Surveys in Mathematics and its Applications

ISSN 1842-6298 (electronic), 1843-7265 (print)
 Volume 17 (2022), 431 -- 445

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APPLICATION OF THE LITTLEWOOD-PALEY METHOD TO CALDERON-ZYGMUND OPERATORS

Mykola Yaremenko

Abstract. In this article, we establish the conditions for the pseudo-differential operator T under which this operator can be represented in convolution form with the singular kernel that satisfies |∂ xα ∂ zβ k(x,z)| ≤ Aβα(L)|z|-l-m-|β |-L for all z ≠ 0, and all multi-indices α, β and L ≥ 0 such that l+m+|β |+L > 0. Also, applying the Littlewood-Paley method, we show the inverse: if a is a symbol such that |∂ xα ∂ ξβ a(x,ξ )| ≤ Aβα(1-|ξ |)(|β |-|α |)δ for some 0 ≤ δ <1, then T(g)(x)= ⟨a(x, ●)ĝ(●)exp(2π ix ●)⟩ defines a bounded pseudo-differential operator L2(Rl) ↦ L2(Rl).
We establish the necessary and sufficient conditions on the kernel K under which there exists a bounded operator T : L2 (Rl) → L2 (Rl). Finally, we establish the necessary and sufficient conditions in terms of the operator T : Lp(Rl) → Lp(Rl) under which a nonnegative Borel measure μ is absolutely continuous dμ (x)=ω(x)dx ω ∈ Ap.

2020 Mathematics Subject Classification: 46B70, 43A15, 43A22, 44A05, 44A10, 44A45.
Keywords: harmonic analysis, singular integrals, Littlewood-Paley method, Calderon-Zygmund operator, dyadic decomposition.

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Mykola Yaremenko
Department of Differential Equations, The National Technical University of Ukraine,
"Igor Sikorsky Kyiv Polytechnic Institute", Kyiv, Ukraine.
email: math.kiev@gmail.com

http://www.utgjiu.ro/math/sma