Analysis of joint spectral multipliers on Lie groups of polynomial growth

Martini, Alessio

doi:10.5802/aif.2721

Cited by 35 publications

(51 citation statements)

References 37 publications

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“…We define the lifted operators on G = G 1 × ⋯ × G M by

{scriptL}_{1}^{#} = L_{1} \otimes 1 \otimes ⋯ \otimes 1

{scriptL}_{2}^{#} = 1 \otimes L_{2} \otimes 1 \otimes ⋯ \otimes 1

, …,

{scriptL}_{M}^{#} = 1 \otimes ⋯ \otimes 1 \otimes L_{M}

. According to, eg, previous literatures, there is a unique spectral decomposition E , which we call the joint spectral resolution of

L_{1}, ⋯, L_{M}

, such that for all Borel subsets A ∈ G , E is a projection on L 2 ( G ) and for any Borel subsets A ℓ ∈ G ℓ , ℓ = 1,…, M , we have

E (A_{1} \times ⋯ \times A_{M}) = E_{{scriptL}_{1}} (A_{1}) \otimes ⋯ \otimes E_{{scriptL}_{M}} (A_{M}) .

Then the joint spectral multipliers can be written as

m (L_{1}^{#}, ⋯, L_{M}^{#}) = \int_{R_{+}^{M}}^{\infty} m (ξ_{1}, ⋯, ξ_{M}) d E (ξ_{1}, ⋯, ξ_{M}) .

The convolution of 2 measurable functions f and g on G can be defined by

f * g (x) = \int_{G} f (y_{1}, ⋯, y_{M}) g

…”

Section: The Boundedness On Lebesgue Spacesmentioning

confidence: 99%

“…After that, plenty of authors have been focusing on investigating the boundedness properties of Fourier multipliers, for instance, Besov, 4 Christ et al, 5 Lu et al, 6 Tomita, 7 Yabuta, 8 Grafakos et al, [9][10][11] Chen et al, 12 Noi, 13 Yang et al, 14 and Zhao et al 15,16 Since Fourier multipliers are special cases of spectral multipliers, it is very natural to consider whether there exist some similar properties for other kinds of spectral multipliers. After the forerunner work made by Wendel,17 there appear a number of studies that focus on investigating multiplier theorems for spectral multipliers on some kinds of nilpotent Lie groups, for example, Alexopoulos, 18 Chen et al, 19 Gong and Yan, 20 Christ, 21 Christ and Müller, 22 Duong, 23 Kolomoitsev, 24 Lin, 25 Martini, 26 Mauceri and Meda, 27 Michele and Mauceri, 28 and Pini. 29 According to these studies, we introduce the original version of the Hörmander multiplier theorem for spectral multipliers on stratified groups as follows.…”

Section: Introductionmentioning

confidence: 99%

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Multilinear and multiparameter spectral multipliers on stratified groups

Fang

Zhao

2018

Math Methods in App Sciences

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As far as we know, the study of multilinear spectral multipliers on nilpotent Lie groups is a very new research work. There is even no study of Hörmander‐type multiplier theorem for multilinear and multiparameter spectral multipliers on nilpotent Lie groups. In this paper, on product spaces of stratified groups G = G1 × ⋯ × GM, we prove Hörmander‐type multiplier theorems for multilinear and multiparameter spectral multipliers from Lp1false(Gfalse)×⋯×LpNfalse(Gfalse) to Lr(G) with 1⩽p1,⋯,pN,r⩽∞, from Lϖ1p1false(Gfalse)×⋯×LϖNpNfalse(Gfalse) to Lϖrfalse(Gfalse) with 1

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“…We define the lifted operators on G = G 1 × ⋯ × G M by

{scriptL}_{1}^{#} = L_{1} \otimes 1 \otimes ⋯ \otimes 1

{scriptL}_{2}^{#} = 1 \otimes L_{2} \otimes 1 \otimes ⋯ \otimes 1

, …,

{scriptL}_{M}^{#} = 1 \otimes ⋯ \otimes 1 \otimes L_{M}

. According to, eg, previous literatures, there is a unique spectral decomposition E , which we call the joint spectral resolution of

L_{1}, ⋯, L_{M}

, such that for all Borel subsets A ∈ G , E is a projection on L 2 ( G ) and for any Borel subsets A ℓ ∈ G ℓ , ℓ = 1,…, M , we have

E (A_{1} \times ⋯ \times A_{M}) = E_{{scriptL}_{1}} (A_{1}) \otimes ⋯ \otimes E_{{scriptL}_{M}} (A_{M}) .

Then the joint spectral multipliers can be written as

m (L_{1}^{#}, ⋯, L_{M}^{#}) = \int_{R_{+}^{M}}^{\infty} m (ξ_{1}, ⋯, ξ_{M}) d E (ξ_{1}, ⋯, ξ_{M}) .

The convolution of 2 measurable functions f and g on G can be defined by

f * g (x) = \int_{G} f (y_{1}, ⋯, y_{M}) g

…”

Section: The Boundedness On Lebesgue Spacesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Multilinear and multiparameter spectral multipliers on stratified groups

Fang

Zhao

2018

Math Methods in App Sciences

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“…(2.12) Proposition 6, together with the Plancherel formula for the Euclidean Fourier transform and the orthogonality properties (2.12) of the Laguerre functions, allows us to compute the Plancherel measure associated to the system (2.2) of commuting operators in the sense of [27,28].…”

Section: )mentioning

confidence: 99%

“…Let us return to the initial setting of a homogeneous sublaplacian L on a two-step stratified group G. The proof of Theorem 2 is reduced, by a standard argument (see, for example, [28,Theorem 4.6]) based on the Calderón-Zygmund theory of singular integral operators, to the following L 1 -estimate for the convolution kernel K F (L) of the operator F (L) corresponding to a compactly supported multiplier F . Proposition 3.…”

Section: Introductionmentioning

confidence: 99%

Spectral multiplier theorems of Euclidean type on new classes of two-step stratified groups

Martini

Müller

2014

Proceedings of the London Mathematical Society

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Abstract. From a theorem of Christ and Mauceri and Meda it follows that, for a homogeneous sublaplacian L on a 2-step stratified group G with Lie algebra g, an operator of the form F (L) is of weak type (1, 1) and bounded on L p (G) for 1 < p < ∞ if the spectral multiplier F satisfies a scale-invariant smoothness condition of order s > Q/2, where Q = dim g + dim [g, g] is the homogeneous dimension of G. Here we show that the condition can be pushed down to s > d/2, where d = dim g is the topological dimension of G, provided that d ≤ 7 or dim[g, g] ≤ 2.

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“…The PhD thesis of Martini, [29] (see also [30,31]), is a treatise of the subject of joint spectral multipliers for general Lie groups of polynomial growth. He proves various Marcinkiewicz-type and Hörmander-type multiplier theorems, mostly with sharp smoothness thresholds.…”

Section: M(λ)d E(λ)mentioning

confidence: 99%