Extremal Problems for Mappings with Generalized Parametric Representation in $${{\mathbb {C}}}^{n}$$

In this paper we are concerned with the family \(\widetilde{S}^t_A(\mathbb {B}^n)\) (\(t\ge 0\)) of normalized biholomorphic mappings on the Euclidean unit ball \(\mathbb {B}^n\) in \({\mathbb {C}}^n\) that can be embedded in normal Loewner chains whose normalizations are given by time-dependent operators \(A\in \widetilde{\mathcal {A}}\), where \(\widetilde{\mathcal {A}}\) is a family of measurable mappings from \([0,\infty )\) into \(L({\mathbb {C}}^n)\) which satisfy certain natural assumptions. In particular, we consider extreme points and support points associated with the compact family \(\widetilde{S}^t_A(\mathbb {B}^n)\), where \(A\in \widetilde{\mathcal {A}}\). We prove that if \(f(z,t)=V(t)^{-1}z+\cdots \) is a normal Loewner chain such that \(V(s)f(\cdot ,s)\in \mathrm{ex}\,\widetilde{S}^s_A(\mathbb {B}^n)\) (resp. \(V(s)f(\cdot ,s)\in \mathrm{supp}\,\widetilde{S}^s_A(\mathbb {B}^n)\)), then \(V(t)f(\cdot ,t)\in \mathrm{ex}\, \widetilde{S}^t_A(\mathbb {B}^n)\), for all \(t\ge s\) (resp. \(V(t)f(\cdot ,t)\in \mathrm{supp}\,\widetilde{S}^t_A(\mathbb {B}^n)\), for all \(t\ge s\)), where V(t) is the unique solution on \([0,\infty )\) of the initial value problem: \(\frac{d V}{d t}(t)=-A(t)V(t)\), a.e. \(t\ge 0\), \(V(0)=I_n\). Also, we obtain an example of a bounded support point for the family \(\widetilde{S}_A^t(\mathbb {B}^2)\), where \(A\in \widetilde{\mathcal {A}}\) is a certain time-dependent operator. We also consider the notion of a reachable family with respect to time-dependent linear operators \(A\in \widetilde{\mathcal {A}}\), and obtain characterizations of extreme/support points associated with these families of bounded biholomorphic mappings on \(\mathbb {B}^n\). Useful examples and applications yield that the study of the family \(\widetilde{S}^t_A(\mathbb {B}^n)\) for time-dependent operators \(A\in \widetilde{\mathcal {A}}\) is basically different from that in the case of constant time-dependent linear operators.     

Regularity for Inhomogeneous Dirichlet Problems of Some Schrödinger Equations on Domains

Let \(n\ge 3, \Omega \) be a bounded, simply connected and semiconvex domain in \({\mathbb {R}}^n\) and \(L_{\Omega }:=-\Delta +V\) a Schrödinger operator on \(L^2 (\Omega )\) with the Dirichlet boundary condition, where \(\Delta \) denotes the Laplace operator and the potential \(0\le V\) belongs to the reverse Hölder class \(RH_{q_0}({\mathbb {R}}^n)\) for some \(q_0\in (\max \{n/2,2\},\infty ]\). Assume that the growth function \(\varphi :\,{\mathbb {R}}^n\times [0,\infty ) \rightarrow [0,\infty )\) satisfies that \(\varphi (x,\cdot )\) is an Orlicz function and \(\varphi (\cdot ,t)\in {\mathbb {A}}_{\infty }({\mathbb {R}}^n)\) (the class of uniformly Muckenhoupt weights). Let \(H_{\varphi ,\,L_{{\mathbb {R}}^n},\,r}(\Omega )\) be the Musielak–Orlicz–Hardy space whose elements are restrictions of elements of the Musielak–Orlicz–Hardy space, associated with \(L_{{\mathbb {R}}^n}:=-\Delta +V\) on \({\mathbb {R}}^n\), to \(\Omega \). In this article, the authors show that the operators \(VL^{-1}_\Omega \) and \(\nabla ^2L^{-1}_\Omega \) are bounded from \(L^1(\Omega )\) to weak-\(L^1(\Omega )\), from \(L^p(\Omega )\) to itself, with \(p\in (1,2]\), and also from \(H_{\varphi ,\,L_{{\mathbb {R}}^n},\,r}(\Omega )\) to the Musielak–Orlicz space \(L^\varphi (\Omega )\) or to \(H_{\varphi ,\,L_{{\mathbb {R}}^n},\,r}(\Omega )\) itself. As applications, the boundedness of \(\nabla ^2{\mathbb {G}}_D\) on \(L^p(\Omega )\), with \(p\in (1,2]\), and from \(H_{\varphi ,\,L_{{\mathbb {R}}^n},\,r}(\Omega )\) to \(L^\varphi (\Omega )\) or to \(H_{\varphi ,\,L_{{\mathbb {R}}^n},\,r}(\Omega )\) itself is obtained, where \({\mathbb {G}}_D\) denotes the Dirichlet Green operator associated with \(L_\Omega \). All these results are new even for the Hardy space \(H^1_{L_{{\mathbb {R}}^n},\,r}(\Omega )\), which is just \(H_{\varphi ,\,L_{{\mathbb {R}}^n},\,r}(\Omega )\) with \(\varphi (x,t):=t\) for all \(x\in {\mathbb {R}}^n\) and \(t\in [0,\infty )\).     

