| Title:
|
Fractional integro-differentiation in harmonic mixed norm spaces on a half-space (English) |
| Author:
|
Avetisyan, K. L. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
42 |
| Issue:
|
4 |
| Year:
|
2001 |
| Pages:
|
691-709 |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this paper some embedding theorems related to fractional integration and differentiation in harmonic mixed norm spaces $h(p,q,\alpha )$ on the half-space are established. We prove that mixed norm is equivalent to a ``fractional derivative norm'' and that harmonic conjugation is bounded in $h(p,q,\alpha )$ for the range $0<p\leq \infty $, $0<q\leq \infty $. As an application of the above, we give a characterization of $h(p,q,\alpha )$ by means of an integral representation with the use of Besov spaces. (English) |
| Keyword:
|
embedding theorems |
| Keyword:
|
integral representations |
| Keyword:
|
conjugation |
| Keyword:
|
projections |
| MSC:
|
26A33 |
| MSC:
|
31B05 |
| MSC:
|
31B10 |
| MSC:
|
31C05 |
| idZBL:
|
Zbl 1090.31500 |
| idMR:
|
MR1883378 |
| . |
| Date available:
|
2009-01-08T19:17:50Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119285 |
| . |
| Reference:
|
[1] Benedek A., Panzone R.: The spaces $L^P$, with mixed norm.Duke Math. J. 28 (1961), 301-324. MR 0126155 |
| Reference:
|
[2] Bergman S.: Über unendliche Hermitische Formen, die zu einem Bereiche gehören, nebst Anwendungen auf Fragen der Abbildung durch Funktionen von zwei komplexen Veränderlichen.Math. Z. 29 (1929), 641-677. MR 1545028 |
| Reference:
|
[3] Bergman S.: The Kernel Function and Conformal Mapping.Math. Surveys, No. 5, New York, 1950. Zbl 0473.30006, MR 0038439 |
| Reference:
|
[4] Bui Huy Qui: Harmonic functions, Riesz potentials, and the Lipschitz spaces of Herz.Hiroshima Math. J. 9 (1979), 245-295. MR 0529335 |
| Reference:
|
[5] Djrbashian A.E.: The classes $A_\alpha^p$ of harmonic functions in half-spaces and an analogue of M. Riesz' theorem.Izv. Akad. Nauk Arm. SSR, Matematika 22 4 (1987), 386-398 (in Russian); English transl.: Soviet J. Contemp. Math. Anal. (Armenian Academy of Sciences) 22 (1987), no. 4, 74-85. MR 0931892 |
| Reference:
|
[6] Djrbashian A.E., Karapetyan A.H.: Integral inequalities between conjugate pluriharmonic functions in multidimensional domains.Izv. Akad. Nauk Arm. SSR, Matematika 23 (1988), 3 216-236 (in Russian); English transl.: Soviet J. Contemp. Math. Anal. (Armenian Academy of Sciences) 23 (1988), no. 3, 20-42. MR 0976482 |
| Reference:
|
[7] Djrbashian A.E., Shamoyan F.A.: Topics in the Theory of $A_\alpha^p$ Spaces.Teubner-Texte zur Math., b. 105, Teubner, Leipzig, 1988. MR 1021691 |
| Reference:
|
[8] Djrbashian M.M.: On canonical representation of functions meromorphic in the unit disk.Dokl. Akad. Nauk Arm. SSR 3 (1945), 3-9 (in Russian). |
| Reference:
|
[9] Djrbashian M.M.: On the representation problem of analytic functions.Soobshch. Inst. Matem. Mekh. Akad. Nauk Arm. SSR 2 (1948), 3-40 (in Russian). |
| Reference:
|
[10] Djrbashian M.M., Djrbashian A.E.: Integral representation for some classes of functions in a half-plane.Dokl. Akad. Nauk SSSR 285 (1985), 547-550 (in Russian). MR 0821337 |
| Reference:
|
[11] Fefferman C., Stein E.M.: $H^p$ spaces of several variables.Acta Math. 129 (1972), 137-193. MR 0447953 |
| Reference:
|
[12] Flett T.M.: Mean values of power series.Pacific J. Math. 25 (1968), 463-494. Zbl 0162.10002, MR 0229807 |
| Reference:
|
[13] Flett T.M.: Inequalities for the $p$th mean values of harmonic and subharmonic functions with $p\le 1$.Proc. London Math. Soc. (3) 20 (1970), 249-275. MR 0257387 |
| Reference:
|
[14] Flett T.M.: On the rate of growth of mean values of holomorphic and harmonic functions.Proc. London Math. Soc. (3) 20 (1970), 749-768. Zbl 0211.39203, MR 0268388 |
| Reference:
|
[15] Flett T.M.: The dual of an inequality of Hardy and Littlewood and some related inequalities.J. Math. Anal. Appl. 38 (1972), 746-765. Zbl 0246.30031, MR 0304667 |
| Reference:
|
[16] Holmstedt T.: Interpolation of quasi-normed spaces.Math. Scand. 26 (1970), 177-199. Zbl 0193.08801, MR 0415352 |
| Reference:
|
[17] Ramey W.C., Yi H.: Harmonic Bergman functions on half-spaces.Trans. Amer. Math. Soc. 348 (1996), 633-660. Zbl 0848.31004, MR 1303125 |
| Reference:
|
[18] Ricci F., Taibleson M.: Boundary values of harmonic functions in mixed norm spaces and their atomic structure.Annali Scuola Nor. Sup. - Pisa, Ser. IV, 10 (1983), 1-54. Zbl 0527.30040, MR 0713108 |
| Reference:
|
[19] Shamoyan F.A.: Applications of Djrbashian's integral representations to the certain problems of analysis.Dokl. Akad. Nauk SSSR 261 (1981), 3 557-561 (in Russian). MR 0638930 |
| Reference:
|
[20] Shamoyan F.A.: Some remarks on the parametric representation of Nevanlinna-Djrbashian classes.Mat. Zametki 52 (1992), 1 128-140 (in Russian); English transl.: Math. Notes 52 (1993), no. 1-2, 727-737. MR 1187723 |
| Reference:
|
[21] Shi J.H.: Inequalities for the integral means of holomorphic functions and their derivatives in the unit ball of $\Bbb C^n$.Trans. Amer. Math. Soc. 328 (1991), 619-637. MR 1016807 |
| Reference:
|
[22] Stein E.M.: Singular Integrals and Differentiability Properties of Functions.Princeton Univ. Press, Princeton, New Jersey, 1970. Zbl 0281.44003, MR 0290095 |
| Reference:
|
[23] Taibleson M.: On the theory of Lipschitz spaces of distributions on Euclidean $n$-space, I. Principal properties.J. Math. Mech. 13 (1964), 407-479. MR 0163159 |
| Reference:
|
[24] Yi H.: Harmonic little Bloch functions on half-spaces.Math. Japonica 47 (1998), 21-28. Zbl 0924.31003, MR 1606287 |
| Reference:
|
[25] Avetisyan K.L.: Fractional integration and integral representations in weighted classes of harmonic functions.Analysis Math. 26 (2000), 161-174. Zbl 0997.30030, MR 1792883 |
| . |
