Bilinear Radial Fourier Multiplier Operator

• The bilinear radial Fourier multiplier operator is defined via a Fourier integral with radial symmetry, enabling analysis through dyadic decomposition and one-dimensional reduction.
• Boundedness results are achieved using uniform Sobolev estimates, yielding dimension-free L²×L² → L¹ bounds and improving upon classical critical index phenomena.
• Extensions to non-Euclidean cases, such as Grushin operators and nilpotent Lie groups, illustrate its broad applicability to sparse domination and weighted inequalities.

A bilinear radial Fourier multiplier operator is a class of bilinear Fourier multipliers exhibiting radial symmetry in frequency variables, with profound connections to the generalized bilinear Bochner–Riesz theory, sparse bounds, square and maximal function estimates, and spectral multiplier theory for subelliptic operators and nilpotent Lie groups. This theory underpins advances in the analysis of bilinear convolution and multiplier operators on $\mathbb{R}^n$ and on non-Euclidean structures such as the Grushin operator and Métivier groups.

1. Definition and Radial Symmetry

A bilinear radial Fourier multiplier operator acts on pairs of functions $f,g:\mathbb{R}^n\to\mathbb{C}$ via

$$T_\sigma(f,g)(x) = \int_{\mathbb{R}^n}\int_{\mathbb{R}^n} e^{2\pi i x \cdot (\xi + \eta)}\, \sigma(\xi,\eta)\, \widehat{f}(\xi)\,\widehat{g}(\eta)\,d\xi\,d\eta.$$

where $\sigma(\xi,\eta)$ is a radial symbol: $\sigma(\xi,\eta) = \Sigma(|\xi|,|\eta|)$ or $\sigma(\xi,\eta) = m_0(|(\xi,\eta)|)$ for $m_0: \mathbb{R}_+ \to \mathbb{C}$, meaning it depends only on the lengths (or combined length) of the frequency variables. The canonical example is the bilinear Bochner–Riesz multiplier, $$\sigma^\delta(\xi,\eta) = \bigl(1-|\xi|^2-|\eta|^2\bigr)_+^\delta$$ (Bernicot et al., 2012, Honzík et al., 14 Jan 2026).

2. Boundedness Results and Critical Index Phenomena

The $$L^2(\mathbb{R}^n)\times L^2(\mathbb{R}^n)\to L^1(\mathbb{R}^n)$$ boundedness of $T_\sigma$ is fundamentally controlled by the smoothness properties of the symbol:

• Dimension-Free Threshold: For bilinear Bochner–Riesz, boundedness holds if and only if $\delta > 0$ — an improvement over the linear theory, for which the critical index is $\delta_c = (n-1)/2$ (Bernicot et al., 2012, Honzík et al., 14 Jan 2026).
• General Symbols: If $\sigma$ admits the uniform Sobolev estimate

$$\sup_{j \in \mathbb{Z}} \| \sigma(2^j\cdot)\, \Phi \|_{L^2_s(\mathbb{R}^{2n})} < \infty, \quad s > 1/2,$$

for a cutoff $\Phi$ on the annulus $|(\xi,\eta)| \in [1/4,4]$, then $T_\sigma: L^2 \times L^2 \to L^1$ is bounded and dimension-free (Honzík et al., 14 Jan 2026). The sharp threshold for the Bochner–Riesz symbol is thus $\delta = 0$.

• Maximal and Square Function Estimates: For order $\delta > \delta_c = (n-1)/2$, maximal and square function variants are bounded on the same $L^p$ scale (with weights); at the endpoint $\delta = \delta_c$ optimal weighted bounds hold for all $(p_1,p_2)$ with $p_1,p_2 > 1$ and $1/p_1 + 1/p_2 = 1/p$ (Choudhary et al., 2022, Jotsaroop et al., 2020).

3. Decomposition and Proof Techniques

The analysis utilizes sophisticated decomposition and orthogonality arguments:

• Dyadic Decomposition: The symbol is split into Littlewood–Paley pieces supported in dyadic annuli, $\sigma = \sum_j \sigma_j$, $$\sigma_j(\xi,\eta) = \sigma(\xi,\eta)\,\Phi(2^{-j}(\xi,\eta))$$ (Honzík et al., 14 Jan 2026, Bernicot et al., 2012).
• One-Dimensional Reduction: Radiality allows reduction to absolute convergence of Fourier series for functions of $|(\xi,\eta)|$; smoothness in $H^{1/2+\epsilon}$ implies summability of associated coefficients.
• Factorization into Linear Multipliers: For each dyadic piece, the bilinear operator splits into sums of products of one-variable linear multipliers acting on each input, with almost-orthogonality controlled by the decay of Fourier coefficients (Honzík et al., 14 Jan 2026).
• Sparse Domination and Weighted Inequalities: Quantitative sparse bounds provide optimal weighted estimates, enabling extrapolation across the full $L^p$ scale (Benea et al., 2016).

