The Stokes Operator with Power Weights Outside the Muckenhoupt Class

Abstract: In this paper, we prove estimates for the Stokes resolvent problem with no-slip boundary conditions on the half space in weighted $L<sup>p$-spaces. The weights we consider are power weights both inside and outside the Muckenhoupt range. Our estimates imply that the corresponding Stokes operator is the generator of a bounded analytic $C_0$-semigroup. We furthermore show that it admits a bounded $H<sup>\infty$-calculus of angle $0.$ This results seems to be new even within the Muckenhoupt range.

• The paper establishes resolvent estimates and sectoriality for the Stokes operator in weighted Lᵖ spaces using explicit kernel bounds for power weights outside the Muckenhoupt range.
• It develops a bounded H∞-calculus to guarantee maximal regularity and analytic semigroup generation despite the failure of classical harmonic analysis tools.
• The methodology employs novel weighted integral and interpolation techniques, yielding optimal range results and new insights into handling boundary singularities in PDEs.

The Stokes Operator with Power Weights Beyond the Muckenhoupt Class

Introduction and Motivation

This work rigorously addresses the behavior of the Stokes operator on the half-space in weighted $L^p$-spaces, focusing specifically on power weights $$w_\gamma(x) = \operatorname{dist}(x, \partial H)^\gamma = x_{n+1}^\gamma$$, $\gamma \in (-p-1, 2p-1) \setminus \{-1, p-1\}$, that lie both inside and, crucially, outside the Muckenhoupt $A_p$ range ($-1 < \gamma < p-1$). Weighted function spaces of this kind arise naturally when analyzing PDEs (and, in particular, stochastic PDEs subject to boundary singularities) where nonuniformities or singular behaviors near the boundary must be accommodated.

A central challenge for analysis outside the $A_p$ class arises from the breakdown of standard harmonic analysis tools, such as the Helmholtz projection and the associated boundedness of singular integral operators. Consequently, the extension of resolvent and functional calculus properties for the Stokes operator with these weights has remained open. The analysis in this paper overcomes this obstacle by developing new resolvent and semigroup estimates for weights even when classical harmonic analysis fails.

Main Results

Resolvent and Sectoriality Estimates

The Stokes resolvent problem is studied in weighted spaces:

$$\lambda u - \Delta u + \nabla p = f, \quad \operatorname{div}u = 0, \quad u|_{\partial H} = 0, \quad f \in L^p_\gamma(H)^{n+1}$$

with $w_\gamma(x) = x_{n+1}^\gamma$, and $1 < p < \infty$.

The first principal result is a uniform estimate for the Stokes resolvent in $L^p_\gamma(H)^{n+1}$ for all $$w_\gamma(x) = \operatorname{dist}(x, \partial H)^\gamma = x_{n+1}^\gamma$$0:

$$w_\gamma(x) = \operatorname{dist}(x, \partial H)^\gamma = x_{n+1}^\gamma$$1

uniformly for $$w_\gamma(x) = \operatorname{dist}(x, \partial H)^\gamma = x_{n+1}^\gamma$$2 in a sector $$w_\gamma(x) = \operatorname{dist}(x, \partial H)^\gamma = x_{n+1}^\gamma$$3, $$w_\gamma(x) = \operatorname{dist}(x, \partial H)^\gamma = x_{n+1}^\gamma$$4. The explicit construction of the solution, using detailed pointwise kernel bounds and precise handling of the weight's singular behavior at the boundary, enables these estimates even in regimes where the Riesz transforms and classical singular integrals fail to be bounded operators.

The corollary is that the abstract Stokes operator $$w_\gamma(x) = \operatorname{dist}(x, \partial H)^\gamma = x_{n+1}^\gamma$$5 is sectorial of angle $$w_\gamma(x) = \operatorname{dist}(x, \partial H)^\gamma = x_{n+1}^\gamma$$6 and generates a bounded analytic $$w_\gamma(x) = \operatorname{dist}(x, \partial H)^\gamma = x_{n+1}^\gamma$$7-semigroup on the solenoidal subspace $$w_\gamma(x) = \operatorname{dist}(x, \partial H)^\gamma = x_{n+1}^\gamma$$8 for all $$w_\gamma(x) = \operatorname{dist}(x, \partial H)^\gamma = x_{n+1}^\gamma$$9 in this extended range.

