Some bounded results of \(\theta (t)\)-type singular integral operators
Let \(B\) be a Banach space in UMD (unconditionality of martingale differences) with an unconditional basis. Let \(L^p_B(\mathbb{R}^n)\) be the space of \(B\)-valued functions, \(1\leq p<\infty\) with the standard \(L^p_B\)-norm. A \(\theta(t)\)-type singular integral operator \(T: L^p_B(\mathbb{R}^n)\to L^p_B(\mathbb{R})\) is defined as \(T(f_j)= \text{p.v. }(k_j* f_j)= (\text{p.v. }k_j\neq f_j)\), where \(K= (k_j)\) satisfies the following conditions: a) \(\|K\|_\infty\leq C\); b) \(|K(x)|\leq C|x|^{-n}\) \((x\in \mathbb{R}^n\setminus \{0\})\), where \(\widehat K\) is the Fourier transform of \(K\) and \(C\) is a constant; c) \(|K(x)- K(x-y)|\leq C\theta({|y|\over|x|})|x|^{-n}\), \(2|y|<|x|\), where \(\theta(t)\) is a nondecreasing function on \([0,+\infty)\) with \(\theta(0)= 0\), \(\theta(2t)\leq C\theta(t)\) and \(\int^1_0{\theta(t)\over t} dt< +\infty\). Main results: \(T\) is weakly bounded in \(L^1_B(\mathbb{R}^n)\) and bounded on \(L^p_B(\mathbb{R}^n)\), \(1< p< +\infty\). \(T\) is bounded from the \(B\)-valued Hardy space \(H^1_B(\mathbb{R}^n)\) to \(L^1_B(\mathbb{R}^n)\) and bounded on \(H^1_B(\mathbb{R}^n)\).
\(\theta(t)\)-type singular integral operator
scientific article; zbMATH DE number 1506299
scientific article