Almost orthogonal operators on the bitorus. II
Let \(x, y \in [0, 2\pi ]\) and let \(N(x, y)\) be a real-valued measurable function. The author considers the following singular integral, defined in principal value, \[ T^{0}f(x,y)=\sum_{h=0}^{\infty}e^{iN(x,y)y'}\varphi_h(y') \sum_{k\geq h}e^{iN(x,y)x'}\varphi_k(x')*f(x,y) \] where \(\varphi_k(t)=2^k\varphi(2^k t)\) and \(\varphi(t)\) is a \(C^\infty\) function supported on \(\{|t| \leq 2\pi \}\) such that \(1/t=\sum_{k=0}^{\infty}\varphi_k(t)\) for \(|t| \leq \pi\), \(t\neq 0\). Roughly speaking \(T^{0}f(x, y)\) is given by \[ \int_{-\pi}^{\pi}\int_{|x'|<|y'|}\frac{e^{iN(x,y)x'}}{x'}\frac{e^{iN(x,y)y'}}{y'}f(x-x', y-y')dx'dy' \] with the cut off \(|x'|<|y'|\) performed in a smooth and dyadic way. Under the basic assumption that \(N(x, y)\) is ''mainly'' a function of \(y\) and the additional assumption that \(N(x, y)\) is nondecreasing in \(x\) for every fixed \(y\), the author gives the result that the operator \(T^0\) satisfies \[ \big|\{|T^0f(x, y)|>\alpha \} \big| \leq c_{\varepsilon} \Big (\frac{\|f\|_2}{\alpha} \Big )^{2-\varepsilon} \] for any \(\varepsilon >0\). It is worth noting that in the paper there is no regularity whatsoever on the dependency of \(N(x, y)\) on the variable \(y\) and just a monotonicity condition on the variable \(x\).
a.e. convergence of double Fourier series
scientific article; zbMATH DE number 6050607
scientific article