Mediterr. J. Math.
Springer International Publishing AG,
part of Springer Nature 2018
Defect Distributions Related to Weakly
Convergent Sequences in Bessel-Type Spaces
Jelena Aleksi´c, Stevan Pilipovi´c and Ivana Vojnovi´c
Abstract. We consider microlocal defect distributions associated to a
weakly convergent sequences u
the space of pseudo-diﬀerential operators with the symbols in (s
Symbols correspond to a weight function Λ determining a quasi-elliptic
symbol. Results are applied to partial diﬀerential equations with sym-
bols related to weights of the type Λ.
Mathematics Subject Classiﬁcation. 46F25, 46F12, 40A30, 42B15.
Keywords. H-distributions (microlocal defect distributions), Weighted
Bessel spaces, Pseudo-diﬀerential operators, Strong convergence.
Our ﬁrst aim in this paper is to study the defect distributions which corre-
spond to the space of quasi-elliptic symbols which are determined by related
weight functions, for example of the form
where m =(m
) ∈ N
≥ 1. In particular, ξ =
is a weight function of this form. We recall the properties of
the spaces of symbols M
, the spaces of multipliers s
and consider such
symbols with the ﬁnite order of regularity and those which vanish at inﬁnity.
Then, by testing weakly convergent sequences in the corresponding weighted
The work presented in this paper is partially supported by Ministry of Education and Sci-
ence, Republic of Serbia, project no. 174024. The third author has partially been supported
by Croatian Science Foundation under the Project no. 9780 WeConMApp.