievement is the demonstration that, as in the Euclidean setting, the norm $H^p(\mathbb{H}^d)$9 is equivalent to the sum of all left-invariant derivatives $BMO(\mathbb{H}^d)$0 of degree up to $BMO(\mathbb{H}^d)$1. This extends Folland’s equivalence for Lebesgue spaces to the full scale of Hardy and $BMO(\mathbb{H}^d)$2 spaces.

Methods and Techniques

Contrary to approaches based on heat kernel estimates, the methodology leverages the explicit noncommutative harmonic analysis on the Heisenberg group, exploiting the group’s representation theory and spectral decomposition. The analysis proceeds via:

• Detailed diagonalization of sub-Laplacian multipliers through the group Fourier transform, connecting their action to families of harmonic oscillator Hamiltonians indexed by the spectral parameter.
• Expansion of Schwartz functions and relevant distributions in the frequency side using Hermite functions and associated Laguerre polynomials, allowing for a biradial Plancherel theorem.
• Sharp control of position-space polynomial multipliers via their action as differential operators in the group variable, as detailed in a crucial lemma that translates spatial multipliers into differentiation in the spectral parameter. This is essential for “counting derivatives” in regularity estimates.

The solution operator for the Schrödinger equation, as well as general sub-Laplacian multipliers, are decomposed into high and low frequency-time regimes, with symbol estimates tailored to control each regime precisely.

Numerical Sharpness and Contradictory Claims

• The regularity exponent and time growth factor in the main inequalities are shown to be sharp in analogy with the Euclidean setting. For instance, the loss of $BMO(\mathbb{H}^d)$3 derivatives cannot, in general, be improved—a fact emphasized by connections to known counterexamples for the wave equation with $BMO(\mathbb{H}^d)$4.
• The paper demonstrates that, for the full range $BMO(\mathbb{H}^d)$5, the fractional Schrödinger evolution on $BMO(\mathbb{H}^d)$6 enjoys boundedness in Hardy spaces and in $BMO(\mathbb{H}^d)$7—significantly broadening previous results which were restricted to $BMO(\mathbb{H}^d)$8 or even smaller ranges for certain $BMO(\mathbb{H}^d)$9.

Practical and Theoretical Implications

The analysis confirms that the mechanism behind the gain or loss of regularity in non-Euclidean settings, such as the Heisenberg group, is strongly governed by the homogeneous dimension and spectral structure. The result has the following implications:

• Sharpness in Sobolev and function space scales: The results provide critical endpoints and off-diagonal mapping properties for dispersive equations and harmonic analysis on stratified nilpotent Lie groups, relevant for PDEs in control theory, CR-geometry, and subelliptic diffusion.
• Robustness of sub-Laplacian functional calculus: The equivalence of Bessel potential and vector field Sobolev spaces for Hardy and $L^p$0 spaces offers a powerful tool for further development of singular integral theory, Littlewood-Paley analysis, and multiplier theorems in noncommutative contexts.

Future Directions

This foundational work opens paths for several lines of investigation:

• Extension to more general homogeneous groups and hypoelliptic operators: The methods appear tractable for adapting to multi-step Carnot groups and more degenerate operators, albeit with technical modifications in spectral analysis.
• Fine endpoint regularity and dispersive estimates: The explicit kernel and multiplier analysis may enable sharp estimates for endpoint spaces and inhomogeneous evolution equations, including Strichartz inequalities and spectral multipliers beyond the sub-Laplacian.
• Characterization of optimality in the wave and half-wave regime: As pointed out in the discussion, the regularity threshold for the half-wave equation on $L^p$1 is not always optimal and merits further study, potentially via microlocal and oscillatory integral methods.

Conclusion

This work provides a precise and comprehensive extension of classical regularity theory for the free Schrödinger equation to the setting of the Heisenberg group, establishing sharp mapping estimates for solution operators and sub-Laplacian Fourier multipliers in $L^p$2 and $L^p$3 spaces for all $L^p$4. The approach, rooted in noncommutative harmonic analysis, yields a functional calculus and Sobolev embeddings that parallel and extend the best known Euclidean results. The implications are significant both for the theoretical development of analysis on Lie groups and the practical study of dispersive PDEs in subelliptic geometries.

Reference:

"Regularity of fractional Schrödinger equations and sub-Laplacian Fourier multipliers on the Heisenberg group" (2603.29588)