In this paper, we study the spectrality of a class of Moran measures $\mu_{\mathcal{P},\mathcal{D}}$ on $\mathbb{R}$ generated by $\{(p_n,D_n)\}_{n=1}^\infty$, where $\mathcal{P} = \{p_n\}_{n=1}^\infty$ is a sequence of positive integers with $p_n > 1$ and $\mathcal{D} = \{D_n\}_{n=1}^\infty$ is a sequence of digit sets of $\mathbb{N}$ with the cardinality $\#D_n \in \{2,3,N_n\}$. We find a countable set $\Lambda \subset \mathbb{R}$ such that the set $\{e^{-2\pi i\lambda x}|\lambda \in \Lambda\}$ 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 $\mathbb{Z}$.
We study a class of Schrödinger-Poisson problems involving the nonlinearity $f(x)|u|^{p-2}u$ ($2 < p < 4$) in $\mathbb{R}^3$. Such problems cannot be studied directly by applying the general Nehari manifold, because $(PS)$ sequence may not be bounded. In this paper, by developing some useful analytical techniques and introducing a novel definition of the Nehari manifold for the auxiliary system of equations, we resolve an open question posed in [S. Kim and J. Seok, Commun. Contemp. Math., 14 (2012)] of whether there exists a sign-changing solution $u_k^\lambda$ changing signs exactly $k$ times to the problem above for the remaining range $2 < p < 4$. Furthermore, the energy of $u_k^\lambda$ is strictly increasing in $k$, as well as some asymptotic behaviors of $u_k^\lambda$ are obtained.
We study the one-dimensional Cauchy problem for a class of degenerate parabolic equations arising in superfast diffusion. For such equations, weak solutions are known to be non-unique. We establish the existence and uniqueness of semigroup solutions and prove the analyticity of the free boundary by means of the maximal regularity theory of Da Prato and Grisvard [14]. Finally, under suitable assumptions, we demonstrate the preservation of spatial convexity for an appropriate power of the solution.
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 $|u| \leq Q$.
In this paper, we study the 3D double-diffusive convection system. Using energy methods and Littlewood-Paley decomposition theory, we establish regularity criteria for the system in Lorentz spaces and sum spaces, respectively.
This paper is devoted to studying some mixed radial-angular estimates for various types of Hausdorff operators and commutators.
In this paper we assume that $L = -\Delta_{\mathbb{H}^n} + V$ 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 $\mathfrak{g}_\lambda^*$-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 $L$.
In this paper, we first establish the characterization of CMO~q,l(R^n) via the boundedness of the commutators of Hardy type operators on mixed Lebesgue space L~q(R^n). Moreover, the characterization of W CMO~q,l(R^n) is also obtained on weak mixed Lebesgue space W L~q(R^n). In addition, we will derive the boundedness of commutators of Hardy type operators generated with the symbol function b ∈ CMO~q,l(R^n) on mixed l-central Morrey space B~q,l(R^n), respectively.
This investigation provides a comprehensive derivation of integral inequalities for subadditive functions within the framework of Katugampola fractional integrals. Using Katugampola fractional integrals, this study extends Hermite–Hadamard inequalities to subadditive functions. It establishes and investigates several variations of these inequalities and fractional integral inequalities for the product of two subadditive functions using Katugampola fractional integrals. Furthermore, we demonstrate that the inequalities presented in this research generalize the previously established inequalities for convex functions.
In this paper, we construct a natural $\mathbb{D}_t$-valued-functional-vector space $\mathcal{L}^2\big(X,\sigma(X),\mu_t\big) $ on a measurable space $(X,\sigma(X))$ up to a $\mathbb{D}_t$-valued probability $\mu_t$, where $\mathbb{D}_t$ is the $t$-scaled hyperbolic numbers for an arbitrarily fixed scale $t \in \mathbb{R}$. We show that this vector space $\mathcal{L}^2\big(X,\sigma(X),\mu_t\big)$ is Banach space over the real field $\mathbb{R}$, equipped with symmetric $\mathbb{D}_t$-valued-positive $\mathbb{R}$-bilinear form for any $t \in \mathbb{R}$. As application, we consider multiplication operators acting on $\mathcal{L}^2\big(X,\sigma(X),\mu_t\big)$.
In this paper, we consider the following Kirchhoff type problem with singular and logarithmic nonlinearity $\begin{cases} -\left(a + b\int_{\Omega}|\nabla u|^2 dx\right)\Delta u = u^{p-1}\ln|u| + \frac{\lambda}{u^\gamma} & \text{in } \Omega,\\ u = 0 & \text{on } \partial\Omega, \end{cases} $ where $\Omega \subset \mathbb{R}^3$ is a bounded domain with smooth boundary, $0 < \gamma < 1$, $4 < p < 6$, $\lambda$ is a positive constant. By using variational method and the critical point theory for nonsmooth functional, we obtain the existence of two positive solutions. In particular, we propose innovative techniques to tackle the challenges arising from the sign-changing property of $u^{p-1}\ln|u|$, which violates both the monotonicity condition and the Ambrosetti-Rabinowitz condition. Moreover, unlike the scenario where $a = 1$ and $b = 0$, the inclusion of the nonlocal term $\left(\int_{\Omega}|\nabla u|^2 dx\right)\Delta u$ brings about extra complexities. A key issue here is the lack of weak continuity in the functional $f: H_0^1(\Omega) \to \mathbb{R}: u \mapsto \left(\int_{\Omega}|\nabla u|^2 dx\right)\nabla u\nabla v $ for any $v \in H_0^1(\Omega)$, a factor that substantially complicates the proof that the limit of a $(PS)_c$ sequence corresponds to a nontrivial solution of the problem.
