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2004•SUT Journal of MathematicsOpen access

Lp estimates for rough parametric Marcinkiewicz integrals

H. M. Al-Qassem

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Abstract

We prove the Lp boundedness of a class of parametric Marcinkiewicz integral operators ℳΩ,hρ when h satisfies a certain integrability condition and Ω belongs to the block space Bq(0,−1/2)(Sn−1) for some q>1, n≥2. Also, we obtain the Lp boundedness for a class of rough parametric Marcinkiewicz integral operators ℳΩ,h,λ*,ρ and ℳΩ,h,Sρ related to the Littlewood-Paley gλ*-function and the area integral S, respectively. Our results are essential improvement and extension of some previously known results.

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We prove the Lp boundedness of a class of parametric Marcinkiewicz integral operators ℳΩ,hρ when h satisfies a certain integrability condition and Ω belongs to the block space Bq(0,−1/2)(Sn−1) for some q>1, n≥2. Also, we obtain the Lp boundedness for a class of rough parametric Marcinkiewicz integral operators ℳΩ,h,λ*,ρ and ℳΩ,h,Sρ related to the Littlewood-Paley gλ*-function and the area integral S, respectively. Our results are essential improvement and extension of some previously known results.

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Available abstract

We prove the Lp boundedness of a class of parametric Marcinkiewicz integral operators ℳΩ,hρ when h satisfies a certain integrability condition and Ω belongs to the block space Bq(0,−1/2)(Sn−1) for some q>1, n≥2. Also, we obtain the Lp boundedness for a class of rough parametric Marcinkiewicz integral operators ℳΩ,h,λ*,ρ and ℳΩ,h,Sρ related to the Littlewood-Paley gλ*-function and the area integral S, respectively. Our results are essential improvement and extension of some previously known results.

Key concepts: Mathematics, Parametric statistics, Space (punctuation), Pure mathematics, Class (philosophy), Extension (predicate logic), Block (permutation group theory), Function (biology)

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