Generalize some norm inequalities of Saitoh (original) (raw)
2011, Kodai Mathematical Journal
Abstract
In this paper we present various norm inequalities in framework Sobolev spaces by using Hö lder's inequality. In particular, we imply some of the corresponding results of Saitoh whose proofs were based on Aronszajn's theory of reproducing kernels.
Loading Preview
Sorry, preview is currently unavailable. You can download the paper by clicking the button above.
References (34)
- K. F. Andersen, Weighted inequalities for iterated convolutions, Proc. Amer. Math. Soc. 127 (1999), 2643-2651.
- N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc. 68 (1950), 337-404.
- J. Burbea, A Dirichlet norm inequality and some inequalities for reproducing kernel spaces, Proc. Amer. Math. Soc. 83 (1981), 279-285.
- J. Burbea, Norm inequalities of exponential type for holomorphic functions, Kodai Math. J. 5 (1982), 339-354.
- M. Cwikel and R. Kerman, On a convolution inequality of Saitoh, Proc. Amer. Math. Soc. 124 (1996), 773-777.
- D. T. Duc and N. D. V. Nhan, On some convolution norm inequalities in weighted L p ðR n ; rÞ spaces and their applications, Math. Inequal. Appl. 11 (2008), 495-505.
- D. T. Duc and N. D. V. Nhan, Some applications of convolution inequalities in weighted L p spaces, Integr. Transform. and Special Funct. 19 (2008), 471-480.
- M. Hegland and J. T. Marti, Numerical computation of least constants for Sobolev inequality, Numer. Math. 48 (1986), 607-616.
- Y. Kametaka, K. Watanabe, A. Nagai and S. Pyatkov, The best constant of Sobolev inequality in an n-dimensional Euclidean space, Proc. Japan Acad. Ser. A Math. Sci. 81 (2005), 57-60.
- Y. Kametaka, H. Yamagishi, K. Watanabe, A. Nagai and K. Takemura, Riemann zeta function, Bernoulli polynomials and the best constant of Sobolev inequality, Sci. Math. Jpn. 65 (2007), 333-359.
- Y. Kametaka, H. Yamagishi, K. Watanabe, A. Nagai and K. Takemura, Discrete Bernoulli polynomials and the best constant of the discrete Sobolev inequality, Funkcial. Ekvac. 51 (2008), 307-327.
- Y. Kametaka, H. Yamagishi, K. Watanabe, A. Nagai and K. Takemura, The best constant of Sobolev inequality on a bounded interval, J. Math. Anal. Appl. 340 (2008), 699-706.
- Y. Kametaka, H. Yamagishi, K. Watanabe, A. Nagai and K. Takemura, The best con- stant of Sobolev inequality corresponding to Dirichlet boundary value problem for ðÀ1Þ M ðd=dxÞ 2M , M. Sci. Math. Jpn. 68 (2008), 299-311.
- D. S. Mitrinovic ´, J. E. Pec ˇaric ´and A. M. Fink, Classic and new inequalities in analysis, Kluwer Academic Publishers, The Netherlands, 1993.
- C. Morosi and L. Pizzocchero, On the constants for some Sobolev imbeddings, J. of Inequal. Appl. 6 (2001), 665-679.
- N. D. V. Nhan and D. T. Duc, Fundamental inequalities for the iterated Laplace convolu- tion in weighted L p spaces and their applications, Integr. Transform. and Special Funct. 19 (2008), 655-664.
- N. D. V. Nhan and D. T. Duc, Fundamental iterated convolution inequalities in weighted L p spaces and their applications, Math. Inequal. Appl. 12 (2009), 487-498.
- N. D. V. Nhan and D. T. Duc, Some convolution inequalities in l p weighted spaces, to appear in Canadian Mathematical Bulletin.
- S. Saitoh, Some inequalities for analytic functions with a finite Dirichlet integral on the unit disc, Math. Ann. 246 (1979), 69-77.
- S. Saitoh, Some inequalities for entire function, Proc. Amer. Math. Soc. 80 (1980), 254-258.
- S. Saitoh, Integral transform in Hilbert spaces, Proc. Japan Acad. Ser. A 58 (1982), 361-364.
- S. Saitoh, A fundamental inequality in the convolution of L 2 functions on the half line, Proc. Amer. Math. Soc. 91 (1984), 285-286.
- S. Saitoh, Hilbert spaces admitting reproducing kernels on the real line and related funda- mental inequalities, Riazi, J. Kar. Math. Assoc. 6 (1984), 25-31.
- S. Saitoh, On the convolution of L 2 functions, Kodai Math. J. 91 (1986), 50-57.
- S. Saitoh, Inequalities in the most simple Sobolev space and convolutions of L 2 functions with weights, Proc. Amer. Math. Soc. 118 (1993), 515-520.
- S. Saitoh, Natural norm inequalities in nonlinear transforms, International Series of Nu- merical Mathematics 123 (1997), 39-52.
- S. Saitoh, Integral transforms, reproducing kernels and their applications, Pitman research notes in mathematics series 369, Addison Wesley Longman, UK, 1997.
- S. Saitoh, Various operators in Hilbert space introduced by transforms, International Journal of Applied Mathematics 1 (1999), 111-126.
- S. Saitoh, Weighted L p -norm inequalities in convolution, Survey on Classical Inequalities, Kluwer Academic Pulishers, Amsterdam, 2000, 225-234.
- S. Saitoh, V. K. Tuan and M. Yamamoto, Reverse weighted L p -norm inequalities in convolutions and stability in inverse problems, J. of Inequal. Pure and Appl. Math. 1 (2000), Article 7.
- S. Saitoh, V. K. Tuan and M. Yamamoto, Reverse convolution inequalities and applications to inverse heat source problems, J. of Inequal. Pure and Appl. Math. 3 (2002), Article 80.
- S. Saitoh, V. K. Tuan and M. Yamamoto, Convolution inequalities and applications, J. of Inequal. Pure and Appl. Math. 4 (2003), Article 50.
- Y. Sawano, H. Fujiwara and S. Saitoh, Real inversion formulas of the Laplace transform on weighted function spaces, Complex Anal. Oper. Theory 2 (2008), 511-521.
- A. Yamada, Saitoh's inequality and Opial's inequality, to appear in Math. Inequal. Appl.. Dinh Thanh Duc Department of Mathematics Quy Nhon University Binh Dinh Vietnam E-mail: ducdinh2002@yahoo.com dinhthanhduc@qnu.edu.vn Nguyen Du Vi Nhan Department of Mathematics Quy Nhon University Binh Dinh Vietnam E-mail: ndvynhan@gmail.com nguyenduvinhan@qnu.edu.vn