Balakrishnan Integral Representation

• The Balakrishnan integral representation is a resolvent-based formula for fractional powers of positive or sectorial operators, providing a canonical approach to defining operators like the fractional Laplacian.
• It extends to nonlinear contexts by representing the fractional p-Laplacian through a modified datum, transferring the nonlinearity from the operator to the input while preserving the integration structure.
• The framework is equivalent to semigroup and extension formulas, offering unified representations for bounded domains and manifolds and facilitating numerical approximation methods.

Searching arXiv for recent and foundational papers relevant to the Balakrishnan integral representation. Balakrishnan integral representation is a resolvent-based formula for fractional powers of operators. In its classical linear form, for a positive or sectorial operator $A$ and $0$A^s$ through an integral involving the resolvent $(t+A)^{-1}$ or, equivalently, the semigroup generated by $A$. In the setting of the Laplacian, this yields a standard representation of the fractional Laplacian. A recent nonlinear analogue replaces the linear datum by the nonlinear vertical difference $\Phi_p(u(x_0)-u(\cdot))$, thereby representing the fractional $p$-Laplacian through the resolvent of the ordinary Laplacian while preserving the characteristic Balakrishnan integration in the spectral parameter $t$ (Teso et al., 2020).

1. Classical operator-theoretic form

For a sectorial, or more generally positive, linear operator $A$ on a Banach space, Balakrishnan’s formula for the fractional power $A^s$, $0

$0

In the Laplacian convention used for $0

$0

and the paper on the fractional $0

$0

for the fractional Laplacian (Teso et al., 2020).

The essential structural feature is the integration of the resolvent response $0

A common misconception is to identify every resolvent integral with a Balakrishnan formula in the strict operator-theoretic sense. The classical formula is specifically tied to fractional powers of linear sectorial operators. Later nonlinear analogues preserve the integral architecture but not the full linear functional-calculus framework.

2. Nonlinear representation for the fractional $0

For $A^s$0 and $A^s$1, the fractional $A^s$2-Laplacian on $A^s$3 is defined for sufficiently smooth $A^s$4 by

$A^s$5

It is the subdifferential of the Gagliardo energy

$A^s$6

and for $A^s$7 it is nonlinear (Teso et al., 2020).

The Balakrishnan-type formula proved for this operator states that for $A^s$8, $A^s$9, $(t+A)^{-1}$0, and $(t+A)^{-1}$1, with the additional condition $(t+A)^{-1}$2 when $(t+A)^{-1}$3, one has

$(t+A)^{-1}$4

where

$(t+A)^{-1}$5

If one writes

$(t+A)^{-1}$6

then the formula becomes

$(t+A)^{-1}$7

The representation is nonlinear in a specific and limited sense. The resolvent $(t+A)^{-1}$8 is still the resolvent of the linear Laplacian, but the input to that resolvent is the nonlinear difference datum $(t+A)^{-1}$9. The paper explicitly emphasizes that $A$0 is not being defined as a fractional power of the local nonlinear operator $A$1 in the sense of linear operator theory; rather, the singular integral defining the nonlocal operator is re-expressed through a linear resolvent acting on nonlinear data. In that precise sense, the formula is a nonlinear Balakrishnan formula.

When $A$2, $A$3, and

$A$4

Consequently,

$A$5

so the nonlinear formula collapses to the classical Balakrishnan formula for $A$6, after the normalization constants are matched.

3. Equivalence with semigroup and extension formulas

The same work develops three equivalent representations of the fractional $A$7-Laplacian: a semigroup formula, an extension formula, and the Balakrishnan formula. The semigroup representation is

$A$8

while the extension representation uses the solution of

$A$9

and yields

$\Phi_p(u(x_0)-u(\cdot))$0

These are proved equivalent to the singular integral definition and to the Balakrishnan representation (Teso et al., 2020).

Representation	Formula type	Linear case $\Phi_p(u(x_0)-u(\cdot))$1
Semigroup	Heat semigroup $\Phi_p(u(x_0)-u(\cdot))$2	Bochner subordination
Extension	Caffarelli–Silvestre-type extension	Classical extension formula
Balakrishnan	Resolvent $\Phi_p(u(x_0)-u(\cdot))$3	Classical Balakrishnan formula

The passage from semigroup to resolvent uses the standard identity

$\Phi_p(u(x_0)-u(\cdot))$4

At the kernel level, the resolvent is written as convolution with $\Phi_p(u(x_0)-u(\cdot))$5, and

$\Phi_p(u(x_0)-u(\cdot))$6

The Balakrishnan formula is then obtained by integrating the semigroup representation against an exponential weight and exploiting the scaling of the resolvent kernel until the singular kernel $\Phi_p(u(x_0)-u(\cdot))$7 emerges.

