Spherical harmonics and a singular integral
Let \(H(x)= \Omega_{| x|}\Bigl({x\over | x|}\Bigr)| x|^{-n}\) (\(x\in \mathbb{R}^ n\), \(x\neq 0\), \(n\geq 2\)) be a (family of) Calderón-Zygmund kernel, where \(\Omega_ r(\xi)\) \((r>0)\) has a spherical harmonic development \(\Omega_ r= \sum^ \infty_{k=1} a_{kr} Y_{kr}(\xi)\) and \(Y_{kr}\) is a spherical harmonic of degree \(k\) normalized by the condition \(\| Y_{kr}\|_{L^ 2(S^{n- 1})}= 1\) with \(S^{n-1}\) being the unit sphere of \(\mathbb{R}^ n\). Let \(a_ k= \sup\{| a_{kr}|: r>0\}\) and define \(T(f)(x)= \text{p.v. } H* f(x)\), \(x\in \mathbb{R}^ n\). The results of this paper are: Theorem 1. Assume \(n\geq 2\) and (1) \(\{a_ k\}^ \infty_{k=1}\) is a square summable sequence, (2) \(\sum^ \infty_{k=1} a_ k k^{- {1\over 2}}= A<\infty\), then \(| \widehat H(y)|\leq cA\) for a constant \(c\) independent of \(H\); consequently, \(T\) is a bounded operator on \(L^ 2(\mathbb{R}^ n)\), \(n\geq 2\), where \(\widehat H\) is the Fourier transform of \(H\) in the following sence \[ \widehat H(y)= \lim_{\varepsilon\to 0^ +} \int_{\varepsilon< | x|< \varepsilon^{-1}} H(x) e^{-ixy} dx. \] Theorem 2. If, in addition to the hypotheses of Theorem 1, \(\sum^ \infty_{k=1} a_ k k^{-{1\over 2}+\delta}= A<\infty\) for some \(\delta> 0\) then \(T\) is a bounded operator on \(L^ p(\mathbb{R}^ n)\) for \(1< p<\infty\). In the proof the spherical harmonics are used.
scientific article; zbMATH DE number 566466