Well-Posedness for Cauchy Problems with Singular Time-Measurable Pseudo-Differential Operators in Quasi-decreasing Weighted $\mathrm{L}_2$-Spaces

Technical Summary: Well-Posedness for Cauchy Problems with Singular Time-Measurable Pseudo-Differential Operators in Quasi-Decreasing Weighted $L^2$-Spaces

Problem Statement
This study addresses the well-posedness of Cauchy problems governed by highly singular, time-measurable pseudo-differential operators (specifically, measurable families of Fourier multipliers). The governing equation is:
$$\begin{cases} \partial_t u(t, x) = \psi(t, -i\nabla)u(t, x) + f(t, x), & (t, x) \in (0, T) \times \mathbb{R}^d, \\ u(0, x) = 0, & x \in \mathbb{R}^d, \end{cases}$$
where $\psi(t, \xi)$ is a complex-valued symbol that is merely Lebesgue-measurable in time and frequency. The central challenge is that these symbols are permitted to exhibit arbitrary blow-up behavior in both time (near $t=0$) and frequency (as $|\xi| \to \infty$). Specifically, the symbols may grow super-exponentially in time and frequency, and they are not required to be locally integrable or continuous.

Standard frameworks for such problems typically rely on operator continuity, boundedness, or specific regularity conditions (e.g., Mikhlin-type conditions) to establish existence and uniqueness in $L^p$ spaces. However, the extreme singularities in this setting destroy the uniform sectoriality and domain stability required by classical evolution family frameworks (such as Kato's theory) and render standard variational or semigroup methods inapplicable. Furthermore, the lack of local integrability prevents the use of standard weak formulations based on test functions.

Methodology
The authors develop a novel theoretical structure that departs from conventional continuity requirements by working within quasi-decreasing weighted $L^2$-spaces. The methodology is characterized by the following elements:

1. Quasi-Decreasing Weights: The analysis employs a weight function $w(t)$ that is strictly positive and quasi-decreasing (i.e., $w(t)/w(s) \le N_w$ for $0 < s < t$). This class generalizes classical non-increasing functions and Muckenhoupt $A_2$ weights, allowing for arbitrary blow-up near the initial time $t=0$ (e.g., $w(t) = \exp(1/t^\alpha)$) while maintaining sufficient integrability properties.
2. Spectral-Limit Solutions: Recognizing that standard weak solutions (defined via pairing with test functions) may not exist due to the lack of local integrability of the symbol, the authors introduce the concept of a spectral-limit solution. This solution is defined as the limit of a sequence of spatial Fourier weak solutions associated with truncated symbols $\psi_n$, where the real and imaginary parts of the symbol are bounded by $n$.
3. Elementary Analytic Tools: Despite the complexity of the singular operators, the proofs rely strictly on elementary methods. The authors avoid advanced functional analytic machinery, instead utilizing Plancherel's theorem, Fubini's theorem, the generalized Minkowski inequality, and the Cauchy–Bunyakovsky–Schwarz inequality.
4. Three Solution Concepts: The paper rigorously defines and relates three notions of solutions:
   • Strong Solutions: Defined via an integral equation in the time domain.
   • Spatial Fourier Weak Solutions: Defined via an integral identity in the frequency domain for test functions with compactly supported Fourier transforms.
   • Spectral-Limit Solutions: Defined as limits of truncated approximations.

Key Contributions and Results

• Existence and Uniqueness for Arbitrary Blow-up: The primary result (Theorem 3.6) establishes the existence and uniqueness of a spectral-limit solution in the weighted space $L^2_{loc}((0, T) \times \mathbb{R}^d, w(t)t^{-2}dtdx)$ for any symbol $\psi$ satisfying a degenerate ellipticity condition ($\Re[\psi] \le 0$) and local time integrability on compact subintervals away from $t=0$. Crucially, no upper bound or regularity is imposed on the symbol's growth in time or frequency.
• Spatial Fourier Weak Solutions under Local Square-Integrability: Under the additional assumption that the symbol satisfies a local square-integrability condition in frequency (Assumption 3.8), the authors prove the existence of a unique spatial Fourier weak solution (Theorem 3.9). This solution satisfies maximal regularity estimates and belongs to the weighted space $L^2_{loc}((0, T) \times \mathbb{R}^d, w(t)t^{-2}dtdx)$.
• Strong Solutions under Spectral Compatibility: To establish the existence of a unique strong solution (Theorem 3.12), the framework requires the stronger condition of Assumption 3.11 (compatibility with a spectral function $\phi$) in addition to Assumption 3.8. Specifically, the real part of the symbol must be bounded by a non-negative spectral function ($\Re[\psi] \le -\phi$), and the ratio $|\psi|/\phi$ must satisfy a weighted integrability condition. Under these conditions, a unique strong solution exists in the intersection of weighted $L^2$ spaces and the spectral potential space $H^\phi_{2,loc}$.
• Maximal Regularity Estimates: The paper derives sharp a priori estimates (Theorem 3.12) that control the solution's norm and the norm of the operator applied to the solution. These estimates depend on the quasi-decreasing constant $N_w$ and are independent of the spatial dimension $d$.
• Handling Non-Integrable Symbols: The framework successfully solves evolutionary equations driven by fractional Laplacians of any negative order and symbols with super-exponential growth (e.g., $\psi(t, \xi) = -t^\alpha e^{t^\beta} |\xi|^\gamma e^{|\xi|^\delta}$), provided the initial data is zero.
• Limitations on Initial Data: The authors explicitly note that the restriction to homogeneous initial data ($u(0, x) = 0$) is a structural necessity. If the symbol is not integrable near $t=0$, the propagator $\exp(\int_0^t \psi(r, \xi) dr)$ becomes ill-defined, rendering non-zero initial data ill-posed without further integrability constraints.

Significance and Claims
The paper claims to provide a "novel concept of solvability" that fundamentally departs from conventional frameworks by bypassing the traditional continuity bottlenecks of pseudo-differential operators. Its significance lies in:

1. Extending Solvability Regimes: It demonstrates that existence and uniqueness can be secured even in the complete absence of operator continuity and local integrability of the symbol, a scenario where classical $L^p$ theories (relying on Mikhlin-type theorems) fail.
2. Connecting Probability and Analysis: By addressing generators of stochastic processes (Feller and Lévy processes) that may have highly irregular symbols, the work bridges the gap between the rigorous machinery of harmonic analysis and the probabilistic study of jump processes with singular characteristics.
3. Generality of the Framework: The methodology is not bound by specific blow-up rates; it accommodates arbitrary levels of irregularity and blow-up in both time and frequency, as illustrated by the "toy models" involving fractional Laplacians of negative order.

The authors emphasize that while the mathematical tools employed are elementary, the results are powerful, offering a robust theory for evolution equations driven by operators that were previously considered analytically intractable due to their singular nature.