Let $L:=-\Delta +V$ be a Schrödinger operator on ${ℝ}^n$ with $n\ge 3$ and $V\ge 0$ satisfying ${\Delta }^{-1}V\in L^{\infty }\left({ℝ}^n\right)$. Assume that $\varphi :{ℝ}^n\times [0,\infty )\to [0,\infty )$ is a function such that $\varphi (x,·)$ is an Orlicz function, $\varphi (·,t)\in {𝔸}_{\infty }\left({ℝ}^n\right)$ (the class of uniformly Muckenhoupt weights). Let $w$ be an $L$-harmonic function on ${ℝ}^n$ with $0<C_1\le w\le C_2$, where $C_1$ and $C_2$ are positive constants. In this article, the author proves that the mapping $H_{\varphi ,L}\left({ℝ}^n\right)\ni f\mapsto wf\in H_{\varphi }\left({ℝ}^n\right)$ is an isomorphism from the Musielak-Orlicz-Hardy space associated with $L$, $H_{\varphi ,L}\left({ℝ}^n\right)$, to the Musielak-Orlicz-Hardy space $H_{\varphi }\left({ℝ}^n\right)$ under some assumptions on $\varphi$. As applications, the author further...

We study the boundedness of the one-sided operator $g{⁺}_{\lambda ,\phi }$ between the weighted spaces $L^p\left(M¯w\right)$ and $L^p\left(w\right)$ for every weight w. If λ = 2/p whenever 1 < p < 2, and in the case p = 1 for λ > 2, we prove the weak type of $g{⁺}_{\lambda ,\phi }$. For every λ > 1 and p = 2, or λ > 2/p and 1 < p < 2, the boundedness of this operator is obtained. For p > 2 and λ > 1, we obtain the boundedness of $g{⁺}_{\lambda ,\phi }$ from $L^p\left({\left(M¯\right)}^{[p/2]+1}w\right)$ to $L^p\left(w\right)$, where ${\left(M¯\right)}^k$ denotes the operator M¯ iterated k times.

We discuss the existence of the current $g^{k}T$, $k\in \mathbb{N}$ for positive and closed currents $T$ and unbounded plurisubharmonic functions $g$. Furthermore, a new type of weighted Lelong number is introduced under the name of weight $k$ Lelong number.

Let G be a locally compact group, let (φ,ψ) be a complementary pair of Young functions, and let $L^{\phi }\left(G\right)$ and $L^{\psi }\left(G\right)$ be the corresponding Orlicz spaces. Under some conditions on φ, we will show that for a Banach $L^{\phi }\left(G\right)$-submodule X of $L^{\psi }\left(G\right)$, the multiplier space $Ho{m}_{L^{\phi }\left(G\right)}(L^{\phi }\left(G\right),X*)$ is a dual Banach space with predual $L^{\phi }\left(G\right)\bullet X:=\overline{span}ux:u\in L^{\phi }\left(G\right),x\in X$, where the closure is taken in the dual space of $Ho{m}_{L^{\phi }\left(G\right)}(L^{\phi }\left(G\right),X*)$. We also prove that if $\phi$ is a Δ₂-regular N-function, then $C{v}_{\phi }\left(G\right)$, the space of convolutors of $M^{\phi }\left(G\right)$, is identified with the dual of a Banach algebra of functions on G...

Our aim in this paper is to establish Trudinger’s inequality on Musielak-Orlicz-Morrey spaces $L^{\Phi ,\kappa }\left(G\right)$ under conditions on $\Phi$ which are essentially weaker than those considered in a former paper. As an application and example, we show Trudinger’s inequality for double phase functionals $\Phi (x,t)=t^{p\left(x\right)}+a\left(x\right)t^{q\left(x\right)}$, where $p(·)$ and $q(·)$ satisfy log-Hölder conditions and $a(·)$ is nonnegative, bounded and Hölder continuous.

We study the duality theory of the weighted multi-parameter Triebel-Lizorkin spaces ${\dot{F}}_p^{\alpha ,q}(\omega ;{ℝ}^{n_1}\times {ℝ}^{n_2})$. This space has been introduced and the result $${\left({\dot{F}}_p^{\alpha ,q}(\omega ;{ℝ}^{n_1}\times {ℝ}^{n_2})\right)}^{*}={\mathrm{CMO}}_p^{-\alpha ,q^{\text{'}}}(\omega ;{ℝ}^{n_1}\times {ℝ}^{n_2})$$ for $0<p\le 1$ has been proved in Ding, Zhu (2017). In this paper, for $1<p<\infty$, $0<q<\infty$ we establish its dual space ${\dot{H}}_p^{\alpha ,q}(\omega ;{ℝ}^{n_1}\times {ℝ}^{n_2})$.

In this paper, characterizations of the embeddings between weighted Copson function spaces ${\mathrm{Cop}}_{p_1,q_1}(u_1,v_1)$ and weighted Cesàro function spaces ${\mathrm{Ces}}_{p_2,q_2}(u_2,v_2)$ are given. In particular, two-sided estimates of the optimal constant $c$ in the inequality $$\begin{array}{cc}\hfill d({\int }_0^{\infty }& {\left({\int }_0^{t}f{\left(\tau \right)}^{p_2}{v}_2\left(\tau \right)\mathrm{d}\tau \right)}^{q_2/p_2}{u}_2\left(t\right)\mathrm{d}t{)}^{1/q_2}\hfill \\ & \le c{\left({\int }_0^{\infty }{\left({\int }_t^{\infty }f{\left(\tau \right)}^{p_1}{v}_1\left(\tau \right)\mathrm{d}\tau \right)}^{q_1/p_1}{u}_1\left(t\right)\mathrm{d}t\right)}^{1/q_1},\hfill \end{array}d$$ where $p_1,p_2,q_1,q_2\in (0,\infty )$, $p_2\le q_2$ and $u_1$, $u_2$, $v_1$, $v_2$ are weights on $(0,\infty )$, are obtained. The most innovative part consists of the fact that possibly different parameters $p_1$ and $p_2$ and possibly different inner weights $v_1$ and $v_2$ are allowed. The proof is based on the combination of duality techniques...

