Heating of the Beurling operator: Sufficient conditions for the two-weight case

S. Petermichl; J. Wittwer

Displaying similar documents to “Heating of the Beurling operator: Sufficient conditions for the two-weight case”

Composition in ultradifferentiable classes

Armin Rainer, Gerhard Schindl (2014)

Studia Mathematica

Similarity:

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 $^{(ω)}$ and the Roumieu space $^{ω}$ as intersection and union of spaces $^{(M)}$ and $^{M}$ for associated weight sequences, respectively.

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 linear bound in A₂ for Calderón-Zygmund operators: a survey

Michael Lacey (2011)

Banach Center Publications

Similarity:

For an L²-bounded Calderón-Zygmund Operator T acting on $L ² (ℝ^{d})$ , and a weight w ∈ A₂, the norm of T on L²(w) is dominated by $C_{T} {| | w | |}_{A ₂}$ . The recent theorem completes a line of investigation initiated by Hunt-Muckenhoupt-Wheeden in 1973 (MR0312139), has been established in different levels of generality by a number of authors over the last few years. It has a subtle proof, whose full implications will unfold over the next few years. This sharp estimate requires that the A₂ character of the weight can...

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.

Beurling algebra analogues of theorems of Wiener-Lévy-Żelazko and Żelazko

S. J. Bhatt, P. A. Dabhi, H. V. Dedania (2009)

Studia Mathematica

Similarity:

Let 0 < p ≤ 1, let ω: ℤ → [1,∞) be a weight on ℤ and let f be a nowhere vanishing continuous function on the unit circle Γ whose Fourier series satisfies $\sum_{n \in ℤ} {| f ̂ (n) |}^{p} ω (n) < \infty$ . Then there exists a weight ν on ℤ such that $\sum_{n \in ℤ} {| \hat{(1 / f)} (n) |}^{p} ν (n) < \infty$ . Further, ν is non-constant if and only if ω is non-constant; and ν = ω if ω is non-quasianalytic. This includes the classical Wiener theorem (p = 1, ω = 1), Domar theorem (p = 1, ω is non-quasianalytic), Żelazko theorem (ω = 1) and a recent result of Bhatt and Dedania (p = 1). An...

Inequalities involving heat potentials and Green functions

Neil A. Watson (2015)

Mathematica Bohemica

Similarity:

We take some well-known inequalities for Green functions relative to Laplace’s equation, and prove not only analogues of them relative to the heat equation, but generalizations of those analogues to the heat potentials of nonnegative measures on an arbitrary open set $E$ whose supports are compact polar subsets of $E$ . We then use the special case where the measure associated to the potential has point support, in the following situation. Given a nonnegative supertemperature on an open set...

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.

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...

The weighted Hardy spaces associated to self-adjoint operators and their duality on product spaces

Suying Liu, Minghua Yang (2018)

Czechoslovak Mathematical Journal

Similarity:

Let $L$ be a non-negative self-adjoint operator acting on $L^{2} (ℝ^{n})$ 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 $H_{L, w}^{p} (ℝ^{n} \times ℝ^{n})$ , $0 < p \leq 1$ associated to $L$ . Based on the atomic decomposition, we show the dual relationship between $H_{L, w}^{1} (ℝ^{n} \times ℝ^{n})$ and ${BMO}_{L, w} (ℝ^{n} \times ℝ^{n})$ .

Weighted norm estimates for the maximal operator of the Laguerre functions heat diffusion semigroup

R. Macías, C. Segovia, J. L. Torrea (2006)

Studia Mathematica

Similarity:

We obtain weighted $L^{p}$ boundedness, with weights of the type $y^{δ}$ , δ > -1, for the maximal operator of the heat semigroup associated to the Laguerre functions, ${ℒ_{k}^{α}}_{k}$ , when the parameter α is greater than -1. It is proved that when -1 < α < 0, the maximal operator is of strong type (p,p) if p > 1 and 2(1+δ)/(2+α) < p < 2(1+δ)/(-α), and if α ≥ 0 it is of strong type for 1 < p ≤ ∞ and 2(1+δ)/(2+α) < p. The behavior at the end points of the intervals where there is strong...

