Miyachi–Peral Fixed-Time Estimates

• Miyachi–Peral estimates are quantitative bounds that specify the minimal regularity loss needed for the wave propagator on L^p spaces across varied geometric contexts.
• They leverage Fourier analysis, complex interpolation, and microlocal techniques to rigorously establish sharp spectral multiplier theorems.
• These estimates underpin applications in well-posedness, endpoint Strichartz analysis, and extend wave multiplier theory to nonelliptic and rough-coefficient operators.

Miyachi-Peral-type fixed-time estimates refer to quantitative regularity bounds for the wave propagator (such as $\cos(t\sqrt{L})$) acting on $L^p$ spaces, quantifying the precise loss of regularity ("smoothing" exponent) necessary for the operator to remain bounded at fixed time. The archetypal context is the study of the wave equation on $\mathbb{R}^d$ or on more general manifolds and differential operators, especially those lacking uniform ellipticity, such as sub-Laplacians on sub-Riemannian spaces or operators with nonsmooth coefficients. These estimates are tightly connected to sharp spectral multiplier theorems, microlocal analysis, and the geometry of the underlying space.

1. Classical Miyachi-Peral Theorem and Sharp Smoothing Indices

The original results of Peral and Miyachi established sharp conditions for the $L^p$ boundedness, with loss of derivatives, of the fixed-time wave propagator on $\mathbb{R}^d$. For $A=\sqrt{-\Delta}$, the estimate

$$\|(1 + A)^{-a} e^{itA} f\|_{L^p} \leq C_{d,p,a}\langle t\rangle^{a} \|f\|_{L^p}$$

holds if and only if $a \geq s_p := (d-1)|1/p - 1/2|$, for $1 < p < \infty$ (Frey et al., 2020, Kriegler, 2012). This threshold is sharp; no better smoothing is admissible. Equivalently, the operator $(I-\Delta)^{-s_p/2} e^{it \sqrt{-\Delta}}$ is bounded on $L^p$ at fixed times if and only if $s_p \leq \alpha$.

The necessity is demonstrated by explicit evaluation on plane wave data. Sufficiency hinges on complex interpolation, Fourier integral operator theory, and deep properties of the Fourier transform.

2. Geometric Generalizations: Sub-Laplacians and Sub-Riemannian Settings

For sub-Laplacians $\mathscr L$ in divergence form on manifolds $M^n$ with a bracket-generating Hamiltonian $H$, Miyachi-Peral-type bounds hold in a manner exactly analogous to the Euclidean case (Martini et al., 2018). The central result is

$$\|(1 + \mathscr L)^{-\alpha/2} \cos(t\sqrt{\mathscr L})\|_{L^p \rightarrow L^p} \lesssim (1 + |t|)^\alpha \implies \alpha \geq (n-1)|1/p - 1/2|$$

uniformly for all $1 < p < \infty$. This remains valid regardless of the step or the Hausdorff dimension $Q$, provided the bracket-generating (Hörmander) condition holds.

The proof relies on a Fourier integral operator parametrix for the wave propagator, properties of the sub-Riemannian geodesic flow, and the stationary phase method to extract the optimal lower bound on $\alpha$. Nondegeneracy of the sub-Riemannian exponential map is critical, ensuring microlocal control on oscillatory integral kernels. Notably, the threshold for $\alpha$ does not worsen even when the underlying geometry is highly degenerate or $Q \gg n$—the necessity remains "Euclidean-type".

3. Extensions to Variable Coefficient and Nonsmooth Operators

Frey and Portal (Frey et al., 2020) showed that for operators $$\mathcal{L} = -\sum_{j=1}^d a_{j+d}(x_j) \partial_j (a_j(x_j) \partial_j)$$ with merely $C^{0,1}$ (Lipschitz) coefficients, Miyachi-Peral-type fixed-time $L^p$ estimates still hold, under the same threshold $s_p$. Specifically,

$$\| (I+\mathcal{L})^{-s_p/2} e^{it \sqrt{\mathcal{L}}} f\|_{L^p} \leq C_{p,d,a_j,t} \|f\|_{L^p}$$

for all $1<p<\infty$, $s_p = (d-1)|1/p-1/2|$.

The sufficiency is achieved not by classical parametrix constructions but via adaptations of tent spaces $T^{p,2}(\mathbb{R}^d)$, Hardy-Sobolev spaces $H^{p,s}_{\mathcal{L}}$, and a wave-packet transform tailored to the Lipschitz structure. Central to the analysis are algebraic factorization properties and boundedness criteria for the associated (now, non-smooth) Fourier integral operators.

This demonstrates that the critical regularity required for boundedness of the wave propagator does not increase even for rough coefficients, provided certain structural and ellipticity conditions are preserved.

