Existence of solutions to the Poisson equation in $L_p$-weighted spaces

Joanna Rencławowicz; Wojciech M. Zajączkowski

Displaying similar documents to “Existence of solutions to the Poisson equation in $L_{p}$ -weighted spaces”

The Properties of the Weighted Space $H_{2, α}^{k} (Ω)$ and Weighted Set $W_{2, α}^{k} (Ω, δ)$

V.A. Rukavishnikov, E.V. Matveeva, E.I. Rukavishnikova (2018)

Communications in Mathematics

Similarity:

We study the properties of the weighted space $H_{2, α}^{k} (Ω)$ and weighted set $W_{2, α}^{k} (Ω, δ)$ for boundary value problem with singularity.

Natural boundary value problems for weighted form laplacians

Wojciech Kozłowski, Antoni Pierzchalski (2008)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Similarity:

The four natural boundary problems for the weighted form Laplacians $L = a d δ + b δ d, a, b > 0$ acting on polynomial differential forms in the $n$ -dimensional Euclidean ball are solved explicitly. Moreover, an algebraic algorithm for generating a solution from the boundary data is given in each case.

Existence of solutions to the (rot,div)-system in $L_{p}$ -weighted spaces

Wojciech M. Zajączkowski (2010)

Applicationes Mathematicae

Similarity:

The existence of solutions to the elliptic problem rot v = w, div v = 0 in a bounded domain Ω ⊂ ℝ³, ${v \cdot n ̅ |}_{S} = 0$ , 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.

Solutions to the equation div u = f in weighted Sobolev spaces

Katrin Schumacher (2008)

Banach Center Publications

Similarity:

We consider the problem div u = f in a bounded Lipschitz domain Ω, where f with $\int_{Ω} f = 0$ is given. It is shown that the solution u, constructed as in Bogovski’s approach in [1], fulfills estimates in the weighted Sobolev spaces $W_{w}^{k, q} (Ω)$ , where the weight function w is in the class of Muckenhoupt weights $A_{q}$ .

The representation of multi-hypergraphs by set intersections

Stanisław Bylka, Jan Komar (2007)

Discussiones Mathematicae Graph Theory

Similarity:

This paper deals with weighted set systems (V,,q), where V is a set of indices, $\subset 2^{V}$ and the weight q is a nonnegative integer function on . The basic idea of the paper is to apply weighted set systems to formulate restrictions on intersections. It is of interest to know whether a weighted set system can be represented by set intersections. An intersection representation of (V,,q) is defined to be an indexed family $R = {(R_{v})}_{v \in V}$ of subsets of a set S such that $| ⋂_{v \in E} R_{v} | = q (E)$ for each E ∈ . A necessary condition...

Disjoint hypercyclic powers of weighted translations on groups

Liang Zhang, Hui-Qiang Lu, Xiao-Mei Fu, Ze-Hua Zhou (2017)

Czechoslovak Mathematical Journal

Similarity:

Let $G$ be a locally compact group and let $1 \leq p < \infty .$ Recently, Chen et al. characterized hypercyclic, supercyclic and chaotic weighted translations on locally compact groups and their homogeneous spaces. There has been an increasing interest in studying the disjoint hypercyclicity acting on various spaces of holomorphic functions. In this note, we will study disjoint hypercyclic and disjoint supercyclic powers of weighted translation operators on the Lebesgue space $L^{p} (G)$ in terms of the weights. Sufficient...

Isometric composition operators on weighted Dirichlet space

Shi-An Han, Ze-Hua Zhou (2016)

Czechoslovak Mathematical Journal

Similarity:

We investigate isometric composition operators on the weighted Dirichlet space $𝒟_{α}$ with standard weights ${(1 - | z |}^{2})^{α}$ , $α > - 1$ . The main technique used comes from Martín and Vukotić who completely characterized the isometric composition operators on the classical Dirichlet space $𝒟$ . We solve some of these but not in general. We also investigate the situation when $𝒟_{α}$ is equipped with another equivalent norm.

The spectral decomposition of weighted shifts and the $A^{p}$ condition

Earl Berkson, T. A. Gillespie (1990)

Colloquium Mathematicae

Similarity:

Lipschitz continuity in Muckenhoupt 𝓐₁ weighted function spaces

Dorothee D. Haroske (2011)

Banach Center Publications

Similarity:

We study continuity envelopes of function spaces $B_{p, q}^{s} (ℝ ⁿ, w)$ and $F_{p, q}^{s} (ℝ ⁿ, w)$ 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.

Weighted sub-Bergman Hilbert spaces

Maria Nowak, Renata Rososzczuk (2014)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

Similarity:

We consider Hilbert spaces which are counterparts of the de Branges-Rovnyak spaces in the context of the weighted Bergman spaces $A_{α}^{2}$ , $- 1 < α < \infty$ . These spaces have already been studied in [8], [7], [5] and [1]. We extend some results from these papers.

Essential norms of weighted composition operators between Hardy spaces $H^{p}$ and $H^{q}$ for 1 ≤ p,q ≤ ∞

R. Demazeux (2011)

Studia Mathematica

Similarity:

We complete the different cases remaining in the estimation of the essential norm of a weighted composition operator acting between the Hardy spaces $H^{p}$ and $H^{q}$ for 1 ≤ p,q ≤ ∞. In particular we give some estimates for the cases 1 = p ≤ q ≤ ∞ and 1 ≤ q < p ≤ ∞.

