By comprehensive utilizing of the geometry structure of 2D Burgers equation and the stochastic noise, we find the decay properties of the solution to the stochastic 2D Burgers equation with Dirichlet boundary conditions. Consequently, the expected ergodicity for this turbulence model is established.

Let $\{X_{v}:v\in\mathbb{Z}^d\}$ be i.i.d. random variables. Let $S(\pi)=\sum_{v\in\pi}X_v$ be the weight of a self-avoiding lattice path $\pi$. Let \[M_n=\max\{S(\pi):\pi\text{ has length }n\text{ and starts from the origin}\}.\] We are interested in the asymptotics of $M_n$ as $n\to\infty$. This model is closely related to the first passage percolation when the weights $\{X_v:v\in\mathbb{Z}^d\}$ are non-positive and it is closely related to the last passage percolation when the weights $\{X_v,v\in\mathbb{Z}^d\}$ are non-negative. For general weights, this model could be viewed as an interpolation between first passage models and last passage models. Besides, this model is also closely related to a variant of the position of right-most particles of branching random walks. Under the two assumptions that $\exists\alpha>0$, $E(X_0^{+})^d(\log^{+}X_0^{+})^{d+\alpha}<+\infty$ and that $E[X_0^{-}]<+\infty$, we prove that there exists a finite real number $M$ such that $M_n/n$ converges to a deterministic constant $M$ in $L^{1}$ as $n$ tends to infinity. And under the stronger assumptions that $\exists\alpha>0$, $E(X_0^{+})^d(\log^{+}X_0^{+})^{d+\alpha}<+\infty$ and that $E[(X_0^{-})^4]<+\infty$, we prove that $M_n/n$ converges to the same constant $M$ almost surely as $n$ tends to infinity.

In this article, we consider a modified version of minimal $L^2$ integrals on sublevel sets of plurisubharmonic functions related to modules at boundary points, and obtain a concavity property of the modified version. As applications, we give characterizations for the concavity degenerating to linearity on open Riemann surfaces and on fibrations over open Riemann surfaces.

Let $\pi:(X,T)\rightarrow (Y,S)$ be a factor map between two topological dynamical systems, and $\mathcal{F}$ a Furstenberg family of $\mathbb{Z}$. We introduce the notion of $relative broken$ $\mathcal{F}-sensitivity$. Let $\mathcal{F}_{s}$ (resp. $\mathcal{F}_{\rm pubd},\mathcal{F}_{\rm inf}$) be the families consisting of all syndetic subsets (resp. positive upper Banach density subsets, infinite subsets). We show that for a factor map $\pi:(X,T)\rightarrow (Y,S)$ between transitive systems, $\pi$ is relatively broken $\mathcal{F}$-sensitive for $\mathcal{F}=\mathcal{F}_{s}\ \text{or}\ \mathcal{F}_{\rm pubd}$ if and only if there exists a relative sensitive pair which is an $\mathcal{F}$-recurrent point of $(R_\pi, T^{(2)})$; is relatively broken $\mathcal{F}_{\rm inf}$-sensitive if and only if there exists a relative sensitive pair which is not asymptotic. For a factor map $\pi:(X,T)\rightarrow (Y,S)$ between minimal systems, we get the structure of relative broken $\mathcal{F}$-sensitivity by the factor map to its maximal equicontinuous factor.

This research paper deals with an extension of the non-central Wishart introduced in 1944 by Anderson and Girshick, that is the non-central Riesz distribution when the scale parameter is derived from a discrete vector. It is related to the matrix of normal samples with monotonous missing data. We characterize this distribution by means of its Laplace transform and we give an algorithm for generating it. Then we investigate, based on the method of the moment, the estimation of the parameters of the proposed model. The performance of the proposed estimators is evaluated by a numerical study.

Let $G$ be a symplectic or orthogonal group defined over a finite field with odd characteristic and let $D\leq G$ be a Sylow $2$-subgroup. In this paper, we classify the essential $2$-subgroups and determine the essential $2$-rank of the Frobenius category $F_D(G)$. Together with the results of An–Dietrich and Cao–An–Zeng, this completes the work of essential subgroups and essential ranks of classical groups.