Let $u$ be a holomorphic function and $\varphi$ a holomorphic self-map of the open unit disk $𝔻$ in the complex plane. We provide new characterizations for the boundedness of the weighted composition operators $u{C}_{\varphi }$ from Zygmund type spaces to Bloch type spaces in $𝔻$ in terms of $u$, $\varphi$, their derivatives, and ${\varphi }^n$, the $n$-th power of $\varphi$. Moreover, we obtain some similar estimates for the essential norms of the operators $u{C}_{\varphi }$, from which sufficient and necessary conditions of compactness of $u{C}_{\varphi }$ follows immediately. ...

Let $R$ be a unital $*$-ring. For any $a,s,t,v,w\in R$ we define the weighted $w$-core inverse and the weighted dual $s$-core inverse, extending the $w$-core inverse and the dual $s$-core inverse, respectively. An element $a\in R$ has a weighted $w$-core inverse with the weight $v$ if there exists some $x\in R$ such that $awxvx=x$, $xvawa=a$ and ${\left(awx\right)}^{*}=awx$. Dually, an element $a\in R$ has a weighted dual $s$-core inverse with the weight $t$ if there exists some $y\in R$ such that $ytysa=y$, $asaty=a$ and ${\left(ysa\right)}^{*}=ysa$. Several characterizations of weighted $w$-core invertible and weighted dual $s$-core...

Let Φ be a concave function on (0,∞) of strictly critical lower type index $p_{\Phi }\in (0,1]$ and $\omega \in A_{\infty }^{loc}\left(ℝⁿ\right)$ (the class of local weights introduced by V. S. Rychkov). We introduce the weighted local Orlicz-Hardy space $h_{\omega }^{\Phi }\left(ℝⁿ\right)$ via the local grand maximal function. Let $\rho \left(t\right)\equiv t^{-1}/{\Phi }^{-1}\left(t^{-1}\right)$ for all t ∈ (0,∞). We also introduce the BMO-type space $bm{o}_{\rho ,\omega }\left(ℝⁿ\right)$ and establish the duality between $h_{\omega }^{\Phi }\left(ℝⁿ\right)$ and $bm{o}_{\rho ,\omega }\left(ℝⁿ\right)$. Characterizations of $h_{\omega }^{\Phi }\left(ℝⁿ\right)$, including the atomic characterization, the local vertical and the local nontangential maximal function characterizations, are...

Let ϕ: ℝ → ℝ₊ ∪ 0 be an even convex continuous function with ϕ(0) = 0 and ϕ(u) > 0 for all u > 0 and let w be a weight function. u₀ and v₀ are defined by u₀ = supu: ϕ is linear on (0,u), v₀=supv: w is constant on (0,v) (where sup∅ = 0). We prove the following theorem. Theorem. Suppose that ${\Lambda }_{\varphi ,w}(0,\infty )$ (respectively, ${\Lambda }_{\varphi ,w}(0,1)$) is an order continuous Lorentz-Orlicz space. (1) ${\Lambda }_{\varphi ,w}$ has normal structure if and only if u₀ = 0 (respectively, ${\int }^{v₀}\varphi \left(u₀\right)·w<2andu₀<\infty ).$(2) ${\Lambda }_{\varphi ,w}$ has weakly normal structure if and only if ${\int }_0^{v₀}\varphi \left(u₀\right)·w<2$.

We prove that the associate space of a generalized Orlicz space $L^{\phi (·)}$ is given by the conjugate modular ${\phi }^{*}$ even without the assumption that simple functions belong to the space. Second, we show that every weakly doubling $\Phi$-function is equivalent to a doubling $\Phi$-function. As a consequence, we conclude that $L^{\phi (·)}$ is uniformly convex if $\phi$ and ${\phi }^{*}$ are weakly doubling.

The Rademacher sums are investigated in the Cesàro spaces $Ce{s}_p$ (1 ≤ p ≤ ∞) and in the weighted Korenblyum-Kreĭn-Levin spaces $K_{p,w}$ on [0,1]. They span l₂ space in $Ce{s}_p$ for any 1 ≤ p < ∞ and in $K_{p,w}$ if and only if the weight w is larger than $tlog{₂}^{p/2}(2/t)$ on (0,1). Moreover, the span of the Rademachers is not complemented in $Ce{s}_p$ for any 1 ≤ p < ∞ or in $K_{1,w}$ for any quasi-concave weight w. In the case when p > 1 and when w is such that the span of the Rademacher functions is isomorphic to l₂, this span is...

Let $L$ be a non-negative self-adjoint operator acting on $L^2\left({ℝ}^n\right)$ satisfying a pointwise Gaussian estimate for its heat kernel. Let $w$ be an $A_r$ weight on ${ℝ}^n\times {ℝ}^n$, $1<r<\infty$. In this article we obtain a weighted atomic decomposition for the weighted Hardy space $H_{L,w}^p({ℝ}^n\times {ℝ}^n)$, $0<p\le 1$ associated to $L$. Based on the atomic decomposition, we show the dual relationship between $H_{L,w}^1({ℝ}^n\times {ℝ}^n)$ and ${\mathrm{BMO}}_{L,w}({ℝ}^n\times {ℝ}^n)$.