On the 2-adic valuation of the cardinality of elliptic curves over finite extensions of $$\mathbb {F}_{\varvec{q}} $$

In this paper we study the difference between the 2-adic valuations of the cardinalities \( \# E( \mathbb {F}_{q^k} ) \) and \( \# E( \mathbb {F}_q ) \) of an elliptic curve E over \( \mathbb {F}_q \). We also deduce information about the structure of the 2-Sylow subgroup \( E[ 2^\infty ]( \mathbb {F}_{q^k} ) \) from the exponents of \( E[ 2^\infty ]( \mathbb {F}_q ) \).     

Map determined by rank-$$\varvec{}$$ matrice for relatively mall $$\varvec{}$$

Let n and s be integers such that \(1\le s<\frac{n}{2}\), and let \(M_n(\mathbb {K})\) be the ring of all \(n\times n\) matrices over a field \(\mathbb {K}\). Denote by \([\frac{n}{s}]\) the least integer m with \(m\ge \frac{n}{s}\). In this short note, it is proved that if \(g:M_n(\mathbb {K})\rightarrow M_n(\mathbb {K})\) is a map such that \(g\left( \sum _{i=1}^{[\frac{n}{s}]}A_i\right) =\sum _{i=1}^{[\frac{n}{s}]}g(A_i)\) holds for any \([\frac{n}{s}]\) rank-s matrices \(A_1,\ldots ,A_{[\frac{n}{s}]}\in M_n(\mathbb {K})\), then \(g(x)=f(x)+g(0)\), \(x\in M_n(\mathbb {K})\), for some additive map \(f:M_n(\mathbb {K})\rightarrow M_n(\mathbb {K})\). Particularly, g is additive if \(char\mathbb {K}\not \mid \left( [\frac{n}{s}]-1\right) \).     

A Sharp Weak-Type $$(\infty ,\infty )$$ Inequality for the Hilbert Transform

The paper is devoted to sharp weak type \((\infty ,\infty )\) estimates for \({\mathcal {H}}^{\mathbb {T}}\) and \({\mathcal {H}}^{\mathbb {R}}\), the Hilbert transforms on the circle and real line, respectively. Specifically, it is proved that

$$\begin{aligned} \left\| {\mathcal {H}}^{\mathbb {T}}f\right\| _{W({\mathbb {T}})}\le \Vert f\Vert _{L^\infty ({\mathbb {T}})} \end{aligned}$$

and

$$\begin{aligned} \left\| {\mathcal {H}}^{\mathbb {R}}f\right\| _{W({\mathbb {R}})}\le \Vert f\Vert _{L^\infty ({\mathbb {R}})}, \end{aligned}$$

where \(W({\mathbb {T}})\) and \(W({\mathbb {R}})\) stand for the weak-\(L^\infty \) spaces introduced by Bennett, DeVore and Sharpley. In both estimates, the constant \(1\) on the right is shown to be the best possible.

