We study continuity envelopes of function spaces $B_{p,q}^s(ℝⁿ,w)$ and $F_{p,q}^s(ℝⁿ,w)$ where the weight belongs to the Muckenhoupt class ₁. This essentially extends partial forerunners in [13, 14]. We also indicate some applications of these results.

The existence of solutions to the elliptic problem rot v = w, div v = 0 in a bounded domain Ω ⊂ ℝ³, ${v·n̅|}_S=0$, S = ∂Ω in weighted $L_p$-Sobolev spaces is proved. It is assumed that an axis L crosses Ω and the weight is a negative power function of the distance to the axis. The main part of the proof is devoted to examining solutions of the problem in a neighbourhood of L. The existence in Ω follows from the technique of regularization.

We study the properties of the weighted space $H_{2,\alpha }^k\left(\Omega \right)$ and weighted set $W_{2,\alpha }^k(\Omega ,\delta )$ for boundary value problem with singularity.

We present several continuous embeddings of the critical Besov space $B_p^{n/p,\rho }\left(ℝⁿ\right)$. We first establish a Gagliardo-Nirenberg type estimate ${\left|\right|u\left|\right|}_{{Ḃ}_{q,w_r}^{0,\nu }}\le Cₙ{(1/(n-r))}^{1/q+1/\nu -1/\rho }{(q/r)}^{1/\nu -1/\rho }{\left|\right|u\left|\right|}_{{Ḃ}_p^{0,\rho }}^{(n-r)p/nq}{\left|\right|u\left|\right|}_{{Ḃ}_p^{n/p,\rho }}^{1-(n-r)p/nq}$, for 1 < p ≤ q < ∞, 1 ≤ ν < ρ ≤ ∞ and the weight function $w_r\left(x\right)=1/{\left(\right|x|}^r)$ with 0 < r < n. Next, we prove the corresponding Trudinger type estimate, and obtain it in terms of the embedding $B_p^{n/p,\rho }\left(ℝⁿ\right)\hookrightarrow B_{\Phi ₀,w_r}^{0,\nu }\left(ℝⁿ\right)$, where the function Φ₀ of the weighted Besov-Orlicz space $B_{\Phi ₀,w_r}^{0,\nu }\left(ℝⁿ\right)$ is a Young function of the exponential type. Another point of interest is to embed $B_p^{n/p,\rho }\left(ℝⁿ\right)$ into the weighted Besov...

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 ${\dot{H}}_p^{\alpha ,q}(\omega ;{ℝ}^{n_1}\times {ℝ}^{n_2})$.

CONTENTS1. Introduction.......................................................................52. Notation and auxiliary results............................................93. Statement of the problem (1.1)-(1.3)..............................204. The problem (3.14).........................................................225. Auxiliary results in $D_{\vartheta }$...............................................346. Existence of solutions of (3.14) in $H_{\mu }^k\left(D_{\vartheta }\right)$............417. Green function................................................................528....

We characterize stability under composition of ultradifferentiable classes defined by weight sequences M, by weight functions ω, and, more generally, by weight matrices , and investigate continuity of composition (g,f) ↦ f ∘ g. In addition, we represent the Beurling space ${}^{\left(\omega \right)}$ and the Roumieu space ${}^{\omega }$ as intersection and union of spaces ${}^{\left(M\right)}$ and ${}^M$ for associated weight sequences, respectively.

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...

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.

For a function $f\in L_{loc}^p\left(ℝⁿ\right)$ the notion of p-mean variation of order 1, ${₁}^p(f,ℝⁿ)$ is defined. It generalizes the concept of F. Riesz variation of functions on the real line ℝ¹ to ℝⁿ, n > 1. The characterisation of the Sobolev space $W^{1,p}\left(ℝⁿ\right)$ in terms of ${₁}^p(f,ℝⁿ)$ is directly related to the characterisation of $W^{1,p}\left(ℝⁿ\right)$ by Lipschitz type pointwise inequalities of Bojarski, Hajłasz and Strzelecki and to the Bourgain-Brezis-Mironescu approach.

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.

Using partial derivatives $\partial f/\partial z$ and $\partial f/\partial \overline{z}$, we introduce Besov spaces of polyanalytic functions in the open unit disk, as well as in the upper half-plane. We then prove that the dilatations of functions in certain weighted polyanalytic Besov spaces converge to the same functions in norm. When restricted to the open unit disk, we prove that each polyanalytic function of degree $q$ can be approximated in norm by polyanalytic polynomials of degree at most $q$.

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 ${\mathrm{BMO}}_{L,w}({ℝ}^n\times {ℝ}^n)$.

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 $u$ 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. ...