Continuity properties of the Laguerre operator and its propagator

Technical Summary: Continuity properties of the Laguerre operator and its propagator

Problem Statement
This work investigates the well-posedness of the Cauchy problem associated with the general form of the Laguerre operator, $E_{\rho,c} = \rho E + c$, acting on the positive orthant $\mathbb{R}^d_+$. The study focuses on the continuity properties of the associated propagator $e^{zE}$ (and its generalizations) within the framework of global Pilipović spaces and their duals (spaces of ultradistributions). The authors aim to determine the conditions under which these operators define topological isomorphisms or continuous injections, specifically contrasting the behavior on "G-type" spaces (classical Gelfand-Shilov type) versus "P-type" spaces (defined via iterates of the Laguerre operator). A secondary objective is to establish explicit connections between the Laguerre propagator and various fractional integral transforms, including the fractional Fourier transform and the fractional Hankel-Clifford transform.

Methodology
The analysis relies on spectral methods utilizing the orthonormal basis of Laguerre functions $\{l_n\}_{n \in \mathbb{N}_0^d}$ on $L^2(\mathbb{R}^d_+)$. The core methodology involves:

1. Sequence Space Characterization: The authors characterize the test function spaces $G_\alpha(\mathbb{R}^d_+)$ and their duals via the decay/growth rates of the Laguerre coefficients $\{a_n\}$. This allows the operator continuity problems to be reduced to continuity problems on specific sequence spaces $\ell_{\alpha}(\mathbb{N}_0^d)$ and $\ell'_{\alpha}(\mathbb{N}_0^d)$.
2. Kernel Representations: For cases where $\text{Re}(z) < 0$, the propagator is represented as an integral operator with a kernel derived from the Hille-Hardy formula involving modified Bessel functions. This representation is extended to the case $\text{Re}(z) = 0$ (unitary case) using Abel's theorem and limit arguments.
3. Radial Symmetry and Change of Variables: A crucial technique involves mapping problems on the positive orthant $\mathbb{R}^d_+$ to problems on the full space $\mathbb{R}^d$ via the change of variables $x_j = r_j^2$. This establishes a topological isomorphism between P-spaces on $\mathbb{R}^d_+$ and radially symmetric subspaces of global Pilipović spaces (specifically those related to the harmonic oscillator).
4. Duality and Transpose Operators: The study extends results from test functions to ultradistributions by utilizing the transpose and inverse of the isomorphisms established between radial and even function spaces.

Key Contributions and Results

• Continuity of the Propagator: The paper establishes precise conditions for the continuity of the mapping $e^{zE}$ on P-spaces $G_\alpha(\mathbb{R}^d_+)$ and $G'_{\alpha}(\mathbb{R}^d_+)$:

  • If $0 < \alpha < 1$, the propagator is a topological isomorphism for all $z \in \mathbb{C} \setminus \{0\}$.
  • If $\text{Re}(z) = 0$, the propagator is a topological isomorphism for all $\alpha > 0$.
  • If $\alpha > 1$ and $\text{Re}(z) < 0$, the mapping is a continuous injection but not surjective.
  • If $\alpha > 1$ and $\text{Re}(z) > 0$, the mapping is not continuous (and in fact, not well-defined) on these spaces.
  • These results are generalized to the operator $E^r_{\rho,c}$, showing that well-posedness depends on the relationship between the parameter $\alpha$ and the exponent $r$ (specifically, well-posedness holds if $\alpha < 1/r$).

• Connection to Integral Transforms: The authors prove that the Laguerre propagator $e^{itE}$ is essentially the fractional power of the Hankel-Clifford transform, denoted $I_{e^{it}, 0}$. Specifically, for $f \in \mathcal{S}(\mathbb{R}^d_+)$, $e^{itE}f = I_{e^{it}, 0}f$. This connection allows the transfer of continuity properties from the Laguerre operator to the fractional Hankel-Clifford transform.

• Relation to the Harmonic Oscillator: By restricting global Pilipović spaces to radially symmetric elements, the paper demonstrates that the Cauchy problem for the Laguerre operator on $\mathbb{R}^d_+$ is equivalent to the Cauchy problem for a specific Schrödinger-type equation involving the harmonic oscillator on $\mathbb{R}^d$ (restricted to radial/even subspaces). This confirms that the findings regarding well-posedness in P-spaces on the orthant align with the results for the harmonic oscillator found in [19].

• Ultradistributional Interpretation: The paper provides a rigorous interpretation of the change of variables and operator actions when initial data are ultradistributions, utilizing the duality of the isomorphisms between radial and even spaces.

Significance
The paper highlights the critical role of Pilipović spaces on positive orthants in the study of Laguerre-type operators. It clarifies that while classical Gelfand-Shilov spaces (G-type) may fail to provide well-posedness for certain time-dependent equations involving these operators (particularly when $\text{Re}(z) > 0$ and $\alpha$ is large), the broader class of P-spaces offers a suitable framework for well-posedness under specific parameter constraints.

Furthermore, the work unifies the theory of the Laguerre operator with fractional integral transforms. By identifying the propagator as a fractional power of the Hankel-Clifford transform, the authors provide a new perspective on the continuity of these transforms and their extensions to ultradistribution spaces. The results serve to bridge the gap between the analysis of operators on the full space $\mathbb{R}^d$ (harmonic oscillator) and those on the positive orthant $\mathbb{R}^d_+$ (Laguerre operator), showing that the radial symmetry of solutions in the former corresponds directly to the domain of the latter.