$\mathfrak{L}_{\nu}$ operator is an important extrinsic differential operator of divergence type and has profound geometric settings. In this paper, we consider the clamped plate problem of $\mathfrak{L}^{2}_{\nu}$ operator on a bounded domain of the complete Riemannian manifolds. A general formula of eigenvalues of $\mathfrak{L}^{2}_{\nu}$ operator is established. Applying this general formula, we obtain some estimates for the eigenvalues with higher order on the complete Riemannian manifolds. As several fascinating applications, we discuss this eigenvalue problem on the complete translating solitons, minimal submanifolds on the Euclidean space, submanifolds on the unit sphere and projective spaces. In particular, we get a universal inequality with respect to the $\mathcal{L}_{II}$ operator on the translating solitons. Usually, it is very difficult to get universal inequalities for weighted Laplacian and even Laplacian on the complete Riemannian manifolds. Therefore, this work can be viewed as a new contribution to universal estimate.

How to analyze flocking behaviors of a multi-agent system with local interaction functions is a challenging problem in theory. Motsch and Tadmor in 2011 also stressed the significance to assume that the interaction function is rapidly decaying or cut-off at a finite distance (cf. Motsch and Tadmor in J. Stat. Phys. 2011). In this paper, we study the flocking behavior of a Cucker–Smale type model with compactly supported interaction functions. Using properties of a connected stochastic matrix, together with an elaborate analysis on perturbations of a linearized system, we obtain a sufficient condition imposed only on model parameters and initial data to guarantee flocking. Moreover, it is shown that the system achieves flocking at an exponential rate.

Let $\mathcal A$ be a unital C$^*$-algebra and $\mathcal B$ a unital C$^*$-algebra with a faithful trace $\tau$. Let $n$ be a positive integer. We give the definition of weakly approximate diagonalization (with respect to $\tau$) of a unital homomorphism $\phi: \mathcal{A} \to M_n(\mathcal{B})$. We give an equivalent characterization of McDuff II$_1$ factors. We show that, if $\mathcal A$ is a unital nuclear C$^*$-algebra and $\mathcal B$ is a type II$_1$ factor with faithful trace $\tau$, then every unital $*$-homomorphism $\phi: \mathcal A \to M_n(\mathcal B)$ is weakly approximately diagonalizable. If $\mathcal{B}$ is a unital simple infinite dimensional separable nuclear C$^*$-algebra, then any finitely many elements in $M_n(\mathcal B)$ can be simultaneously weakly approximately diagonalized while the elements in the diagonals can be required to be the same.

This is a continuation of our previous work (Ann. Sc. Norm. Super. Pisa Cl. Sci.,20, 1295–1324, 2020). Let $(\Sigma,g)$ be a closed Riemann surface, where the metric $g$ has conical singularities at finite points. Suppose $\mathbf{G}$ is a group whose elements are isometries acting on $(\Sigma,g)$. Trudinger–Moser inequalities involving $\mathbf{G}$ are established via the method of blow-up analysis, and the corresponding extremals are also obtained. This extends previous results of Chen (Proc. Amer. Math. Soc., 1990), Iula–Manicini (Nonlinear Anal., 2017), and the authors (2020).

In this paper, with $(\Sigma,g)$ being a closed Riemann surface, we analyze the possible concentration behavior of a heat flow related to the Trudinger–Moser energy. We obtain a long time existence for the flow. And along some sequence of times $t_k\rightarrow +\infty$, we can deduce the convergence of the flow in $H^2(\Sigma)$. Furthermore, the limit function is a critical point of the Trudinger–Moser functional under certain constraint.

In this paper, the notion of $C$-semigroup of continuous module homomorphisms on a complete random normal (RN) module is introduced and investigated. The existence and uniqueness of solution to the Cauchy problem with respect to exponentially bounded $C$-semigroups of continuous module homomorphisms in a complete RN module are established.

