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Non-Isotropic Potential Theoretic Inequality

Year 2018, , 325 - 338, 29.06.2018
https://doi.org/10.17776/csj.436027

Abstract

In this paper, the new weighted inequalities were derived by β-distance which is similar to the given inequality for the potential operator defined in [1]. The results presented here would provide extensions of those given in earlier works.

References

  • [1]. Adams, D., Traces of potentials arising from traslation invariant operators, Ann. Scuola Norm. Sup. Pisa, 25 (1971) 203-217.
  • [2]. Besov, O.V., Lizorkin, P.I., The L^{p} estimates of a certain class of non-isotropic singular integrals, Dokl. Akad. Nauk, SSSR, 69 (1960) 1250-1253.
  • [3]. Garcia-Cuerva, J., Martell, J.M., Two-weight norm inequalies for maximal operator and fractional integrals on non-homogeneous spaces, Indiana Univ. Math. J., 50-3 (2001) 1241-1280.
  • [4]. Hedberg, L., On certain convolution inequalities. Proc. Amer. Math. Soc. 36 (1972) 505-510.
  • [5]. Kufner, A., O. John and S. Fucik, Function spaces, Academia, Prague and Noordhoff, Leyden 1977.
  • [6]. Levitan, B.M., Generelized Translation Operators and Some of Their Applications, Nauka, Moscow, 1962; English translation: Israel Program for Scientific Translation 1964.
  • [7]. Morrey, C.B., On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc. 43 (1938) 126-166.
  • [8]. Ragusa, M. A., Catania, and P. Zamboni, Sant'agata-Messina, A Potential Theoretic Inequality, Czechoslovak Mathematical Journal, 51-126 (2001) 55-65.
  • [9]. Samko, S.G., Kilbas, A.A., and Marichev, O.I., Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach, Linghorne 1993.
  • [10]. Sarikaya, M.Z., Yıldırım, H., The restriction and the continuity properties of potentials depending on λ-distance, Turk. J. Math., 30-3 (2006) 263-275.
  • [11]. Sarikaya, M.Z., Yıldırım, H., On the β-spherical Riesz potential generated by the β-distance, Int. Journal of Contemp. Math. Sciences, 1, No. 1-4 (2006) 85-89.
  • [12]. Sarikaya, M.Z., Yıldırım, H., On the non-isotropic fractional integrals generated by the λ-distance, Selçuk Journal of Appl. Math., 7-1 (2005) 17-23.
  • [13]. Sarikaya, M.Z., Yıldırım, H., Ozkan, U. M., Norm inequalities with non-isotropic kernels, Int. Journal of Pure and Applied Mathematics, 31-3 (2006) 337-343.
  • [14]. Schechter, M., Spectra of Partial Differential Operators. North Holland, 1986.
  • [15]. Stein, E.M., Singular Integrals and Differentiability Properties of Functions, Princeton Uni. Press, Princeton, New Jersey, 1970.
  • [16]. Welland, G. V., Weighted norm inequalities for fractional integral. Proc. Amer. Math. Soc., 51 (1975) 143-148.
  • [17]. Yıldırım, H., On generalization of the quasi homogeneous Riesz potential, Turk. J. Math., 29 (2005) 381-387.

İzotropik Olmayan Potansiyel Teorik Eşitsizlik

Year 2018, , 325 - 338, 29.06.2018
https://doi.org/10.17776/csj.436027

Abstract

Bu
yazıda, [1] 'de tanımlanan potansiyel operatör için verilen eşitsizliğe benzer
β-mesafesi ile türetilen yeni ağırlıklı eşitsizlikler elde edilmiştir. Burada
sunulan sonuçlar daha önceki çalışmalarda ki verilenleri destekler.

References

  • [1]. Adams, D., Traces of potentials arising from traslation invariant operators, Ann. Scuola Norm. Sup. Pisa, 25 (1971) 203-217.
  • [2]. Besov, O.V., Lizorkin, P.I., The L^{p} estimates of a certain class of non-isotropic singular integrals, Dokl. Akad. Nauk, SSSR, 69 (1960) 1250-1253.
  • [3]. Garcia-Cuerva, J., Martell, J.M., Two-weight norm inequalies for maximal operator and fractional integrals on non-homogeneous spaces, Indiana Univ. Math. J., 50-3 (2001) 1241-1280.
  • [4]. Hedberg, L., On certain convolution inequalities. Proc. Amer. Math. Soc. 36 (1972) 505-510.
  • [5]. Kufner, A., O. John and S. Fucik, Function spaces, Academia, Prague and Noordhoff, Leyden 1977.
  • [6]. Levitan, B.M., Generelized Translation Operators and Some of Their Applications, Nauka, Moscow, 1962; English translation: Israel Program for Scientific Translation 1964.
  • [7]. Morrey, C.B., On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc. 43 (1938) 126-166.
  • [8]. Ragusa, M. A., Catania, and P. Zamboni, Sant'agata-Messina, A Potential Theoretic Inequality, Czechoslovak Mathematical Journal, 51-126 (2001) 55-65.
  • [9]. Samko, S.G., Kilbas, A.A., and Marichev, O.I., Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach, Linghorne 1993.
  • [10]. Sarikaya, M.Z., Yıldırım, H., The restriction and the continuity properties of potentials depending on λ-distance, Turk. J. Math., 30-3 (2006) 263-275.
  • [11]. Sarikaya, M.Z., Yıldırım, H., On the β-spherical Riesz potential generated by the β-distance, Int. Journal of Contemp. Math. Sciences, 1, No. 1-4 (2006) 85-89.
  • [12]. Sarikaya, M.Z., Yıldırım, H., On the non-isotropic fractional integrals generated by the λ-distance, Selçuk Journal of Appl. Math., 7-1 (2005) 17-23.
  • [13]. Sarikaya, M.Z., Yıldırım, H., Ozkan, U. M., Norm inequalities with non-isotropic kernels, Int. Journal of Pure and Applied Mathematics, 31-3 (2006) 337-343.
  • [14]. Schechter, M., Spectra of Partial Differential Operators. North Holland, 1986.
  • [15]. Stein, E.M., Singular Integrals and Differentiability Properties of Functions, Princeton Uni. Press, Princeton, New Jersey, 1970.
  • [16]. Welland, G. V., Weighted norm inequalities for fractional integral. Proc. Amer. Math. Soc., 51 (1975) 143-148.
  • [17]. Yıldırım, H., On generalization of the quasi homogeneous Riesz potential, Turk. J. Math., 29 (2005) 381-387.
There are 17 citations in total.

Details

Primary Language English
Journal Section Natural Sciences
Authors

Merve Esra Yıldırım

Abdullah Akkurt

Hüseyin Yıldırım

Publication Date June 29, 2018
Submission Date September 1, 2017
Acceptance Date May 9, 2018
Published in Issue Year 2018

Cite

APA Yıldırım, M. E., Akkurt, A., & Yıldırım, H. (2018). Non-Isotropic Potential Theoretic Inequality. Cumhuriyet Science Journal, 39(2), 325-338. https://doi.org/10.17776/csj.436027