2026-03-18
Hongya Gao, Ruiqi Lv, Qianqian Liu
This paper deals with minimizers for noncoercive integral functional of the type $\mathcal{J}(v) = \int_{\Omega} a(x)|\nabla v(x)|^p dx - \int_{\Omega} f(x)v(x)dx,\quad v\in W_0^{1,p}(\Omega), $ with $1 < p < n, 0 < a(x) \leq \beta, a.e. \Omega$ and $\frac{1}{a(x)}$ and $f(x)$ belong to some Lebesgue or Marcinkiewicz spaces. It is shown by Weierstrass Theorem that such a functional has a minimizer in a larger space $W_0^{1,q}(\Omega)$ for an appropriate exponent $1 < q < p$. Some regularity properties are given by using Stampacchia Lemma. This paper also considers regularizing effect of an interplay between the coefficient of zero order term and the datum in noncoercive integral functional of the type $ \mathcal{I}(v)=\int_{\Omega} a(x)|\nabla v(x)|^p dx + \int_{\Omega} b(x)|v(x)|^p dx - \int_{\Omega} f(x)v(x)dx,\quad v\in W_0^{1,p}(\Omega). $ It is shown that, even if $0 < b(x)$ and $f(x)$ belong only to $L^1(\Omega)$, the interplay $ |f(x)| \leq 2Q b(x) $ implies the existence of a minimizer $u\in W_0^{1,q}(\Omega)$ satisfying $|u| \leq Q$.
2026-03-18
Zhiyong Wang, Jing Yu
In this paper we assume that $L = −∆_{\mathbb{H}^n}+ V$ is a Schrödinger operator on the Heisenberg group $\mathbb{H}^n,$ where the nonnegative potential $V$ belongs to the reverse Hölder class $B_{Q/2}.$ We introduce the Littlewood-Paley $\mathfrak{g}$-functions, the Lusin area functions and the $\mathfrak{g}^∗_λ$-functions generated by the heat semigroup $\{e^{−tL}\}_{t>0}$ and the Poisson semigroup $\{e^{−t\sqrt{L}}\}_{t>0},$ respectively. By means of the reproducing formulas and the regularity properties of semigroups, we establish several square function characterizations of the Hardy space $H_L^1(\mathbb{H}^n)$ associated with $L.$
2026-03-18
Yali Zheng, Yingqing Xiao
In this paper, we study the spectrality of a class of Moran measures $\mu_{\mathcal{P},\mathcal{D}}$ on $\mathbb{R}$ generated by $\{(p_n, D_n)\}^∞_{n=1},$ where $\mathcal{P} = \{p_n\}^∞_{n=1}$ is a sequence of positive integers with $p_n > 1$ and $\mathcal{D}= \{D_n\}^∞_n=1$ is a sequence of digit sets of $\mathbb{N}$ with the cardinality #$D_n ∈ \{2, 3, N_n\}.$ We find a countable set $Λ ⊂ \mathbb{R}$ such that the set $\{e^{−2πiλx}|λ ∈ Λ\}$ is an orthonormal basis of $L^2 (\mu_{\mathcal{P},\mathcal{D}})$ under some conditions. As an application, we show that when $\mu_{\mathcal{P},\mathcal{D}}$ is absolutely continuous, $\mu_{\mathcal{P},\mathcal{D}}$ not only is a spectral measure, but also its support set tiles $\mathbb{R}$ with $\mathbb{Z}.$
2026-01-18
2026-01-18
Ji Li, Heping Liu, Lizhong Peng, Lixin Yan
2026-01-14
Ting Chen, Wenchang Sun
We study the bilinear fractional integral considered by Kenig and Stein, where linear combinations of variables with matrix coefficients are involved. Under more general settings, we give a complete characterization of the corresponding parameters for which the bilinear fractional integral is bounded from $L^{p_1}(\mathbb{R}^{n_1}) × L^{p_2}(\mathbb{R}^{n_2})$ to $L^q(\mathbb{R}^m)$.
2026-01-14
Naijia Liu, Minxing Shen, Liang Song, Lixin Yan
In this article we consider a modification of the Stein's spherical maximal operator of complex order $\alpha$ on $\mathbb{R}^n$: $\mathfrak{M}_{[1,2]}^{\alpha} f(x) = \sup\limits_{t \in [1,2]} \left| \frac{1}{\Gamma(\alpha)} \int_{|y| \leq 1} \left( 1 - |y|^2 \right)^{\alpha - 1} f(x - ty) dy \right|.$ We show that when $n \geq 2$, suppose $\|\mathfrak{M}_{[1,2]}^{\alpha} f\|_{L^q(\mathbb{R}^n)} \leq C \|f\|_{L^p(\mathbb{R}^n)}$ holds for some $\alpha \in \mathbb{C}$, $p, q \geq 1$, then we must have that $q \geq p$ and $$\operatorname{Re} \alpha \geq \sigma_n(p, q) := \max \left\{ \frac{1}{p} - \frac{n}{q},\; \frac{n+1}{2p} - \frac{n-1}{2} \left( \frac{1}{q} + 1 \right),\; \frac{n}{p} - n + 1 \right\}.$$ Conversely, we show that $\mathfrak{M}_{[1,2]}^{\alpha}$ is bounded from $L^p(\mathbb{R}^n)$ to $L^q(\mathbb{R}^n)$ provided that $q \geq p$ and $\operatorname{Re} \alpha > \sigma_2(p, q)$ for $n = 2$; and $\operatorname{Re} \alpha > \max \left\{ \sigma_n(p, q),\; 1/(2p) - (n-2)/(2q) - (n-1)/4 \right\}$ for $n > 2$. The range of $\alpha$, $p$ and $q$ is almost optimal in the case when either $n = 2$, or $\alpha = 0$, or $(p, q)$ lies in certain regions for $n > 2$.
2026-01-09
Yanchang Han, Yongsheng Han, Ji Li, Chaoqiang Tan
The objective of this paper is to establish the local Hardy space in the Dunkl setting, which pertains to the geometric framework defined by both the Euclidean metric and the Dunkl metric, the latter being influenced by finite reflection groups. This study leverages the weak local wavelet decomposition in $L^2$ space and the theory of nonhomogeneous singular integral operators as pivotal components.
2025-09-28
Yingxin Sun
Let $\Omega$ be a domain with a hole containing the origin in $\mathbb{R}^2$ and $u$ be a solution to the problem where $\partial^{\pm}\Omega$ represents the outer and inner boundaries of $\Omega,$ respectively, $c$ is a constant. Let ${\mu}_k$ denote the $k{\rm th}$ Neumann eigenvalue of the Laplacian on $\Omega$ and${\Omega}_h$ is the hole. We establish that if $\mu
2025-09-28
Hui Liu, Ni Xiang, Lina Zheng
We first consider the a priori estimates to a class of general parabolic $(k,l)-$ Hessian quotient type equations of the form with $0{\le}1 or has interior gradient estimates and Pogorelov type estimates. As an application, we prove Liouville type theorems for these equations.
2025-09-14
Sergey Volosivets, Yulia Krotova
For complex-valued functions $f \in L^1(\mathbb{R}^2_+)$, where $\mathbb{R}_+ := [0,\infty)$ we give sufficient conditions under which the double cosine or cosine-sine Fourier transform of $f$ belongs to a generalized Lipschitz class defined by the mixed modulus of smoothness of orders $m,n \in \mathbb{N} = \{1,2,\cdots \}$ in uniform metric. The sharpness of these conditions is established under some restriction for non-negative functions.