Pointwise Multiplication on Vector-Valued Function Spaces with Power Weights

Meyries, Martin; Veraar, Mark

doi:10.1007/s00041-014-9362-1

Cited by 34 publications

(57 citation statements)

References 51 publications

(125 reference statements)

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“…So we may restrict ourselves to the case = ℂ. Recall from [36,Proposition 3.4] that ∕ is a bounded linear operator from , ( ℝ,…”

Section: Results On the Whole Real Linementioning

confidence: 99%

“…The following result is proved in [, Proposition 3.2 and 3.7] by a direct application of Proposition . Proposition Let X be a UMD space, let

p \in (1, \infty)

k \in {double-struckN}_{0}

, and let

w \in A_{p}

.…”

Section: Weighted Function Spacesmentioning

confidence: 96%

“…Proof of Theorem We only consider

s \geq 0

. The case

s < 0

follows from a duality argument using [, Proposition 3.5]. By Lemma it is enough to prove

∥ {bold1}_{{double-struckR}_{+}^{d}} f {true∥}_{H^{s, p} false({double-struckR}^{d}, w_{γ}; X false)} ≲_{s, p, d, γ, X} {∥ f ∥}_{H^{s, p} false({double-struckR}^{d}, w_{γ}; X false)}

for an arbitrary

f \in S true({double-struckR}^{d} true) \otimes X

.…”

Section: Pointwise Multiplication With 1r+dmentioning

confidence: 99%

“…Proof As in [, Proposition 3.4] one checks the following equivalence of extended norms on

{scriptS}^{'} false(double-struckR; X false)

0true {false∥ f false∥}_{H^{s, p} (R, w_{γ}; X)} ≂_{s, γ, p, X} {false∥ f false∥}_{H^{s - k, p} (R, w_{γ}; X)} + ∥ \partial^{k} {f false∥}_{H^{s - k, p} (R, w_{γ}; X)} ≂_{s, γ, p, X} 0true \sum_{j = 0}^{k} {false∥ \partial^{j} f false∥}_{H^{s - k, p} (R, w_{γ}; X)} .

Let

f \in H^{s, p} true(double-struckR, w_{γ}; X true) \cap W_{loc, 0}^{k + 1, 1} false(double-struckR; X false)

. Using , Lemma and Theorem we find

\begin{matrix} true \\ right {∥ {bold1}_{R_{+}} f ∥}_{H^{s, p} false(double-struckR, w_{γ}; X false)} & ≲_{s, p, γ, X} & left {true∥ 1_{{double-struckR}_{+}} f true∥}_{H^{s - k,}} \end{matrix}

…”

Section: Application To Interpolation Theory and The First Derivativementioning

confidence: 99%

“…For (1) we denote by the realization of on ( ℝ + , ; ) with domain 1, 0 ( ℝ + , ; ) and by the realization of − on ′ ( ℝ + , ′ ; * ) with domain 1, ′ ( ℝ + , ′ ; * ) . Recall that [ ( ℝ + , ; )] * = ′ ( ℝ + , ′ ; * ) with respect to the natural pairing (see [36,Proposition 3.5]), being reflexive as a UMD space (see [20,Theorem 4.3.3]). Integration by parts (see Lemma 6.9 below) yields ⊂ * .…”

Section: Fractional Domain Spacesmentioning

confidence: 99%

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Complex interpolation with Dirichlet boundary conditions on the half line

Lindemulder

Meyries

Veraar

2018

Mathematische Nachrichten

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We prove results on complex interpolation of vector‐valued Sobolev spaces over the half‐line with Dirichlet boundary condition. Motivated by applications in evolution equations, the results are presented for Banach space‐valued Sobolev spaces with a power weight. The proof is based on recent results on pointwise multipliers in Bessel potential spaces, for which we present a new and simpler proof as well. We apply the results to characterize the fractional domain spaces of the first derivative operator on the half line.

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“…So we may restrict ourselves to the case = ℂ. Recall from [36,Proposition 3.4] that ∕ is a bounded linear operator from , ( ℝ,…”

Section: Results On the Whole Real Linementioning

confidence: 99%

“…The following result is proved in [, Proposition 3.2 and 3.7] by a direct application of Proposition . Proposition Let X be a UMD space, let

p \in (1, \infty)

k \in {double-struckN}_{0}

, and let

w \in A_{p}

.…”

Section: Weighted Function Spacesmentioning

confidence: 96%

“…Proof of Theorem We only consider

s \geq 0

. The case

s < 0

follows from a duality argument using [, Proposition 3.5]. By Lemma it is enough to prove

∥ {bold1}_{{double-struckR}_{+}^{d}} f {true∥}_{H^{s, p} false({double-struckR}^{d}, w_{γ}; X false)} ≲_{s, p, d, γ, X} {∥ f ∥}_{H^{s, p} false({double-struckR}^{d}, w_{γ}; X false)}

for an arbitrary

f \in S true({double-struckR}^{d} true) \otimes X

.…”

Section: Pointwise Multiplication With 1r+dmentioning

confidence: 99%

“…Proof As in [, Proposition 3.4] one checks the following equivalence of extended norms on

{scriptS}^{'} false(double-struckR; X false)

0true {false∥ f false∥}_{H^{s, p} (R, w_{γ}; X)} ≂_{s, γ, p, X} {false∥ f false∥}_{H^{s - k, p} (R, w_{γ}; X)} + ∥ \partial^{k} {f false∥}_{H^{s - k, p} (R, w_{γ}; X)} ≂_{s, γ, p, X} 0true \sum_{j = 0}^{k} {false∥ \partial^{j} f false∥}_{H^{s - k, p} (R, w_{γ}; X)} .

Let

f \in H^{s, p} true(double-struckR, w_{γ}; X true) \cap W_{loc, 0}^{k + 1, 1} false(double-struckR; X false)

. Using , Lemma and Theorem we find

\begin{matrix} true \\ right {∥ {bold1}_{R_{+}} f ∥}_{H^{s, p} false(double-struckR, w_{γ}; X false)} & ≲_{s, p, γ, X} & left {true∥ 1_{{double-struckR}_{+}} f true∥}_{H^{s - k,}} \end{matrix}

…”

Section: Application To Interpolation Theory and The First Derivativementioning

confidence: 99%

Section: Fractional Domain Spacesmentioning

confidence: 99%

See 3 more Smart Citations

Complex interpolation with Dirichlet boundary conditions on the half line

Lindemulder

Meyries

Veraar

2018

Mathematische Nachrichten

Self Cite

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Stability analysis for stationary solutions of the Mullins–Sekerka flow with boundary contact

Garcke

Rauchecker

2022

Mathematische Nachrichten

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We first give a complete linearized stability analysis around stationary solutions of the Mullins-Sekerka flow with 90 • contact angle in two space dimensions. The stationary solutions include flat interfaces, as well as arcs of circles. We investigate the different stability behaviour in dependence of properties of the stationary solution, such as its curvature and length, as well as the curvature of the boundary of the domain at the two contact points. We show that the behaviour changes in terms of these parameters, ranging from exponential stability to instability.We also give a first result on nonlinear stability for curved boundaries.

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