Geodesic
Analog of an inequality of Bohr for integrals of functions from $L^{p}(R^{n})$. II
Problemy analiza, Tome 3 (2014) no. 2, pp. 32-51

Voir la notice de l'article provenant de la source Math-Net.Ru

Let $p\in(2,+\infty],$ $n\ge1$ and $\Delta=(\Delta_1,\ldots,\Delta_n),$ $\Delta_k>0,$ $1\le k\le n.$ It is proved that for functions $\gamma(t)\in L^p(R^n)$ spectrum of which is separated from each of $n$ the coordinate hyperplanes on the distance not less than $\Delta_k,$ $1\le k\le n$ respectively, the inequality is valid: $$\left\|\int\limits_{E_t}\gamma(\tau)\,d\tau\right\| _{L^{\infty}(R^n)}\le C^n(q)\left[\prod_{k=1}^n\frac{1} {\Delta_k^{1/q}}\right]\left\|\gamma(\tau)\right\|_{L^p(R^n)},$$ where $t=(t_1,\ldots,t_n)\in R^n,$ $E_t=\{\tau\,|\,\tau=(\tau_1,\ldots,\tau_n)\in R^n,$ $\tau_j\in[0,t_j],$ if $t_j\ge0,$ and $\tau_j\in[t_j,0],$ if $t_j<0,\ 1\le j\le n\},$ and the constant $C(q)>0,$ $\displaystyle\frac{1}{p}+ \frac{1}{q}=1$ does not depend on $\gamma(\tau)$ and vector $\Delta$.
Keywords: inequality of Bohr. 
B. F. Ivanov. Analog of an inequality of Bohr for integrals of functions from $L^{p}(R^{n})$. II. Problemy analiza, Tome 3 (2014) no. 2, pp. 32-51. http://geodesic.mathdoc.fr/item/PA_2014_3_2_a2/
@article{PA_2014_3_2_a2,
     author = {B. F. Ivanov},
     title = {Analog of an inequality of {Bohr} for integrals of~functions from~$L^{p}(R^{n})${.~II}},
     journal = {Problemy analiza},
     pages = {32--51},
     year = {2014},
     volume = {3},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/PA_2014_3_2_a2/}
}
TY  - JOUR
AU  - B. F. Ivanov
TI  - Analog of an inequality of Bohr for integrals of functions from $L^{p}(R^{n})$. II
JO  - Problemy analiza
PY  - 2014
SP  - 32
EP  - 51
VL  - 3
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/PA_2014_3_2_a2/
LA  - en
ID  - PA_2014_3_2_a2
ER  - 
%0 Journal Article
%A B. F. Ivanov
%T Analog of an inequality of Bohr for integrals of functions from $L^{p}(R^{n})$. II
%J Problemy analiza
%D 2014
%P 32-51
%V 3
%N 2
%U http://geodesic.mathdoc.fr/item/PA_2014_3_2_a2/
%G en
%F PA_2014_3_2_a2

[1] Ivanov B. F., “Analog of an inequality of Bohr for integrals of functions from $L^p(R^n)$”, The Issues of Analysis, 3 (21):2 (2014), 16–34 | DOI | MR

[2] Bohr H., “Un théoréme général sur l'intégration d'un polynome trigonométrigue”, Comptes Rendus De L'Academie des sciences, 200:15 (1935), 1276–1277

[3] Ivanov B. F., “On a generalization of an inequality of Bohr”, The Issues of Analysis, 2 (20):2 (2013), 21–57 | DOI | MR | Zbl

[4] Gelfand I. M., Shilov G. E., The generalized functions and the operations over them, iss. 1, PhM, M., 1959, 470 pp.

[5] Vladimirov V. S., Generalized functions in mathematical physics, Nauka, M., 1979, 318 pp. | MR