Let $q\in(0,\infty]$ and $\varphi$ be a Musielak-Orlicz function with uniformly lower type $p_{\varphi}^-\in(0,\infty)$ and uniformly upper type $p_{\varphi}^+\in(0,\infty)$. In this article, the authors establish various real-variable characterizations of the Musielak-Orlicz-Lorentz Hardy space $H^{\varphi,q}(\mathbb{R}^n)$, respectively, in terms of various maximal functions, finite atoms, and various Littlewood-Paley functions. As applications, the authors obtain the dual space of $H^{\varphi,q}(\mathbb{R}^n)$ and the summability of Fourier transforms from $H^{\varphi,q}(\mathbb{R}^n)$ to the Musielak-Orlicz-Lorentz space $L^{\varphi,q}(\mathbb{R}^n)$ when $q\in(0,\infty)$ or from the Musielak-Orlicz Hardy space $H^{{\varphi}}({\mathbb{R}^n})$ to $L^{\varphi,\infty}(\mathbb{R}^n)$ in the critical case. These results are new when $q\in(0,\infty)$ and also essentially improve the existing corresponding results (if any) in the case $q=\infty$ via removing the original assumption that $\varphi$ is concave. To overcome the essential obstacles caused by both that $\varphi$ may not be concave and that the boundedness of the powered Hardy-Littlewood maximal operator on associated spaces of Musielak-Orlicz spaces is still unknown, the authors make full use of the obtained atomic characterization of $H^{\varphi,q}(\mathbb{R}^n)$, the corresponding results related to weighted Lebesgue spaces, and the subtle relation between Musielak-Orlicz spaces and weighted Lebesgue spaces.

In this paper we prove that isometries with respect to the Kobayashi metric between certain domains having boundary points at which the boundary is infinitely flat extend continuously to the boundary. The strategy is to reestablish the Gehring-Hayman-type Theorem for these complex domains. Furthermore, the regularity of boundary extension map is given.

We completely characterize the boundedness of area operators from the Bergman spaces A^p_α(??_n) to the Lebesgue spaces L^q(??_n) for all $0＜p,q＜∞$. For the case $n=1$, some partial results were previously obtained by Wu in [Wu, Z.: Volterra operator, area integral and Carleson measures, Sci. China Math., 54, 2487–2500 (2011)]. Especially, in the case $q＜p$ and $q＜s$, we obtain some characterizations for the area operators to be bounded. We solve the cases left open there and extend the results to $n$-complex dimension.

In this paper, we construct a superfermionic representation as well as a vertex representation for twisted general linear affine Lie superalgebras. We also establish a module isomorphism between them, which generalizes the super boson-fermion correspondence of type $B$ given by Kac-van de Leur. Based on this isomorphism, we determine explicitly the irreducible components of these two representations. Particularly, we obtain in this way two kinds of systematic construction of level $1$ irreducible integrable highest weight modules for twisted general linear affine Lie superalgebras.

In this article, we study deformations of conjugate self-dual Galois representations. The study is twofold. First, we prove an R=T type theorem for a conjugate self-dual Galois representation with coefficients in a finite field, satisfying a certain property called rigid. Second, we study the rigidity property for the family of residue Galois representations attached to a symmetric power of an elliptic curve, as well as to a regular algebraic conjugate self-dual cuspidal representation.

In the spirit of Morse homology initiated by Witten and Floer, we construct two $\infty$-categories $\mathcal{A}$ and $\mathcal{B}$. The weak one $\mathcal{A}$ comes out of the Morse-Smale pairs and their higher homotopies, and the strict one $\mathcal{B}$ concerns the chain complexes of the Morse functions. Based on the boundary structures of the compactified moduli space of gradient flow lines of Morse functions with parameters, we build up a weak $\infty$-functor $\mathcal{F}: \mathcal{A}\rightarrow \mathcal{B}$. Higher algebraic structures behind Morse homology are revealed with the perspective of defects in topological quantum field theory.

In this article, we construct free centroid hom-associative algebras and free centroid hom-Lie algebras. We also construct some other relatively free centroid hom-associative algebras by applying the Gröbner–Shirshov basis theory for (unital) centroid hom-associative algebras. Finally, we prove that the "Poincaré-Birkhoff-Witt theorem" holds for certain type of centroid hom-Lie algebras over a field of characteristic 0, namely, every centroid hom-Lie algebra such that the eigenvectors of the map $\beta$ linearly generates the whole algebra can be embedded into its universal enveloping centroid hom-associative algebra, and the linear basis of the universal enveloping algebra does not depend on the multiplication table of the centroid hom-Lie algebra under consideration.

We study the topology of closed, simply-connected, $6$-dimensional Riemannian manifolds of positive sectional curvature which admit isometric actions by ${\rm SU}(2)$ or ${\rm SO}(3)$. We show that their Euler characteristic agrees with that of the known examples, i.e., $S^6$, $\mathbb{C P}^3$, the Wallach space ${\rm SU}(3)/T^2$ and the biquotient ${\rm SU}(3)//T^2$. We also classify, up to equivariant diffeomorphism, certain actions without exceptional orbits and show that there are strong restrictions on the exceptional strata.

