Triebel-Lizorkin space, singular kernels, generalized Marcinkiewicz integrals, $ L^p $ estimates, extrapolation,

Abstract

We study the almost everywhere pointwise convergence of the Boussinesq operator along sequences $ \{t_n\}_{n = 1}^\infty $ with $ \lim\limits_{n\rightarrow \infty} t_n = 0 $ in one dimension. We obtain a characterization of convergence almost everywhere when $ \{t_n\}\in l^{r, \infty}(\mathbb{N}) $ for all $ f\in H^s(\mathbb{R}) $ provided $ 0 < s < \frac12 $.

Keywords

• boussinesq operator
• pointwise convergence
• sobolev space
• lorentz space