Mixed Lebesgue–Besov Spaces

• Mixed Lebesgue–Besov spaces are Banach/quasi-Banach function spaces that combine direction-dependent integrability with multi-scale smoothness.
• They utilize Littlewood–Paley decompositions to capture anisotropic scaling and embed both isotropic and dominating mixed smoothness scenarios.
• Applications include critical regularity in Navier–Stokes, analysis of singular SDEs, and high-dimensional approximation techniques.

A mix Lebesgue–Besov space is a Banach or quasi-Banach function space that fuses the anisotropic scaling and integrability structures of mixed-norm Lebesgue spaces with the multi-scale smoothness architecture of Besov spaces. Such spaces are designed to encode direction-dependent integrability (using separate $L^{p_i}$ norms in each variable) and to measure regularity across frequency scales, often via Littlewood–Paley decompositions. The class encompasses both isotropic and dominating mixed smoothness scenarios and underlies recent advances in partial differential equations (PDE), stochastic PDEs, and approximation theory, particularly where anisotropic or sparse-grid effects dominate. Notable applications include global well-posedness results for Navier–Stokes equations and the analysis of singular SDEs and Fokker–Planck equations with distributional or rough drifts (Aurazo-Alvarez et al., 20 Mar 2025, Fitoussi et al., 17 Dec 2025, Bahrii, 18 Oct 2025).

1. Core Definitions and Functional Structure

The canonical mixed-norm Lebesgue space, $L^{\mathbf p}(\mathbb R^d)$ for $\mathbf p = (p_1,\ldots,p_d)$, is given by

$$\|f\|_{L^{\mathbf p}} = \left(\int_{\mathbb R} \cdots \left(\int_{\mathbb R} |f(x_1,\ldots,x_d)|^{p_1}\,dx_1\right)^{\frac{p_2}{p_1}}\cdots dx_{d}\right)^{\frac{1}{p_d}}$$

with component-wise iterations, making $L^{\mathbf p}$ a Banach space for admissible $\mathbf p$ (Aurazo-Alvarez et al., 20 Mar 2025).

Given homogeneous or inhomogeneous Littlewood–Paley decompositions $\{\varphi_j\}$, the mixed-norm Besov space $\dot B^s_{\mathbf p, q}(\mathbb R^d)$ is defined by

$$\|f\|_{\dot B^s_{\mathbf p,q}} = \left( \sum_{j\in\mathbb Z} 2^{jsq} \|\Delta_j f\|_{L^{\mathbf p}}^q \right)^{1/q}$$

for $q<\infty$, and by a supremum over $j$ for $q = \infty$. The polynomials are quotiented out for homogeneity.

Fourier-side analogues, such as $\dot F^s_{\mathbf p, q}$, are formulated using $\|\varphi_j \widehat{f}\|_{L^{\mathbf p}}$. On domains, mixed-smoothness Besov spaces can also be constructed via moduli of continuity and finite difference characterizations, often using $L^{\mathbf p}$ increments in cylinder or product directions (Kudryavtsev, 2021, Nikolaev et al., 7 Mar 2024).

2. Key Analytical Properties

Banach Space Structure and Embeddings

Mix Lebesgue–Besov spaces are Banach for $q<\infty$ or for the critical exponent $s = \sum_i(1/p_i-1)$ with $q=1$ (Aurazo-Alvarez et al., 20 Mar 2025). Embedding properties generalize standard isotropic cases:

• Bernstein-type inequalities relate derivatives and mixed norms when Fourier supports are localized.
• If $s_{2}-s_{1} = \sum_i(1/p_i - 1/q_i)$ and $\mathbf p \leq \mathbf q$, the embedding

$$\dot B^{s_2}_{\mathbf p, q_2} \hookrightarrow \dot B^{s_1}_{\mathbf q, q_1}$$

is continuous for $q_2 \leq q_1$.

