Analysis in Theory and Applications (ATA), used to have the title Approximation Theory and its Applications (from 1984 to 2002), publishes research results in broad areas of approximation including Dynamic Systems, Geometric Analysis, Harmonic Analysis, Numerical Approximation, Signal Analysis, Special Functions, Differential Equations, Wavelets, and Approximation Theory, as well as their applications. The journal is supervised by Office of Education of Jiangsu Province and sponsored by Nanjing University.
Copyright (c) 2026 Analysis in Theory and Applications
Preface: Special Issue Dedicated to the Memory of Prof. Donggao Deng’s 90th Birthday (Part II)
Preface: Special Issue Dedicated to the Memory of Prof. Donggao Deng’s 90th Birthday (Part I)
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 $L^q(\mathbb{R}^m)$.
Copyright (c) 2025 Analysis in Theory and Applications
$L^p$ → $L^q$ Estimates for Stein’s Spherical Maximal Operators
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 $n > 2$.
Weak Boundedness for Commutators of $n$-Dimensional Hardy-Type Operators on Mixed Herz Spaces
In this paper, we obtain the boundedness for the commutators of $n$-dimensional Hardy-type operators from mixed Herz spaces to weak mixed Herz spaces, where the symbol functions are in the intersection of the mixed central bounded mean oscillation space $\text{CBMO}_{\vec{q'}}(\mathbb{R}^n)$ and the weak mixed central bounded mean oscillation space $W_{\vec{q}}(\mathbb{R}^n)$.
Weak mixed central bounded mean oscillation space
Signal and Image Recovery with Scale and Signed Permutation Invariant Sparsity-Promoting Functions
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.
Long Time Behavior of the Modified One-Dimensional Derivative Schrödinger Equation
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.
Matrix Weights, Maximal Operators, Calderón–Zygmund Operators, and Besov–Triebel–Lizorkin-Type Spaces — A Survey
The primary purpose of this survey is threefold. First, the authors recall some histories and present some recent developments of matrix weights, in which the authors not only improve some known results on the intrinsic properties of matrix weights, but also establish some new ones. Then the authors summarize matrix-weighted inequalities associated with various operators, such as Hardy–Littlewood-type maximal operators and Calderόn–Zygmund operators. Finally, the authors overview matrix-weighted function spaces, including matrix-weighted Sobolev, BMO, and Besov–Triebel–Lizorkin-type spaces. Several open questions on these subjects are also presented.
We present a new type of universal series, termed $\delta$-universal series, for which the sum of squared coefficients satisfies $$\sum\limits_{k=0}^{\infty} |a_k|^2 < \delta$$ 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.
A Signal Separation Method Based on Instantaneous Frequency Embedded Continuous Wavelet Transform and Short-Time Fourier Transform
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.
signal separation operator based on IF–embedded continuous wavelet transform
Singular Integral and Local Hardy Spaces in Dunkl Setting
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.
Nonhomogeneous Dunkl-Calderón-Zygmund singular integral
Weighted Boundedness of Toeplitz Type Operator Associated to General Singular Integral Operator Satisfying a Variant of Hörmander’s Condition
In this paper, we establish the weighted sharp maximal function inequalities for the Toeplitz type operator associated to the general singular integral operator satisfying a variant of Hörmander's condition. As an application, we obtain the boundedness of the operator on weighted Lebesgue and Morrey spaces.
Copyright (c) 2025 COPYRIGHT: © Global Science Press
Liouville Type Theorems for a Class of General Parabolic Hessian Quotient Type Equations
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.
Parabolic $(k, l)$-Hessian quotient type equations
Variable Exponent Central Morrey Estimates for Multilinear Bochner-Riesz Operators
We prove the boundedness of multilinear Bochner-Riesz operator on central Morrey spaces with variable exponents. Moreover, we also show that the variable exponent Lipschitz commutator of Bochner-Riesz operator are bounded on above spaces. As applications, we also obtain the bounds of the commutators generated by $\lambda$-central BMO functions and Bochner-Riesz operators on central Morrey spaces with variable exponents.
central Morrey space with variable exponent
A Rigidity Result for the Schiffer Conjecture on Domain with a Hole
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 $\mu< {\mu}_8,$ then $\Omega$ is an annulus.