We consider multilinear commutators with vector symbol $~b = (b1, b2, ..., bm)$ defined by $T~b f(x) = ∫_{R^n} ∏_{i=1}^m (b_i(x) - b_i(y)) K(x,y) f(y) dy,$ where m ∈ N, K is a kernel satisfying the standard Calderón-Zygmund estimates and $b_i ∈ BMO(R^n), i = 1,2,...,m.$ Applying some properties of variable exponents and the generalized BMO norms, we prove the boundedness of the operators T~b on the generalized Herz spaces, where all the three main indices are variable.
We consider the almost everywhere pointwise convergence of the Boussinesq operator along sequences ${t_k}_{k=1}^∞$ with $lim_{k→∞} t_k = 0$ in higher dimensions. We obtain a characterization of Boussinesq maximal estimate when ${t_k}_{k=1}^∞$ belongs to Lorentz space for all $f ∈ H^s(R^n)$ with appropriate s.
We study the long time behavior of the modified one-dimensional deriva- tive Schrödinger equation $(D_t - F(D))u = D_x(|u|^2u)$, where $F(\xi)$ is a nonnegative second order constant coefficient elliptic symbol. For any smooth initial datum of size $\varepsilon \ll 1$, we prove that the solution is global-in-time, combining the vector fields method and a semiclassical analysis method introduced by Delort. Moreover, we present the pointwise decay estimates and the large time asymptotic formulas of the solution.
The sharp range of the parameters $(p,β)$ is determined for the boundedness of a wave operator from the weighted Hardy space $H^p_β(M^n)$ to the Lebesgue space $L^p(M^n)$ on an n-dimensional $(n ≥ 2),$ compact, connected Riemannian manifold $M^n$ without boundary. This extends previously known results on Euclidean spaces $R^n$ and compact Lie groups.
We present a new type of universal series, termed delta-universal series, for which the sum of squared coefficients satisfies (infinity )& sum; (k=0) |a(k)|(2) < delta for an arbitrarily small delta > 0. We establish a version of Seleznev's theorem within this framework. To construct such delta-universal series, we develop a variation of Mergelyan's theorem.
In this article we consider a modification of the Stein's spherical maximal operator of complex order a on Rn: M-[1,2](alpha) f (x) = sup (t is an element of [1,2])| 1/ Gamma(alpha) integral (|y|<= 1) (1-|y| (2))(alpha-1) f(x-ty)dy|. We show that when n >= 2, suppose ||M-[1,2](alpha) f || L-q (R-n) <= C || f|| L-p(R-n) holds for some alpha is an element of C, p, q >= 1, then we must have that q >= p and Re alpha >= sigma(n) (p, q) := max {1/p-n/ q , n + 1 /2p-n-1/2 (1/q+1), n/p - n+1 } Conversely, we show that Ma[1,2] is bounded from L-p(R-n) to (LRn)-R-q() provided that q >= p and Re alpha > sigma(2)(p,q) for n = 2; and Re alpha > max {sigma(n)(p, q), 1/(2p)-(n-2)/(2q)-(n-1)/4} for n > 2. The range of a, 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.
Modeling a non-stationary, multicomponent signal as a superposition of frequency components, each with a well-defined instantaneous frequency (IF), is crucial for extracting information, such as the underlying dynamics hidden within the signal. The synchrosqueezing transform (SST) has emerged as an alternative to empirical mode decomposition (EMD) for separating non-stationary signals. However, because the SST estimates the IFs of all frequency components based on a single phase transformation, its accuracy can be limited. To address this, SST variants based on the IF-embedded short-time Fourier transform (IFE-STFT) and the IF-embedded continuous wavelet transform (IFE-CWT) were developed. More recently, a direct time-frequency method called the signal separation operation (SSO) was introduced for multicomponent signal separation. SSO bypasses the second step of the two-step SST method for component recovery and is based on variants of the STFT or CWT. In this paper, we propose a direct signal separation method by combining the SSO method with IFE-CWT and IFE-STFT, creating the IFE-CWT-based SSO (IWSSO) and the IFE-STFT-based SSO (IFSSO). Both IWSSO and IFSSO directly separate multicomponent signals without the squeezing operation inherent in SST. Our algorithms and techniques yield more accurate instantaneous frequency estimates and signal separation than conventional SSO or SST methods.
Sparse signal recovery has been a cornerstone of advancements in data processing and imaging. Recently, the squared ratio of ℓ_1 to ℓ_2 norms, (ℓ_1/ℓ_2)^2, has been introduced as a sparsity-prompting function, showing superior performance compared to traditional ℓ_1 minimization, particularly in challenging scenarios with high coherence and dynamic range. This paper explores the integration of the proximity operator of (ℓ_1/ℓ_2)^2 and ℓ_1/ℓ_2 into efficient optimization frameworks, including the Accelerated Proximal Gradient (APG) and Alternating Direction Method of Multipliers (ADMM). We rigorously analyze the convergence properties of these algorithms and demonstrate their effectiveness in compressed sensing and image restoration applications. Numerical experiments highlight the advantages of our proposed methods in terms of recovery accuracy and computational efficiency, particularly under noise and high-coherence conditions.