Scaling and Criticality

Under anisotropic dilation $f_\lambda(x) = f(\lambda x)$,

$$\|f_\lambda\|_{\dot B^s_{\mathbf p, q}} = \lambda^{s - \sum_{i=1}^d 1/p_i} \|f\|_{\dot B^s_{\mathbf p, q}}$$

implying that the critical regularity for scaling-invariant PDEs (e.g., Navier–Stokes) is $s_{\mathrm{crit}} = -1 + \sum_{i=1}^d 1/p_i$ (Aurazo-Alvarez et al., 20 Mar 2025).

Product, Convolution, and Duality Estimates

Mix Lebesgue–Besov spaces admit:

• Product rules: For regularity $\lambda \geq 0$, $f \in B^\lambda_{l_1,\infty}$ and $g\in B^\lambda_{l_2,1}$, the product satisfies

$$\|f g\|_{B^\lambda_{\ell,\infty}} \leq C \|f\|_{B^\lambda_{l_1,\infty}}\|g\|_{B^\lambda_{l_2,1}}$$

analogously for negative regularity (crucial for SDEs and Fokker–Planck analysis) (Bahrii, 18 Oct 2025, Fitoussi et al., 17 Dec 2025).

• Young convolution-type bounds: For Besov indices and $1+1/\ell = 1/\ell_1 + 1/\ell_2$,

$$\|f * g\|_{B^\gamma_{\ell,m}} \leq C \|f\|_{B^{\gamma-\delta}_{\ell_1, m_1}}\|g\|_{B^{\delta}_{\ell_2, m_2}}$$

useful for semigroup smoothing and Duhamel expansions.

• Duality: $$\langle f, g \rangle \leq \|f\|_{B^\gamma_{\ell,m}} \|g\|_{B^{-\gamma}_{\ell',m'}}$$ with $1/\ell+1/\ell'=1$.

Interpolation

The real and complex interpolation frameworks extend, with

$$\left( \dot B^{s_0}_{\mathbf p, q_0}, \dot B^{s_1}_{\mathbf p, q_1} \right)_{\theta, r} = \dot B^{(1-\theta) s_0 + \theta s_1}_{\mathbf p, r}$$

providing flexibility for regularity transfers and mixed-norm scaling (Aurazo-Alvarez et al., 20 Mar 2025, Fitoussi et al., 17 Dec 2025).

3. Comparison to Isotropic and Dominating Mixed Besov Spaces

Isotropic vs. Mixed Smoothness

Isotropic Besov $B^s_{p,q}(\mathbb R^d)$ spaces impose equal smoothness and integrability in all directions. Mixed/dominating mixed Besov spaces $S^s_{p,q}B(\mathbb R^d)$, defined via coordinate-wise tensor-product Littlewood–Paley projections, distinguish between directions (Nguyen et al., 2016): $$\|f\|_{S^s_{p,q}B} = \left(\sum_{\mathbf k\in \mathbb N_0^d} 2^{|\mathbf k|_1 s q} \| \Delta_{\mathbf k} f \|_{L^p}^q\right)^{1/q}$$ where $\Delta_{\mathbf k}$ is a product frequency block.

Embeddings between isotropic and dominant-mixed scales depend on both $s$ and $(p, q)$. There are precise, sharp ranges where one space embeds into the other, but in general the scales are distinct except in one dimension or away from the critical lines (Nguyen et al., 2016).

Difference Characterizations

On product manifolds or bounded domains, mixed Besov norms can be equivalently described by mixed finite differences (moduli of continuity), with explicit norm equivalences between the Littlewood–Paley decomposition and difference-based moduli (Nikolaev et al., 7 Mar 2024, Kudryavtsev, 2021). This allows tools such as extension theorems, regularity criteria for random fields, and criteria for partial regularity of solutions to stochastic PDEs.

4. Parameter Regimes, Criticality, and Examples

Mix Lebesgue–Besov spaces admit classification and parametrization by smoothness $s$, integrability vector $\mathbf p$, summability $q$, and (for time-dependent problems) by mixed space-time $L^r$ rituals, e.g., $$L^r\left([0, T]; B^\beta_{p, q}(\mathbb R^d)\right)$$ (Fitoussi et al., 17 Dec 2025, Bahrii, 18 Oct 2025).