Estimates for Parabolic Schrödinger Operators with Certain Nonnegative Potentials
In this paper, the parabolic Schrödinger operator $\mathcal{P}={\partial}_t-\triangle+V(x)$ on $\mathbb{R}^{n+1}$ is considered, where $n\ge3,$ nonnegative potential $V$ belongs to the reverse Hölder class $RH_q$ with $q \ge n /2$. The $L^p$ boundedness of operators $V^{\alpha}\mathcal{P}^{-\beta},$ $V^{\alpha}{\nabla}\mathcal{P}^{-\beta}$ and their adjoint operators are established.
Separation of Sequences and Multipliers in the Space of Tempered Distributions
We consider the notions of $v$-separation and $(N,v)$-separation for increasing sequences that tend to infinity. We study several of the connections between the properties of a multiplier in the space $S'(\mathbb{R})$ and in other related spaces and the separation properties of the sequence of its zeros. We also prove that a distributional division problem $$Fh=f,$$always has tempered solutions $h$ for any tempered data $f$ if and only if the non integrable function $1/F$ admits regularizations that are tempered, and that this holds if and only if the pseudofunction $\mathcal{P} f (1/F)$ is tempered.
Double Cosine and Cosine-Sine Fourier Transforms and Generalized Lipschitz Classes in Uniform Metric
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.
double cosine and cosine-sine Fourier transform
On Noveley Inequalities for Convex Stochastic Processes on Time Scale Utilizing Quantum Operators
We aim to present a innovation extension of the mean-square stochastic integral of a process on time scale based on a q-operator and fractional q-operator. By presenting the q-convex definition for a stochastic process and using quantum operators, we were able to prove some well-known inequalities such as the HermitHadamardand the Jensen for the processes.
Multiplicity of Solutions for an Integro-Differential Equation with Trudinger-Moser Subcritical or Critical Nonlinearities
This paper study a general nonlocal problem characterized by the equation: \[ -\mathcal{L}_K u + V(x)u = P(x)h(u) \] in $\mathbb{R}$. Here, $\mathcal{L}_K$ represents a nonlocal integro-differential operator, and $h$ is a nonlinear term displaying subcritical or critical growth of Trudinger-Moser type. We initially prove the existence of a ground state solution by employing variational methods and innovative analytical approaches. Furthermore, through the application of constrained variational methods, Brouwer degree theory, and a quantitative deformation lemma, we establish the existence of a sign-changing solution with minimal energy, surpassing the energy of the ground state solution.
Two-Weight Inequality of Calderón-Zygmund Operator on Non-Homogeneous Metric Measure Space
Let $(\mathcal{X}, d, \mu)$ be a non-homogeneous metric measure space satisfying both the upper doubling and the geometrically doubling conditions. In this paper, we study weighted inequalities of the Calder\'{o}n-Zygmund operator on $(\mathcal{X}, d, \mu)$. Specifically, for $1 < p < \infty$, we identify sufficient conditions for the weight on one side, which guarantee the existence of another weight in the other side, so that the weighted $L^p$ inequality holds. We deal with this problem by developing a vector-valued theory for Calder\'{o}n-Zygmund operators on the non-homogeneous metric measure spaces which is interesting in its own right.
On Non-Linearly Coupled Schrodinger Systems
Abstract. In this paper we study the following nonlinear Schr\"{o}dinger system: \[ \begin{cases} -\Delta u + a(x) u = \lambda(x) f(u, v), & x \in \mathbb{R}^N, \\ -\Delta v + b(x) v = \lambda(x) g(u, v), & x \in \mathbb{R}^N, \\ u(x) \to 0, & v(x) \to 0 \end{cases} \] as $ |x| \to \infty $. Here, $ a, b, \lambda \in C(\mathbb{R}^N, \mathbb{R}) $ are all non-periodic in $ x_i $ for $ i = 1, \cdots, N, N \geq 3, f, g \in C(\mathbb{R}^2, \mathbb{R}) $. We show that this system has infinitely many solutions with small negative energies and infinitely many large-energy solutions. To the best of our knowledge, there is no corresponding result about such a Schr\"{o}dinger system.
Decomposition Formulas of Kampé de Fériets Double Hypergeometric Functions
This paper is sequel to the authors paper [18]. By using the generalized Burchnall-Chaundy operator method, the authors are aiming at deriving certain decomposition formulas for some interesting special cases of Kamp\u00e9 de F\u00e9riets series of double hypergeometric series $F^{p:q;k}_{l:m;n}$.
