Oscillation and variation for the Hilbert transform
For the Hilbert transform \[ Hf(x)= \text{p.v. }{1\over\pi} \int_{\mathbb{R}} t^{-1} f(x-t) dt,\tag{1} \] one defines \[ H_\varepsilon f(x)= {1\over\pi} \int_{|t|>\varepsilon} t^{-1}f(x- t) dt. \] In this paper, the authors consider the oscillation and variation of the family \(\{H_\varepsilon\}\) as \(\varepsilon\) goes to zero, which gives extra information on their convergence, as well as an estimate on the number of \(\lambda\)-jumps they can have. For each fixed decreasing sequence \((t_i)\to 0\), one defines the oscillation operator \[ {\mathcal O}(H_* f)(x)= \Biggl(\sum^\infty_{i=1} \sup_{t_{i+1}\leq \varepsilon_{i+1}< \varepsilon_i\leq t_i}|H_{\varepsilon_i}f(x)- H_{\varepsilon_{i+1}} f(x)|^2\Biggr)^{1/2} \] and the variation operator \[ {\mathcal V}_q(H_* f)(x)= \sup\Biggl(\sum^\infty_{i=1}|H_{\varepsilon_i}f(x)- H_{\varepsilon_{i+1}} f(x)|^q\Biggr)^{1/q}, \] where the supremum is taken over all decreasing sequence \((\varepsilon_i)\) going to zero. One also defines \[ V_k(H_* f)(x)= \sup \Biggl(\sum_{2^{-k}< \varepsilon_{i+1}< \varepsilon_i\leq 2^{-k+1}}|H_{\varepsilon_i} f(x)- H_{\varepsilon_{i+1}} f(x)|^2\Biggr)^{1/2}, \] where the supremum is taken over all decreasing sequences. Define the ``short variation'' operator \[ S_V(H_* f)(x)= \Biggl(\sum_{k\in\mathbb{Z}} (V_k(H_*)f(x))^2\Biggr)^{1/2}. \] The following is one of the main theorems in this reviewed paper: Theorem. For \(1< p< \infty\) and \(q> 2\), all operators \({\mathcal O}(H_* f)\), \({\mathcal V}_q(H_*f)\) and \(S_V(H_*f)\) are bounded on \(L^p(\mathbb{R})\). When \(p= 1\), and \(q> 2\), all the above operators are of weak type (1,1).
scientific article; zbMATH DE number 1820754