Geodesic
Transformations of multipliers for pseudodifferential operators in $L_p$
Sbornik. Mathematics, Tome 14 (1971) no. 3, pp. 399-416 
K. Tel'ner. Transformations of multipliers for pseudodifferential operators in $L_p$. Sbornik. Mathematics, Tome 14 (1971) no. 3, pp. 399-416. http://geodesic.mathdoc.fr/item/SM_1971_14_3_a5/
@article{SM_1971_14_3_a5,
     author = {K. Tel'ner},
     title = {Transformations of multipliers for pseudodifferential operators in~$L_p$},
     journal = {Sbornik. Mathematics},
     pages = {399--416},
     year = {1971},
     volume = {14},
     number = {3},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1971_14_3_a5/}
}
TY  - JOUR
AU  - K. Tel'ner
TI  - Transformations of multipliers for pseudodifferential operators in $L_p$
JO  - Sbornik. Mathematics
PY  - 1971
SP  - 399
EP  - 416
VL  - 14
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/SM_1971_14_3_a5/
LA  - en
ID  - SM_1971_14_3_a5
ER  - 
%0 Journal Article
%A K. Tel'ner
%T Transformations of multipliers for pseudodifferential operators in $L_p$
%J Sbornik. Mathematics
%D 1971
%P 399-416
%V 14
%N 3
%U http://geodesic.mathdoc.fr/item/SM_1971_14_3_a5/
%G en
%F SM_1971_14_3_a5

Voir la notice de l'article provenant de la source Math-Net.Ru

By a transformation of multipliers we mean the operation assigning to each pseudodifferential (ps.d.) operator $K$ with symbol $K(\xi,x)$, i.e. $$ (Ku)(x)=\int_{\mathbf R^m}K(\xi,x)e^{i\langle\xi,x\rangle}\widehat u(\xi)\,d\xi, $$ a new ps.d. operator $\Phi K$ with symbol $\varphi(\xi,x)K(\xi,x)$, i.e. $$ (\Phi Ku)(x)=\int_{\mathbf R^m}\varphi(\xi,x)K(\xi,x)e^{i\langle\xi,x\rangle}\widehat u(\xi)\,d\xi. $$ Here $\mathbf R^m$ is $m$-dimensional Euclidean space; $x$ and $\xi$ are points in $\mathbf R^m$; $\langle\xi,x\rangle=\xi_1x_1+\dots+\xi_mx_m$; $\widehat u$ is the Fourier transform of $u$. There are given two criteria for the transformation $K\to\Phi K$ to preserve the continuity of ps.d. operators in the spaces $L_p(\mathbf R^m)$. As a corollary there are obtained conditions for the boundedness of ps.d. operators (or singular integral operators) in $ L_p$. Bibliography: 12 titles.

[1] M. Sh. Birman, “Dvoinye operatornye integraly Stiltesa”, Problemy matematicheskoi fiziki, no. 2, LGU, 1967, 26–60 | MR

[2] M. Sh. Birman, M. Z. Solomyak, “Ob otsenkakh singulyarnykh chisel integralnykh operatorov. II”, Vestnik LGU, 1967, no. 13, 21–28 | MR | Zbl

[3] M. Sh. Birman, M. Z. Solomyak, “Ob otsenkakh singulyarnykh chisel integralnykh operatorov. III. Operatory v neogranichennykh oblastyakh”, Vestnik LGU, 1969, no. 1, 35–48 | MR | Zbl

[4] M. Sh. Birman, M. Z. Solomyak, “Zamechaniya o yadernosti integralnykh operatorov i ob ogranichennosti psevdodifferentsialnykh operatorov”, Izv. VUZov, Matematika, 1969, no. 9(88), 11–17 | MR | Zbl

[5] K. Telner, “Dvoinye operatornye integraly v energeticheskom prostranstve”, Vestnik LGU, 1970, no. 13, 69–82 | MR | Zbl

[6] J. Peetre, “Applications de la theorie des espaces d'interpolation dans l'analyse harmonique”, Ricerche di Mathematica, XV:1 (1956), 3–36 | MR

[7] W. Littman, Ch. McCarthy, N. Riviere, “$L_p$-multiplier theorems”, Studia Math., 30:2 (1968), 193–217 | MR | Zbl

[8] S. G. Mikhlin, Mnogomernye singulyarnye integraly i integralnye uravneniya, Fizmatgiz, Moskva, 1963

[9] L. Khermander, Psevdodifferentsialnye operatory, Mir, Moskva, 1967 | MR

[10] V. M. Kagan, “Ob ogranichennosti psevdodifferentsialnykh operatorov v $L_p$”, Izv. VUZov, Matematika, 1968, no. 6(73), 35–44 | MR | Zbl

[11] S. L. Sobolev, Nekotorye primeneniya funktsionalnogo analiza v matematicheskoi fizike, Fizmatgiz, Moskva, 1953

[12] M. Sh. Birman, M. Z. Solomyak, “Kusochno-polinomialnye priblizheniya funktsii klassov $W_P^a$”, Matem. sb., 73(115) (1967), 331–355 | MR | Zbl