Estimates for tail functions under Riesz transforms in Grand Lebesgue Spaces

Abstract: We study the tail behaviour of measurable functions under generalized Riesz-type operators in the framework of Grand Lebesgue Spaces. By exploiting the connection between the growth of $L<sup>p$ norms and the Young--Fenchel transform, we derive explicit tail estimates from suitable $L<sup>p$ bounds. We also present model examples and apply the abstract result to the classical Riesz transforms, showing how the $L<sup>p$ growth of the operator interacts with the intrinsic tail behaviour of the input function.

• The paper establishes explicit exponential tail bounds for Riesz-type transforms in Grand Lebesgue Spaces using Young–Fenchel transforms.
• It rigorously connects Lp norm growth with tail asymptotics through detailed analysis of radial functions and singular behavior.
• The methodology refines operator norm estimates, offering practical insights for harmonic analysis and risk assessment in stochastic systems.

Tail Estimates for Riesz Transforms in Grand Lebesgue Spaces

Introduction and Background

The paper "Estimates for tail functions under Riesz transforms in Grand Lebesgue Spaces" (2603.29564) develops explicit upper bounds for the tails of measurable functions subjected to generalized Riesz-type operators, employing the analytic structure of Grand Lebesgue Spaces (GLS). A central focus is on how $L^p$ norm growth, intrinsic to such operators, dictates tail behavior, leveraging Young–Fenchel transforms. The work advances classical harmonic analysis by integrating an operator-centric approach within GLS, enabling tighter control over function tails in contexts where traditional $L^p$ spaces may be insufficient.

Grand Lebesgue Spaces and Natural Functions

Grand Lebesgue Spaces $G\psi(a,b)$, defined via norm

$$\|f\|_{G\psi(a,b)} = \sup_{p\in(a,b)} \frac{\|f\|_p}{\psi(p)},$$

provide a flexible scale of spaces that encode $L^p$ integrability profiles through generating functions $\psi$. The function $\psi[g](p) = \|g\|_p$ serves as a "natural function," explicitly characterizing the $L^p$ growth of $g$. This framework generalizes classical spaces such as $L^{p),\theta}(\Omega)$, allowing sharper detection of integration and tail behavior across a continuum of $L^p$0 values. Examples given in the paper—subgaussian generating functions $L^p$1 for probabilistic measures and power-type $L^p$2—are paradigmatic of applications ranging from probability theory to harmonic analysis.

Tail Functions and Stein’s Identity

The tail function,

$L^p$3

is fundamental to quantifying the distribution of large values of $L^p$4. Stein's identity connects the moments and tails,

$L^p$5

establishing a bridge between $L^p$6 norms and tail asymptotics. The framework reveals that controlling $L^p$7 profile implicitly constrains tail behavior, a principle leveraged extensively in this work.

Explicit Examples: Radial Functions and Tail Asymptotics

Through explicit computations with radially symmetric functions possessing jump discontinuities on $L^p$8, the paper demonstrates the operational intricacies of $L^p$9 norm and tail asymptotics:

• For $G\psi(a,b)$0, the $G\psi(a,b)$1 norm is

$G\psi(a,b)$2

valid for $G\psi(a,b)$3. The tail asymptotic satisfies

$G\psi(a,b)$4

• For $G\psi(a,b)$5, the $G\psi(a,b)$6 norm is

$G\psi(a,b)$7

for $G\psi(a,b)$8, with corresponding tail asymptotic

$G\psi(a,b)$9

These results explicitly characterize how radial decay and singularity at the origin manifest in tail regimes and operator action.

