Neves, J. S.:
On decompositions in generalised Lorentz-Zygmund spaces
Bollettino dell'Unione Matematica Italiana Serie 8 4-B (2001), fasc. n.1, p. 239-267, Unione Matematica Italiana (English)
pdf (570 Kb), djvu (349 Kb). | MR1821406 | Zbl 1178.46029
Sunto
Il lavoro presenta diverse caratterizzazioni degli spazi Lorentz-Zygmund generalizzati (GLZ) $L_{p, q; \mathbf{\alpha}}(R)$, con $p, q \in (0, +\infty]$, $m \in \mathbb{N}$, $\mathbf{\alpha}\in \mathbb{R}^{m}$ e $(R, \mu)$ spazio misurato con misura $\mu(R)$ finita. Dato uno spazio misurato $(R, \mu)$ e $\mathbf{\alpha} \in \mathcal{R}^{m}_{-}$ , otteniamo representazioni equivalenti per la (quasi-) norma dello spazio GLZ $L_{\infty, \infty; \mathbf{\alpha}} (R)$. Inoltre, se $(R, \mu)$ è uno spazio misurato con misura finita e $\mathbf{\alpha} \in \mathcal{R}^{m}_{+}$, viene presentata in termini di decomposizioni una norma equivalente per lo spazio $L_{1, 1; \mathbf{\alpha}}(R)$. Si dimostra che le norme equivalenti considerate per $L_{\infty, \infty; \mathbf{\alpha}}(R)$, con $(R, \mu)$ uno spazio a misura finita, e la norma di decomposizione in $L_{1, 1; \mathbf{\alpha}}(R)$ possono essere utilizzate per ottenere semplici dimostrazioni di alcuni risultati di estrapolazione concernenti questi spazi.
Referenze Bibliografiche
[2]
C. BENNETT-
K. RUDNICK,
On Lorentz-Zygmund spaces,
Dissertationes Math. (Rozprawy Mat.),
175 (
1980), 1-72. |
MR 576995 |
Zbl 0456.46028[3]
C. BENNETT-
R. SHARPLEY,
Interpolation of Operators, volume
129 of
Pure and Applied Mathematics,
Academic Press, New York,
1988. |
MR 928802 |
Zbl 0647.46057[4]
D. E. EDMUNDS-
P. GURKA-
B. OPIC,
Double exponential integrability of convolution operators in generalised Lorentz-Zygmund spaces,
Indiana Univ. Math. J.,
44 (
1995), 19-43. |
MR 1336431 |
Zbl 0826.47021[5]
D. E. EDMUNDS-
P. GURKA-
B. OPIC,
On embeddings of logarithmic Bessel potential spaces,
J. Funct. Anal.,
146 (
1997), 116-150. |
MR 1446377 |
Zbl 0934.46036[6]
D. E. EDMUNDS-
P. GURKA-
B. OPIC,
Norms of embeddings of logarithmic Bessel potential spaces,
Proc. Amer. Math. Soc.,
126 (8) (
1998), 2417-2425. |
MR 1451796 |
Zbl 0895.46020[7]
D. E. EDMUNDS-
M. KRBEC,
On Decomposition in Exponential Orlicz Spaces,
Math. Nachr.,
213 (
2000), 77-88. |
MR 1755247 |
Zbl 0971.46019[8]
D. E. EDMUNDS-
H. TRIEBEL,
Function Spaces, Entropy Numbers and Differential Operators, volume
120 of
Cambridge Tracts in Mathematics,
Cambridge University Press,
1996. |
MR 1410258 |
Zbl 0865.46020[9] A. FIORENZA-M. KRBEC, On decompositions in $L(\log L)^{\alpha}$, Preprint n. 129, Academy of Sciences of the Czech Republic, 1998.
[10]
P. GURKA-
B. OPIC,
Global limiting embeddings of logarithmic Bessel potential spaces,
Math. Inequal. Appl.,
1 (
1998) 565-584. |
MR 1646690 |
Zbl 0934.46034[11]
B. JAWERTH-
M. MILMAN,
Extrapolation theory with applications,
Mem. Amer. Math. Soc.,
89 (440) (
1991). |
MR 1046185 |
Zbl 0733.46040[12]
A. KUFNER-
O. JOHN-
S. FUČÍK,
Function Spaces,
Noordhoff International Publishing, Leyden,
Academia, Publishing House of the Czechoslovak Academy of Sciences, Prague,
1977. |
MR 482102 |
Zbl 0364.46022[14]
M. MILMAN,
Extrapolation and optimal decomposition, volume
1580 of
Lecture Notes in Mathematics,
Springer Verlag, Berlin, Heidelberg,
1994. |
MR 1301774 |
Zbl 0852.46059[15]
W. RUDIN,
Real and Complex Analysis,
Mcgraw-Hill Book Co., Singapore, 3rd edition,
1986. |
Zbl 0278.26001[16]
E. M. STEIN,
Singular Integrals and Differentiability Properties of Functions,
Princeton University Press, Princeton, New Jersey,
1970. |
MR 290095 |
Zbl 0207.13501[17]
A. TORCHINSKY,
Real Variable Methods in Harmonic Analysis, volume
123 of
Pure and Applied Mathematics,
Academic Press, San Diego,
1986. |
MR 869816 |
Zbl 0621.42001[18]
H. TRIEBEL,
Interpolation Theory, Function Spaces, Differential Operators, volume
18 of
North-Holland Mathematical Library. VEB Deutscher Verlag der Wissenschaften, Berlin 1978;
North-Holland Publishing Co.,
1978. |
Zbl 0387.46032[19]
H. TRIEBEL,
Approximation numbers and entropy numbers of embeddings of fractional Besov-Sobolev spaces in Orlicz spaces,
Proc. London Math. Soc. (3),
66 (3) (
1993), 589-618. |
MR 1207550 |
Zbl 0792.46024[22]
A. ZYGMUND,
Trigonometric Series, volume II,
Cambridge University Press, Cambridge, 2nd edition,
1959. |
MR 107776 |
Zbl 0085.05601