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

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,

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[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.