     

On Lipschitz continuity of solutions of hyperbolic Poisson’s equation

In this paper, we investigate solutions of the hyperbolic Poisson equation \(\Delta _{h}u(x)=\psi (x)\), where \(\psi \in L^{\infty }(\mathbb {B}^{n}, {\mathbb R}^n)\) and

$$\begin{aligned} \Delta _{h}u(x)= (1-|x|^2)^2\Delta u(x)+2(n-2)\left( 1-|x|^2\right) \sum _{i=1}^{n} x_{i} \frac{\partial u}{\partial x_{i}}(x) \end{aligned}$$

is the hyperbolic Laplace operator in the n-dimensional space \(\mathbb {R}^n\) for \(n\ge 2\). We show that if \(n\ge 3\) and \(u\in C^{2}(\mathbb {B}^{n},{\mathbb R}^n) \cap C(\overline{\mathbb {B}^{n}},{\mathbb R}^n )\) is a solution to the hyperbolic Poisson equation, then it has the representation \(u=P_{h}[\phi ]-G_{ h}[\psi ]\) provided that \(u\mid _{\mathbb {S}^{n-1}}=\phi \) and \(\int _{\mathbb {B}^{n}}(1-|x|^{2})^{n-1} |\psi (x)|\,d\tau (x)<\infty \). Here \(P_{h}\) and \(G_{h}\) denote Poisson and Green integrals with respect to \(\Delta _{h}\), respectively. Furthermore, we prove that functions of the form \(u=P_{h}[\phi ]-G_{h}[\psi ]\) are Lipschitz continuous.

     

On $$L^p$$-Boundedness of Pseudo-Differential Operators of Sjöstrand’s Class

We extended the known result that symbols from modulation spaces \(M^{\infty ,1}(\mathbb {R}^{2n})\), also known as the Sjöstrand’s class, produce bounded operators in \(L^2(\mathbb {R}^n)\), to general \(L^p\) boundedness at the cost of loss of derivatives. Indeed, we showed that pseudo-differential operators acting from \(L^p\)-Sobolev spaces \(L^p_s(\mathbb {R}^n)\) to \(L^p(\mathbb {R}^n)\) spaces with symbols from the modulation space \(M^{\infty ,1}(\mathbb {R}^{2n})\) are bounded, whenever \(s\ge n|1/p-1/2|.\) This estimate is sharp for all \(1< p<\infty \).     

Subconvexity for sup-norms of cusp forms on $$\mathrm{PGL}(n)$$

Let F be an \(L^2\)-normalized Hecke Maaß cusp form for \(\Gamma _0(N) \subseteq {\mathrm{SL}}_{n}({\mathbb {Z}})\) with Laplace eigenvalue \(\lambda _F\). If \(\Omega \) is a compact subset of \(\Gamma _0(N)\backslash {\mathrm{PGL}}_n/\mathrm{PO}_{n}\), we show the bound \(\Vert F|_{\Omega }\Vert _{\infty } \ll _{ \Omega } N^{\varepsilon } \lambda _F^{n(n-1)/8 - \delta }\) for some constant \(\delta = \delta _n> 0\) depending only on n.     

On the image,characterization, and automatic continuity of $$\varvec{(\sigma }, \varvec{\tau }$$ )-derivations

The first main theorem of this paper asserts that any \((\sigma , \tau )\)-derivation d, under certain conditions, either is a \(\sigma \)-derivation or is a scalar multiple of (\(\sigma - \tau \)), i.e. \(d = \lambda (\sigma - \tau )\) for some \(\lambda \in \mathbb {C} \backslash \{0\}\). By using this characterization, we achieve a result concerning the automatic continuity of \((\sigma , \tau \))-derivations on Banach algebras which reads as follows. Let \(\mathcal {A}\) be a unital, commutative, semi-simple Banach algebra, and let \(\sigma , \tau : \mathcal {A} \rightarrow \mathcal {A}\) be two distinct endomorphisms such that \(\varphi \sigma (\mathbf e )\) and \(\varphi \tau (\mathbf e )\) are non-zero complex numbers for all \(\varphi \in \Phi _\mathcal {A}\). If \(d : \mathcal {A} \rightarrow \mathcal {A}\) is a \((\sigma , \tau )\)-derivation such that \(\varphi d\) is a non-zero linear functional for every \(\varphi \in \Phi _\mathcal {A}\), then d is automatically continuous. As another objective of this research, we prove that if \(\mathfrak {M}\) is a commutative von Neumann algebra and \(\sigma :\mathfrak {M} \rightarrow \mathfrak {M}\) is an endomorphism, then every Jordan \(\sigma \)-derivation \(d:\mathfrak {M} \rightarrow \mathfrak {M}\) is identically zero.     