Maged G. Bin-Saad, Anvar Hasanov & Mamasali Turaev
Erdös Type Inequality for Lorentz Polynomials
An elementary, but very useful tool for proving inequalities for polynomials with restricted zeros is the Bernstein or Lorentz representation of polynomials. In the present paper, we give two classes of Lorentz polynomials, for which the Erd\u00f6s-type inequality holds.
Existence of Solutions for Fractional Differential Equations Involving Two Riemann-Liouville Fractional Orders
In this work, we study existence and uniqueness of solutions for multi-point boundary value problem of nonlinear fractional differential equations with two fractional derivatives. By using the variety of fixed point theorems, such as Banach\u2019s fixed point theorem, Leray-Schauder\u2019s nonlinear alternative and Leray-Schauder\u2019s degree theory, the existence of solutions is obtained. At the end, some illustrative examples are discussed.
Some Characterizations of Bloch Functions
We define Bloch-type functions of $\mathcal{C}^1(\mathbb{D})$ on the unit disk of complex plane $\mathbb{C}$ and characterize them in terms of weighted Lipschitz functions. We also discuss the boundedness of a composition operator $C_{\phi}$\u00a0acting between two Bloch-type spaces. These obtained results generalize the corresponding known ones to the setting of $\mathcal{C}^1(\mathbb{D})$.
Weighted Boundedness of Commutators of Generalized Calderόn-Zygmund Operators
$[b,T]$ denotes the commutator of generalized Calder\u00f3n-Zygmund operators $T$ with Lipschitz function $b$, where $b \u2208 \rm{Lip}_\u03b2(R^n)$, $(0<\u03b2\u22641)$ and $T$ is a $\u03b8(t)$\u2212type Calder\u00f3n-Zygmund operator. The commutator $[b,T]$ generated by $b$ and $T$ is defined by\u00a0 $$[b,T] f(x)=b(x)Tf(x)\u2212T(bf)(x)=\u222bk(x,y)(b(x)\u2212b(y))f(y)dy.$$ In this paper, the authors discuss the boundedness of the commutator $[b,T]$ on weighted Hardy spaces and weighted Herz type Hardy spaces and prove that $[b,T]$ is bounded from $H^p(\u03c9^p)$ to $L^q(\u03c9^q)$, and from $H\dot{K}^{\u03b1,p}_{q_1}(\u03c9_1,\u03c9^{q_1}_2)$ to $\dot{K}^{\u03b1,p}_{q_2}(\u03c9_1,\u03c9^{q_2}_2)$. The results extend and generalize the well-known ones in [7].
Domain of Euler Mean in the Space of Absolutely $p$-Summable Double Sequences with $0< p
In this study, as the domain of four dimensional Euler mean $E(r,s)$ of orders $r$, $s$ in the space $\mathcal{L}_p$ for $0<p<1$, we examine the double sequence space $\varepsilon^{r,s}_p$ and some properties of four dimensional Euler mean. We determine the $\u03b1$- and $\u03b2(bp)$-duals of the space $\varepsilon^{r,s}_p$, and characterize the classes $(\varepsilon^{r,s}_p:\mathcal{M}_u)$,\u00a0$(\varepsilon^{r,s}_p:\mathcal{C}_{bp})$\u00a0and $(\varepsilon^{r,s}_p:\mathcal{L}_q)$ of four dimensional matrix transformations, where $1\u2264q<\u221e$. Finally, we shortly emphasize on the Euler spaces of single and double sequences, and note some further suggestions.
Medine Ye\u015filkayagil & Feyzi Ba\u015far
Hopf Bifurcation of a Nonresident Computer Virus Model with Delay
In this paper, a delayed nonresident computer virus model with graded infection rate is considered in which the following assumption is imposed: latent computers have lower infection ability than infectious computers. With the aid of the bifurcation theory, sufficient conditions for stability of the infected equilibrium of the model and existence of the Hopf bifurcation are established. In particular, explicit formulae which determine direction and stability of the Hopf bifurcation are derived by means of the normal form theory and the center manifold reduction for functional differential equations. Finally, a numerical example is given in order to show the feasibility of the obtained theoretical findings.
Zizhen Zhang, Yougang Wang & Massimiliano Ferrara