Bessel Pairs and Weighted Inequalities

• Bessel pairs are pairs of weight functions defined by the existence of a positive solution to a Bessel-type ODE, leading to Hardy, Hardy–Rellich, or Rellich inequalities.
• They employ methodological tools like Picone identities, ground-state transforms, and spherical decompositions to establish weighted inequalities in Euclidean, anisotropic, hyperbolic, and subelliptic contexts.
• The Bessel-pair framework not only certifies inequality validity but also identifies explicit extremals and sharp constants applicable to diverse geometric and operator settings.

Searching arXiv for recent and foundational papers on Bessel pairs in analysis and related contexts. arXiv search: "Bessel pairs Hardy Rellich inequalities"

A Bessel pair most commonly denotes a pair of weights, usually written $(V,W)$ or $(W,H)$, whose admissibility is characterized by the existence of a positive solution of an associated Bessel-type ordinary differential equation, and whose principal analytic consequence is a Hardy-, Hardy–Rellich-, or Rellich-type inequality. In the recent arXiv literature, this notion is developed in the Euclidean $L^2$ setting recalled from Ghoussoub–Moradifam, extended to $L^p$, anisotropic elliptic operators, hyperbolic space, Baouendi–Grushin structures, Carnot-group settings, and abstract vector-field frameworks; at the same time, several papers use the word “pair” in Bessel-related contexts that are not Bessel pairs in this sense (Ruzhansky et al., 2021, Cossetti et al., 13 Nov 2025).

1. Definition through positive solutions of Bessel-type equations

In the classical Euclidean $L^2$ formulation recalled in the recent unified treatment, one considers positive radial weights $V,W$ for which

$$\int_{B_R} V(x)\,|\nabla u|^2\,dx \ge \int_{B_R} W(x)\,|u|^2\,dx \qquad \forall u\in C^\infty(B_R),$$

and Ghoussoub–Moradifam’s criterion is the positivity of a radial ODE solution: $$y''(r)+\Big(\frac{N-1}{r}+\frac{V_r(r)}{V(r)}\Big)y'(r)+\frac{W(r)}{V(r)}y(r)=0 \quad \text{on }(0,R].$$ The same principle is recast abstractly as the defining mechanism of a Bessel pair: the weight pair is admissible precisely when the associated differential equation has a positive solution (Cossetti et al., 13 Nov 2025).

For Grushin-type identities, the notion is stated explicitly. A pair $(V,W)$ of $C^1$-functions is a Bessel pair on $(W,H)$0, $(W,H)$1, if

$(W,H)$2

admits a positive solution $(W,H)$3 on $(W,H)$4. A pair $(W,H)$5 is a $(W,H)$6-dimensional Bessel pair if

$(W,H)$7

admits a positive solution $(W,H)$8 on $(W,H)$9. This $L^2$0-dimensional form is the one used directly for the Baouendi–Grushin geometry of homogeneous dimension $L^2$1 in the case $L^2$2 (Ganguly et al., 2024).

The $L^2$3 extension replaces the linear equation by a quasilinear one. In the anisotropic framework,

$L^2$4

is the radial ODE whose positive solvability characterizes the pair $L^2$5. The paper presenting this formulation emphasizes the logical chain

$L^2$6

and explicitly identifies this with the Ghoussoub–Moradifam Bessel-pair philosophy (Ruzhansky et al., 2021).

2. Hardy inequalities and the Bessel-pair mechanism

The first major role of Bessel pairs is the generation of weighted Hardy inequalities. In the anisotropic $L^2$7 setting, one works on a bounded domain with smooth boundary, with a symmetric, uniformly positive definite matrix $L^2$8, anisotropic norm

$L^2$9

and a quasi-norm $L^p$0. The basic inequality has the form

$L^p$1

and for $L^p$2 this specializes to

$L^p$3

Theorem 2.2 in that paper gives a sufficient criterion: if $L^p$4 and $L^p$5 are positive radially symmetric functions, $L^p$6,

$L^p$7

and

$L^p$8

then the Hardy inequality holds for all complex-valued $L^p$9 (Ruzhansky et al., 2021).

The same principle is formulated more abstractly in a vector-field setting. If there exist an open set $L^2$0, a vector field $L^2$1, weights $L^2$2, $L^2$3, a constant $L^2$4, and a nonzero real function $L^2$5 satisfying

$L^2$6

then

$L^2$7

There is also a non-directional version with $L^2$8 in place of $L^2$9. In both formulations, $V,W$0, when admissible, is the candidate maximiser (Cossetti et al., 13 Nov 2025).

A central feature of the newer literature is that Bessel pairs do not merely certify inequalities: they often identify explicit extremals or extremal candidates. For the model choice

$V,W$1

the unified paper obtains

$V,W$2

and in homogeneous settings proves sharp constants for power-law and logarithmic Hardy inequalities, as well as Gaussian-weighted variants and annular-domain improvements (Cossetti et al., 13 Nov 2025).

