THE CONVOLUTION IN ANISOTROPIC BESOV SPACES

  • N.T. Tleukhanova L.N. Gumilyov Eurasian National University
  • K.K. Sadykova L.N. Gumilyov Eurasian National University

Abstract

We study the boundedness of the convolution operator in Nikol'skii-Besov anisotropic spaces $B_{\bf p\boldsymbol\tau}^{\boldsymbol\alpha\bf q}$. These spaces are constructed on the basis of anisotropic Lorentz spaces $L_{\bf p\boldsymbol\tau}$, where $\bf p$ и $\boldsymbol\tau$ are vector parameters. The properties of anisotropic Nikol'skii-Besov spaces are investigated.The main goal of the paper is to solve the following problem: let $f$ and $g$ be functions from some classes of the Nikol'skii-Besov space scale. It is necessary to determine which space belongs to their convolution $f*g$. We prooved the inequality of different Nikol'skii metrics for trigonometric polynomials with spectrum in binary blocks in anisotropic Lorentz spaces $L_{\bf p\boldsymbol\tau}$. Conditions are obtained in terms of the corresponding vector parameters $\boldsymbol\alpha$, $\bf p$, $\bf q$, $\boldsymbol\tau$, $\bf r$, $\boldsymbol\mu$, $\boldsymbol\beta$, $\boldsymbol\eta$, $\bf h$, $\boldsymbol\nu$, $\boldsymbol\gamma$, $\boldsymbol\xi$, which are necessary and sufficient conditions for embeddings\begin{equation*}B_{\bf r\boldsymbol\mu}^{\boldsymbol{\beta\eta}}*B_{\bf h\boldsymbol\nu}^{\boldsymbol{\gamma\xi}}\hookrightarrow B_{\bf p\boldsymbol\tau}^{\boldsymbol\alpha\bf q}.\end{equation*}This statement is an analogue of O'Neil inequality for Lorentz spaces. In particular, the classical O'Neil inequality follows from the proved results. The obtained criterion is generalized by the results of Burenkov and Batyrov, who considered this problem in Besov spaces with scalar parameters.

References

[1] Bennett C. , Sharpley R., "Interpolation of Operators", Pure and Applied Mathematics 129, Boston, MA, Academic Press,
INC (1988): 469.
[2] Brézis H., Wainger S., "A note on limiting cases of Sobolev embeddings and convolution inequalities" Comm. Partial Differential Equations vol. 5, no. 7 (1980): 773-789.
[3] Hörmander L., "The analysis of linear partial differential operators I" Distribution theory and Fourier analysis Reprint
of the second edition / Berlin: Classics in Mathematics, Springer-Verlag (2013): 440.
[4] O'Neil R., "Convolution operators and L(p, q) spaces" Duke Math. J. 30 (1963): 129-142.
[5] Yap L. Y.H., "Some remarks on convolution operators and l(p, q) spaces" Duke Math. J. 36 (1969): 647-658.
[6] Hunt R. A., "On L(p, q) spaces", Enseignement Math. vol. 12, no. 2 (1966): 249-276.
[7] Nursultanov E., Tikhonov S., "Convolution inequalities in Lonentz spaces" J. Fourier Anal. Appl… 17 (2011): 486-505.
[8] Blozinski A. P., "On a convolution theorem for L(p, q) spaces" Trans. Amer. Math. Soc. 164 (1972): 255-265.
[9] Nursultanov E.D., Tleukhanova N. T., "O multiplikatorah kratnyh ryadov Furry [Multipliers of Multiple Fourier Series]"
Proc. Steklov Inst. Math. 227 (1999): 231-236.
[10] Tleukhanova N. T., Sadykova K. K. "O"Neil-type inequalities for convolutions in anisotropic Lorestz spaces" Eurasian
Mathematical Journal vol. 10, no. 3 (2019): 68-83.
[11] Nursultanov E., Tikhonov S., Tleukhanova N., "Norm inequalities for convolution operators" C. R. Acad. Sci. Paris vol.
I, no. 347 (2009): 1385-1388.
[12] Nursultanov E. Tikhonov S., Tleukhanova N. "Norm convolution inequalities in Lebesgue spaces" Rev. Mat. Iberoam
vol. 34, no. 2 (2018): 811-838.
[13] Heil C. "An introduction to weighted Wiener amalgams. In Wavelets and their applications" Allied Publishers, New
Delphi (2003): 183-216.
[14] Kaminska A., "On convolution operator in Orlicz spaces", Rev. Mat. Univ. Complutense 2 (1989): 157-178.
[15] Kerman R. A., "Convolution theorems with weights" Trans. Amer. Math. Soc. vol. 280, no. 1 (1983): 207-219.
[16] Kerman R., Sawyer E., "Convolution algebras with weighted rearrangement-invariant norm", Studia Math. vol. 108, no.
2 (1994): 103-126.
[17] Nursultanov E. Tikhonov S. "Weighted norm inequalities for convolution and Riesz potential" Potential Analysis vol.
42, no. 2 (2015): 435-456.
[18] Sampson G., Naparstek A., Drobot V., "(Lp, Lq) mapping properties of convolution transforms" Studia Math. vol. 55,
no. 1 (1976): 41-70.
[19] Golovkin K.K., Solonnikov V. A., "Ocenki integral"nyh operatorov v translyacionno-invariantnyh normah [Estimates of
integral operators in translation-invariant norms]", Tr. LIAN 70 (1964): 47-58.
[20] Golovkin K.K., Solonnikov V. A., "Ocenki integral'nyh operatorov v translyacionno-invariantnyh normah. II [Estimates of
integral operators in translation-invariant norms. III" Tr. MIAN 92 (1966): 5-30.
[21] Batyrov B. E., Burenkov V. I., "Estimates for convolutions in Nikol'skii-Besov spaces", Dokl. Akad. Nauk vol. 330, no. 1
(1993): 9-11.
[22] Bui H., "Weighted Young's inequality and convolution theorems on weighted Besov spaces" Math. Nachr. 170 (1994):
25-37.
[23] Golovkin K.K., Solonnikov V.A., "Ob ocenkah operatorov svertki [Estimates of convolution operators|" Zap. Naučn.
Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 7 (1968): 6-86.
[24] Sadykova K.K., Tleukhanova N.T., "Estimates of the norm of the convolution operator in anisotropic Besov spaces with
the dominated mixed derivative" Bulletin ofthe Karaganda University-Mathematics, vol. 95, no. 3 (2019): 51-59.
[25] Bekmaganbetov K., Nursultanov E., "Interpolation of Besov and Lizorkin-Triebel spaces" Analysis Mathemat-
ica, 35 (2009): 169-188.
Published
2020-06-26
How to Cite
TLEUKHANOVA, N.T.; SADYKOVA, K.K.. THE CONVOLUTION IN ANISOTROPIC BESOV SPACES. Journal of Mathematics, Mechanics and Computer Science, [S.l.], v. 106, n. 2, p. 18-30, june 2020. ISSN 2617-4871. Available at: <https://bm.kaznu.kz/index.php/kaznu/article/view/759>. Date accessed: 23 sep. 2020. doi: https://doi.org/10.26577/JMMCS.2020.v106.i2.02.
Keywords Young-O’Neil inequality, anisotropic Besov spañes, convolution operator