In this paper, the entropy of discrete Heisenberg group actions is considered. Let $\alpha$ be a discrete Heisenberg group action on a compact metric space $X$. Two types of entropies, $\widetilde{h}(\alpha)$ and $h(\alpha)$ are introduced, in which $\widetilde{h}(\alpha)$ is defined in Ruelle's way and $h(\alpha)$ is defined via the natural extension of $\alpha$. It is shown that when $X$ is the torus and $\alpha$ is induced by integer matrices then $\widetilde{h}(\alpha)$ is zero and $h(\alpha)$ can be expressed via the eigenvalues of the matrices.

In this paper, we propose a class of robust independence tests for two random vectors based on weighted integrals of empirical characteristic functions. By letting weight functions be probability density functions of a class of special distributions, the proposed test statistics have simple closed forms and do not require moment conditions on the random vectors. Moreover, we derive the asymptotic distributions of the test statistics under the null hypothesis. The proposed testing method is computationally feasible and easy to implement. Based on a data-driven bandwidth selection method, Monte Carlo simulation studies indicate that our tests have a relatively good performance compared with the competitors. A real data example is also presented to illustrate the application of our tests.

Using methods from complex analysis in one variable, we define an integral operator that solves $\bar\partial$ with supnorm estimates on product domains in $\mathbb{C}^n$.

We introduce notions of continuous orbit equivalence and its one-sided version for countable left Ore semigroup actions on compact spaces by surjective local homeomorphisms, and characterize them in terms of the corresponding transformation groupoids and their operator algebras. In particular, we show that two essentially free semigroup actions on totally disconnected compact spaces are continuously orbit equivalent if and only if there is a canonical abelian subalgebra preserving $C^*$-isomorphism between the associated transformation groupoid $C^*$-algebras. We also give some examples of orbit equivalence, consider the special case of semigroup actions by homeomorphisms and relate continuous orbit equivalence of semigroup actions to that of the associated group actions.

In this paper, the dynamics (including shadowing property, expansiveness, topological stability and entropy) of several types of upper semi-continuous set-valued maps are mainly considered from differentiable dynamical systems points of view. It is shown that (1) if $f$ is a hyperbolic endomorphism then for each $\varepsilon>0$ there exists a $C^1$-neighborhood $\mathcal{U}$ of $f$ such that the induced set-valued map $F_{f,\mathcal{U}}$ has the $\varepsilon$-shadowing property, and moreover, if $f$ is an expanding endomorphism then there exists a $C^1$-neighborhood $\mathcal{U}$ of $f$ such that the induced set-valued map $F_{f,\mathcal{U}}$ has the Lipschitz shadowing property; (2) when a set-valued map $F$ is generated by finite expanding endomorphisms, it has the shadowing property, and moreover, if the collection of the generators has no coincidence point then $F$ is expansive and hence is topologically stable; (3) if $f$ is an expanding endomorphism then for each $\varepsilon>0$ there exists a $C^1$-neighborhood $\mathcal{U}$ of $f$ such that $h(F_{f,\mathcal{U}}, \varepsilon)=h(f)$; (4) when $F$ is generated by finite expanding endomorphisms with no coincidence point, the entropy formula of $F$ is given. Furthermore, the dynamics of the set-valued maps based on discontinuous maps on the interval are also considered.

In this paper, we derive the optimal Cauchy-Schwarz inequalities on a class of Hilbert and Krein modules over a Clifford algebra, which heavily depend on the Clifford algebraic structure. The obtained inequalities further lead to very general uncertainty inequalities on these modules. Some new phenomena arise, due to the non-commutative nature, the Clifford-valued inner products and the Krein geometry. Taking into account applications, special attention is given to the Dirac operator and the Howe dual pair ${\rm Pin}(m)\times \mathfrak{osp}(1|2)$. Moreover, it is surprisingly to find that the recent highly non-trivial uncertainty relation for triple observables is indeed a direct consequence of our Cauchy-Schwarz inequality. This new observation leads to refined uncertainty relations in terms of the Wigner-Yanase-Dyson skew information for mixed states and other generalizations. These show that the obtained uncertainty inequalities on Clifford modules can be considered as new uncertainty relations for multiple observables.

