Anti-Wick and Weyl quantization on ultradistribution spaces

Pilipović, Stevan; Prangoski, Bojan

doi:10.1016/j.matpur.2014.04.011

Cited by 20 publications

(17 citation statements)

References 12 publications

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“…where A = a(x, D), f is a given test function in our setting and F [u] is a nonlinear term given by a suitable infinite sum of powers of u. In [21] we investigated the class of operators of [23] in the context of the Weyl and the Anti Wick calculus while in the recent paper [8], we considered the case of linear equations and proved a result of hypoellipticity via the construction of a parametrix. To treat semilinear equations, we need to adopt a more sophisticated method based on suitable commutators and nonlinear estimates.…”

Section: Introductionmentioning

confidence: 99%

Semilinear pseudodifferential equations in spaces of tempered ultradistributions

Cappiello

Pilipović

Prangoski

2016

Journal of Mathematical Analysis and Applications

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We study a class of semilinear elliptic equations on spaces of tempered ultradistributions of Beurling and Roumieu type. Assuming that the linear part of the equation is an elliptic pseudodifferential operator of infinite order with a sub-exponential growth of its symbol and that the non linear part is given by an infinite sum of powers of u with sub-exponential growth with respect to u, we prove a regularity result in the functional setting of the quoted ultradistribution spaces for a weak Sobolev type solution u. * Corresponding author: marco.cappiello@unito.it order, is used also here but with a more advanced technique since we consider infinite series both in the case of symbols and in the case of nonlinear terms. Examples of our elliptic symbols are a(x, ξ) = e c (x,ξ) 1/m , m > 1, c ∈ R, whereas for what concerns the nonlinear terms we can consider F [u] = β c β P β (x)u |β| where P β (x) are ultrapolynomials of the form γc γ x γ /γ! m , m > 1 and c β are suitable complex numbers tending rapidly to zero. Actually, we will consider nonlinear terms also with the additional assumption that u ∈ H s (R d ), s > d/2. With this we have plenty of examples, for example F [u] = P (x) cos u or F [u] = P (x)e u k , k ∈ Z + , where P has sub-exponential growth of the order related to the order of the growth of the symbol. In this way we can analyse elliptic operators A of infinite order and sub-exponential growth as well as nonlinear terms with sub-exponential growth, not considered in the literature, which shows an intrinsic connection of the pseudodifferential calculus of [23] with the spaces of ultradistributions.The paper is organised as follows. In the next Section 1 we introduce the main tools involved in the paper and state the main result, namely Theorem 1.2. Section 2 contains some examples of elliptic operators and nonlinear terms which motivate our analysis and for which Theorem 1.2 holds. In Section 3 we refine some results about the pseudodifferential operators studied in [23] and we prove some precise estimates for the norms of some composed operators which will be instrumental in the proof of Theorem 1.2. Finally, Section 4 is devoted to the proof of the theorem which will be divided in two parts, one corresponding to the proof of the decay properties of the solution and the other related to its regularity. Notation and the main theoremBefore stating our results, let us fix some notation and introduce the functional setting where they are obtained. In the sequel, the sets of integer, non-negative integer, positive integer, real and complex numbers are denoted as standard by Z, N, Z + , R, C. We denote x = (1 + |x| 2 ) 1/2 forFixed B > 0, we shall denote by Q c B the set of all (x, ξ) ∈ R 2d for which we have x ≥ B or ξ ≥ B. Finally, for s ∈ R, we shall denote by H s (R d ) the Sobolev space of all u ∈ S ′ (R d ) for which ξ sû (ξ) ∈ L 2 (R d ), whereû denotes the Fourier transform of u. Following [14], in the sequel we shall consider sequences M p of positive numbers such that M 0 = M 1 = 1 and satisfying...

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Section: Introductionmentioning

confidence: 99%

Semilinear pseudodifferential equations in spaces of tempered ultradistributions

Cappiello

Pilipović

Prangoski

2016

Journal of Mathematical Analysis and Applications

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“…For a ∈ Γ * ,∞ Ap,ρ (R 2d ), we denote by A a its anti-Wick quantisation. By [16,Theorem 3.2], there exists a ∈ Γ * ,∞ Ap,ρ (R 2d ) and a * -regularising operator T such that b w = A a + T . By a careful inspection of the proof of the quoted result, one can find the explicit construction of a; it is given as follows.…”

Section: 1mentioning

confidence: 99%

Spectral asymptotics for infinite order pseudo-differential operators

2018

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We study spectral properties of a class of global infinite order pseudodifferential operators and obtain the asymptotic behaviour of the spectral counting functions of such operators. Unlike their finite order counterparts, their spectral asymptotics are not of power-log-type but of log-type. The ultradistributional setting of such operators of infinite order makes the theory more complex so that the standard finite order global Weyl calculus cannot be used in this context.

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“…The authors gave there general functional definitions and proved fundamental results on convolvability and the convolution of Roumieu ultradistributions in a way analogous to the known general approaches of Chevalley and Schwartz in case of distributions. For other aspects of the theory, see, e.g., [1,2,4,6,8,20,24]. See also the recent article [21] for results concerning the quasianalytic case.…”

Section: Introductionmentioning

confidence: 99%

A sequential approach to the convolution of Roumieu ultradistributions

Mincheva-Kaminska

2020

Adv. Oper. Theory

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We consider several general sequential conditions for convolvability of two Roumieu ultradistributions on $${\mathbb {R}}^d$$ R d in the space $${\mathcal {D}}'^{\{M_p\}}$$ D ′ { M p } and prove that they are equivalent to the convolvability of these ultradistributions in the sense of Pilipović and Prangoski. The discussed conditions, based on two classes $${{\mathbb {U}}}^{\{M_p\}}$$ U { M p } and $$\overline{{\mathbb {U}}}^{\{M_p\}}$$ U ¯ { M p } of approximate units and corresponding sequential conditions of integrability of Roumieu ultradistributions, are analogous to the known convolvability conditions in the space $${\mathcal {D}}'$$ D ′ of distributions and in the space $${\mathcal {D}}'^{(M_p)}$$ D ′ ( M p ) of ultradistributions of Beurling type. Moreover, the following property of the convolution and ultradifferential operator P(D) of class $$\{M_p\}$$ { M p } is proved: if $$S, T \in \mathcal{D}'^{\{M_{p }\}}({\mathbb {R}}^d)$$ S , T ∈ D ′ { M p } ( R d ) are convolvable, then $$\begin{aligned} P(D)(S*T) = (P(D)S)*T = S*(P(D)T). \end{aligned}$$ P ( D ) ( S ∗ T ) = ( P ( D ) S ) ∗ T = S ∗ ( P ( D ) T ) .

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Anti-Wick and Weyl quantization on ultradistribution spaces

Cited by 20 publications

References 12 publications

Semilinear pseudodifferential equations in spaces of tempered ultradistributions

Semilinear pseudodifferential equations in spaces of tempered ultradistributions

Spectral asymptotics for infinite order pseudo-differential operators

A sequential approach to the convolution of Roumieu ultradistributions

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