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Let 
$\mathcal{M}$ and G denote, respectively, the maximal operator and the geometric maximal operator associated with the dyadic lattice on 
$\mathbb{R}^d$.
(i) We prove that for any 0 < p < ∞, any weight w on $\mathbb{R}^d$ and any measurable f on $\mathbb{R}^d$, we have Fefferman–Stein-type estimate
For each p, the constant e1/p is the best possible.
$$\begin{equation*}
||G(f)||_{L^p(w)}\leq e^{1/p}||f||_{L^p(\mathcal{M}w)}.
\end{equation*}
$$
(ii) We show that for any weight w on $\mathbb{R}^d$ and any measurable f on $\mathbb{R}^d$,
and prove that the constant e is optimal.
$$\begin{equation*}
\int_{\mathbb{R}^d} G(f)^{1/\mathcal{M}w}w\mbox{d}x\leq e\int_{\mathbb{R}^d} |f|^{1/w}w\mbox{d}x
\end{equation*}
$$
We provide ℕ-filtrations on the negative part Uq(
$\mathfrak{n}$−) of the quantum group associated to a finite-dimensional simple Lie algebra 
$\mathfrak{g}$, such that the associated graded algebra is a skew-polynomial algebra on $\mathfrak{n}$−. The filtration is obtained by assigning degrees to Lusztig's quantum PBW root vectors. The possible degrees can be described as lattice points in certain polyhedral cones. In the classical limit, such a degree induces an ℕ-filtration on any finite-dimensional simple $\mathfrak{g}$-module. We prove for type An, Cn, B3, D4 and G2 that a degree can be chosen such that the associated graded modules are defined by monomial ideals, and conjecture that this is true for any $\mathfrak{g}$.
In this paper, we study the multiplicity of solutions for the following problem:
where α ≥ 2, Ω is a smooth bounded domain in 
$$\begin{equation*}
\begin{cases}
-\Delta u-\Delta(|u|^{\alpha})|u|^{\alpha-2}u=g(x,u)+\theta h(x,u), \ \ x\in \Omega,\\
u=0, \ \ x\in \partial\Omega,
\end{cases}
\end{equation*}$$
${\mathbb{R}}$N, θ is a parameter and g, h ∈ C(
$\bar{\Omega}$ × ${\mathbb{R}}$). Under the assumptions that g(x, u) is odd and locally superlinear at infinity in u, we prove that for any j ∈ 
$\mathbb{N}$ there exists ϵj > 0 such that if |θ| ≤ ϵj, the above problem possesses at least j distinct solutions. Our results generalize some known results in the literature and are new even in the symmetric situation.
Let 
${\mathcal C}[{\mathcal X}]$ be any class of operators on a Banach space 
${\mathcal X}$, and let 
${Holo}^{-1}({\mathcal C})$ denote the class of operators A for which there exists a holomorphic function f on a neighbourhood 
${\mathcal N}$ of the spectrum σ(A) of A such that f is non-constant on connected components of 
${\mathcal N}$ and f(A) lies in 
${\mathcal C}$. Let 
${{\mathcal R}[{\mathcal X}]}$ denote the class of Riesz operators in 
${{\mathcal B}[{\mathcal X}]}$. This paper considers perturbation of operators 
$A\in\Phi_{+}({\mathcal X})\Cup\Phi_{-}({\mathcal X})$ (the class of all upper or lower [semi] Fredholm operators) by commuting operators in 
$B\in{Holo}^{-1}({\mathcal R}[{\mathcal X}])$. We prove (amongst other results) that if πB(B) = ∏mi = 1(B − μi) is Riesz, then there exist decompositions 
${\mathcal X}=\oplus_{i=1}^m{{\mathcal X}_i}$ and 
$B=\oplus_{i=1}^m{B|_{{\mathcal X}_i}}=\oplus_{i=1}^m{B_i}$ such that: (i) If λ ≠ 0, then 
$\pi_B(A,\lambda)=\prod_{i=1}^m{(A-\lambda\mu_i)^{\alpha_i}} \in\Phi_{+}({\mathcal X})$ (resp., 
$\in\Phi_{-}({\mathcal X})$) if and only if 
$A-\lambda B_0-\lambda\mu_i\in\Phi_{+}({\mathcal X})$ (resp., 
$\in\Phi_{-}({\mathcal X})$), and (ii) (case λ = 0) 
$A\in\Phi_{+}({\mathcal X})$ (resp., 
$\in\Phi_{-}({\mathcal X})$) if and only if 
$A-B_0\in\Phi_{+}({\mathcal X})$ (resp., 
$\in\Phi_{-}({\mathcal X})$), where B0 = ⊕mi = 1(Bi − μi); (iii) if 
$\pi_B(A,\lambda)\in\Phi_{+}({\mathcal X})$ (resp., 
$\in\Phi_{-}({\mathcal X})$) for some λ ≠ 0, then 
$A-\lambda B\in\Phi_{+}({\mathcal X})$ (resp., 
$\in\Phi_{-}({\mathcal X})$).
Given a complete hereditary cotorsion pair 
$(\mathcal{X}, \mathcal{Y})$, we introduce the concept of 
$(\mathcal{X}, \mathcal{X} \cap \mathcal{Y})$-Gorenstein projective modules and study its stability properties. As applications, we first get two model structures related to Gorenstein flat modules over a right coherent ring. Secondly, for any non-negative integer n, we construct a cofibrantly generated model structure on Mod(R) in which the class of fibrant objects are the modules of Gorenstein injective dimension ≤ n over a left Noetherian ring R. Similarly, if R is a left coherent ring in which all flat left R-modules have finite projective dimension, then there is a cofibrantly generated model structure on Mod(R) such that the cofibrant objects are the modules of Gorenstein projective dimension ≤ n. These structures have their analogous in the category of chain complexes.
Let 
$\mathbb{K}$ be a field and S = ${\mathbb{K}}$[x1, . . ., xn] be the polynomial ring in n variables over the field $\mathbb{K}$. For every monomial ideal I ⊂ S, we provide a recursive formula to determine a lower bound for the Stanley depth of S/I. We use this formula to prove the inequality sdepth(S/I) ≥ size(I) for a particular class of monomial ideals.
Let (A, 
${\mathfrak{m}$) be a Cohen–Macaulay local ring of dimension d and let I ⊆ J be two ${\mathfrak{m}$-primary ideals with I a reduction of J. For i = 0,. . .,d, let eiJ(A) (eiI(A)) be the ith Hilbert coefficient of J (I), respectively. We call the number ci(I, J) = eiJ(A) − eiI(A) the ith relative Hilbert coefficient of J with respect to I. If GI(A) is Cohen–Macaulay, then ci(I, J) satisfy various constraints. We also show that vanishing of some ci(I, J) has strong implications on depth GJn(A) for n ≫ 0.
Let (M, g(t)) be a compact Riemannian manifold and the metric g(t) evolve by the Ricci flow. In the paper, we prove that the eigenvalues of geometric operator −Δφ + 
$\frac{R}{2}$ are non-decreasing under the Ricci flow for manifold M with some curvature conditions, where Δφ is the Witten Laplacian operator, φ ∈ C2(M), and R is the scalar curvature with respect to the metric g(t). We also derive the evolution of eigenvalues under the normalized Ricci flow. As a consequence, we show that compact steady Ricci breather with these curvature conditions must be trivial.