4. Extensions: Bochner–Riesz, Maximal Functions, and Non-Euclidean Cases

Bochner–Riesz Operators

For $$\sigma^\delta(\xi,\eta) = (1-|\xi|^2-|\eta|^2)_+^\delta$$, the operator $T^\delta$ is bounded $L^2 \times L^2 \to L^1$ iff $\delta > 0$, and more generally, $T^\lambda$ is bounded from $(L^2)^m$ to $L^{2/m}$ iff $\lambda > m/2 - 1$ in the $m$-linear case (Honzík et al., 14 Jan 2026).

Maximal and Square Functions

Maximal variants, $$B_*^\delta(f,g)(x) = \sup_{R>0} |B_R^\delta(f,g)(x)|$$, and Stein-type square functions possess analogous boundedness regimes, subject to slight increases in the threshold index ($\alpha > \alpha_+(p_1,p_2) + 1/2$ for maximal functions) (Choudhary et al., 2021, Jeong et al., 2019).

Grushin Operators and Lie Groups

Analogues for the Grushin operator $\mathcal{L} = -\Delta_{x'} - |x'|^2 \Delta_{x''}$ and on Métivier groups preserve the piecewise structure of critical indices, with thresholds given by the topological dimension $d$ or homogeneous dimension $Q$ depending on region and support assumptions (Bagchi et al., 24 May 2025, Bagchi et al., 6 Apr 2025). Except for endpoints, the critical index mirrors the Euclidean case with $n \to d$.

5. Weighted, Vector-Valued, and Endpoint Properties

Weighted inequalities for bilinear Bochner–Riesz operators at the critical index $\alpha = n - 1/2$ are established for all weights in the multilinear Muckenhoupt class $A_{(p_1,p_2)}$, with extrapolation extending results to all $p_1,p_2 > 1$ (Jotsaroop et al., 2020, Choudhary et al., 2022). Sparse form domination facilitates new quantitative bounds on mixed-norm and vector-valued extensions (Benea et al., 2016).

Endpoint phenomena feature sharp failure of weak-type $(1,1;1/2)$ at the critical index for maximal and square function operators (Choudhary et al., 2022, Jeong et al., 2019), and some loss at the extreme points for subelliptic settings.

6. Generalizations and Open Problems

The bilinear radial Fourier multiplier theory, supported by dimension-free Coifman–Meyer bounds and bilinear Stein–Tomas restriction arguments, enables a robust extension to:

• Non-Euclidean Settings: Operators associated with stratified Lie groups, subelliptic, and degenerate Laplacians (Grushin, Métivier), with topological dimension or group geometry determining smoothness thresholds (Bagchi et al., 24 May 2025, Bagchi et al., 6 Apr 2025).
• Multilinear and Spherical Maximal: The machinery applies to multilinear symbols and generalized spherical averages (Choudhary et al., 2021, Grafakos et al., 2018).
• Open Directions: These include sharp endpoint estimates, strong-type bounds at critical indices, explicit characterization of weighted norm dependence, and extension to non-Banach $(p<1)$ regimes and restriction-type estimates for two-step and higher-step groups.

Summary Table: Bilinear Radial Multiplier Boundedness (Dimension-Free Regime)

Operator/Symbol	Threshold for $L^2 \times L^2 \to L^1$	Dimension Dependence
$(1-|\xi|^2-|\eta|^2)_+^\delta$	$\delta > 0$	None (dimension-free)
General $\sigma$ (Sobolev condition)	$s > 1/2$ for $\sigma(2^j\cdot) \Phi$	None (Coifman–Meyer)
Linear Bochner–Riesz	$\delta > (n-1)/2$	Explicit ($n$ appears)

The bilinear radial Fourier multiplier framework affords precise control over boundedness and regularity in both Euclidean and non-Euclidean settings, and continues to be the locus for advances in sparse bounds, vector-valued inequalities, and weighted theory for multi-parameter multipliers (Honzík et al., 14 Jan 2026, Bernicot et al., 2012, Benea et al., 2016, Choudhary et al., 2021, Bagchi et al., 24 May 2025, Bagchi et al., 6 Apr 2025).