Spherical Fourier Multiplier

• Spherical Fourier multipliers are operators that act diagonally on the spectral decomposition of functions on symmetric and homogeneous spaces using spherical harmonics.
• They generalize classical Fourier multipliers by employing holomorphic extensions and Hörmander-type smoothness conditions to achieve Lp boundedness in non-Euclidean contexts.
• These multipliers are key in spectral theory and approximation, enabling efficient computation and applications in fields such as signal processing and cosmology.

A spherical Fourier multiplier is an operator acting diagonally on the spectral decomposition of functions defined on symmetric spaces, compact manifolds, or homogeneous spaces, with respect to their spherical harmonic or spherical function basis. Such operators generalize classical Fourier multipliers from Euclidean harmonic analysis to settings with richer geometric and representation-theoretic structure, enabling the extension of multiplier theorems and $L^p$ mapping properties to non-Euclidean, curved, or noncommutative environments.

1. Definitions and General Characterization

Given a function space equipped with a spectral decomposition in spherical harmonics or spherical functions, a spherical Fourier multiplier $T_m$ acts by multiplying each spectral component by a corresponding scalar $m(\lambda)$. On the unit sphere $S^d$, for example, one writes

$Tf(x) = \sum_{k=0}^\infty m(k) f_k(x),$

where $f_k$ denotes the projection of $f$ onto the harmonics of degree $k$ (Jordão et al., 2014, Denson, 30 Oct 2024). For more general Gelfand pairs $(G,K)$, the spherical Fourier transform $\mathcal{F}f$ is defined via integration against spherical functions $\varphi$,

$$\mathcal{F}f(\varphi) = \int_G f(x)\varphi(x^{-1})\, dx,$$

and the multiplier operator is

$$T_m f(x) = \int_{S^+} m(\varphi)\mathcal{F}f(\varphi)\varphi(x)\,d\mu(\varphi).$$

In the setting of noncompact symmetric spaces, Damek–Ricci spaces, or spaces with Jacobi analysis, $m(\lambda)$ typically acts on the spectral variable $\lambda$ of the spherical/Jacobi transform (Johansen, 2011).

The principal analytic problem is to characterize conditions on $m$ which guarantee $L^p$-boundedness (or weak-type $(1,1)$) of $T_m$ and to relate the boundedness of spherical multipliers to associated multiplier problems on Euclidean (flat) spaces (Gupta et al., 2017, Denson, 30 Oct 2024).

2. $L^p$-Boundedness, Hörmander-Type Conditions, and Transference Theorems

Sharp $L^p$-boundedness criteria for spherical Fourier multipliers are typically established via analogs of the Hörmander–Mihlin multiplier theorem. A prominent approach requires that the multiplier $m(\lambda)$ (or $m(\varphi)$) extends holomorphically into a strip or tube domain and satisfies uniform smoothness estimates on derivatives: $$\forall\, \lambda = x+iy\in \Omega_1,\quad \left|\frac{d^a}{dx^a} m(x+iy)\right|\le C_{a,y}(1+|x|)^{-a} \quad(0\le a\le N)$$ for a suitable region $\Omega_1$ in $\mathbb{C}$ (Johansen, 2011). Under such conditions, $T_m$ is bounded on $L^p$ for $1 < p < \infty$ and often also of weak type $(1,1)$. These results generalize the work of Clerc–Stein and Stanton–Tomas for noncompact symmetric spaces.

Transference theorems play a fundamental role in connecting spherical Fourier multiplier theory with classical Euclidean multiplier results. In the context of compact symmetric spaces $U/K$, a two-sided transference theorem establishes that operators $T_m$ are bounded on $L^p(U/K)$ if and only if the corresponding multipliers are bounded on $L^p(i\mathfrak{p})$, where $i\mathfrak{p}$ is the tangent space at the identity coset (Gupta et al., 2017). These mechanisms generalize deLeeuw's classical result on multipliers for $\mathbb{T}$ and $\mathbb{R}$.

In the case of product symmetric spaces $X_1\times X_2$, Marcinkiewicz-type multiplier theorems provide precise mixed smoothness conditions and localization in the complex domain under which boundedness holds (Meda et al., 2019).

3. Connections to Jacobi Analysis and Special Function Transforms

In many rank-one symmetric spaces, spherical functions can be parametrized as Jacobi functions

$$\varphi^{(\alpha,\beta)}_\lambda(t) = {}_2F_1\left(\frac{\alpha+\beta+1-i\lambda}{2},\frac{\alpha+\beta+1+i\lambda}{2};\alpha+1;- \sinh^2t\right)$$

where suitable choices of $\alpha,\beta$ correspond to the geometric multiplicities of the space in question (Johansen, 2011). This identification enables a "Jacobi analysis approach" to multiplier problems, unifying their treatment across noncompact symmetric, Damek–Ricci, and even Heckman–Opdam (hypergeometric) settings. The corresponding multiplier theorems become portable between frameworks by varying these parameters.

