Weighted and unweighted regularity of bilinear pseudo-differential operators with symbols in general Hörmander classes

Abstract: This paper investigates the boundedness of bilinear pseudo-differential operators with symbols in the Hörmander class $BS_{\varrho,δ}<sup>m(\mathbb{R}<sup>n)$ in the previously unexplored regime $0 \leq \varrho &lt; δ&lt; 1$. We establish boundedness from $$H^{p(\mathbb{R}ⁿ⁾} \times H<sup>q(\mathbb{R}<sup>n)$$ to $L<sup>r(\mathbb{R}<sup>n)$ (with $L<sup>r$ replaced by $\mathrm{BMO}$ when $p=q=r=\infty$) under the probably optimal condition on the order $$m \leq m_\varrho(p,q) - \frac{n\max{δ-\varrho,0}}{\max{r,2}},$$ where $m_\varrho(p,q)$ is the critical order in the case $0\leqδ\leq\varrho&lt;1.$ Furthermore, we develop refined pointwise estimates via sharp maximal functions, establishing that for $$m \leq -n(1-\varrho)(\frac{1}{\min{r_1,2}}+ \frac{1}{\min{r_2,2}})$$ with $1&lt;r_{1},r_{2}&lt;\infty$, the bilinear operators satisfy $$M^\sharp T_a(f_1,f_2)(x) \lesssim \mathcal{M}_{\vec{r}}(f_1,f_2)(x).$$ This extends the parameter range from the restrictive condition $0 \leq δ\leq \varrho &lt; 1$ to the general setting $0 \leq \varrho \leq 1$, $0 \leq δ&lt; 1$ with $δ&gt; \varrho$ permitted, and generalizes previous results of Park and Tomita to distinct exponent pairs. Consequently, we obtain weighted norm inequalities for bilinear pseudo-differential operators under multilinear $A_{\vec{p},(\vec{r},\infty)}$ weights.

• The paper proves boundedness of bilinear operators from Hardy spaces to Lᵣ spaces under new order conditions in the regime 0 ≤ ρ < δ < 1.
• It employs intricate symbol decompositions, Calderón–Vaillancourt-type estimates, and sharp L² and endpoint analyses to extend classical results.
• Maximal function techniques yield novel weighted norm inequalities, broadening the applicability of multilinear and non-convolution operator theory.

Weighted and Unweighted Regularity of Bilinear Pseudo-Differential Operators with Symbols in General Hörmander Classes

Introduction and Context

The theory of pseudo-differential operators forms a core part of harmonic analysis and partial differential equations, tracing back to Hörmander's foundational framework for linear operators. While the mapping properties of linear pseudo-differential operators with symbols in Hörmander classes $S^m_{\varrho,\delta}$ are well understood, the multilinear extension, and in particular the bilinear case, presents significantly more complex phenomena. Previous results established sharp $L^p$ boundedness under restrictive conditions—typically $0 \leq \delta \leq \varrho < 1$—and for symbols of order at most $m_\varrho(p)$. However, for the regime $0 \leq \varrho < \delta < 1$, boundedness and order optimality were largely unresolved, especially in the context of general Hörmander classes.

This paper provides a comprehensive analysis of the boundedness properties, regularity, pointwise estimates, and weighted norm inequalities for bilinear pseudo-differential operators with symbols in $BS^m_{\varrho,\delta}(\mathbb{R}^n)$, extending the scope of classical results to previously inaccessible parameter regimes.

Main Results and Methods

Critical Order Boundedness for Bilinear Operators

The central result addresses the boundedness of bilinear operators $T_a$ mapping $H^p(\mathbb{R}^n) \times H^q(\mathbb{R}^n)$ to $L^r(\mathbb{R}^n)$ (with $BMO$ as endpoint) for symbols in $L^p$0. The key theorem establishes that boundedness holds under

$L^p$1

where $L^p$2 denotes the critical order in the previously studied case $L^p$3. Notably, this generalizes earlier work (e.g., by Miyachi and Tomita [MiyachiTomita2013, MiyachiTomita2020]) both by treating the less explored regime $L^p$4 and by removing symmetry and integrability restrictions on the exponents.

The proof architecture leverages intricate symbol decompositions, interpolation arguments, and kernel estimates, utilizing advanced harmonic analysis tools such as (i) sharp kernel regularity, (ii) atomic decompositions on Hardy spaces, and (iii) the generalization of almost-orthogonality principles.

