Geodesic
On weighted generalized functions associated with quadratic forms
Problemy analiza, Tome 5 (2016) no. 2, pp. 52-68 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

In this article we consider certain types of weighted generalized functions associated with nondegenerate quadratic forms. Such functions and their derivatives are used for constructing fundamental solutions of iterated ultra-hyperbolic equations with the Bessel operator and for constructing negative real powers of ultra-hyperbolic operators with the Bessel operator.
Keywords: weighted generalized function; quadratic form; ultra-hyperbolic operator; Bessel operator. 
@article{PA_2016_5_2_a4,
     author = {E. L. Shishkina},
     title = {On weighted generalized functions associated with quadratic forms},
     journal = {Problemy analiza},
     pages = {52--68},
     year = {2016},
     volume = {5},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/PA_2016_5_2_a4/}
}
TY  - JOUR
AU  - E. L. Shishkina
TI  - On weighted generalized functions associated with quadratic forms
JO  - Problemy analiza
PY  - 2016
SP  - 52
EP  - 68
VL  - 5
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/PA_2016_5_2_a4/
LA  - en
ID  - PA_2016_5_2_a4
ER  - 
%0 Journal Article
%A E. L. Shishkina
%T On weighted generalized functions associated with quadratic forms
%J Problemy analiza
%D 2016
%P 52-68
%V 5
%N 2
%U http://geodesic.mathdoc.fr/item/PA_2016_5_2_a4/
%G en
%F PA_2016_5_2_a4
E. L. Shishkina. On weighted generalized functions associated with quadratic forms. Problemy analiza, Tome 5 (2016) no. 2, pp. 52-68. http://geodesic.mathdoc.fr/item/PA_2016_5_2_a4/

[1] Kipriyanov I. A., Ivanov L. A., “Riezs potentials on the Lorentz spaces”, Mat. sb., 130(172):4(8) (1986), 465–474 | MR | Zbl

[2] Klyuchantsev M. I., “Fractional order integrals and singular boundary problems”, Differ. Uravn., 12:6 (1976), 983–990 | MR | Zbl

[3] Guliyev V., Hasanov J. J., “Necessary and sufficient conditions for the boundedness of $B$-Riesz potential in the $B$-Morrey spaces”, Journal of Mathematical Analysis and Applications, 347:1 (2012), 113-122 | DOI | MR

[4] Aliev I. A., Gadzhiev A. D., “Weighted estimates of multidimensional singular integrals generated by the generalized shift operator”, Mat. Sb., 183:9 (1992), 45–66 | MR

[5] Lyakhov L. N., Shishkina E. L., “Inversion of general Riesz $B$-potentials with homogeneous characteristic in weight classes of functions”, Doklady Mathematics, 79:3 (2009), 377–381 | DOI | MR | Zbl

[6] Shishkina E. L., “Inversion of integral of $B$-potential type with density from $\Phi_\gamma$”, Journal of Mathematical Sciences, 160:1 (2009), 95–102 | DOI | MR | Zbl

[7] Kipriyanov I. A., Kononenko V. I., “Fundamental solutions of some singular equations in partial derivatives”, Differ. Uravn., 5:8 (1969), 1470–1483 | MR | Zbl

[8] Stempak K., Ciaurri O., “Transplantation and multiplier theorems for Fourier–Bessel expansions”, Trans. Amer. Math. Soc., 358 (2006), 4441–4465 | DOI | MR | Zbl

[9] Yildiri H., Sarikaya M. Z., Sermin O., “The solutions of the $n$-dimensional Bessel diamond operator and the Fourier-Bessel transform of their convolution”, Proc. Indian Acad. Sci. (Math. Sci.), 114:4, November (2003), 375–387 | DOI | MR

[10] Kipriyanov I. A., Singular Elliptic Boundary Value Problems, Nauka, M., 1997 (in Russian) | MR | Zbl

[11] Hörmander L., The Analysis of Linear Partial Differential Operators, v. I, Springer-Verlag, 2003 | MR | Zbl

[12] Gelfand I. M., Shilov G. E., Generalised Functions, Academic Press, 1964 | MR

[13] Lyakhov L. N., Shishkina E. L., “Weighted mixed spherical means and singular ultrahyperbolic equation”, Analysis (Germany), 36:2 (2016), 65–70 | DOI | MR | Zbl

[14] Lyakhov L. N., Polovinkin I. P., Shishkina E. L., “Formulas for the solution of the Cauchy problem for a singular wave equation with Bessel time operator”, Doklady Mathematics, 90:3 (2014), 737–742 | DOI | MR | Zbl

[15] Lyakhov L. N., Polovinkin I. P., Shishkina E. L., “On a Kipriyanov problem for a singular ultrahyperbolic equation”, Differential Equations, 50:4 (2014), 513–525 | DOI | MR | Zbl

[16] Shishkina E. L., “Boundedness of potential operators with hyperbolic distance”, Abstracts of the 8th International Workshop AMADE-2015 (Minsk, Belarus, September 14–19, 2015), Institute of Mathematics, National Academy of Sciences of Belarus, 90