Open Access
Subscription Access
Open Access
Subscription Access
Landau-Kolmogorov and Gagliardo-Nirenberg Inequalities for Differential Operators in Lorentz Spaces
Subscribe/Renew Journal
In this paper, we establish some Landau-Kolmogorov inequalities and Gagliardo-Nirenberg inequalities for di?erential operators generated by polynomials. We illustrate the relation between ||P(D)f||N? and ||f||N?, ||Dm(P(D)f)||N? as follows
||P(D)f||N? K1(E)||f||N? + K2(E)||Dm(P(D)f)||N?
for all E > 0, where ||.||N? is the norm in Lorentz spaces N?(R). The corresponding inequalities in Lp(Rn) is also obtained.
Keywords
Lorentz Spaces, Fourier Transform, Landau-Kolmogorov Inequality, Gagliardo-Nirenberg Inequaly, Generalized Functions.
Subscription
Login to verify subscription
User
Font Size
Information
- H. H. Bang, A remark on the Kolmogorov-Stein inequality, J. Math. Analysis Appl., 203 (1996), 861–867.
- H. H. Bang, On inequalities of Bohr and Bernstein, J. Inequalities and Applications, 7(2002), 349–366.
- H. H. Bang and V. N. Huy, Some extensions of the Kolmogorov-Stein inequality, Vietnam J. Math., 43(1)(2015), 173–179.
- S. N. Bernstein, Collected works, Vol. 1. Moscow: Akad Nauk SSSR; 1952 (Russian).
- H. Bohr, Ein allgemeiner Satz u¨ber die Integration eines trigonometrischen Polynoms, Prace Matem.-Fiz., 43(1935), 273–288.
- Z. Ditzian, A Kolmogorov-type inequality, Mathematical Proceedings of the Cambridge Philosophical Society, 136(2004), 657–663.
- A. N. Kolmogorov, On inequalities between the upper bounds of the successive derivatives of an arbitrary function on an in?nitive interval, Ucen .Zap. Moskov. Gos. Univ. Mat., 30(1939), 3-13, Amer. Math. Soc. Transl., 1 (4)(1949), 1–19.
- E. Landau, Ungleichungen fu¨r zweimal di?erenzierbare Funktionen, Proc. London Math. Soc., 13 (1913), 43–49.
- A. A. Markov, On a question by D. I. Mendeleev, Zap. Imp. Akad. Nauk SPb., 62 (1890), 1–24.
- V. H. Nguyen. Sharp weighted Sobolev and Gagliardo-Nirenberg inequalities on halfspaces via mass transport and consequences, Proc. Lond. Math. Soc., 111(2015), 127–148.
- S. M. Nikolskii, Approximation of Functions of Several Variables and Imbedding Theorems, Nauka, Moscow, 1977.
- L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa. 13(1959), 115–162.
- G. E. Shilov, On inequalities between derivatives, Sb. Nauchn. Stud. Rabot Univ., 1 (1937), 17–27.
- M. S. Steigerwalt and A. J. White, Some function spaces related to Lp, Proc. London. Math. Soc., 22(1971), 137–163.
- E. M. Stein, Functions of exponential type, Ann. Math., 65 (1957), 582–592.
Abstract Views: 232
PDF Views: 0