Vol 89, No 5 (2025)

We prove existence and uniqueness of a solution of the Dirichlet problem for separately $(\alpha, \beta)$-harmonic functions on $\mathbb D^n$ with boundary data in $C(\mathbb T^n)$ using $(\alpha, \beta)$-Poisson kernel $P_{\alpha, \beta} (z, \zeta)$. A characterization by hypergeometric functions of separately $(\alpha, \beta)$-harmonic functions which are also $m$-homogeneous is given, it is used to obtain series expansion of separately $(\alpha, \beta)$-harmonic functions. Basic $H^p$ theory of such functions is developed: integral representations by measures and $L^p$ functions on $\mathbb T^n$, norm and weak$^\ast$ convergence at the distinguished boundary $\mathbb T^n$. Weak $(1,1)$-type estimate for a restricted non-tangential maximal function $M_{A, B}^{\mathrm{NT}}$ is derived. We show that slice functions $u(z_1, …, z_k, \zeta_{k+1}, …, \zeta_n)$, where some of the variables are fixed, belong in the appropriate space of separately $(\alpha', \beta')$-harmonic functions of $k$ variables. We prove a Fatou type theorem on a.e. existence of restricted non-tangential limits for these functions and a corresponding result for unrestricted limit at a point in $\mathbb T^n$. Our results extend earlier results for $(\alpha, \beta)$-harmonic functions in the disc and for $n$-harmonic functions in $\mathbb D^n$.

This paper gives some characterizations of the boundedness of the maximal or non-linear commutator of the $p$-adic fractional maximal operator $ \mathcal{M}_{\alpha}^p$ with the symbols belong to the $p$-adic BMO spaces on (variable) Lebesgue spaces and Morrey spaces over $p$-adic field, by which some new characterizations of BMO functions are obtained in the $p$-adic field context. Meanwhile, some equivalent relations between the $p$-adic BMO norm and the $p$-adic (variable) Lebesgue or Morrey norm are given.

We perform homogenization of a thin spatial network of quantum waveguides with small nodules (the Dirichlet problem for the Laplace operator). In contrast to a problem with the Neumann boundary condition, the low-frequency but situated far away from the coordinate origin range of the spectrum of the Dirichlet problem is characterized by localization of the corresponding eigenfunctions near angular joints of ligaments or at the ligaments themselves in accordance with distribution of eigenvalues in the discrete spectra (surely non-empty) of model problems in junctions of semi-infinite cylindrical quantum waveguides of various shapes. The behavior of eigenvalues and eigenfunctions in the mid-frequency range of the spectrum depend crucially on the phenomenon of threshold resonances in the above-mentioned junctions as well as the relation between small parameters, namely, the period of distribution of the nodules and their diameter, comparable in order but bigger than diameter of the ligaments. We consider concrete cases of rectangular and circular cross-sections and formulate open questions.

A coupled system consisting of a sweeping process and an evolution maximal monotone inclusion is considered. The values of moving set of the sweeping process are prox-regular sets that depend on time and state of the system. The right-hand side of the sweeping process contains the sum of two multivalued time- and state-dependent perturbations with different semicontinuity properties. The perturbation in the right-hand side of maximal monotone inclusion is a single-valued function. A solution to the sweeping process is a right continuous function of bounded variation (a BV-solution). A solution to the maximal monotone inclusion is an absolutely continuous function. A theorem on existence of a solution to this system is proved, and when the perturbations are convex, a theorem on compactness of the solution set is established.