Weighted norm estimates and L<sub>p</sub>-spectral independence of linear operators

Kunstmann, Peer Christian; Vogt, Hendrik

doi:10.4064/cm109-1-11

Cited by 6 publications

(5 citation statements)

References 13 publications

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“…We want to remark that a generalized Gaussian (q 0 , q 1 )-estimate of order m > 1 is known to be a powerful tool to establish further properties for semigroups or their generators. For the technique of generalized Gaussian estimates we refer in particular to [43,Section 2], [14,17,39], and the recent series of papers [5][6][7][8]. In case q 0 = q 1 bounds of the form (3) for z = t real are referred to as Gaffney-Davies estimates.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…For more on this technique we refer to [14,17,36,39,43,[47][48][49], see also the recent series of papers [5][6][7][8]. Here we just mention the following.…”

Section: Remark 37mentioning

confidence: 98%

“…But the given estimates shall be sufficient for the applications in Section 5. We also refer to [14,17,22,39,43] for the kind of arguments used in the proof and for converse statements.…”

Section: Corollary 32mentioning

confidence: 99%

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On maximal regularity of typeLp–Lqunder minimal assumptions for elliptic non-divergence operators

Kunstmann

2008

Journal of Functional Analysis

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We prove maximal regularity of type L p -L q for operators in non-divergence form with complex-valued measurable coefficients on R n . For a certain range of q, which depends on dimension and the order of the operators, this is done under the sole assumption that they generate an analytic semigroup in L q . Thus we give, for this class of operators and this range of q, a positive answer to Brézis' question whether generation of an analytic semigroup entails maximal L p -regularity. For other values of q we give several additional assumptions. The proof relies on a result on maximal regularity under the assumption of suitable off-diagonal bounds due to S. Blunck and the author, which we improve here. These off-diagonal bounds are obtained by a modification of Davies' technique suited to cope with operators in non-divergence form, and they also imply new results on the scale of L q -spaces in which a given operator generates an analytic semigroup. We thus obtain a completely new approach to maximal L p -regularity for operators of this kind, in particular for those whose highest order coefficients belong to VMO. Moreover, we obtain extensions of recent results due to Kim and Krylov for operators with measurable coefficients depending on one coordinate.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

“…For more on this technique we refer to [14,17,36,39,43,[47][48][49], see also the recent series of papers [5][6][7][8]. Here we just mention the following.…”

Section: Remark 37mentioning

confidence: 98%

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On maximal regularity of typeLp–Lqunder minimal assumptions for elliptic non-divergence operators

Kunstmann

2008

Journal of Functional Analysis

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“…[6,Lemma 6.3] and the proof of [6, Theorem 1.7]). The author noticed Karrmann's result via reference [7].…”

Section: Lemma 35 Let T P and K T Be As In Proposition 34 Then Tmentioning

confidence: 90%

Kernel estimates and Lp-spectral independence of generators of C0-semigroups

Shindoh

2008

Journal of Functional Analysis

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After the appearance of W. Arendt's result that "Gaussian estimate of a semigroup implies the L p -spectral independence of the generator," various generalizations have been obtained. This paper shows that a certain kernel estimate of a semigroup implies the L p -spectral independence of the generator, generalizing the case of upper Gaussian estimate and "Gaussian estimate of order α ∈ (0, 1] [S. Miyajima, H. Shindoh, Gaussian estimates of order α and L p -spectral independence of generators of C 0 -semigroups, Positivity 11 (1) (2007) 15-39], Definition 3.1." The proof uses S. Karrmann's result about the L p -spectral independence and B.A. Barnes' theorem about the spectrum of integral operators. As an application, the L p -spectral independence of −[(−Δ) α + V ] (α ∈ (0, 1]) for a suitable V is proved with the help of a recent result by V. Liskevich, H. Vogt and J. Voigt [V. Liskevich, H. Vogt, J. Voigt, Gaussian bounds for propagators perturbed by potentials, J. Funct. Anal. 238 (2006) 245-277].

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“…The problem of the p-independence of the spectrum of generators of semigroups on L p was raised by B. Simon in [33] for Schrödinger semigroups in R d and has been studied in many papers, see for example [4,10,11,15,18,19,31,32,33,34] and references therein.…”

Section: Introductionmentioning

confidence: 99%

On the boundedness and compactness of weighted Green operators of second-order elliptic operators

Pinchover¹

2016

Preprint

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For a given second-order linear elliptic operator L which admits a positive minimal Green function, and a given positive weight function W , we introduce a family of weighted Lebesgue spaces L p (φp) with their dual spaces, where 1 ≤ p ≤ ∞. We study some fundamental properties of the corresponding (weighted) Green operators on these spaces. In particular, we prove that these Green operators are bounded on L p (φp) for any 1 ≤ p ≤ ∞ with a uniform bound. We study the existence of a principal eigenfunction for these operators in these spaces, and the simplicity of the corresponding principal eigenvalue. We also show that such a Green operator is a resolvent of a densely defined closed operator which is equal to (−W −1 )L on C ∞ 0 , and that this closed operator generates a strongly continuous contraction semigroup. Finally, we prove that if W is a (semi)small perturbation of L, then for any 1 ≤ p ≤ ∞, the associated Green operator is compact on L p (φp), and the corresponding spectrum is p-independent.

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Weighted norm estimates and L_p-spectral independence of linear operators

Cited by 6 publications

References 13 publications

On maximal regularity of typeLp–Lqunder minimal assumptions for elliptic non-divergence operators

On maximal regularity of typeLp–Lqunder minimal assumptions for elliptic non-divergence operators

Kernel estimates and Lp-spectral independence of generators of C0-semigroups

On the boundedness and compactness of weighted Green operators of second-order elliptic operators

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Weighted norm estimates and Lp-spectral independence of linear operators

Cited by 6 publications

References 13 publications

On maximal regularity of typeLp–Lqunder minimal assumptions for elliptic non-divergence operators

On maximal regularity of typeLp–Lqunder minimal assumptions for elliptic non-divergence operators

Kernel estimates and Lp-spectral independence of generators of C0-semigroups

On the boundedness and compactness of weighted Green operators of second-order elliptic operators

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Weighted norm estimates and L_p-spectral independence of linear operators