On weighted Calderón-Zygmund singular integrals and applications

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Ahmed Loulit


This paper studies some weighted norm inequalities related to some Calderon-Zygmund singular integrals. Applications to the Sobolev-Gagliardo-Nirenberg inequality, differential forms, and the potential equation du = f are given.


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Loulit, A. 2022. On weighted Calderón-Zygmund singular integrals and applications. Journal of Innovative Applied Mathematics and Computational Sciences. 2, 1 (Feb. 2022), 1–26.
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J. Bourgain and H. Brezis, On the equation divY = f and application to control of phases, J. Amer. Math. Soc. 16(2) (2003), 393–426.

J. S. Bradley, Hardy inequalities with mixed norms, Canad. Math. Bull. 21 (1978), 405–408.

H.-Q. M. Bui, Weighted Besov and Triebel spaces, Interpolation by the real method, Hiroshima Math. J. 12(3) (1982), 581–605.

H.-Q. M. Bui, M. Palusznsky and M.H. Taibleson, A maximal function characterization of weighted Besov-Lipschitz and Triebel-Lizorkin spaces, Studia Math. 119 (1996), 219–246.

H.-Q. M. Bui, M. Palusznsky and M.H. Taibleson. M, Characterization of the Besov-Lipschitz and Triebel-Lizorkin spaces, The case q < 1, J. Fourier Anal. Appl. 3 (1997), 837–846.

A. P. Calderón, and A. Torchinsky, Parabolic maximal functions associated with a distribution, Adv. Math. 16 (1975), 1–64.

A. P. Calderón, and A. Torchinsky, Parabolic maximal functions associated with a distribution, II, Adv. Math. 24 (1977), 101–171.

D. Chang, S. Krantz and E. Stein, Hp theory on a smooth domain in Rn and elliptic boundary value problems, J. Funct. Anal. 114(1) (1993), 286–347.

M. Costabel, A. McIntosh and R.J. Taggart, Potential maps, Hardy spaces, and tent spaces on special Lipschitz domains, Pub. Math. 57(2) (2013), 295–331.

C. Fefferman and E. M. Stein, Hp spaces of several variables, Acta Math. 129 (1972), 137–193.

M. Frazier and B. Jawerth, A Discrete Transform and Decomposition of Distribution Spaces, J. Funct. Anal. 93(1) (1989), 34–170.

M. Frazier and B. Jawerth, Decomposition of Besov spaces, Indiana Univ. Math. J. 34 (4) (1985), 777–799.

M. Frazier and B. Jawerth, The j-transform and applications to distribution spaces In: M. Cwikel, J. Peetre, Y. Sagher, H. Wallin (eds) Function Spaces and Applications. Lecture Notes in Mathematics, vol. 1302, Springer, Berlin, Heidelberg, 1988.

J. Garcia-Cuerva and J. L. Rubio De Francia, Weighted norm inequalities and Related Topics, Vol. 116, North-Holland Mathematics Studies, 1985.

F. Gürbüz and A. Loulit, Lp Smoothness on Weighted Besov-Triebel-Lizorkin Spaces in terms of Sharp Maximal Functions, J. Math. 2021 (2021), Article ID 8104815, 9 pages.

F. Gürbüz, Product Generalized Local Morrey Spaces and Commutators of Multi-Sublinear Operators Generated by Multilinear Calderón-Zygmund Operators and Local Campanato Functions, Filomat, 35 (9) 2021, 2849–2868.

F. Gürbüz, Sublinear operators with rough kernel generated by Calderón-Zygmund operators and their commutators on generalized Morrey spaces, Math. Notes. 101 (2017), 429–442.

W. Grant and Z. Shiying, e-Families of operators in Triebel-Lizorkin and tent spaces, Can. J. Math. 47 (5), (1995), 1095-1120.

H. P. Heinig, Weighted estimates for classical operators, In: M. Krbec, A. Kufner, and J. Rákosník, (eds.) Nonlinear Analysis, Function Spaces and Applications, Proceedings of the Spring School held in Litomy˜sl, 1986. Vol. 3 BSB B. G. Teubner-Texte Math. 93, Teubner, Verlagsgesellschaft, Leipzig, 1986. pp. 31-53.