3. Rellich, Hardy–Rellich, and second-order identities

The second major development is the passage from first-order Hardy inequalities to second-order identities and inequalities. In the anisotropic Euclidean framework, the Rellich side uses a second-order Picone identity and a positive function $V,W$3 satisfying

$V,W$4

Under this condition,

$V,W$5

and for $V,W$6,

$V,W$7

The same paper recovers the standard sharp Rellich inequality

$V,W$8

for $V,W$9 (Ruzhansky et al., 2021).

For Baouendi–Grushin operators, Bessel pairs become the engine of exact ground-state transform identities. If $$\int_{B_R} V(x)\,|\nabla u|^2\,dx \ge \int_{B_R} W(x)\,|u|^2\,dx \qquad \forall u\in C^\infty(B_R),$$0 is a $$\int_{B_R} V(x)\,|\nabla u|^2\,dx \ge \int_{B_R} W(x)\,|u|^2\,dx \qquad \forall u\in C^\infty(B_R),$$1-dimensional Bessel pair on $$\int_{B_R} V(x)\,|\nabla u|^2\,dx \ge \int_{B_R} W(x)\,|u|^2\,dx \qquad \forall u\in C^\infty(B_R),$$2 with positive solution $$\int_{B_R} V(x)\,|\nabla u|^2\,dx \ge \int_{B_R} W(x)\,|u|^2\,dx \qquad \forall u\in C^\infty(B_R),$$3, then for $$\int_{B_R} V(x)\,|\nabla u|^2\,dx \ge \int_{B_R} W(x)\,|u|^2\,dx \qquad \forall u\in C^\infty(B_R),$$4,

$$\int_{B_R} V(x)\,|\nabla u|^2\,dx \ge \int_{B_R} W(x)\,|u|^2\,dx \qquad \forall u\in C^\infty(B_R),$$5

with an analogous radial identity. The right-hand side is nonnegative, so the identity yields a sharp inequality and an explicit remainder term. The special pairs

$$\int_{B_R} V(x)\,|\nabla u|^2\,dx \ge \int_{B_R} W(x)\,|u|^2\,dx \qquad \forall u\in C^\infty(B_R),$$6

produce the classical Hardy and Rellich constants in Grushin space (Ganguly et al., 2024).

On the hyperbolic space $$\int_{B_R} V(x)\,|\nabla u|^2\,dx \ge \int_{B_R} W(x)\,|u|^2\,dx \qquad \forall u\in C^\infty(B_R),$$7, the same strategy is pushed to second order. The relevant abstract notion is again a Bessel pair, now combined with the radial measure structure. The paper constructs

$$\int_{B_R} V(x)\,|\nabla u|^2\,dx \ge \int_{B_R} W(x)\,|u|^2\,dx \qquad \forall u\in C^\infty(B_R),$$8

and the weight

$$\int_{B_R} V(x)\,|\nabla u|^2\,dx \ge \int_{B_R} W(x)\,|u|^2\,dx \qquad \forall u\in C^\infty(B_R),$$9

For radial functions,

$$y''(r)+\Big(\frac{N-1}{r}+\frac{V_r(r)}{V(r)}\Big)y'(r)+\frac{W(r)}{V(r)}y(r)=0 \quad \text{on }(0,R].$$0

with an explicit nonnegative remainder; the full nonradial identity is then obtained by spherical harmonics. The paper states that the constants are jointly sharp for $$y''(r)+\Big(\frac{N-1}{r}+\frac{V_r(r)}{V(r)}\Big)y'(r)+\frac{W(r)}{V(r)}y(r)=0 \quad \text{on }(0,R].$$1 (Berchio et al., 2021).

4. Geometric settings and model operators

Bessel-pair methods extend well beyond the standard Laplacian. In the anisotropic framework, the standard Laplacian corresponds to $$y''(r)+\Big(\frac{N-1}{r}+\frac{V_r(r)}{V(r)}\Big)y'(r)+\frac{W(r)}{V(r)}y(r)=0 \quad \text{on }(0,R].$$2, $$y''(r)+\Big(\frac{N-1}{r}+\frac{V_r(r)}{V(r)}\Big)y'(r)+\frac{W(r)}{V(r)}y(r)=0 \quad \text{on }(0,R].$$3, and $$y''(r)+\Big(\frac{N-1}{r}+\frac{V_r(r)}{V(r)}\Big)y'(r)+\frac{W(r)}{V(r)}y(r)=0 \quad \text{on }(0,R].$$4. The same paper gives explicit quasi-norms for degenerate or subelliptic settings, including the Baouendi–Grushin operator

$$y''(r)+\Big(\frac{N-1}{r}+\frac{V_r(r)}{V(r)}\Big)y'(r)+\frac{W(r)}{V(r)}y(r)=0 \quad \text{on }(0,R].$$5

the Heisenberg gauge

$$y''(r)+\Big(\frac{N-1}{r}+\frac{V_r(r)}{V(r)}\Big)y'(r)+\frac{W(r)}{V(r)}y(r)=0 \quad \text{on }(0,R].$$6

and analogous constructions on the Engel and Cartan groups. In each case, the Hardy inequality is produced by the same Bessel-pair mechanism (Ruzhansky et al., 2021).