4. Function Space Refinements: Hardy, Tent, and DS$_\beta$ Spaces

Classical results are formulated in $L^p$ and Hardy spaces $\mathcal{H}^1$. Recent developments have extended Miyachi-Peral-type estimates to $β$-dimensional stable measure spaces $DS_\beta(\mathbb{R}^n)$ (Basak et al., 14 Dec 2025). For $T_b^t$ the wave multiplier,

$$\| T_b^t \mu \|_{L^p} \leq C t^{-n/p'} \|\mu\|_{DS_\beta}$$

holds in several regimes, with the threshold $b_p = (n+1)/2 - 1/p$ and suitable conditions on $\beta$ and $p$. For $p \geq 2$, $\beta \in (0,n]$, and for $1 \leq p < 2$, $\beta > n-1$ (or $>2/p$ in $n=2$), the bound is achieved.

The definition of $DS_\beta$ measures involves atomic decompositions with cancellation and heat extension conditions, interpolating between Hardy space ($\beta = n$) and the full measure space ($\beta \to 0$). Proofs for $p<2$ exploit refined two-scale decompositions and Sobolev-type $L^2$ control.

This framework allows endpoint estimates for data beyond classical function spaces, refining and extending known Hardy space theory for wave multipliers.

5. Analytic Tools: Paley-Littlewood, Transference, and Functional Calculus

Spectral multiplier results for the wave equation are often established via Paley–Littlewood decompositions, boundedness of analytic families, and transference principles for regularized groups (Kriegler, 2012). For general self-adjoint or $0$-sectorial operators $L$ that admit a suitable Paley-Littlewood decomposition and satisfy polynomial growth bounds

$$\|(1+L)^{-B} e^{itL}\|_{X \to X} \leq C \langle t \rangle^a$$

the corresponding fixed-time wave multiplier

$$\|(1+L)^{-a} e^{itL} \|_{X \to X} \leq C \langle t \rangle^{a}$$

follows, mirroring the Euclidean case. This framework allows deduction of Miyachi-Peral-type fixed-time estimates in settings as general as Lie groups (e.g., Heisenberg), under off-diagonal heat or Poisson kernel bounds.

Functional calculus and complex interpolation play key roles in transferring $L^2\to L^2$ and end-point $L^1\to$ BMO or $L^1\to L^\infty$ estimates to the entire $L^p$ scale.

6. Comparative Summary and Sharpness of Bounds

A crucial theme across all settings (Euclidean, sub-Riemannian, nonsmooth) is the universality of the lower bound $\alpha \geq (n-1)|1/p - 1/2|$ for fixed-time $L^p$ wave propagator estimates. This threshold is sharp and dictated by underlying dispersive geometry and the nature of oscillations in the wave kernel. In no generalization considered—whether increasing the step in Carnot groups, reducing regularity of coefficients, or weakening cancellation assumptions on data—is it possible to extend $p$ or reduce $\alpha$ beyond the Euclidean threshold (Martini et al., 2018, Frey et al., 2020, Basak et al., 14 Dec 2025).

The table below summarizes the sharp regularity thresholds in several model contexts:

Context	Regularity Threshold ($\alpha$ or $b$)	Reference
Euclidean $\mathbb{R}^n$	$(n-1)\left|1/p - 1/2\right|$	(Martini et al., 2018)
Sub-Laplacians on bracket-generating $M$	same as Euclidean	(Martini et al., 2018)
Lipschitz-structured variable coeff.	same as above	(Frey et al., 2020)
$\beta$-dim. stable measures	$b_p = (n+1)/2 - 1/p$	(Basak et al., 14 Dec 2025)

This universality indicates that the critical loss of derivatives in $L^p$ estimates for wave equations is a geometric and algebraic invariant, resistant to increased degeneracy or roughness in the operator or the data.

7. Applications, Corollaries, and Ongoing Directions

Miyachi-Peral-type fixed-time bounds have immediate implications for well-posedness, smoothing, and regularity properties of wave equations, including those with measure data (Basak et al., 14 Dec 2025). They underpin endpoint Strichartz bounds, guide spectral multiplier theory for sub-Laplacians, and inform the design of microlocal analysis in nonelliptic or subelliptic geometries.

Recent work demonstrates that these bounds refine classical inequalities by extending optimal wave estimates to broader classes of functions and measures, revealing new phenomena when tracking cancellation, atomicity, or fractal-like properties of measures. The continued focus is on weakening regularity requirements on both operators and data while preserving sharp bounds dictated by the geometry and oscillatory analysis. Further generalizations include noncompact settings, semigroups on abstract Banach spaces, and analysis on manifolds with nontrivial microgeometry or singularities.

These developments further the central theme that the fixed-time regularity loss for the wave propagator is fundamentally constrained by the underlying geometric dispersion, as initially encapsulated by the Miyachi-Peral theorem.