Landau-Kolmogorov and Gagliardo-Nirenberg Inequalities for Differential Operators in Lorentz Spaces

Authors

  • Vu Nhat Huy Department of Mathematics, Hanoi University of Science, Vietnam National University, 334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam
  • Ngoc Huy Nguyen

DOI:

https://doi.org/10.18311/jims/2022/25986

Keywords:

Lorentz Spaces, Fourier Transform, Landau-Kolmogorov Inequality, Gagliardo-Nirenberg Inequaly, Generalized Functions.

Abstract

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.

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References

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.

Published

2022-08-23

How to Cite

Huy, V. N., & Nguyen, N. H. (2022). Landau-Kolmogorov and Gagliardo-Nirenberg Inequalities for Differential Operators in Lorentz Spaces. The Journal of the Indian Mathematical Society, 89(3-4), 317–332. https://doi.org/10.18311/jims/2022/25986