Additionally, the multiplier approach extends naturally to the analysis of fractional integration operators ("Riesz potentials") via multipliers of the form $m_a(\lambda) = (\lambda^2 + \rho^2)^{-a/2}$, giving near-optimal $L^p\to L^q$ estimates that are dictated by the scaling behavior of the convolution kernel near the origin.

4. Eigenvalue Decay, Approximation, and Applications

Spherical Fourier multipliers play a critical role in controlling the behavior of integral operators, particularly in spectral theory and approximation on the sphere. For Mercer-type kernels $K(x,y)$ with spherical harmonic expansions

$$K(x,y) = \sum_k \sum_j a_{k,j} Y_{k,j}(x) Y_{k,j}(y),$$

the regularity of $K$ can be quantitatively linked to the decay rate of its eigenvalues via abstract Hölder-type conditions on families of multiplier operators. One achieves eigenvalue decay of order $O(n^{-1-\beta/m})$ given a $\beta$-Hölder condition for a suitable family of averaging operators, with $m$ the sphere's dimension (Jordão et al., 2014).

These results govern approximation rates, regularity transfer, and are crucial in applications such as potential theory, tomography (weighted Radon transforms), and the analysis of positive integral operators, including their spectral localization properties.

5. Spherical Multipliers in Representation Theory and Noncommutative Settings

The spherical Fourier multiplier formalism extends to noncommutative and bundle-valued contexts. On the unit sphere in $\mathbb{C}^n$, spectral multipliers for the Kohn Laplacian acting on differential forms are governed by sharp critical indices (e.g., $s > n-\frac{1}{2}$ for boundedness), where representation theory of $U(n)$ and detailed decomposition into irreducibles (with explicit eigenvalue formulas) underpins the weighted Plancherel estimates needed for the multiplier theorem (Casarino et al., 2015).

Analogous advances occur on quaternionic spheres, where a joint decomposition into quaternionic spherical harmonics indexed by two quantum numbers $(h,m)$ yields a functional calculus for subelliptic operators. Here, Mihlin–Hörmander multiplier theorems are established with critical indices reflecting the sub-Riemannian geometry, and explicit formulas for the kernels of projections onto each spectral subspace are achieved (Ahrens et al., 2016, Ariyo et al., 3 Sep 2025).

In the context of homogeneous groups and motion groups (e.g., quaternionic Heisenberg group), multipliers are constructed as functions on the spectral parameter space associated with the commutative algebra of $K$-bi-invariant functions, where $K$ is a compact group of automorphisms. The spherical Fourier multiplier $T_{\mathcal{M}}$ acts by $$\hat{T_{\mathcal{M}} f}(\lambda, n) = \mathcal{M}(\phi_{\lambda, n})\hat{f}(\lambda, n)$$, with $L^p$ boundedness inferred from kernel and square function estimates (Ariyo et al., 3 Sep 2025).

6. Schatten-von Neumann Properties, Compactness, and Operator Ideals

For compact Gelfand pairs, the discrete nature of the spherical spectrum allows spherical multipliers to be classified by operator ideals. Under summability conditions $m \in \ell^p(S^+)$, the multiplier operator $T_m$ belongs to the Schatten–von Neumann class $S_p(L^2(G))$, with explicit bounds: $\|T_m\|_{S_p} \leq C \|m\|_{\ell^p(S^+)}$ for $1 < p < \infty$, and $T_m$ is trace class if $m \in \ell^1(S^+)$ (Mensah et al., 12 Jul 2024). Such fine operator-theoretic classifications have implications for localization, approximation, and regularity of functions in both harmonic analysis and applications to partial differential equations.

7. Computational and Applied Aspects

Practical computation of spherical Fourier multipliers is addressed via rapid algorithms for transforming between spherical harmonic and bivariate Fourier series representations (Slevinsky, 2017), via fast and backward-stable transforms allowing efficient spectral methods and the application of multipliers in high-dimensional and data-centric contexts.

In applications ranging from signal processing to cosmology (Samushia, 2019), the precise control afforded by spherical Fourier multipliers over spectral localization, smoothing, and mode selection (including new Fourier bases tailored to survey geometry) provides crucial efficiency, decorrelation, and statistical advantages.

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Summary Table: Spherical Fourier Multiplier Criteria and Settings

Setting	Multiplier Condition Type	Key Boundedness Result
Spheres ($S^d$)	Smoothness/decay of cosine transform (Denson, 30 Oct 2024)	$L^p$ uniform boundedness via $C_p(m)$
Noncompact symmetric, Damek–Ricci	Holomorphy, strip derivative bounds (Johansen, 2011)	$L^p$ + weak-type $(1,1)$
Product symmetric spaces	Marcinkiewicz multi-index (Meda et al., 2019)	$L^p$ boundedness for $N_j>(n_j+3)/2$
Compact Gelfand pairs	Summability $\ell^p$ (Mensah et al., 12 Jul 2024)	Schatten–von Neumann classification

Spherical Fourier multipliers thus constitute a central structural tool in the spectral analysis of geometric and group-invariant operators. Their rigorous characterization, boundedness, and spectral properties make them indispensable in modern harmonic analysis, representation theory, geometric analysis, and computational mathematics.