Existence of solutions to the nonstationary Stokes system in $H_{- μ}^{2, 1}$ , μ ∈ (0,1), in a domain with a distinguished axis. Part 2. Estimate in the 3d case

W. M. Zajączkowski (2007)

Applicationes Mathematicae

Similarity:

We examine the regularity of solutions to the Stokes system in a neighbourhood of the distinguished axis under the assumptions that the initial velocity v₀ and the external force f belong to some weighted Sobolev spaces. It is assumed that the weight is the (-μ )th power of the distance to the axis. Let $f \in L_{2, - μ}$ , $v ₀ \in H_{- μ} ¹$ , μ ∈ (0,1). We prove an estimate of the velocity in the $H_{- μ}^{2, 1}$ norm and of the gradient of the pressure in the norm of $L_{2, - μ}$ . We apply the Fourier transform with respect to the variable along...

Hardy's theorem for the helgason Fourier transform on noncompact rank one symmetric spaces

S. Thangavelu (2002)

Colloquium Mathematicae

Similarity:

Let G be a semisimple Lie group with Iwasawa decomposition G = KAN. Let X = G/K be the associated symmetric space and assume that X is of rank one. Let M be the centraliser of A in K and consider an orthonormal basis $Y_{δ, j} : δ \in K ̂ ₀, 1 \leq j \leq d_{δ}$ of L²(K/M) consisting of K-finite functions of type δ on K/M. For a function f on X let f̃(λ,b), λ ∈ ℂ, be the Helgason Fourier transform. Let $h_{t}$ be the heat kernel associated to the Laplace-Beltrami operator and let $Q_{δ} (i λ + ϱ)$ be the Kostant polynomials. We establish the following...

Koecher-Maass series of a certain half-integral weight modular form related to the Duke-Imamoḡlu-Ikeda lift

Hidenori Katsurada, Hisa-aki Kawamura (2014)

Acta Arithmetica

Similarity:

Let k and n be positive even integers. For a cuspidal Hecke eigenform h in the Kohnen plus space of weight k - n/2 + 1/2 for Γ₀(4), let f be the corresponding primitive form of weight 2k-n for SL₂(ℤ) under the Shimura correspondence, and Iₙ(h) the Duke-Imamoḡlu-Ikeda lift of h to the space of cusp forms of weight k for Spₙ(ℤ). Moreover, let $ϕ_{I ₙ (h), 1}$ be the first Fourier-Jacobi coefficient of Iₙ(h), and $σ_{n - 1} (ϕ_{I ₙ (h), 1})$ be the cusp form in the generalized Kohnen plus space of weight k - 1/2 corresponding to...

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 $q$ .

Some weighted norm inequalities for a one-sided version of $g *_{λ}$

L. de Rosa, C. Segovia (2006)

Studia Mathematica

Similarity:

We study the boundedness of the one-sided operator $g ⁺_{λ, φ}$ between the weighted spaces $L^{p} (M ¯ w)$ and $L^{p} (w)$ 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 $g ⁺_{λ, φ}$ . 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 $g ⁺_{λ, φ}$ from $L^{p} ({(M ¯)}^{[p / 2] + 1} w)$ to $L^{p} (w)$ , where ${(M ¯)}^{k}$ denotes the operator M¯ iterated k times.

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...

Lacunary series in $Q_{K}$ spaces

Hasi Wulan, Kehe Zhu (2007)

Studia Mathematica

Similarity:

Under mild conditions on the weight function K we characterize lacunary series in the so-called $_{K}$ spaces.

Controlling products of currents by higher powers of plurisubharmonic functions

Ahmad K. Al Abdulaali, Hassine El Mir (2020)

Czechoslovak Mathematical Journal

Similarity:

We discuss the existence of the current $g^{k} T$ , $k \in ℕ$ 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.

Tykhonov well-posedness of a heat transfer problem with unilateral constraints

Mircea Sofonea, Domingo A. Tarzia (2022)

Applications of Mathematics

Similarity:

We consider an elliptic boundary value problem with unilateral constraints and subdifferential boundary conditions. The problem describes the heat transfer in a domain $D \subset ℝ^{d}$ and its weak formulation is in the form of a hemivariational inequality for the temperature field, denoted by $𝒫$ . We associate to Problem $𝒫$ an optimal control problem, denoted by $𝒬$ . Then, using appropriate Tykhonov triples, governed by a nonlinear operator $G$ and a convex $\tilde{K}$ , we provide results concerning the well-posedness...

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.