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Uspekhi Mat. Nauk, 2005, Volume 60, Issue 6(366), Pages 139–156 (Mi umn1680)  

This article is cited in 9 scientific papers (total in 9 papers)

Inequalities of Gagliardo–Nirenberg type and estimates for the moduli of continuity

V. I. Kolyada

Karlstads University

Abstract: In this paper a study is made of multiplicative inequalities of Gagliardo–Nirenberg type that connect partial moduli of continuity and partial derivatives of functions with respect to a fixed variable in different Lorentz norms. The main results are expressed by estimates of the form
$$ (\int_\delta^\infty[h^{-\theta r}\omega_j^r(f;h)_{p,s}]^s \frac{dh}h)^{1/s}\le c\|f\|_{p_0,s_0}^{1-\theta}[\delta^{-r}\omega_j^r(f;\delta)_{p_1,s_1}]^\theta, $$
where $0<\theta<1$,
$$ \frac1p=\frac{1-\theta}{p_0}+\frac{\theta}{p_1} , \qquad \frac1s=\frac{1-\theta}{s_0}+\frac{\theta}{s_1} , $$
and the exponents $p_i$ and $s_i$ satisfy certain conditions. In particular, these estimates imply optimal inequalities involving Besov norms and Lorentz norms. The limit case $p_1=s_1=1$ and estimates in terms of total variation are also studied.

DOI: https://doi.org/10.4213/rm1680

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English version:
Russian Mathematical Surveys, 2005, 60:6, 1147–1164

Bibliographic databases:

UDC: 517.51
MSC: Primary 26D99; Secondary 26A15, 46E35, 46E30
Received: 12.09.2005

Citation: V. I. Kolyada, “Inequalities of Gagliardo–Nirenberg type and estimates for the moduli of continuity”, Uspekhi Mat. Nauk, 60:6(366) (2005), 139–156; Russian Math. Surveys, 60:6 (2005), 1147–1164

Citation in format AMSBIB
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\by V.~I.~Kolyada
\paper Inequalities of Gagliardo--Nirenberg type and estimates for the moduli of continuity
\jour Uspekhi Mat. Nauk
\yr 2005
\vol 60
\issue 6(366)
\pages 139--156
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\adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?2005RuMaS..60.1147K}
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\transl
\jour Russian Math. Surveys
\yr 2005
\vol 60
\issue 6
\pages 1147--1164
\crossref{https://doi.org/10.1070/RM2005v060n06ABEH004285}
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\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-33646431134}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Chen Jiecheng, Li Hongliang, “A kind of estimate of difference norms in anisotropic weighted Sobolev-Lorentz spaces”, J. Inequal. Appl., 2009, 161405, 22 pp.  crossref  mathscinet  zmath  isi
    2. Barza S., Kolyada V., Soria J., “Sharp constants related to the triangle inequality in Lorentz spaces”, Trans. Amer. Math. Soc., 361:10 (2009), 5555–5574  crossref  mathscinet  zmath  isi
    3. Kolyada V., Soria J., “Hölder type inequalities in Lorentz spaces”, Ann. Mat. Pura Appl. (4), 189:3 (2010), 523–538  crossref  mathscinet  zmath  isi
    4. Kolyada V.I., Pérez Lázaro F.J., “Inequalities for partial moduli of continuity and partial derivatives”, Constr. Approx., 2011  crossref  mathscinet  isi
    5. A. I. Parfenov, “Otsenka pogreshnosti obobschennoi formuly M. A. Lavrenteva normoi drobnogo prostranstva Soboleva”, Sib. elektron. matem. izv., 10 (2013), 335–377  mathnet
    6. D.D.. Haroske, Hans Triebel, “Some recent developments in the theory of function spaces involving differences”, J. Fixed Point Theory Appl, 2013  crossref  mathscinet  isi
    7. Proc. Steklov Inst. Math., 284 (2014), 263–279  mathnet  crossref  crossref  isi  elib
    8. A. I. Parfenov, “Series in a Lipschitz perturbation of the boundary for solving the Dirichlet problem”, Siberian Adv. Math., 27:4 (2017), 274–304  mathnet  crossref  crossref  elib
    9. Fiorenza A., Formica M.R., Roskovec T., Soudsky F., “Gagliardo-Nirenberg Inequality For Rearrangement-Invariant Banach Function Spaces”, Rend. Lincei-Mat. Appl., 30:4 (2019), 847–864  crossref  isi
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