# On the density of a special class of Lizorkin functions in a weighted Lebesgue space $L^{\gamma}_p$

### Abstract

We study the class of test functions $\Phi^{+}_{\gamma},$ constructed on the principle of Lizorkin spaces by means of mixed Fourier–Kipriyanov–Katrakhov transform. Initially, such classes of functions, constructed on the basis of a mixed Fourier–Bessel transform, were investigated by L. N. Lyakhov. The spaces introduced by him could not take into account “odd” orders of singular derivatives. But the latter appeared to be fundamentally necessary in the problems of determining the fundamental solutions of differential equations (ordinary and in partial derivatives). The integral Kipriyanov–Katrakhov transform (belonging to the class of Bessel transforms) is adapted to work with singular differential operators of the type $D^{2m+k}_B\frac{\partial^k}{\partial x^k}B^m_x,$ where k takes values 0 or 1, $B^m_x$ is a singular differential Bessel operator and the order of differentiation is 2m. The spaces of the basic functions that represent the images of the mixed Fourier–Kipriyanov–Katrakhov transform of functions vanishing at the origin and infinity are considered in this paper. We study the possibility of approximating functions from weighted Lebesgue classes $L^{\gamma}_p$ with power weight $\Pi|x_i|^{\gamma_i},$ namely, the density theorem $\Phi^{+}_{\gamma}$ in the Lebesgue function space $L^{\gamma}_p$.

### References

[1]

Lizorkin P. I., “$L_p^r(\Omega)$ spaces. Extension and embedding theorems,” Sov. Math., Dokl., 3, 1053–1057 (1962).

[2]

Lizorkin P. I., “Operators related to fractional differentiation and classes of differentiable functions,” Proc. Steklov Inst. Math., 117, 251–286 (1972).

[3]

Samko S. G., “Generalized Riesz potentials and hypersingular integrals, their symbols and inversion,” Sov. Math., Dokl., 18, 97–101 (1977).

[4]

Samko S. G., “Generalized Riesz potentials and hypersingular integrals with homogeneous characteristics, their symbols and inversion,” Proc. Steklov Math. Inst., 156, 173–243 (1980).

[5]

Samko S. G., Hypersingular Integrals and Their Applications, Taylor & Francis, London (2002). (Anal. Methods Spec. Functions; V. 5).

[6]

Lyakhov L. N., “On a class of hypersingular integrals,” Sov. Math., Dokl., 42, No. 3, 765–769 (1991).

[7]

Lyakhov L. N. and Raykhelgauz L. B., “Even and odd Fourier–Bessel transformations and some singular differential equations,” in: Analytic Methods of Analysis and Differential Equations (AMADE-2009), Camb. Sci. Publ., 2012, pp. 107–112.

[8]

Lyakhov L. N. and Polovinkina M. V., “The space of weighted Bessel potentials,” Proc. Steklov Math. Inst., 250, 178–182 (2005)

[9]

Kipriyanov I. A. and Katrakhov V. V., “On a class of one-dimensional singular pseudodifferential operators,” Math. USSR, Sb., 33, 43–61 (1977).

[10]

Katrakhov V. V., “Transmutation operators and pseudodifferential operators,” Sib. Math. J., 21, 64–73 (1980).

[11]

Katrakhov V. V. and Lyakhov L. N., “Full Fourier–Bessel transform and the algebra of singular pseudodifferential operators,” Differ. Equ., 47, No. 5, 681–695 (2011).

[12]

Lyakhov L. N. and Roshchupkin S. A., “A priori estimates for solutions of singular В-elliptic pseudodifferential equations with Bessel ∂B-operators,” J. Math. Sсi. 2014. V. 196, No. 4. P. 563–571.

[13]

Lyakhov L. N., “On the symbol of the integral operator of the B-potential type with a single characteristic,” Dokl. Math., 54, No. 3, 852–856 (1996).

[14]

Kipriyanov I. A., Singular Elliptic Boundary Value Problems [in Russian], Nauka, Moscow (1997).

[15]

Levitan B. M., “Fourier series and integrals expansion in Bessel functions [in Russian],” Usp. Mat. Nauk, 6, No. 2, 102–143 (1951).

[16]

Lyakhov L. N. and Roshchupkin S. A., “Full Fourier–Bessel transform for some fundamental functional classes [in Russian],” Nauch. Vedom. Belgorod. Gos. Univ., No. 11, 85–92 (2013).

[17]

Kipriyanov I. A. and Klyuchantsev M. I., “Singular integrals generated by a general translation operator, I,” Sib. Math. J., 11, 787–804 (1971).

*Mathematical notes of NEFU*, 25(4), pp. 60-73. doi: https://doi.org/10.25587/SVFU.2018.100.20554.

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