J. Janson and M.H. Taibleson, I teoremi di rappresentazione di Calderón, Rend. Sem. Mat. Univ. Politec. Torino. 39 (1981), 27-35.

V. M Kokilashvili, On Hardy’s inequality in weigted spaces, Soobshch. Akad. Nauk Grusin. SSR. 96 (1979), 37-40. MathSciNet

M. Krbec, B. Opic, L. Pick and J. Rakosnik, Some recent results on Hardy type operators in weighted function spaces and related topics, in: Function Spaces, Differential Operators and Nonlinear Analysis (Friedrichroda, 1992), Teubner-Texte Math. Vol. 133, Teubner, Stuttgart 1993. pp. 158-184.

A. Kufner and L.-E. Persson, Weighted Inequalities of Hardy Type, World Scientific Publishing Co., Inc., River Edge, New Jersey, 2003.

A. Kufner,L.-E. Persson and A. Wedestig, A study of some constants characterizing the weighted Hardy inequality, Banach Center, Publ. Orlicz Centenary, Vol. 64 (2004), 135-146.

A. Kufner and H. Triebel, Generalizations of Hardy’s inequality, Confer. Sem. Mat. Univ. Bari 156 (1978), 1-21.

A. Kufner, L. Maligranda and L.-E. Persson, The Hardy Inequality. About its History and Some Related Results, Vydavatelsky Servis, Plsen, 2007. URN: urn:nbn:se:ltu:diva-16695

A. Kufner and A. Opic, Hardy-Type Inequalities, Harlow, Essex, England: Longman Scientific & Technical. 1990.

L. Lanzani and E. Stein, A note on div curl inequalities, Math. Res. Lett. 12(1) (2005), 57-61.

A. Loulit, Weighted estimates for L1-vector fields, Proc. Amer. Math. Soc. 142 (2014), 4171-4179.

I. Mitrea and M. Mitrea, A remark on the regularity of the div-curl system, Proc. Amer. Math. Soc. 137 (2009), 1729-1733.

B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207-226.

U. Neri, Singular integrals and Sobolev spaces, In: Singular Integrals. Lecture Notes in Mathematics, vol. 200. Springer, Berlin, Heidelberg, 1971.

S. V. Rychkov, Littlewood-Paley Theory and Function Spaces with A_p^loc Weights, Math. Nachr. 224 (2001), 145-180.

J. V. Schafttingen, Limitind Fractional and Lorentz Spaces Estimates of differential forms, Proc. Amer. Math. Soc. 138(1) (2010), 235-240.

W. Sickel and H. Triebel, Holder inequalities and sharp embeddings in function spaces of B^s_p,q and F^s_p ,q type, Z. Anal. Anwend. 14(1) (1995), 105-140.

W. Yuan, W. Sickel and D. Yang , Morrey and Campanato Meet Besov, Lizorkin and Triebel, Lecture Notes in Mathematics, Vol. 2005, Springer-Verlag, Berlin, 2010.

E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series, Vol. 43. Monographs in Harmonic Analysis, III. Princeton University Press, Princeton, NJ, 1993.

E. M. Stein, II. Singular Integrals, In Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, Vol. 30, 26-53. Princeton: Princeton University Press, 1971.

J-O. Strömberg and A. Torchinsky, Weighted Hardy Spaces, Lecture Notes in Mathematics, Vol. 1381, Springer-Verlag, Berlin, 1989.

H. Triebel,Theory of Function Spaces II, Birkhïauser Verlag, Basel, 1992.

M. Troyanov, On The Hodge Decomposition in Rn, Moscow Math. J. 9(4) (2009), 899-926.

C. Young-Kum , Continuous Characterezation of The Treibel-Lizorking spaces and Fourier Multipliers Bull, Korean Math. Soc. 47(4) (2010), 839-857.