The boundedness of the Marcinkiewicz integrals associated with Schrödinger operator from the localized Morrey-Campanato space to the localized Morrey-Campanato-BLO space is established. Similar results for the Marcinkiewicz integrals associated with the Schrödinger operator with rough kernels were also investigated.

We establish the continuity of bilinear fractional maximal commutator with Lipschitz symbols in Sobolev spaces, both in the global and local cases. The main ingredients of proving the main results are some new pointwise estimates for the weak derivatives of the above commutators. The continuity result in global case answers a question originally motivated by Wang and Liu in [Math. Inequal. Appl., 2022, 25(2): 573–600].

In this paper, we mainly prove some conjectural congruences of Z.-H. Sun involving Almkvist–Zudilin numbers

Let p > 3 be a prime. If p ≡ 3 (mod 4), then

It is proved that every sufficiently large odd number can be expressed as a sum of one prime, four prime cubes and 9 powers of 2. This improves the previous result for k = 15. Also, when k = 27, every sufficiently large even integer can be represented in a sum of eight cubes of primes and k powers of 2. This is an improvement of previous result for k = 28.

We show that a subcategory of the m-cluster category of type

is isomorphic to a category consisting of arcs in an (n − 2)m-gon with two central (m − 1)-gons in its interior. We show that the mutation of colored quivers and m-cluster-tilting objects is compatible with the flip of an (m + 2)-angulation. In the final part of this paper, we detail an example of a quiver of type

{\stackrel{\sim }{D}}_7

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.

The concept of Whittaker modules for finite-dimensional Lie algebras was introduced by Kostant [Invent. Math., 1978, 48(2): 101–184], where he studied the Whittaker modules with respect to nonsingular Whittaker functions in full detail. However, the Whittaker modules are not considered for the singular case, which turns out to be more complicated. In this paper, we study the Whittaker modules for the Lie algebra

, the first Lie algebra that admits singular Whittaker functions. We investigate the submodule structure of the singular universal Whittaker modules and obtain a full description of all Whittaker vectors. Consequently, we determine all maximal submodules and provide a complete classification of all simple Whittaker modules of singular type in terms of quotient modules of the universal Whittaker modules with two parameters.

In this paper, we obtain the notion of an extended Rota-Baxter operator that generalizes the Rota-Baxter operator and the modified Rota-Baxter operator on Leibniz algebras, and give examples and general properties of extended Rota-Baxter operators. Next, we define a representation theory of extended Rota-Baxter Leibniz algebras and construct a cohomology theory. Furthermore, we justify this cohomology theory by interpreting lower degree cohomology groups as formal deformations of extended Rota-Baxter Leibniz algebras. Finally, we study non-abelian extensions of an extended Rota-Baxter Leibniz algebra by another extended Rota-Baxter Leibniz algebra, and we consider abelian extensions of extended Rota-Baxter Leibniz algebras as a particular case of non-abelian extensions.

This paper is concerned with ergodic properties of inhomogeneous Markov processes. Since the transition probabilities depend on initial times, the existing methods to obtain invariant measures for homogeneous Markov processes are not applicable straightforwardly. We impose some appropriate conditions under which invariant measure families for inhomogeneous Markov processes can be studied. Specifically, the existence of invariant measure families is established by either a generalization of the classical Krylov–Bogolyubov method or a Lyapunov criterion. Moreover, the uniqueness and exponential ergodicity are demonstrated under a contraction assumption of the transition probabilities on a large set. Finally, three examples, including Markov chains, diffusion processes and storage processes, are analyzed to illustrate the practicality of our method.