We prove a new Stein-Weiss type inequality of the formfor any positive functions f is an element of L p(& Ropf;n-m), g is an element of Lq '(Rn) , and p, q ' is an element of (1, infinity), alpha, beta, mu such that n-mn1p+1q '+alpha+beta+mu n=2n-mn . As a result, we prove the existence of optimizers for such inequality and introduce a family of Stein-Weiss inequalities with partial weight in & Ropf;m. Furthermore, we consider the reverse version of the above inequality. Our approach relied on the Hardy inequality on the & Ropf;n-m & times; & Ropf;n and rearrangement argument, which allows us to study whether the best constant in this inequality is attained. We also prove asymptotic estimates and the necessary conditions of existence for the corresponding Euler-Lagrange equation.
Motivated by the work of Li and Wang [Calc. Var. Partial Differ. Equ., 62, Paper No. 135, 24 pp. (2023)], we investigate a length-preserving and an area-preserving locally constrained inverse curvature flows for ℓ-convex Legendre curves. Under the length-preserving flow, an ℓ-convex Legendre curve of algebraic length L0 deforms into a point if L0 = 0 and into a circle if L0 > 0. Under the area-preserving flow, an ℓ-convex Legendre curve of algebraic area A0 > 0 evolves to a circle. As applications of these two flows, some geometric inequalities for ℓ-convex Legendre curves are established.
Tutte’s 3-flow conjecture states that every 4-edge-connected graph admits a nowhere-zero 3-flow. A Cayley graph Cay(G, S) is quasi-abelian if the symbol S is a union of conjugacy classes of the group G. In this paper, we give a characterization of connected quasi-abelian Cayley graphs of non-abelian groups and of degree 3 or 5. Combining known results, it follows that every quasi-abelian Cayley graph of degree at least 2 admits a nowhere-zero 3-flow unless it is a disjoint union of copies of the circular ladder CL2ℓ+1 or the Möbius ladder ML2ℓ for an integer ℓ ≥ 1, which confirms Tutte’s 3-flow conjecture for quasi-abelian Cayley graphs.
In this paper, we show that every non-compact Riemann surface X admits a conformal tiling by rectangles, in such a way that every vertex belongs to finitely many rectangles.
We study the remainders of Lindelöf p-spaces and the properties of the class of charming spaces. It is proved that if X is a nowhere locally compact space with remainder Y, then X is a locally Lindelöf p-space if and only if every (or some) remainder Y of X is a Lindelöf p-space outside of some compact subspace F of Y. Some propositions and corollaries related to this statement are presented. It is shown that if g: X → Y is a perfect mapping, then X is a 2-charming space if and only if Y is a 2-charming space. It is also proved that if X is a nowhere locally compact space with a remainder Y and X is a locally Lindelöf p-space with a compactification bX such that the remainder Y = bX X is locally perfect, then Y is a Lindelöf Σ-space and X is an s-space. Some related propositions are proved. In addition, some properties of Lindelöf Σ-spaces are investigated.
