Application of the Littlewood-Paley method to Calderon-Zygmund operators
Mykola Yaremenko · DOAJ (DOAJ: Directory of Open Access Journals) · 2022
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.