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Mat. Sb., 2001, Volume 192, Number 7, Pages 51–72 (Mi msb579)  

This article is cited in 15 scientific papers (total in 15 papers)

Embeddings of fractional Sobolev spaces and estimates of Fourier transforms

V. I. Kolyada

I. I. Mechnikov Odessa National University

Abstract: Fractional anisotropic Sobolev–Liouville spaces $L_p^{r_1,…,r_n}(\mathbb R^n)$ are investigated for $1\leqslant p<\infty$ and positive $r_k$. For functions in these spaces estimates of norms in modified spaces of Lorentz and Besov kinds, defined in terms of iterative rearrangements, are established. These estimates are used to prove inequalities for the Fourier transforms of functions in $L_1^{r_1,…,r_n}$.
This paper continues works of the author in which similar issues have been discussed for integer $r_k$.
The methods used in the paper are based on estimates of iterative rearrangements. This approach enables one to simplify proofs and at the same time to obtain stronger results. In particular, the analysis of the limit case $p=1$ becomes much easier.


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English version:
Sbornik: Mathematics, 2001, 192:7, 979–1000

Bibliographic databases:

UDC: 517.5
MSC: 46E35, 42B10
Received: 02.11.2000

Citation: V. I. Kolyada, “Embeddings of fractional Sobolev spaces and estimates of Fourier transforms”, Mat. Sb., 192:7 (2001), 51–72; Sb. Math., 192:7 (2001), 979–1000

Citation in format AMSBIB
\by V.~I.~Kolyada
\paper Embeddings of fractional Sobolev spaces and estimates of Fourier transforms
\jour Mat. Sb.
\yr 2001
\vol 192
\issue 7
\pages 51--72
\jour Sb. Math.
\yr 2001
\vol 192
\issue 7
\pages 979--1000

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    This publication is cited in the following articles:
    1. Kolyada V.I., Pérez F.J., “Estimates of difference norms for functions in anisotropic Sobolev spaces”, Math. Nachr., 267 (2004), 46–64  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    2. Pérez F.J., “Embedding theorems for anisotropic Lipschitz spaces”, Studia Math., 168:1 (2005), 51–72  crossref  mathscinet  zmath  isi
    3. Pérez Lázaro J.P., “A note on extreme cases of Sobolev embeddings”, J. Math. Anal. Appl., 320:2 (2006), 973–982  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    4. Krbec M., Schmeisser H.-J., “Critical imbeddings with multivariate rearrangements”, Studia Math., 181:3 (2007), 255–284  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    5. Pérez Lázaro J.P., “Embeddings for anisotropic besov spaces”, Acta Math. Hungar., 119:1-2 (2008), 25–40  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    6. Tikhonov S., “Weak type inequalities for moduli of smoothness: the case of limit value parameters”, J. Fourier Anal. Appl., 2009  crossref  mathscinet  isi  scopus  scopus  scopus
    7. Esfahani A., Pastor A., “Instability of solitary wave solutions for the generalized BO-ZK equation”, J. Differential Equations, 247:12 (2009), 3181–3201  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus  scopus
    8. Hansen M., Vybiral J., “The Jawerth-Franke embedding of spaces with dominating mixed smoothness”, Georgian Math. J., 16:4 (2009), 667–682  mathscinet  zmath  isi
    9. Simonov B., Tikhonov S., “Sharp Ul'yanov-type inequalities using fractional smoothness”, Journal of Approximation Theory, 162:9 (2010), 1654–1684  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    10. Algervik R., Kolyada V.I., “On Fournier-Gagliardo Mixed Norm Spaces”, Ann Acad Sci Fenn Math, 36:2 (2011), 493–508  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    11. Esfahani A., “Remarks on solitary waves of the generalized two dimensional Benjamin-Ono equation”, Applied Mathematics and Computation, 218:2 (2011), 308–323  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    12. Kolyada V.I., “Iterated rearrangements and Gagliardo-Sobolev type inequalities”, J Math Anal Appl, 387:1 (2012), 335–348  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    13. G. A. Akishev, “Estimates for Kolmogorov widths of the Nikol'skii — Besov — Amanov classes in the Lorentz space”, Proc. Steklov Inst. Math. (Suppl.), 296, suppl. 1 (2017), 1–12  mathnet  crossref  mathscinet  isi  elib
    14. Esfahani A., Pastor A., Bona J.L., “Stability and Decay Properties of Solitary-Wave Solutions To the Generalized Bo-Zk Equation”, Adv. Differ. Equat., 20:9-10 (2015), 801–834  mathscinet  zmath  isi
    15. Kolyada V.I., “Embedding Theorems For Sobolev and Hardy-Sobolev Spaces and Estimates of Fourier Transforms”, Ann. Mat. Pura Appl., 198:2 (2019), 615–637  crossref  mathscinet  zmath  isi  scopus
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