Characterization of the Boundedness of the Discrete Fractional Maximal Operator Between Weighted Lorentz Sequence Spaces

2026-08-26T07:27:40+00:00

In this paper, we give the complete characterization of the boundedness of the discrete fractional maximal operator $M_\gamma $, defined on $\mathbb{Z}^n$, $n\in\mathbb{N}$, $\gamma\in [0,n)$, between the classical Lorentz sequence spaces $\lambda_p(v)$ and $\lambda_q(w)$, where $0<p,q<\infty$, $\{v(m)\}$ and $\{w(m)\}$ are non-negative sequences defined on $\mathbb{N}$. We first obtain a sharp upper estimate for the nonincreasing rearrangement of the discrete fractional maximal operator $M_\gamma x$, with a supremal operator involving a discrete Hardy operator and the nonincreasing rearrangement of the sequence $\{x(k)\}$, and we show that this estimate is sharp by showing a lower estimate for special sequences. Using this result, we can reduce the characterization of the boundedness of the discrete fractional maximal operator $ M_\gamma$ between the discrete weighted Lorentz sequence spaces to the boundedness of the discrete supremum operator between weighted Lebesgue sequence spaces, restricted to nonincreasing sequences. This problem was investigated in the recent paper by the authors, which enables us to obtain the final characterization.

On Bredon Cohomology with Mackey Functor in Morava $K$-Theory

2026-10-06T09:07:56+00:00

Utilizing the algebraic framework of Mackey functors, we analyze the cochain complex and explicitly compute the cellular differentials governed by restrictions and transfer maps within Morava $K$-theory $K(n)^*$. We focus on the geometric construction of the octahedral surface 2-complex, isolating purely equivariant phenomena driven by stabilizer jumps. By analyzing the complex directly over the Morava ring $K(n)_*[x,y]$, we determine the zero-th and first codifferentials. We show that while the underlying space is geometrically trivial in degree one, the equivariant chromatic classes persist due to the cluster's sector geometry. Finally, we provide a complete description of the first Bredon cohomology group $H_{Q_8}^1(X; \underline{K(n)}^*)$ expressed as a quotient of the coefficient ring. These results contribute to the structural classification of equivariant cohomology rings for finite $p$-groups with concentrated even degrees.

Bulletin of L.N. Gumilyov Eurasian National University. Mathematics, computer science, mechanics series

Characterization of the Boundedness of the Discrete Fractional Maximal Operator Between Weighted Lorentz Sequence Spaces

On Bredon Cohomology with Mackey Functor in Morava $K$-Theory