Bilinear embedding in Orlicz spaces for divergence-form operators with complex coefficients

Abstract: We prove a bi-sublinear embedding for semigroups generated by non-smooth complex-coefficient elliptic operators in divergence form and for certain mutually dual pairs of Orlicz-space norms. This generalizes a result by Carbonaro and Dragičević from power functions to more general Young functions that still behave like powers. To achieve this, we generalize a Bellman function constructed by Nazarov and Treil.

• The paper establishes a dimension-free bilinear embedding estimate in Orlicz spaces that extends classical Lp results to operators with complex coefficients.
• It introduces a generalized Bellman function tailored to Young functions, ensuring sharp control over the gradient of heat semigroups in diverse Orlicz settings.
• The results imply improved endpoint estimates for divergence-form elliptic operators, highlighting the pivotal role of p-ellipticity and precise Orlicz index bounds in harmonic analysis.

Bilinear Embedding in Orlicz Spaces for Divergence-Form Operators with Complex Coefficients

Overview and Motivation

This paper addresses Orlicz-space bilinear embedding inequalities for gradient flows generated by complex-coefficient, divergence-form elliptic operators on $\mathbb{R}^d$. It extends classical results for $\text{L}^p$ spaces—in particular, the work of Carbonaro and Dragičević—to a broad scale of Orlicz spaces defined by Young functions exhibiting near-power behavior. The main technical innovation is a generalized Bellman function adapted to Orlicz norms, allowing the authors to prove a dimension-free bi-sublinear estimate that covers both $\text{L}^p$ and certain endpoint-type Orlicz spaces, including Zygmund and superposition spaces.

The motivation comes from the centrality of bilinear embeddings in harmonic analysis and PDE theory, where such inequalities both reflect endpoint boundedness phenomena (complementing operator theory) and underpin quantitative estimates for related singular integrals and functional calculi. While the real-coefficient case is well-understood, complex coefficients introduce substantial technical obstacles—most estimates for real coefficients fail without significant modification.

Function Spaces: Young Functions and Orlicz Norms

Let $\Phi: [0,\infty) \to [0,\infty)$ be a Young function, and ${\Phi_*}$ its convex conjugate; both satisfy precise differentiability and growth, including strict monotonicity of the derivatives, essential bounds on the indices $m, M, \tilde{m}, \tilde{M}$ (related to Matuszewska-Orlicz indices), and doubling conditions. These requirements ensure the Orlicz spaces are reflexive, possess good duality properties, and allow for the sharp estimates developed in the sequel.

The analysis includes classical power-law settings ($\Phi(s) = s^p/p, {\Phi_*}(s)=s^q/q$), Zygmund-type ($\Phi(s) = s^r\log(s+e)$), and nontrivial superpositions of powers, establishing the generality of the results.

Divergence-Form Elliptic Operators with Complex Coefficients

The primary analytic object is the divergence-form elliptic operator $L_A = -\text{div}(A\nabla)$, where $A: \mathbb{R}^d \rightarrow \mathbb{C}^{d\times d}$ is a bounded measurable matrix function. The operator need not possess any regularity beyond the coefficient boundedness and $p$-ellipticity, a gradated condition interpolating between coercivity for real coefficients and full complex ellipticity. The $p$-ellipticity condition, in the regime $p = \tilde{M} + 1$, is key in controlling the behavior of the associated heat semigroups and their gradients.

Main Theorem and Its Consequences

The central result is the following dimension-free Orlicz-space bilinear embedding:

$$\int_{0}^{\infty}\int_{\mathbb{R}^d}\left|\nabla T_t^A f(x)\right| \left|\nabla T_t^B g(x)\right|\,dx\,dt \leq C \|f\|_{\Phi}\|g\|_{{\Phi_*}}$$

where $C$ is explicit, depending only on the ellipticity and doubling constants:

$C = 40\,C_p(A,B)\,D(\Phi),$

with $C_p(A,B)$ involving the maximal anisotropy and minimal $p$-ellipticity across the coefficients $A$, $B$, and $D(\Phi)$ only involving quantities derived from $\Phi$ (see equations (2.15)–(2.16) in the paper (2110.15812)).

