On singular integrals depending on three parameters (Q648290)

Let \(L_{2\pi}\) be the class of all real functions \(f(s,t)\), \(2\pi\) periodic separately in each variable, Lebesgue-integrable in the square \(Q=\langle -\pi,\pi \rangle \times \langle -\pi,\pi \rangle\), where \(\langle -\pi,\pi \rangle \times \langle -\pi,\pi \rangle\) is an arbitrary region such as the closed or semi-closed or open square region in \(\mathbb{R}^2\). In the paper, the authors investigate the pointwise convergence of \(L(f; x, y, \lambda)\) to \(f(x_0, y_0)\) in the space \(L_{2\pi}\), by the three parameter family of singular operators with radial kernel of the form \[ L(f;x,y,\lambda) = \int_{-\pi}^{\pi}\int_{-\pi}^{\pi}f(s,t)K\big ( \sqrt{(s-x)^2 + (t-y)^2};\lambda \big )dsdt, \] where \((x,y) \in Q\) and \(\lambda \in \Lambda \subset \mathbb{R}\). Assume that the kernel \(K(\sqrt{s^2+t^2}; \lambda): \mathbb{R}^2 \times \Lambda \rightarrow \mathbb{R}\) belongs to class \(\mathcal{A}\), if the following conditions are satisfied. (a) \(K(\sqrt{s^2+t^2}; \lambda)\) is a \(2\pi\) periodic function, even defined for all \((s,t)\) on \(Q\) and \(\lambda \in \Lambda\) (where \(\Lambda\) is a given set of numbers with accumulation point \(\lambda_0\)), measurable with respect to \((s,t)\) for each fixed \(\lambda \in \Lambda\), (b) \(||K(\cdot;\lambda)||_{L^1} \leq M <\infty\) for every \(\lambda \in \Lambda\), where \(M\) is independent on \(\lambda\), (c) \(\lim_{\lambda \rightarrow \lambda_0}K(\delta; \lambda) = 0\) for every \(\delta >0\), (d) \(\lim_{(x,y,\lambda) \rightarrow (x_0,y_0,\lambda_0)}\int_{-\pi}^{\pi}\int_{-\pi}^{\pi} K\big (\sqrt{(s-x)^2 + (t-y)^2}; \lambda \big )dsdt = 1\), (e) \(\lim_{\lambda \rightarrow \lambda_0} \big [ \sup_{\delta \leq \sqrt{s^2+t^2}}|K(\sqrt{s^2+t^2}; \lambda)| \big ] = 0\) for \(\delta > 0\) \( (\lambda \in \Lambda)\). Let \((x_0, y_0)\) be a generalized Lebesgue point of function \(f(s,t)\in L_{2\pi}^1\), i.e., \((x_0, y_0)\) satisfies the condition \[ \lim_{(h,r) \rightarrow (0,0)} \frac{1}{h^{\alpha + 1}r^{\alpha + 1}} \int_{0}^h \int_{0}^r |f(s+x_0, t+y_0) - f(x_0, y_0)|dsdt = 0, \;(0 \leq \alpha <1). \] Then the main result is \[ \lim_{(x,y,\lambda) \rightarrow (x_0,y_0,\lambda_0)}L(f;x,y,\lambda) = f(x_0,y_0), \] if \((x,y,\lambda)\) tends to \((x_0,y_0,\lambda_0)\) on any set \(Z\) on which the function \[ \int x_0-\delta^{x_0+\delta}\int_{y_0 -\delta}^{y_0+\delta}K\big (\sqrt{(s-x)^2 + (t-y)^2};\lambda \big )|d(s-x_0)^{(\alpha +1)}(t-y_0)^{(\alpha +1)}| \] is bounded.