The penalized variable selection methods are often used to select the relevant covariates and estimate the unknown regression coefficients simultaneously, but these existing methods may fail to be consistent for the setting with highly correlated covariates. In this paper, the semi-standard partial covariance (SPAC) method with Lasso penalty is proposed to study the generalized linear model with highly correlated covariates, and the consistencies of the estimation and variable selection are shown in high-dimensional settings under some regularity conditions. Some simulation studies and an analysis of colon tumor dataset are carried out to show that the proposed method performs better in addressing highly correlated problem than the traditional penalized variable selection methods.

In this paper, we study the projectively Ricci-flat general $(\alpha, \beta)$-metrics within to a spray framework and also bring out the rich variety of behaviour displayed by an important projective invariant. Projective Ricci curvature is one of the essential projective invariant in Finsler geometry which has been introduced by Z. Shen. The projective Ricci curvature is defined as Ricci curvature of a projective spray associated with a given spray $G$ on $M^{n}$ with a volume form $dV$ on $M^{n}$.

In this paper, we establish a weighted maximal $L_2$ estimate of operator-valued Bochner-Riesz means. The proof is based on noncommutative square function estimates and a sharp weighted noncommutative Hardy-Littlewood maximal inequality.

Given a simple graph $G$ and a proper total-$k$-coloring $\phi$ from $V(G)\cup E(G)$ to $\{1,2,\ldots,k\}$. Let $f(v)=\phi(v)\prod_{uv\in E(G)}\phi(uv)$. The coloring $\phi$ is neighbor product distinguishing if $f(u)\neq f(v)$ for each edge $uv\in E(G)$. The neighbor product distinguishing total chromatic number of $G$, denoted by $\chi_{\Pi}^{\prime\prime}(G)$, is the smallest integer $k$ such that $G$ admits a $k$-neighbor product distinguishing total coloring. Li et al. conjectured that $\chi_{\Pi}^{\prime\prime}(G)\leq \Delta(G)+3$ for any graph with at least two vertices. Dong et al. showed that conjecture holds for planar graphs with maximum degree at least 10. By using the famous Combinatorial Nullstellensatz, we prove that if $G$ is a planar graph without 5-cycles, then $\chi_{\Pi}^{\prime\prime}(G)\leq$ max$\{\Delta(G)+2,12\}$.

The distributional properties of a multi-dimensional continuous-state branching process are determined by its cumulant semigroup, which is defined by the backward differential equation. We provide a proof of the assertion of Rhyzhov and Skorokhod (Theory Probab. Appl., 1970) on the uniqueness of the solutions to the equation, which is based on a characterization of the process as the pathwise unique solution to a system of stochastic equations.

In this paper, we prove that for each positive $k\equiv 1$ mod $m$ there exists a $P$-symmetric $km\tau$-periodic solution $x_k$ for asymptotically linear $m\tau$-periodic Hamiltonian systems, which are nonautonomous and endowed with a $P$-symmetry. If the $P$-symmetric Hamiltonian function is semi-positive, one can prove, under a new iteration inequality of the Maslov-type $P$-index, that $x_{k_1}$ and $x_{k_2}$ are geometrically distinct for $k_1/k_2\geq(2n+1)m+1$; and $x_{k_1},x_{k_2}$ are geometrically distinct for $k_1/k_2\geq m+1$ provided $x_{k_1}$ is non-degenerate.

The goal of this paper is to deal with a new dynamic system called a differential evolution hemivariational inequality (DEHVI) which couples an abstract parabolic evolution hemivariational inequality and a nonlinear differential equation in a Banach space. First, by applying surjectivity result for pseudomonotone multivalued mappins and the properties of Clarke's subgradient, we show the nonempty of the solution set for the parabolic hemivariational inequality. Then, some topological properties of the solution set are established such as boundedness, closedness and convexity. Furthermore, we explore the upper semicontinuity of the solution mapping. Finally, we prove the solution set of the system (DEHVI) is nonempty and the set of all trajectories of (DEHVI) is weakly compact in $C(I,X)$.

A diffeomorphism is non-uniformly partially hyperbolic if it preserves an ergodic measure with at least one zero Lyapunov exponent. We prove that a $C^{1}$-smooth $\mathbb{Z}^d$-action has the quasi-shadowing property if one of the generators is $C^{1+\alpha}\,(\alpha>0)$ non-uniformly partially hyperbolic.