Key technical features:

• The embedding is dimension-free: the constant $C$ does not depend on $d$.
• It extends bilinear embedding estimates from the classical $\text{L}^p$ context to general Orlicz spaces whenever the Young function closely mimics power growth.
• In the $\text{L}^p$ case, the estimate recovers the result of Carbonaro and Dragičević with constant matching up to a factor of $4$.
• The result cannot be obtained by interpolation, as many Orlicz spaces covered (such as superpositions with small powers) do not satisfy the Marcinkiewicz endpoint integral conditions required for restricted weak-type methods (see discussion around [Cianchi, 1998]).

Bellman Function Approach and Generalized Convexity

The proof leverages a highly nontrivial generalization of the Nazarov-Treil Bellman function, tailored to the pair of Young functions $(\Phi,{\Phi_*})$. The construction ensures:

• $\mathfrak{X}(u, v)$ is piecewise defined, matches continuously on a specific critical hypersurface in $\mathbb{C}^2$, and is of class $C^1$.
• Positivity and growth/upper bounds compatible with Orlicz modulars and their quasinorms.
• Crucially, it verifies a lower bound on the generalized Hessian (a matrix differential form arising in heat flow energy estimates) that establishes a type of strong convexity adapted to the Orlicz structure and the $p$-ellipticity of $A, B$.

The analysis of the generalized Hessian, including all delicate cross-terms, and the manipulation of Orlicz indices, is nontrivial; in particular, ensuring the 'hidden term' (in the sense of Nazarov-Treil) properly mediates between the off-diagonal and main terms is specific to the Orlicz context.

Technical Implications

The result significantly broadens the set of spaces and differential operators admitting a bilinear embedding estimate:

• It covers cases where the Orlicz space is strictly between $\text{L}^p$ and $\text{L}^q$ (with $q$ conjugate to $p$), including Zygmund and certain log-bumped spaces.
• Endpoint-type spaces that cannot be reached by classical real or complex interpolation fall within the scope when the associated Young functions remain within the controlled range of indices.
• The estimate is highly robust with respect to small perturbations in the Orlicz structure, as the constants $C_p(A,B)$ and $D(\Phi)$ depend only on limiting behaviors, not specific profile parameters.

From a functional analytic perspective, the result underlines the near-optimality of $p$-ellipticity as the structural condition for contractivity and norm estimates of heat semigroups with non-selfadjoint, non-real coefficients.

Theoretical and Practical Outlook

The approach opens several new avenues:

• Harmonic analysis: It provides machinery for endpoint and critical estimates for Riesz transforms, Schrödinger semigroups, and integral operators associated with complex elliptic PDEs, enabling sharp control in regimes unattainable via $\text{L}^p$ methods.
• Extensibility: The Bellman function construction suggests possible extension to further nonlinear and vector-valued settings in Orlicz and Musielak-Orlicz spaces.
• Multilinear estimates: The scope for triangulating to multilinear embeddings (as in [Carbonaro-Dragičević-Kovač-Škreb]) is outlined, although the lack of a natural generalization of convex conjugation beyond pairs restricts progress.
• Limitations: The necessity of strict lower and upper bounds on the Orlicz indices precludes endpoint spaces such as $\text{L}^1$ or $\text{L}^\infty$; counterexamples and dimension blowup phenomena confirm this is not a technical artifact.

One salient aspect is that, whereas interpolation and restricted weak-type arguments only recover the sharp constants for 'distant' Orlicz spaces, the Bellman function method provides uniform control over the constants even in nearly degenerate or limiting cases.

Conclusion

This paper provides a comprehensive framework for bilinear embeddings in Orlicz spaces for divergence-form operators with complex coefficients (2110.15812). The novel adaptation of the Bellman function method to the Orlicz context answers several open questions on endpoint and quasi-endpoint operator estimates for heat semigroups with complex structure. The techniques and results stand to significantly influence future developments in the analysis of degenerate and non-selfadjoint PDEs, advanced harmonic analysis, and functional calculus in nonstandard Banach spaces.