Table: Representative Spaces and Criticality

Space Type	Parameters	Key Criticality/Remarks
$\dot B^{s}_{\mathbf p, q}$	$\mathbf p, q, s$	$s_{\mathrm{crit}} = -1 + \sum 1/p_i$ (Navier–Stokes)
$B^{(a, \beta)}_{p_1, p_2}$	$a, \beta, p_1, p_2$	Anisotropic smoothness—factorizes for product test functions
$L^r([0, T]; B^\beta_{p, q})$	$r, p, q, \beta$	Used for parabolic/kinetic equations with time-space interplay

Notable anisotropic examples include:

• $\mathbf p = (q, q, q)$, $d=3$, $s=-1 + 3/q$ yields scaling-critical initial data for Navier–Stokes not covered by $BMO^{-1}$ (Aurazo-Alvarez et al., 20 Mar 2025).
• $\mathbf p = (2, \infty, 2)$ with $s = 0$ captures $L^\infty$-control in one direction and $L^2$ in others.

5. Applications in Analysis and Probability

Mix Lebesgue–Besov spaces are instrumental in:

• Navier–Stokes Equations: Forming critical classes for global well-posedness outside $BMO^{-1}$ and classical isotropic Besov frameworks. Key tools include Bernstein inequalities, Bony paraproducts, and fixed-point contraction in function spaces such as

$$Z = C([0, \infty); \dot B^s_{\mathbf p, q}) \cap L^1([0, \infty); \dot B^{s+2}_{\mathbf p, q})$$

(Aurazo-Alvarez et al., 20 Mar 2025).

• Stochastic Differential Equations (SDEs) and Fokker–Planck Equations: Weak well-posedness and density regularity under minimal spatial smoothness of the interaction kernel or drift, extending below $-1$ in regularity using product rules for Besov spaces and heat-kernel smoothing (Bahrii, 18 Oct 2025, Fitoussi et al., 17 Dec 2025).
• Random Fields and SPDEs: Mixed-norm regularity captures anisotropic temporal and spatial regularity of the Brownian sheet and solutions to parabolic equations, with Kolmogorov criteria adapted to mixed Besov spaces (Nikolaev et al., 7 Mar 2024). Young convolutions propagate regularity via kernel integration.
• Fractional Calculus: Sharp operator norm bounds for anisotropic Riesz and Marchaud operators rely on precise mixed Besov–Lebesgue structure, with exact constants and optimal parameter regimes (Ostrovsky et al., 2015).

6. Methodological Tools and Wavelet Characterizations

Wavelet and tensor-product spline bases (e.g., hyperbolic wavelets) provide atomic and frame decompositions adapted to mixed-norm or dominating mixed smoothness spaces (Nguyen et al., 2016). Such bases are crucial in high-dimensional approximation, sparse-grid methods, and the numerical analysis of anisotropic operators. Mixed-norm function extension operators allow transfer of boundary values and traces within domains satisfying geometric regularity conditions (Kudryavtsev, 2021).

7. Significance, Open Directions, and Further Remarks

Mix Lebesgue–Besov spaces unify and generalize many known regularity classes, addressing the demands of anisotropic analysis in high-dimensional settings, PDEs with singularities or critical scaling, and stochastic processes with directional regularity disparities. Their precise embeddings, interpolation properties, and operator estimates provide the technical infrastructure necessary for current advances in both deterministic and stochastic models.

A plausible implication is that further developments in sparse approximation, compressive sensing, and deep learning in high dimensions may profit from the nuanced expressivity of mix Lebesgue–Besov frameworks, particularly in regimes requiring simultaneous control over oscillatory and spatially localized features. The theory continues to evolve, especially in the context of new PDE paradigms and stochastic models where classical Sobolev or isotropic Besov spaces prove insufficient (Aurazo-Alvarez et al., 20 Mar 2025, Fitoussi et al., 17 Dec 2025, Nguyen et al., 2016).