Let Ω be a domain of $(\mathbb{R}^n)$ with n ≥ 2 and p(·) be a local Lipschitz funcion in Ω with 1 < p(x) < ∞ in Ω. We build up an interior quantitative second order Sobolev regularity for the normalized p(·)-Laplace equation -Δ_p(·)^Nu = 0 in Ω as well as the corresponding inhomogeneous equation -Δ_p(·)^Nu=f in Ω with f ∈ C⁰(Ω). In particular, given any viscosity solution u to -Δ_p(·)^Nu= 0 in Ω, we prove the following:
(i) in dimension $n=2$, for any subdomain $U \Subset \Omega$ and any $\beta \geq 0$, one has $|D u|^\beta D u \in L_{\text {loc }}^{2+\delta}(U)$ with a quantitative upper bound, and moreover, the map $\left(x_1, x_2\right) \rightarrow|D u|^\beta\left(u_{x_1},-u_{x_2}\right)$ is quasiregular in $U$ in the sense that
$\left|D\left[|D u|^\beta D u\right]\right|^2 \leq-C \operatorname{det} D\left[|D u|^\beta D u\right] \quad$ a.e. in $U$.
(ii) in dimension $n \geq 3$, for any subdomain $U \Subset \Omega$ with $\inf _U p(x)>1$ and $\sup _U p(x)<3+\frac{2}{n-2}$, one has $D^2 u \in L_{\text {loc }}^{2+\delta}(U)$ with a quantitative upper bound, and also with a pointwise upper bound
$\left|D^2 u\right|^2 \leq-C$ $\sum\limits_{1 \le i < j \le n} {} $ $\left[u_{x_i x_j} u_{x_j x_i}-u_{x_i x_i} u_{x_j x_j}\right]$ a.e. in $U$.
Here constants $\delta>0$ and $C \geq 1$ are independent of $u$. These extend the related results obtaind by Adamowicz-Hästö [Mappings of finite distortion and PDE with nonstandard growth. Int. Math. Res. Not. IMRN, 10, 1940-1965 (2010)] when $n=2$ and $\beta=0$.

The first passage time has many applications in fields like finance, econometrics, statistics, and biology. However, explicit formulas for the first passage density have only been obtained for a few cases. This paper derives an explicit formula for the first passage density of Brownian motion with two-sided piecewise continuous boundaries which may have some points of discontinuity. Approximations are used to obtain a simplified formula for estimating the first passage density. Moreover, the results are also generalized to the case of two-sided general nonlinear boundaries. Simulations can be easily carried out with Monte Carlo method and it is demonstrated for several typical two-sided boundaries that the proposed approximation method offers a highly accurate approximation of first passage density.

Let $R$ be a unitary ring and $a, b\in R$ with $ab=0$. We find the $2/3$ property of Drazin invertibility: if any two of $a, b$ and $a+b$ are Drazin invertible, then so is the third one. Then, we combine the $2/3$ property of Drazin invertibility to characterize the existence of generalized inverses by means of units. As applications, the need for two invertible morphisms used by You and Chen to characterize the group invertibility of a sum of morphisms is reduced to that for one invertible morphism, and the existence and expression of the inverse along a product of two regular elements are obtained, which generalizes the main result of Mary and Patrício (2016) about the group inverse of a product.

In this paper, we show the scattering of the radial solution for the nonlinear Schrödinger equation with combined power-type and Choquard-type nonlinearities \begin{align*} {\rm i}u_t+\Delta u=\lambda_1|u|^{p_1-1}u+\lambda_2(I_\alpha\ast|u|^{p_2})|u|^{p_2-2}u. \end{align*} in the energy space $H^1\left(\mathbb{R}^N\right)$ for $\lambda_1\lambda_2=-1$. We establish a scattering criterion for radial solution together with Morawetz estimate which implies the scattering theory. Results show that the defocusing perturbation terms does not determine the scattering solution in energy space.

In this paper, we characterize weighted composition operators that preserve frames on the weighted Hardy spaces in the unit disk. In particular, we obtain the symbol properties of the bounded invertible weighted composition operators. Moreover, we establish the equivalence between bounded invertible operators and frame-preserving operators. Furthermore, we show that weighted composition operator preserves frames if and only if it preserves the Riesz bases property. Additionally, we investigate the weighted composition operators that preserve tight or normalized tight frames on the Dirichlet space.