On homogeneous hypersurfaces in $${\mathbb {C}}^3$$

We consider a family \(M_t^n\), with \(n\geqslant 2\), \(t>1\), of real hypersurfaces in a complex affine n-dimensional quadric arising in connection with the classification of homogeneous compact simply connected real-analytic hypersurfaces in  \({\mathbb {C}}^n\) due to Morimoto and Nagano. To finalize their classification, one needs to resolve the problem of the embeddability of \(M_t^n\) in  \({\mathbb {C}}^n\) for \(n=3,7\). In our earlier article we showed that \(M_t^7\) is not embeddable in  \({\mathbb {C}}^7\) for every t and that \(M_t^3\) is embeddable in  \({\mathbb {C}}^3\) for all \(1<t<1+10^{-6}\). In the present paper, we improve on the latter result by showing that the embeddability of \(M_t^3\) in fact takes place for \(1<t<\sqrt{(2+\sqrt{2})/3}\). This is achieved by analyzing the explicit totally real embedding of the sphere \(S^3\) in \({\mathbb {C}}^3\) constructed by Ahern and Rudin. For \(t\geqslant {\sqrt{(2+\sqrt{2})/3}}\), the problem of the embeddability of \(M_t^3\) remains open.     

Properties of singular integral operators $$\varvec{S}_{\varvec{\alpha ,\beta }}$$

For \(\alpha , \beta \in L^{\infty } (S^1),\) the singular integral operator \(S_{\alpha ,\beta }\) on \(L^2 (S^1)\) is defined by \(S_{\alpha ,\beta }f:= \alpha Pf+\beta Qf\), where P denotes the orthogonal projection of \(L^2(S^1)\) onto the Hardy space \(H^2(S^1),\) and Q denotes the orthogonal projection onto \(H^2(S^1)^{\perp }\). In a recent paper, Nakazi and Yamamoto have studied the normality and self-adjointness of \(S_{\alpha ,\beta }\). This work has shown that \(S_{\alpha ,\beta }\) may have analogous properties to that of the Toeplitz operator. In this paper, we study several other properties of \(S_{\alpha ,\beta }\).     

$${\varvec{}}$$-Jorda homomorphisms ad pseudo $${\varvec{}}$$-Jorda homomorphisms o Baach algebras

Let \(n\in \mathbb {N}\), A and B be Banach algebras and let B be a right A-module. We say that a linear mapping \(\varphi :A\longrightarrow B\) is a pseudo n-Jordan homomorphism if there exists an element \(w\in A\) such that \(\varphi (a^nw)=\varphi (a)^n\cdot w\), for every \(a\in A\) and \(n\ge \) 2. In this paper, among other things, we show that under some conditions if a linear mapping \(\varphi \) is a (pseudo) n-Jordan homomorphism, then it is a (pseudo) \((n + 1)\)-Jordan homomorphism. Additionally, we investigate automatic continuity of surjective pseudo n-Jordan homomorphisms under some conditions.     

A gradient estimate on graphs with the $$\varvec{}$$-condition

Let \(\omega \) be an unbounded radial weight on \(\mathbb {C}^d\), \(d\ge 1\). Using results related to approximation of \(\omega \) by entire maps, we investigate Volterra type and weighted composition operators defined on the growth space \(\mathcal {A}^\omega (\mathbb {C}^d)\). Special attention is given to the operators defined on the growth Fock spaces.     

$$L^{p}$$-Norms a nd Mahler’s Measure of Poly nomials o n the n-Dime nsio nal Torus

We prove Nikol’skii type inequalities that, for polynomials on the n-dimensional torus \(\mathbb {T}^n\), relate the \(L^p\)-norm with the \(L^q\)-norm (with respect to the normalized Lebesgue measure and \(0 <p <q < \infty \)). Among other things, we show that \(C=\sqrt{q/p}\) is the best constant such that \(\Vert P\Vert _{L^q}\le C^{\text {deg}(P)} \Vert P\Vert _{L^p}\) for all homogeneous polynomials P on \(\mathbb {T}^n\). We also prove an exact inequality between the \(L^p\)-norm of a polynomial P on \(\mathbb {T}^n\) and its Mahler measure M(P), which is the geometric mean of |P| with respect to the normalized Lebesgue measure on \(\mathbb {T}^n\). Using extrapolation, we transfer this estimate into a Khintchine–Kahane type inequality, which, for polynomials on \(\mathbb {T}^n\), relates a certain exponential Orlicz norm and Mahler’s measure. Applications are given, including some interpolation estimates.     