This equivalence is structurally important. It shows that the Balakrishnan representation is not an isolated formal identity but one face of a three-way correspondence: $\Phi_p(u(x_0)-u(\cdot))$8 The paper describes the underlying mechanism as a splitting procedure in which the nonlinearity is moved into the datum while the linear Laplacian supplies the semigroup, extension operator, and resolvent.

4. Normalization and limiting regimes

A central issue is the choice of the normalization constant $\Phi_p(u(x_0)-u(\cdot))$9. The paper selects it so that the operator depends continuously on both $p$0 and $p$1, in particular so that

$p$2

and

$p$3

The proposed normalization is

$p$4

For $p$5, this recovers the standard constant of the fractional Laplacian,

$p$6

The corresponding Balakrishnan prefactor then becomes compatible with the known linear constant (Teso et al., 2020).

The same normalization governs the limiting behavior of the Balakrishnan formula. As $p$7, the weight $p$8 concentrates near $p$9, while the constant $t$0 behaves like $t$1. Combined with the singular-integral asymptotics, this yields the local limit $t$2. As $t$3, the paper states that the natural limit under this normalization is the zero operator for sufficiently regular functions.

These limits also clarify the status of the nonlinear representation. It is calibrated to interpolate continuously between nonlocal and local regimes and between nonlinear and linear exponents. That continuity is one of the main reasons the normalization problem is not merely conventional.

5. Spectral-type operators on domains and manifolds

The semigroup and Balakrishnan formulas motivate spectral-type definitions on bounded domains. If $t$4 is smooth and $t$5 denotes the Dirichlet heat semigroup, the paper defines

$t$6

For $t$7, this is exactly the spectral fractional Laplacian generated by the Dirichlet Laplacian; for $t$8, the paper presents it as a natural nonlinear generalization (Teso et al., 2020).

A corresponding domain Balakrishnan formula is also proposed: $t$9 The equivalence between the semigroup and Balakrishnan definitions on bounded domains is explicitly left as an open problem.

The paper further proposes a manifold version. On a Riemannian manifold $A$0, with $A$1 the Laplace–Beltrami operator and $A$2 its heat semigroup, the same philosophy leads to

$A$3

Several consequences are attached to this framework. The paper proposes alternative characterizations of the seminorm

$A$4

through both semigroup and Balakrishnan expressions. It also proposes numerical schemes based on quadrature in $A$5 together with discretization of resolvent problems $A$6. The paper notes that these inherit parallelization advantages known for the linear spectral fractional Laplacian.

A second common misconception is that a spectral-type fractional $A$7-Laplacian on a domain must coincide with the restricted operator obtained by extending $A$8 by zero outside $A$9 and applying the whole-space definition. The paper states explicitly that the spectral-type construction differs from that restricted operator.

6. Related usages and generalized transform analogues

The term “Balakrishnan integral representation” also appears in a distinct arithmetic-geometric setting. For elliptic curves over imaginary quadratic fields, the quadratic Chabauty literature uses Coleman integrals

$A^s$0

to form symmetric bilinear functions

$A^s$1

and then represents a $A^s$2-adic height pairing as

$A^s$3

In that paper, this quadratic expression in Coleman integrals is explicitly called the Balakrishnan integral representation of the height pairing, and it underlies the quadratic Chabauty function

$A^s$4

used to determine integral points (Jha, 2023).

This arithmetic usage is terminologically related to Balakrishnan’s work through the research lineage of Balakrishnan and collaborators, but it is conceptually different from the classical operator-theoretic Balakrishnan formula for fractional powers. The shared theme is representational: a nontrivial object is encoded through integrals or quadratic expressions built from more elementary analytic data.

A further neighboring development arises in generalized fractional calculus. A bicomplex Prabhakar-function study establishes Mellin–Barnes, Laplace, and Mellin integral representations such as

$A^s$5

and

$A^s$6

That paper explicitly presents these transform formulas as structurally close to Balakrishnan-type formulas, because the kernels behave like scalar resolvent-type functions and suggest operator substitutions of the form $A^s$7 in a sectorial calculus (Sharma et al., 18 May 2026).

This suggests a broader encyclopedic picture. In the strict sense, the Balakrishnan integral representation belongs to resolvent-based fractional powers of operators. In a wider sense, it functions as a template: one integrates a resolvent or a transform kernel against a singular weight to reconstruct a fractional or nonlocal object. The nonlinear fractional $A^s$8-Laplacian formula is the clearest direct extension of that template, because it preserves the Balakrishnan integration in $A^s$9 while relocating the nonlinearity from the operator to the datum.