The unified 2025 treatment pushes the same idea to Carnot groups, Heisenberg/Greiner operators, and Baouendi–Grushin operators. It formulates $$y''(r)+\Big(\frac{N-1}{r}+\frac{V_r(r)}{V(r)}\Big)y'(r)+\frac{W(r)}{V(r)}y(r)=0 \quad \text{on }(0,R].$$7-radial inequalities of the form

$$y''(r)+\Big(\frac{N-1}{r}+\frac{V_r(r)}{V(r)}\Big)y'(r)+\frac{W(r)}{V(r)}y(r)=0 \quad \text{on }(0,R].$$8

and then specializes to homogeneous, logarithmic, Gaussian, annular, cylindrical, and boundary-sensitive settings, with explicit candidate maximisers such as

$$y''(r)+\Big(\frac{N-1}{r}+\frac{V_r(r)}{V(r)}\Big)y'(r)+\frac{W(r)}{V(r)}y(r)=0 \quad \text{on }(0,R].$$9

in the annular case (Cossetti et al., 13 Nov 2025).

A plausible implication is that Bessel pairs now function as a unifying analytic template rather than a purely radial Euclidean device: the core requirement remains the existence of a positive solution of a one-dimensional or effectively radial model equation, while the surrounding geometry may be elliptic, subelliptic, or hyperbolic.

5. Structural methods: Picone identities, ground states, and spherical decomposition

Across these papers, the method is strikingly stable. In first-order problems, one combines a positive solution of the Bessel-type equation with a Picone identity and the divergence theorem. In second-order problems, one uses a second-order Picone identity or a ground-state transform. In the Grushin setting, the Bessel pair yields an identity in which the quadratic form equals a potential term plus a nonnegative remainder, and spherical harmonics or spherical vector fields then transfer radial identities to full operator inequalities (Ganguly et al., 2024).

In hyperbolic space, the passage from radial to nonradial functions is achieved by the decomposition

$(V,W)$0

which allows the radial Bessel-pair identity to be applied mode by mode. The paper emphasizes that the radial operators retain the same constants as the full inequalities, which is crucial for sharpness analysis (Berchio et al., 2021).

The unified framework abstracts the algebraic step behind these constructions into vector-valued nonnegativity identities for $(V,W)$1. Equality forces the relevant vector fields to coincide almost everywhere, which explains why explicit candidate extremals emerge whenever the model equation has an admissible positive solution (Cossetti et al., 13 Nov 2025).

6. Terminological scope and non-equivalent uses of “pair”

The term Bessel pair is not uniform across the broader Bessel literature. Several arXiv papers use paired Bessel structures without referring to the Hardy-theoretic notion.

“Bessel operators on Jordan pairs and small representations of semisimple Lie groups” studies a Jordan pair $(V,W)$2 and a $(V,W)$3-valued second-order differential operator

$(V,W)$4

whose tangentiality to rank orbits characterizes small quotients. Here the Jordan pair is the algebraic object; the paper does not define a Bessel pair of weights (Möllers et al., 2016).

“Strong annihilating pairs for the Fourier-Bessel transform” defines a different kind of pair: measurable sets $(V,W)$5 such that

$(V,W)$6

These are annihilating pairs for the Fourier-Bessel transform, not Bessel pairs in the Hardy–Rellich sense (Ghobber et al., 2010).

Other papers employ still different pair-like structures. The multivariate Hankel paper pairs a Bessel-type differential operator with its kernel $(V,W)$7 through

$(V,W)$8

but explicitly notes that this is not a Bessel pair in the standard variational sense (Zakharov, 2024). The Poincaré–Bessel beam paper uses a zero-order and an $(V,W)$9-th order Bessel pair,

$C^1$0

in a vector-optics construction rather than in inequality theory (Kumar et al., 2023). “Constant Mean Curvature Surfaces For The Bessel Equation” is explicitly not about Bessel pairs; its only “pair-like” structures are a $C^1$1 matrix potential and a fundamental system of two scalar ODE solutions (Mota, 2019).

This suggests a stable terminological distinction. In current analysis and geometric PDE, a Bessel pair is primarily a weight pair characterized by positive solvability of a Bessel-type ODE or PDE and used to produce Hardy- and Rellich-type inequalities. In neighboring literatures, “pair” may instead refer to Jordan pairs, annihilating pairs, operator/kernel pairs, or paired Bessel modes.