Nonlinear Evolution Equations with Infinitely Many Derivatives

We study the nonlinear evolutionary euclidean bosonic string equation

$$\begin{aligned} u_t = \Delta e^{-c \Delta }\,u + U(t,u) , \quad c > 0 \; \end{aligned}$$

on the Euclidean space \({\mathbb {R}}^n\). We interpret the nonlocal operator \(\Delta e^{-c\,\Delta }\) using entire vectors of \(\Delta \) in \(L^2({\mathbb {R}}^n)\). We prove that it generates a bounded holomorphic \(C_0\)-semigroup on \(L^2({\mathbb {R}}^n)\) (so that it also satisfies maximal \(L^p\) regularity) and we show the well-posedness of the corresponding nonlinear Cauchy problem.

     

Weighted local Hardy spaces associated with operators

Let L be a self-adjoint positive operator on \(L^2(\mathbb {R}^n)\). Assume that the semigroup \(e^{-tL}\) generated by \(-L\) satisfies the Gaussian kernel bounds on \(L^2(\mathbb {R}^n)\). In this article, we study weighted local Hardy space \(h_{L,w}^{1}(\mathbb {R}^n)\) associated with L in terms of the area function characterization, and prove their atomic characters. Then, we introduce the weighted local BMO space \(\mathrm{bmo}_{L,w}(\mathbb {R}^n)\) and prove that the dual of \(h_{L,w}^{1}(\mathbb {R}^n)\) is \(\mathrm{bmo}_{L,w}(\mathbb {R}^n)\). Finally a broad class of applications of these results is described.     

Some Examples of Extremal Functions on the Dunkl-Type Fock Space $$F_k(\mathbb {C})$$

In this work, we recall some properties of the generalized Fock space \(F_k(\mathbb {C})\) and the Hilbert space \(H_k(\mathbb {C})\) (defined by the squared norm \(\Vert f\Vert ^2_{H_k(\mathbb {C})}:=\Vert f\Vert ^2_{F_k(\mathbb {C})}+\langle zf',f\rangle _{F_k(\mathbb {C})}\)). Next, we give the best approximation of the bounded operators \(L:F_k(\mathbb {C})\rightarrow H_k(\mathbb {C})\). As applications, we come up with some results regarding the approximate formulas for the difference operator, the Dunkl difference operator and the primitive operator.     

On the Stationary Navier–Stokes Problem in $$\mathbb {R}^3$$: An Approach in Weighted Sobolev Spaces

In this paper, we study the steady-state Navier–Stokes equations in \(\mathbb {R}^3\). First, we establish the existence of very weak solution in \(\varvec{L}^p(\mathbb {R}^3)\) with \(3/2< p < +\infty \) under smallness conditions on the data. A uniqueness result is also given in case the data belong to \(\mathbb {L}^r(\mathbb {R}^3)\cap \mathbb {L}^{3/2}(\mathbb {R}^3)\) with \(3/2<r<3\). We also discuss the case where the data are not necessarily small. In particular, these results enhance those obtained by Bjorland et al. (Commun Partial Differ Equ 26:216–246, 2011), and are in agreement with those obtained by Kim and Kozono (J Math Anal Appl 395(2):486–495, 2012). Second, we prove a result of existence and uniqueness of weak solution in the weighted Sobolev space \(\varvec{W}_0^{1,p}(\mathbb {R}^3)\cap \varvec{W}_0^{1,\,3/2}(\mathbb {R}^3)\) in case of small external forces given by \(\mathrm{div}\mathbb {F}\) with \(\mathbb {F} \in \mathbb {L}^p(\mathbb {R}^3)\cap \mathbb {L}^{3/2}(\mathbb {R}^3)\) and \(1<p<3\).     

