( L p , L q ) Boundedness of the fractional maximal operator associated with the Dunkl operator on the real line

Q: What conditions ensure the boundedness of the fractional maximal operator M β ?

The main theorem establishes necessary and sufficient conditions: for 1 < p < q < ∞, 1/p - 1/q = β/(2α + 4). Additionally, for L_1(K) to weak L_q(K), the condition is 1 - 1/q = β/(2α + 4).

Q: How does the fractional maximal operator behave under dilations?

The paper demonstrates that under a dilation r > 0, the operator acts as M_β f_r (x, t) = r^{-β - 2α + 4/q} M_β f W L_q(K). This behavior is crucial for establishing boundedness criteria.

Q: In what context are the generalized translation operators T (α)(x,t) used?

The operators T (α)(x,t) are employed in defining convolution products within the Laguerre hypergroup framework, crucial for analyzing properties of maximal functions. They facilitate the examination of boundedness conditions via integrative techniques.

Yagub Y. Mammadov

doi:10.1080/10652460903507248

( L p , L q ) Boundedness of the fractional maximal operator associated with the Dunkl operator on the real line

Vagif S. Guliyev andYagub Y. Mammadov and more

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Abstract

Let K = [0, ∞) × R be the Laguerre hypergroup which is the fundamental manifold of the radial function space for the Heisenberg group. In this paper we obtain necessary and sufficient conditions on the parameters for the boundedness of the fractional maximal operator on the Laguerre hypergroup from the spaces L p (K) to the spaces L q (K) and from the spaces L 1 (K) to the weak spaces W L q (K).

Key takeaways

AI generated

This paper establishes conditions for the boundedness of the fractional maximal operator on Laguerre hypergroups.
The main result links boundedness from L p (K) to L q (K) with parameters 1 < p < (2α + 4)/β.
For weak spaces, the condition 1 - 1/q = β/(2α + 4) is necessary and sufficient.
The fractional maximal operator M β is defined on the Laguerre hypergroup and relates to harmonic analysis.
Parameter conditions for boundedness cannot be weakened, ensuring robustness of results.

FAQ's

AI generated

What conditions ensure the boundedness of the fractional maximal operator M β ?add

The main theorem establishes necessary and sufficient conditions: for 1 < p < q < ∞, 1/p - 1/q = β/(2α + 4). Additionally, for L_1(K) to weak L_q(K), the condition is 1 - 1/q = β/(2α + 4).

How does the fractional maximal operator behave under dilations?add

The paper demonstrates that under a dilation r > 0, the operator acts as M_β f_r (x, t) = r^{-β - 2α + 4/q} M_β f W L_q(K). This behavior is crucial for establishing boundedness criteria.

What is the significance of using Laguerre hypergroups in this research?add

Laguerre hypergroups allow for the definition of the fractional maximal function in a generalized framework for harmonic analysis. They enable unique structural properties that facilitate the analysis of boundedness transitions between function spaces.

What numerical thresholds are defined for parameters p and q?add

For boundedness from L_p(K) to L_q(K), the conditions stipulate 1 < p < (2α + 4)/β. Furthermore, for L_1(K) to weak Lebesgue spaces W L_q(K), the condition is 1 - 1/q = β/(2α + 4).

In what context are the generalized translation operators T (α)(x,t) used?add

The operators T (α)(x,t) are employed in defining convolution products within the Laguerre hypergroup framework, crucial for analyzing properties of maximal functions. They facilitate the examination of boundedness conditions via integrative techniques.

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References (9)

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