$L^p$5 and Extreme Endpoint Estimates

Sharp $L^p$6-based estimates are achieved for the regime $L^p$7, leading to boundedness from $L^p$8 to $L^p$9 and $0 \leq \delta \leq \varrho < 1$0 to $0 \leq \delta \leq \varrho < 1$1. The endpoint $0 \leq \delta \leq \varrho < 1$2 mapping is also addressed, with the critical order $0 \leq \delta \leq \varrho < 1$3 shown to be sharp in the regime $0 \leq \delta \leq \varrho < 1$4.

These results are underpinned by variant Calderón–Vaillancourt-type estimates, advanced kernel decomposition, and careful treatment of the lack of symbolic calculus in the $0 \leq \delta \leq \varrho < 1$5 range.

Sharp Maximal and Weighted Inequalities

A significant advance is made in sharp pointwise control of bilinear operators through maximal function techniques. The authors extend maximal function estimates of Park and Tomita to the case where the minimal integrability parameters in the Hardy–Littlewood maximal function can differ between the two input functions. Specifically, for $0 \leq \delta \leq \varrho < 1$6 and $0 \leq \delta \leq \varrho < 1$7, the Fefferman–Stein maximal function of $0 \leq \delta \leq \varrho < 1$8 is controlled by the general maximal operator $0 \leq \delta \leq \varrho < 1$9. This result enables the derivation of weighted norm inequalities with respect to multilinear $m_\varrho(p)$0 weights, following the generalized extrapolation and domination theory for multilinear weights.

This analytical machinery substantially widens the reach of weighted norm theory in the multilinear setting, moving beyond the classic Muckenhoupt $m_\varrho(p)$1 regime and allowing for simultaneous control of mixed-norm input spaces.

Summary Table of Main Boundedness Results

Operator Symbol Class	Mapping Property	Order Condition on $m_\varrho(p)$2
$m_\varrho(p)$3	$m_\varrho(p)$4	$m_\varrho(p)$5
$m_\varrho(p)$6	$m_\varrho(p)$7	$m_\varrho(p)$8
$m_\varrho(p)$9	$0 \leq \varrho < \delta < 1$0	$0 \leq \varrho < \delta < 1$1
$0 \leq \varrho < \delta < 1$2	Weighted $0 \leq \varrho < \delta < 1$3	$0 \leq \varrho < \delta < 1$4 as in the maximal function condition

Discussion and Implications

This work resolves the regularity and weighted theory for bilinear pseudo-differential operators in the general Hörmander class, closing significant gaps in the multilinear and weighted context. Several aspects are particularly noteworthy:

• Relaxed Symbol and Exponent Regimes: The extension to $0 \leq \varrho < \delta < 1$5 broadens the set of admissible symbols, overcoming the technical obstacles related to the lack of symbolic calculus in this case.
• Sharp Weighted Inequalities: The framework for weighted estimates based on general $0 \leq \varrho < \delta < 1$6 classes and mixed maximal functions will likely serve as a template for future developments in multilinear analysis, facilitating weighted norm inequalities for other non-convolution and singular integral operators.
• Pointwise Control and Maximal Function Estimates: The transition from uniform to variable integrability in sharp maximal estimates is a nontrivial innovation with implications for the study of sparse domination, extrapolation, and endpoint theory in multilinear settings.

Future Directions

• Optimality Questions: While sharpness is demonstrated for certain mapping properties, the absolute optimality of the order $0 \leq \varrho < \delta < 1$7 in all regimes remains an open problem, mirroring the situation in linear theory and likely requiring constructions of counterexamples or new approaches to limiting regularity.
• Multilinear Extensions: Extending these techniques and results beyond the bilinear setting to general multilinear operators represents an immediate and challenging direction. The lack of suitable symbolic calculus and almost-orthogonality estimates grows rapidly with arity.
• Applications in Nonlinear PDEs: The mapping properties established in this work have potential applications to nonlinear evolution equations and other problems where bilinear pseudo-differential operators naturally arise.

Conclusion

The paper systematically advances the boundedness, regularity, and weighted theory for bilinear pseudo-differential operators with symbols in general Hörmander classes, especially addressing the challenging regime $0 \leq \varrho < \delta < 1$8. The strong pointwise and weighted norm inequalities are obtained under near-optimal conditions, and the techniques developed—especially those concerning maximal function estimates for mixed exponents—are likely to have wide impact on harmonic analysis and related fields. Further progress on optimality of critical order and multilinear extensions is anticipated as a natural continuation of this line of research.

Reference:

"Weighted and unweighted regularity of bilinear pseudo-differential operators with symbols in general Hörmander classes" (2604.09013)