Bounded $\gamma \in (-p-1, 2p-1) \setminus \{-1, p-1\}$0-Calculus

A crucial advance is establishing that $\gamma \in (-p-1, 2p-1) \setminus \{-1, p-1\}$1 admits a bounded $\gamma \in (-p-1, 2p-1) \setminus \{-1, p-1\}$2-calculus with angle $\gamma \in (-p-1, 2p-1) \setminus \{-1, p-1\}$3 in $\gamma \in (-p-1, 2p-1) \setminus \{-1, p-1\}$4 for all $\gamma \in (-p-1, 2p-1) \setminus \{-1, p-1\}$5, using direct kernel and maximal function techniques tailored to the non-$\gamma \in (-p-1, 2p-1) \setminus \{-1, p-1\}$6 scenario. That is, for all $\gamma \in (-p-1, 2p-1) \setminus \{-1, p-1\}$7 in the relevant holomorphic function class and all weights in the described regime, the operator functional calculus is bounded:

$\gamma \in (-p-1, 2p-1) \setminus \{-1, p-1\}$8

This analysis does not rely on the machinery available for $\gamma \in (-p-1, 2p-1) \setminus \{-1, p-1\}$9 weights (e.g., Mikhlin-type multiplier theorems) but instead develops a strategy based on explicitly representing the resolvent and handling the kernel singularities via intricate weighted integral estimates and interpolation arguments.

Notably, these results generalize known theorems for the Laplacian (as in the analytic framework developed for Dirichlet Laplacians with power weights) but are extended here to the Stokes system, which couples velocity and pressure and presents additional technical complications due to the divergence-free constraint.

Range Optimality and Duality

It is shown that the resolvent and $A_p$0-calculus results are optimal in the sense that the range $A_p$1 cannot be further extended, due to explicit counterexamples concerning semigroup invariance and well-posedness for the associated heat and Stokes equations in weighted spaces. Duality arguments allow passage from the subrange $A_p$2 to the full stated regime.

Technical Methods

The analysis synthesizes several advanced techniques:

• Explicit kernel representations: Starting from the explicit Stokes solution formula in half-space (following [DHP01]), the work provides detailed kernel estimates, which are leveraged to establish norm bounds via careful use of Minkowski and Young inequalities in the weighted context.
• Weighted integral estimates: Nontrivial analysis of exponential integral asymptotics and singularities in the weighted measure is crucial for achieving $A_p$3-independent norm bounds.
• Functional calculus without $A_p$4 assumptions: The H\"ormander-type and multiplier machinery is replaced with a direct approach, interpolating weak and strong norm bounds and exploiting maximal function inequalities without relying on $A_p$5's boundedness of the Hardy-Littlewood maximal operator.
• Stability under duality and interpolation: The passage from dense subspaces to the full Banach range, and from weak-$A_p$6 to strong $A_p$7 estimates using the Marcinkiewicz interpolation theorem, ensures that the operator bounds hold in the target setting.

Implications and Theoretical Consequences

Establishing bounded analytic semigroups and a bounded $A_p$8-calculus for the Stokes operator in weights beyond $A_p$9 significantly impacts the well-posedness and maximal regularity theory of (stochastic) Navier–Stokes equations in domains with boundary singularities or modeling physical effects such as boundary layers. The $-1 < \gamma < p-1$0-calculus implies strong deterministic and stochastic maximal $-1 < \gamma < p-1$1-regularity, essential for nonlinear PDE theory and the analysis of random forcing or boundary roughness in SPDEs.

The results resolve longstanding questions regarding the possibility of developing a functional calculus and maximal regularity framework in weights where harmonic analysis tools are insufficient. This opens further avenues for the analysis of PDEs with nonstandard singularities, especially in problems arising in stochastic analysis and boundary-dominated phenomena.

Future Directions

The framework developed in this work motivates several further research directions:

• Nonlinear and stochastic applications: The setting is suitable for boundary-value problems in stochastic Navier–Stokes equations, where existing techniques fail for general weights.
• Extension to more general domains: While the analysis focuses on the half-space, it suggests the possibility of further extension to rough or exterior domains, possibly by patching asymptotically power-like weights or utilizing localization and perturbation methods.
• Transference to related systems: The methodology may be adapted for boundary-coupled parabolic systems and other elliptic or evolutionary operators lacking $-1 < \gamma < p-1$2-compatible representations.
• Sharpness and counterexamples: Investigating the precise limitations of the weighted $-1 < \gamma < p-1$3-calculus and exploring possible pathological weights that remain outside the developed theory.

Conclusion

This paper fills a significant gap in the theory of the Stokes operator by providing explicit sectoriality, resolvent, semigroup, and functional calculus results in weighted $-1 < \gamma < p-1$4-spaces with power weights outside the Muckenhoupt class (2606.31542). The approach forgoes classical harmonic analysis in favor of direct kernel and weighted analysis, thereby expanding the range of applicable weights far beyond previous barriers and establishing a robust foundation for maximal regularity and analytic semigroup theory in non-standard settings. The results are essential for the further development of analytic and stochastic PDEs with singularity or non-homogeneous features near the boundary.