On weighted generalized functions associated with quadratic forms
Problemy analiza, Tome 5 (2016) no. 2, pp. 52-68.

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

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},
     publisher = {mathdoc},
     volume = {5},
     number = {2},
     year = {2016},
     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
PB  - mathdoc
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
%I mathdoc
%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