A Characterization of Two-Weight Norm Inequality for Littlewood–Paley $$g_{\lambda }^{*}$$-Function

Let \(n\ge 2\) and \(g_{\lambda }^{*}\) be the well-known high-dimensional Littlewood–Paley function which was defined and studied by E. M. Stein,

$$\begin{aligned} g_{\lambda }^{*}(f)(x) =\bigg (\iint _{\mathbb {R}^{n+1}_{+}} \Big (\frac{t}{t+|x-y|}\Big )^{n\lambda } |\nabla P_tf(y,t)|^2 \frac{\mathrm{d}y \mathrm{d}t}{t^{n-1}}\bigg )^{1/2}, \ \quad \lambda > 1, \end{aligned}$$

where \(P_tf(y,t)=p_t*f(y)\), \(p_t(y)=t^{-n}p(y/t)\), and \(p(x) = (1+|x|^2)^{-(n+1)/2}\), \(\nabla =(\frac{\partial }{\partial y_1},\ldots ,\frac{\partial }{\partial y_n},\frac{\partial }{\partial t})\). In this paper, we give a characterization of two-weight norm inequality for \(g_{\lambda }^{*}\)-function. We show that \(\big \Vert g_{\lambda }^{*}(f \sigma ) \big \Vert _{L^2(w)} \lesssim \big \Vert f \big \Vert _{L^2(\sigma )}\) if and only if the two-weight Muckenhoupt \(A_2\) condition holds, and a testing condition holds:

$$\begin{aligned} \sup _{Q : \text {cubes}~\mathrm{in} \ {\mathbb {R}^n}} \frac{1}{\sigma (Q)} \int _{{\mathbb {R}^n}} \iint _{\widehat{Q}} \Big (\frac{t}{t+|x-y|}\Big )^{n\lambda }|\nabla P_t(\mathbf {1}_Q \sigma )(y,t)|^2 \frac{w \mathrm{d}x \mathrm{d}t}{t^{n-1}} \mathrm{d}y < \infty , \end{aligned}$$

where \(\widehat{Q}\) is the Carleson box over Q and \((w, \sigma )\) is a pair of weights. We actually prove this characterization for \(g_{\lambda }^{*}\)-function associated with more general fractional Poisson kernel \(p^\alpha (x) = (1+|x|^2)^{-{(n+\alpha )}/{2}}\). Moreover, the corresponding results for intrinsic \(g_{\lambda }^*\)-function are also presented.

     

Intrinsic Structures of Certain Musielak–Orlicz Hardy Spaces

For any \(p\in (0,\,1]\), let \(H^{\Phi _p}(\mathbb {R}^n)\) be the Musielak–Orlicz Hardy space associated with the Musielak–Orlicz growth function \(\Phi _p\), defined by setting, for any \(x\in \mathbb {R}^n\) and \(t\in [0,\,\infty )\),

$$\begin{aligned}&\Phi _{p}(x,\,t)\\&\quad := {\left\{ \begin{array}{ll} \displaystyle \frac{t}{\log {(e+t)}+[t(1+|x|)^n]^{1-p}}&{} \quad \text {when}\ n(1/p-1)\notin \mathbb N \cup \{0\},\\ \displaystyle \frac{t}{\log (e+t)+[t(1+|x|)^n]^{1-p}[\log (e+|x|)]^p}&{} \quad \text {when}\ n(1/p-1)\in \mathbb N\cup \{0\}, \end{array}\right. } \end{aligned}$$

which is the sharp target space of the bilinear decomposition of the product of the Hardy space \(H^p(\mathbb {R}^n)\) and its dual. Moreover, \(H^{\Phi _1}(\mathbb {R}^n)\) is the prototype appearing in the real-variable theory of general Musielak–Orlicz Hardy spaces. In this article, the authors find a new structure of the space \(H^{\Phi _p}(\mathbb {R}^n)\) by showing that, for any \(p\in (0,\,1]\), \(H^{\Phi _p}(\mathbb {R}^n)=H^{\phi _0}(\mathbb {R}^n) +H_{W_p}^p({{{\mathbb {R}}}^n})\) and, for any \(p\in (0,\,1)\), \(H^{\Phi _p}(\mathbb {R}^n)=H^{1}(\mathbb {R}^n) +H_{W_p}^p({{{\mathbb {R}}}^n})\), where \(H^1(\mathbb {R}^n)\) denotes the classical real Hardy space, \(H^{\phi _0}({{{\mathbb {R}}}^n})\) the Orlicz–Hardy space associated with the Orlicz function \(\phi _0(t):=t/\log (e+t)\) for any \(t\in [0,\infty )\), and \(H_{W_p}^p(\mathbb {R}^n)\) the weighted Hardy space associated with certain weight function \(W_p(x)\) that is comparable to \(\Phi _p(x,1)\) for any \(x\in \mathbb {R}^n\). As an application, the authors further establish an interpolation theorem of quasilinear operators based on this new structure.