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WEIGHTED BOUNDEDNESS OF THE FRACTIONAL MAXIMAL OPERATOR AND RIESZ POTENTIAL GENERATED BY GEGENBAUER DIFFERENTIAL OPERATOR

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Abstract

In the paper we study the weighted L p,ω,λ , L q,ω,λ -boundedness of the fractional maximal operator M α G (G is a fractional maximal operator) and the Riesz potential (G is the Riesz potential) generated by the Gegenbauer differential operator

Key takeaways
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  1. The study focuses on the weighted Lp,ω,λ and Lq,ω,λ-boundedness of fractional maximal operators.
  2. Gegenbauer differential operator plays a crucial role in defining the fractional maximal operator MαG and Riesz potential.
  3. The paper establishes the equivalence of G-fractional maximal functions MαG and Mαµ for weighted inequalities.
  4. It introduces Muckenhoupt-type weight classes vital for proving inverse Hölder's inequality.
  5. The results include weighted boundedness for both G-fractional maximal operator and G-Riesz potential.

References (32)

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  4. A. P. Calderon, Inequalities for the maximal function relative to a metric. Studia Math. 57 (1976), no. 3, 297-306.
  5. R. R. Coifman, C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals. Studia Math. 51 (1974), 241-250.
  6. Y. Ding, S. Z. Lu, Weighted norm inequalities for fractional integral operator with rough kernel. Canad. J. Math. 50 (1998), 29-39.
  7. L. Durand, P. M. Fishbane, L. M. Simmons, Expansion formulas and addition theorems for Gegenbauer functions. J. Mathematical Phys. 17 (1976), no. 11, 1933-1948.
  8. C. Fefferman, E. Stein, Some maximal inequalities. Amer. J. Math. 93 (1971), 107-115.
  9. A. I. Gadziyev, On fractional maximal functions and fractional integrals, generated by Bessel differential operators. Trans. Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci. 20 (2000), no. 4, Math. Mech., 52-63, 265.
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  15. E. J. Ibrahimov, A. Akbulut, The Hardy-Littlewood-Sobolev theorem for Riesz potential generated by Gegenbauer operator. Trans. A. Razmadze Math. Inst. 170 (2016), no. 2, 166-199.
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FAQs

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What are the implications of the G-fractional maximal operator's boundedness?add

The findings highlight that the G-fractional maximal operator exhibits weighted (L p,w,λ , L q,w,λ )-boundedness, which is crucial for applications in harmonic analysis. This indicates significant advancements over previous operators like the fractional integrals.

How does the boundedness of G-Riesz potential compare to previous results?add

The G-Riesz potential's boundedness parallels findings from related potentials, demonstrating strong inequalities under conditions where the parameter α is confined between 0 and 2λ + 1. This reaffirms the utility of Riesz potentials in harmonic analysis.

What defines the class A λ p for weight functions in this context?add

Weight functions ω belong to the class A λ p (R +) if they satisfy specific growth conditions, particularly involving constants that compare local measures. The study finds that sh α u is in A λ p if - (2λ + 1) < α < (2λ + 1)(p - 1).

What key inequalities are proven for the G-fractional maximal operator?add

The study establishes a weighted analogue of the Fefferman-Stein inequality, demonstrating that for certain weight classes, the G-fractional maximal operator retains boundedness across various function spaces. This includes showcasing Chebyshev-type inequalities applicable for all α > 0.

When were the G-maximal functions and their properties formally introduced?add

The G-maximal functions were introduced in a framework established by E. V. Guliyev, who investigated the analogs of Muckenhoupt classes, emphasizing their role in generating fractional maximal functions associated with differential operators starting around 2019.

About the author
Saadat Cafarova
Vanderbilt University, Undergraduate
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