Lp-microlocal regularity for pseudodifferential operators of quasi-homogeneous type†

Garello, Gianluca; Morando, Alessandro

doi:10.1080/17476930903030044

Cited by 9 publications

(19 citation statements)

References 9 publications

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“…Let us define now the M ‐characteristic set of

Q (x, \partial)

{Char}_{M} Q = \{(x, ξ) \in {boldR}_{x}^{2} \times {boldR}_{ξ}^{2} ∖ {0}, q (x, ξ) = 0\} .

We have

{Char}_{M} Q = \{(0, x_{2}, ξ_{1}, ξ_{2}); x_{2} \in boldR, ξ_{1} = ξ_{2}^{2}, ξ_{2} \neq 0\} = {0} \times R \times ⋂_{0 < k < 1} ({boldR}^{2} ∖ X_{k}) .

Let us observe at the end that any set

X_{k}

is M ‐conic, that is for any

ξ \in X_{k}

and

t > 0

we have

t^{1 / M} ξ : = (t ξ_{1}, t^{1 / 2} ξ_{2}) \in X_{k}

. The same clearly holds for

R^{2} ∖ X_{k}

and Char

_{M} Q

, agreeing with [, Definition 4.3]. About the operator

P (x, \partial)

we can notice that for

x^{0} = (0, x_{2}^{0})

, with an arbitrary

x_{2}^{0} \in R

p ()...</p>

…”

Section: Some Examplesmentioning

confidence: 99%

m‐Microlocal elliptic pseudodifferential operators acting on

Garello

Morando

2016

Mathematische Nachrichten

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“…Let us define now the M ‐characteristic set of

Q (x, \partial)

{Char}_{M} Q = \{(x, ξ) \in {boldR}_{x}^{2} \times {boldR}_{ξ}^{2} ∖ {0}, q (x, ξ) = 0\} .

We have

{Char}_{M} Q = \{(0, x_{2}, ξ_{1}, ξ_{2}); x_{2} \in boldR, ξ_{1} = ξ_{2}^{2}, ξ_{2} \neq 0\} = {0} \times R \times ⋂_{0 < k < 1} ({boldR}^{2} ∖ X_{k}) .

Let us observe at the end that any set

X_{k}

is M ‐conic, that is for any

ξ \in X_{k}

and

t > 0

we have

t^{1 / M} ξ : = (t ξ_{1}, t^{1 / 2} ξ_{2}) \in X_{k}

. The same clearly holds for

R^{2} ∖ X_{k}

and Char

_{M} Q

, agreeing with [, Definition 4.3]. About the operator

P (x, \partial)

we can notice that for

x^{0} = (0, x_{2}^{0})

, with an arbitrary

x_{2}^{0} \in R

p ()...</p>

…”

Section: Some Examplesmentioning

confidence: 99%

m‐Microlocal elliptic pseudodifferential operators acting on

Garello

Morando

2016

Mathematische Nachrichten

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“…Let us recall below some basic notions, see [15] for more details. Later on it is set for short T • R n := R n × (R n \ {0}).…”

Section: The Case Of Quasi-homogeneous Equationsmentioning

confidence: 99%

“…More generally, failing of any homogeneity properties, the propagation of the microlocal singularities are described in terms of filter of neighborhoods, introduced in [28] and further developed in [6], [7], [8], [10], [11], [16], [17] [18]. In some previous works of the authors continuity and microlocal properties are considered in Sobolev spaces in L p setting, see [12], [13], [15], [16], [17], [18]. In the present paper we prove a result of propagation of singularities of Fourier Lebesgue type, for partial (pseudo) differential equations, whose symbol satisfies generalized elliptic properties.…”

Section: Introductionmentioning

confidence: 99%

Inhomogeneous microlocal propagation of singularities in Fourier Lebesgue spaces

Garello

Morando

2016

J. Pseudo-Differ. Oper. Appl.

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“…In previous papers, [4], [5], [6], [7], the authors studied the problem of L p and Besov continuity and local regularity for pseudodifferential operators with smooth and non smooth symbols, whose derivatives decay at infinity in non homogeneous way. Particularly in [6], [7] emphasis is given on symbols with quasi-homogeneous decay; in [8] also microlocal properties were studied. Pseudodifferential operators whose smooth symbols have a quasi-homogeneous decay at infinity were first introduced in 1977 in Lascar [9], where their microlocal properties in the L 2 −framework were studied.…”

Section: Introductionmentioning

confidence: 99%