Log-Oscillatory Pseudodifferential Operators with Prabhakar–Fox–Wright Symbols A Non-Classical Symbol Model within the 𝑆𝑚 1,0 Calculus: Boundedness, Parametrix, Well-Posedness, and Numerical Verification

<title>Abstract</title> <p>Let ⟨ξ⟩ = (1 + |ξ|2)1/2 and consider the symbol aΘ(z, ξ) = ⟨ξ⟩m exp iτ sin 􀀀 ω log⟨ξ⟩+ϕ(z) Eγ α,β 􀀀 −λ(z)⟨ξ⟩−ρ 2Ψ2 (a1,A1), (a2,A2) (b1,B1), (b2,B2) ; −μ(z)⟨ξ⟩−κ , combining a logarithmically oscillating phase — of the type arising in weakly hyperbolic Cauchy problems with non-Lipschitz coefficients — with a decay envelope built from the Prabhakar (threeparameter Mittag-Leffler) function Eγ α,β and the Fox–Wright function 2Ψ2. We give a self-contained proof of the symbol-level classification aΘ ∈ Sm 1,0 (non-classical, elliptic of order m) and work out the operator theory of the associated Kohn–Nirenberg operator Op(aΘ) within the classical Sm 1,0 framework. We prove: (i) Op(aΘ) maps S → S and extends to a bounded operator Hs → Hs−m for every s ∈ R; (ii) the composition and formal-adjoint calculus holds, with principal symbols aΘb and aΘ respectively; (iii) ellipticity yields a two-sided parametrix B ∈ Op(S−m) with principal symbol 1/aΘ, giving elliptic regularity; (iv) both the operator and its parametrix are non-classical, inheriting the persistent log-oscillation; and (v) — the main new result of this version — a well-posedness theorem for the natural hyperbolic realization ∂2 t v + Op(Re aΘ)v = 0: no loss of derivatives for Lipschitz coefficients, and for log-Lipschitz coefficients a finite loss whose exponent δ0 is driven only by the phase data τ, |ϕ|LL — provably independent of the Prabhakar/Fox–Wright envelope, which contributes a loss-free bounded drift — with all constants certified numerically at 50 significant digits. Items (i)–(iii) are specializations of the classical Sm 1,0 calculus and are claimed as new only at the level of their verification for this non-classical symbol and the explicit Prabhakar/Fox–Wright form of the leading corrections; the specifically new mathematics is the non-classicality analysis, the exact two-term and reciprocal-coefficient structure, and the envelope-resolved well-posedness theorem. Every analytical claim is cross-checked by direct numerical computation, with fitted exponents matching theory to within 1–4%. A dated literature search (2 July 2026), a structured comparison with the neighbouring frameworks, and the explicitly stated residual limitations of the search are documented in Section 10.</p>