Monotonicity of generalized weighted mean values

Alfred Witkowski (2004)

Colloquium Mathematicae

Similarity:

The author gives a new simple proof of monotonicity of the generalized extended mean values $M (r, s) = \leq {((\int f^{s} d μ) / (\int f^{r} d μ))}^{1 / (s - r)}$ introduced by F. Qi.

Existence of solutions to the (rot,div)-system in L₂-weighted spaces

Wojciech M. Zajączkowski (2009)

Applicationes Mathematicae

Similarity:

The existence of solutions to the elliptic problem rot v = w, div v = 0 in Ω ⊂ ℝ³, ${v \cdot n ̅ |}_{S} = 0$ , S = ∂Ω, in weighted Hilbert spaces is proved. It is assumed that Ω contains an axis L and the weight is a negative power of the distance to the axis. The main part of the proof is devoted to examining solutions in a neighbourhood of L. Their existence in Ω follows by regularization.

Weighted variable $L^{p}$ integral inequalities for the maximal operator on non-increasing functions

C. J. Neugebauer (2009)

Studia Mathematica

Similarity:

Let $B_{p}$ be the Ariõ-Muckenhoupt weight class which controls the weighted $L^{p}$ -norm inequalities for the Hardy operator on non-increasing functions. We replace the constant p by a function p(x) and examine the associated $L^{p (x)}$ -norm inequalities of the Hardy operator.

Existence and regularity of solutions of some elliptic system in domains with edges

Wojciech M. Zajączkowski

Similarity:

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 $D_{ϑ}$ ...............................................346. Existence of solutions of (3.14) in $H_{μ}^{k} (D_{ϑ})$ ............417. Green function................................................................528....

Compactness of composition operators acting on weighted Bergman-Orlicz spaces

Ajay K. Sharma, S. Ueki (2012)

Annales Polonici Mathematici

Similarity:

We characterize compact composition operators acting on weighted Bergman-Orlicz spaces $_{α}^{ψ} = f \in H () : \int_{} ψ (| f (z) |) d A_{α} (z) < \infty$ , where α > -1 and ψ is a strictly increasing, subadditive convex function defined on [0,∞) and satisfying ψ(0) = 0, the growth condition $l i m_{t \to \infty} ψ (t) / t = \infty$ and the Δ₂-condition. In fact, we prove that $C_{φ}$ is compact on $_{α}^{ψ}$ if and only if it is compact on the weighted Bergman space $²_{α}$ .

Note on duality of weighted multi-parameter Triebel-Lizorkin spaces

Wei Ding, Jiao Chen, Yaoming Niu (2019)

Czechoslovak Mathematical Journal

Similarity:

We study the duality theory of the weighted multi-parameter Triebel-Lizorkin spaces ${\dot{F}}_{p}^{α, q} (ω; ℝ^{n_{1}} \times ℝ^{n_{2}})$ . This space has been introduced and the result ${({\dot{F}}_{p}^{α, q} (ω; ℝ^{n_{1}} \times ℝ^{n_{2}}))}^{*} = {CMO}_{p}^{- α, q^{'}} (ω; ℝ^{n_{1}} \times ℝ^{n_{2}})$ for $0 < p \leq 1$ has been proved in Ding, Zhu (2017). In this paper, for $1 < p < \infty$ , $0 < q < \infty$ we establish its dual space ${\dot{H}}_{p}^{α, q} (ω; ℝ^{n_{1}} \times ℝ^{n_{2}})$ .

Polyanalytic Besov spaces and approximation by dilatations

Ali Abkar (2024)

Czechoslovak Mathematical Journal

Similarity:

Using partial derivatives $\partial f / \partial z$ and $\partial f / \partial \bar{z}$ , 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 $q$ .

Remarks on the critical Besov space and its embedding into weighted Besov-Orlicz spaces

Hidemitsu Wadade (2010)

Studia Mathematica

Similarity:

We present several continuous embeddings of the critical Besov space $B_{p}^{n / p, ρ} (ℝ ⁿ)$ . We first establish a Gagliardo-Nirenberg type estimate ${| | u | |}_{Ḃ_{q, w_{r}}^{0, ν}} \leq C ₙ {(1 / (n - r))}^{1 / q + 1 / ν - 1 / ρ} {(q / r)}^{1 / ν - 1 / ρ} {| | u | |}_{Ḃ_{p}^{0, ρ}}^{(n - r) p / n q} {| | u | |}_{Ḃ_{p}^{n / p, ρ}}^{1 - (n - r) p / n q}$ , for 1 < p ≤ q < ∞, 1 ≤ ν < ρ ≤ ∞ and the weight function $w_{r} (x) = 1 / {(| x |}^{r})$ with 0 < r < n. Next, we prove the corresponding Trudinger type estimate, and obtain it in terms of the embedding $B_{p}^{n / p, ρ} (ℝ ⁿ) ↪ B_{Φ ₀, w_{r}}^{0, ν} (ℝ ⁿ)$ , where the function Φ₀ of the weighted Besov-Orlicz space $B_{Φ ₀, w_{r}}^{0, ν} (ℝ ⁿ)$ is a Young function of the exponential type. Another point of interest is to embed $B_{p}^{n / p, ρ} (ℝ ⁿ)$ into the weighted Besov...

Displaying similar documents to “Existence of solutions to the Poisson equation in L p -weighted spaces”

Displaying similar documents to “Existence of solutions to the Poisson equation in $L_{p}$ -weighted spaces”