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Estimates for tail functions under Riesz transforms in Grand Lebesgue Spaces

Maria Rosaria Formica, Eugene Ostrovsky, Leonid Sirota

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 $L^p$ norms and the Young--Fenchel transform, we derive explicit tail estimates from suitable $L^p$ bounds. We also present model examples and apply the abstract result to the classical Riesz transforms, showing how the $L^p$ growth of the operator interacts with the intrinsic tail behaviour of the input function.

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 norms and the Young--Fenchel transform, we derive explicit tail estimates from suitable bounds. We also present model examples and apply the abstract result to the classical Riesz transforms, showing how the growth of the operator interacts with the intrinsic tail behaviour of the input function.

Paper Structure

This paper contains 6 sections, 1 theorem, 116 equations.

Table of Contents

  1. Introduction and preliminaries
  2. Examples: Lp norms and tail asymptotics
  3. Boundedness of Generalized Riesz-type Operators in G psi spaces
  4. Tail estimates for operators with Lp bounds
  5. Application: the Riesz transform
  6. Conclusion

Key Result

Theorem 4.1

Let $f\in L(a,b)$, $1\leq a<b<\infty$. Let $1\leq a_1<b_1<\infty$ and $\zeta\in\Psi_{(a_1,b_1)}$. Assume Let $U$ be a generalized Riesz-type operator satisfying Define and Then Consequently, for every $t>0$, where Equivalently, in the notation of tail-from-norm,

Theorems & Definitions (12)

  • Definition 1.1: Operators with Riesz-type $L^p$ growth
  • Definition 1.2: Grand Lebesgue Spaces
  • Remark 1.3: Iwaniec-Sbordone spaces
  • Remark 1.4
  • Example 2.1
  • Example 2.2
  • Remark 2.3
  • Example 2.4
  • Remark 2.5
  • Theorem 4.1
  • ...and 2 more