The paper considers differentiation properties of density bases formed of bounded open sets. We prove that two quasi-equivalent subbases of some density basis differentiate the same class of non-negative functions. Applications for bases formed of rectangles are discussed.

The paper considers differentiation properties of rare basis of dyadic rectangles corresponding to an increasing sequence of integers $\{\nu_k\}$. We prove that the condition $$ \sup_k(\nu_{k+1}-\nu_k)< \infty $$ is necessary and sufficient for such basis to be equivalent to the full basis of dyadic rectangles.

In 1927 Littlewood constructed a bounded holomorphic function on the unit disc, having no tangential boundary limits almost everywhere. This theorem was the complement of a positive theorem of Fatou (1906), establishing almost everywhere non-tangential convergence of bounded holomorphic functions. There are several generalizations of Littlewood’s theorem whose proofs are based on the specific properties of holomorphic functions. Applying real variable methods, we extend these theorems to general convolution operators.

In this paper, the concept of a g-dimonoid is introduced and the construction of a free g-dimonoid is described. (A g-dimonoid is a duplex satisfying two additional identities.)

We define $\lambda(r)$-convergence, which is a generalization of nontangential convergence in the unit disc. We prove Fatou-type theorems on almost everywhere nontangential convergence of Poisson-Stiltjes integrals for general kernels $\{\varphi_r\}$, forming an approximation of identity. We prove that the bound $$ \limsup_{r\to 1}\lambda(r) \|\varphi_r\|_\infty< \infty $$ is necessary and sufficient for almost everywhere $\lambda(r)$-convergence of the integrals $$ \int_{{\mathbb T}} \varphi_r(t-x)d\mu(t). $$