$B$-Riesz Transforms Generated by Generalized Translate Operator on $HM^p_{q,{\Delta_{\nu}}}$ Hardy-Morrey Spaces

Year 2022, Volume: 5 Issue: 2, 127 - 134, 01.06.2022

Cansu Keskin

https://doi.org/10.33401/fujma.1013757

Abstract

We study the decomposition of Hardy-Morrey spaces via atoms and molecules, which have similar properties of $H^{p}_{\Delta_{\nu}}(\mathbb{R}^{n}_{+})$ Hardy spaces. Then we introduce the $HM^p_{q,{\Delta_{\nu}}}$ boundedness of $ B $-Riesz transforms generated by a generalized translate operator that is associated to Laplace Bessel operator for $0<p\leq 1<q\leq \infty$ with $p\neq q$ through atomic decomposition and molecular characterization.

Keywords

$B$-Riesz transforms, Decomposition theory, Generalized translation operator, Hardy-Morrey spaces, Laplace-Bessel differential operator

Supporting Institution

TUBITAK

Project Number

119N455

Thanks

The author would like to express their sincere thanks to the editor and the anonymous reviewers for their helpful comments and suggestions.

References

• [1] H. Jia, H. Wang, Decomposition of Hardy-Morrey spaces, J. Math. Anal. Appl., 354 (1) (2009), 99-110.
• [2] H. Jia, H. Wang, Singular integral operator, Hardy-Morrey space estimates for multilinear operators and Navier-Stokes equations, Math. Methods Appl. Sci., 33 (2010), 1661-1684.
• [3] A. Akbulut, V. S. Guliyev, T. Noi, Y. Sawano, Generalized Hardy-Morrey spaces, Z. Anal. Anwend., 36 (2) (2017), 129-149.
• [4] K. Ho, Atomic decompositions of weighted Hardy-Morrey spaces, Hokkaido Math. J., 42 (2013), 131-157.
• [5] K. Ho, Atomic decomposition of Hardy-Morrey spaces with variable exponents, Ann. Acad. Sci. Fenn. Math., 40 (2015), 31-62.
• [6] E. M. Stein, Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series, 43. Monographs in Harmonic Analysis, III., Princeton University Press, Princeton, 1993.
• [7] C. Keskin, I. Ekincioglu, V. S. Guliyev, Characterizations of Hardy spaces associated with Laplace-Bessel operators, Analy. Math. Physc., 19 (4) (2019), 2281-2310.
• [8] I. Ekincioglu, The boundedness of high order B-Riesz transformations generated by the generalized shift operator in weighted Lp;w;g -spaces with general weights, Acta Appl. Math., 109 (2010), 591-598.
• [9] I. Ekincioglu, I. K. Ozkin, On high order Riesz Transformations generated by generalized shift operator, Turk. J. Math., 21 (1997), 51-60.
• [10] I. Ekincioglu, A. Serbetci, On the singular integral operators generated by the generalized shift operator, Int. J. App. Math., 1 (1999), 29-38.
• [11] V. S. Guliyev, A. Serbetci, I. Ekincioglu, On boundedness of the generalized B-potential integral operators in the Lorentz spaces, Integral Transforms and Special Functions, 18 (12) (2007), 885-895.
• [12] I. A. Kipriyanov, Fourier-Bessel transformations and imbedding theorems, Trudy Math. Inst. Steklov, 89 (1967), 130-213.
• [13] B. M. Levitan, Bessel function expansions in series and Fourier integrals, Uspekhi Mat. Nauk., (Russian) 6 (2) (1951), 102-143.
• [14] I. Ekincioglu, C. Keskin, R. V. Guliyev, Lipschitz estimates for rough fractional multilinear integral operators on local generalized Morrey spaces, Tbilisi Math. J., 1 (13) (2020), 47-60.
• [15] V. S. Guliyev, I. Ekincioglu, E. Kaya, Z. Safarov, Characterizations for the fractional maximal commutator operator in generalized Morrey spaces on Carnot group, Integral Transforms and Special Functions, 6 (30) (2020), 453-470.
• [16] C. Keskin, Different approach to the decomposition theory of HMp q;Dn Hardy-Morrey spaces, J. Pseudo-Differ. Oper. Appl. 12 (2021), Article ID 54, 14 pages, doi:10.1007/s11868-021-00426-7.
• [17] M. Y. Lee, C. C. Lin, The molecular characterization of weighted Hardy spaces, J. Funct. Anal., 188 (2002), 442-460.
• [18] M. H. Taibleson, G. Weiss, The molecular characterization of certain Hardy spaces, Asterisque, 77 (1980), 67-149.
• [19] D. G. Deng, Y. S. Hang, Hp Theory, Beijing University Press, Beijing, 1992.
• [20] L. N. Lyakhov, Multipliers of the mixed Fourier-Bessel transform, Proc. Steklov Inst. Math., 214 (1997), 234-249.
• [21] I. A. Aliev, Riesz transforms generated by a generalized translation operator, Izv. Acad. Nauk Azerbaijan. SSR Ser. Fiz. Tekhn. Mat. Nauk 8, 1 (1987), 7-13.