Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Properties and applications of partial multiple weights for fractional integrals

Wang Dinghuai, Yin Huicheng

TL;DR

This work introduces partial multiple weights and develops Rubio de Francia extrapolation within the partial Muckenhoupt framework, enabling robust two-weight and off-diagonal extrapolation for fractional operators. The authors prove that a commutator bound $\|[b,I_{\alpha}](f)\|_{L^{q}(u^{q}w^{q})}$ is equivalent to $b\in\mathrm{BMO}$ under precise partial-weight conditions, and they obtain weighted estimates for $I_{\alpha}$ and its commutator using novel factorizations and extrapolation tools. Key contributions include three new Rubio de Francia extrapolation theorems for partial weights, endpoint and off-diagonal estimates for fractional operators, and weighted CK–N inequalities and Fefferman–Phong-type inequalities in the partial-weight setting. The results yield a partial resolution to an open question by Cruz-Uribe on commutators and establish a versatile framework with potential applications in weighted PDEs and harmonic analysis under nonstandard weight structures.

Abstract

In this paper, through the introduction of partial multiple weights, we firstly study the related Rubio de Francia extrapolation theorem within the framework of partial Muckenhoupt classes and further obtain the corresponding extrapolation theorem for two types of off-diagonal estimates. Secondly, we establish some weighted estimates for fractional integrals associated with partial Muckenhoupt weights. As applications, several basic inequalities (including the Fefferman-Phong inequality, the degenerate Poincaré inequality and the Caffarelli-Kohn-Nirenberg inequality) related to partial Muckenhoupt weights are derived. Meanwhile, our results can give the characterization of the commutators of fractional integrals, which yields a partial answer to an open question proposed by D. Cruz-Uribe in the paper [D. Cruz-Uribe, Two weight inequalities for fractional integral operators and commutators, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2017, 25-85].

Properties and applications of partial multiple weights for fractional integrals

TL;DR

This work introduces partial multiple weights and develops Rubio de Francia extrapolation within the partial Muckenhoupt framework, enabling robust two-weight and off-diagonal extrapolation for fractional operators. The authors prove that a commutator bound is equivalent to under precise partial-weight conditions, and they obtain weighted estimates for and its commutator using novel factorizations and extrapolation tools. Key contributions include three new Rubio de Francia extrapolation theorems for partial weights, endpoint and off-diagonal estimates for fractional operators, and weighted CK–N inequalities and Fefferman–Phong-type inequalities in the partial-weight setting. The results yield a partial resolution to an open question by Cruz-Uribe on commutators and establish a versatile framework with potential applications in weighted PDEs and harmonic analysis under nonstandard weight structures.

Abstract

In this paper, through the introduction of partial multiple weights, we firstly study the related Rubio de Francia extrapolation theorem within the framework of partial Muckenhoupt classes and further obtain the corresponding extrapolation theorem for two types of off-diagonal estimates. Secondly, we establish some weighted estimates for fractional integrals associated with partial Muckenhoupt weights. As applications, several basic inequalities (including the Fefferman-Phong inequality, the degenerate Poincaré inequality and the Caffarelli-Kohn-Nirenberg inequality) related to partial Muckenhoupt weights are derived. Meanwhile, our results can give the characterization of the commutators of fractional integrals, which yields a partial answer to an open question proposed by D. Cruz-Uribe in the paper [D. Cruz-Uribe, Two weight inequalities for fractional integral operators and commutators, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2017, 25-85].

Paper Structure

This paper contains 21 sections, 45 theorems, 181 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Open question and main results
  3. Sketch on the proofs of Theorems \ref{['ex-Apqu']}-\ref{['ex-Appqu']}
  4. Sketch on the proof of Theorem \ref{['D-OD']}
  5. Partial multiple weighted estimates for fractional operators
  6. Sketch on the proof of Theorem \ref{['main']}
  7. Properties of partial multiple weights
  8. Characterization of the partial multiple weights
  9. Power weights in partial Muckenhoupt weights
  10. Factorization for partial Muckenhoupt weights
  11. The reverse Hölder property
  12. The relation between $A^{u}_{\vec{p},q}$ and $A_{\vec{p},q}$.
  13. Extrapolation theorems for partial multiple weights
  14. The extrapolation theorem of Rubio de Francia with $A_{p,q}^u$ weights
  15. Extrapolation theorem for two types of off-diagonal estimates
  16. ...and 6 more sections

Key Result

Theorem 1.6

Let $0\leq \beta<\alpha<d$, $1 < p\leq q< \infty$, $\frac{\beta}{d}=\frac{1}{p}-\frac{1}{q}$ and $1\leq s<\frac{d}{\alpha-\beta}$. Then $b\in {\rm BMO}$ if and only if $\|[b,I_{\alpha}](f)\|_{L^{q}(u^{q}w^{q})}\lesssim \|u\|_{M^{\frac{d}{\alpha-\beta}}_{s}(\mathbb{R}^d)}\|f\|_{L^{p}(w^{p})}$ for any

Figures (1)

  • Figure :

Theorems & Definitions (80)

  • Definition 1.5
  • Theorem 1.6
  • Remark 1.7
  • Remark 1.8
  • Remark 1.9
  • Theorem 1.12
  • Theorem 1.13
  • Theorem 1.14
  • Theorem 1.15
  • Theorem 1.16
  • ...and 70 more