Generalized Riesz-type Operators: Boundedness in GLS

The operator $$\|f\|_{G\psi(a,b)} = \sup_{p\in(a,b)} \frac{\|f\|_p}{\psi(p)},$$0 (not necessarily linear) is called of "Riesz type" if its $$\|f\|_{G\psi(a,b)} = \sup_{p\in(a,b)} \frac{\|f\|_p}{\psi(p)},$$1 norm satisfies

$$\|f\|_{G\psi(a,b)} = \sup_{p\in(a,b)} \frac{\|f\|_p}{\psi(p)},$$2

for constants $$\|f\|_{G\psi(a,b)} = \sup_{p\in(a,b)} \frac{\|f\|_p}{\psi(p)},$$3. Such operators encompass classical singular integrals and Riesz transforms, capturing endpoint singularities and norm growth within $$\|f\|_{G\psi(a,b)} = \sup_{p\in(a,b)} \frac{\|f\|_p}{\psi(p)},$$4. The paper rigorously establishes embeddings:

$$\|f\|_{G\psi(a,b)} = \sup_{p\in(a,b)} \frac{\|f\|_p}{\psi(p)},$$5

where $$\|f\|_{G\psi(a,b)} = \sup_{p\in(a,b)} \frac{\|f\|_p}{\psi(p)},$$6, substantiating continuity and boundedness across GLS via precise variational analysis.

An explicit minimization over $$\|f\|_{G\psi(a,b)} = \sup_{p\in(a,b)} \frac{\|f\|_p}{\psi(p)},$$7 for $$\|f\|_{G\psi(a,b)} = \sup_{p\in(a,b)} \frac{\|f\|_p}{\psi(p)},$$8 yields the optimal operator constant:

$$\|f\|_{G\psi(a,b)} = \sup_{p\in(a,b)} \frac{\|f\|_p}{\psi(p)},$$9

with $L^p$0, sharpening estimates and clarifying boundary growth.

Main Tail Estimate: Young–Fenchel Transform

The main theorem formalizes tail bounds for $L^p$1:

$L^p$2

where $L^p$3 with $L^p$4, and $L^p$5 denotes the Young–Fenchel transform. This result synthesizes operator norm growth, function integrability, and their joint effect on tail decay, providing exponential upper bounds tailored by the structural data of $L^p$6 and $L^p$7.

Application: Classical Riesz Transforms

For the $L^p$8-th classical Riesz transform

$L^p$9

the operator norm satisfies

$\psi$0

valid for $\psi$1. The endpoint behavior exhibits a singularity at $\psi$2, $\psi$3, and linear growth as $\psi$4, $\psi$5. The explicit tail estimate for $\psi$6 thus hinges on both the decay profile of $\psi$7 and the operator’s norm growth, exemplified by

$\psi$8

with $\psi$9 constructed from the product of the operator profile and the natural function.

Implications and Outlook

The results provide a general analytic device for quantifying the tail behavior of functions under Riesz-type transformations, effectively bridging $\psi[g](p) = \|g\|_p$0 norms and tail estimates with flexibility afforded by Grand Lebesgue Spaces. This methodology is salient in settings where operators may not be bounded on $\psi[g](p) = \|g\|_p$1 or $\psi[g](p) = \|g\|_p$2 and is particularly relevant for singular integral analysis, ergodic theory, and probabilistic tail control.

Theoretically, the approach clarifies the operational geometry of harmonic analysis, delineating how tail decay and operator singularity interact under varying $\psi[g](p) = \|g\|_p$3-scales. Practically, such tail bounds may inform risk assessment in stochastic PDEs, statistical learning for rare event analysis, and the calibration of operator-theoretic algorithms.

Future developments may extend to multilinear operators, anisotropic spaces, non-Euclidean settings, or situations involving random fields and distributions with heavy tails, leveraging the flexibility of GLS and the power of the Young–Fenchel transform for exponential estimates.

Conclusion

This work systematically elucidates the interplay between operator $\psi[g](p) = \|g\|_p$4 norm growth and tail function estimates within Grand Lebesgue Spaces, deriving sharp exponential bounds for generalized Riesz-type transforms. The explicit linkage via Young–Fenchel transforms provides both theoretical insight and practical bounds, establishing a comprehensive analytic toolkit for tail control in advanced harmonic and functional analysis (2603.29564).