On a theorem of Hardy and Littlewood on the polydisc
Author:
Hong Oh Kim
Journal:
Proc. Amer. Math. Soc. 97 (1986), 403409
MSC:
Primary 32A35; Secondary 30D55
MathSciNet review:
840619
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: We prove the polydisc version of the theorem of Hardy and Littlewood on the fractional integral: If and if , then with where is the fractional integral of of order .
 [1]
Peter
L. Duren, Theory of 𝐻^{𝑝} spaces, Pure and
Applied Mathematics, Vol. 38, Academic Press, New YorkLondon, 1970. MR 0268655
(42 #3552)
 [2]
T.
M. Flett, The dual of an inequality of Hardy and Littlewood and
some related inequalities, J. Math. Anal. Appl. 38
(1972), 746–765. MR 0304667
(46 #3799)
 [3]
Arlene
P. Frazier, The dual space of 𝐻^{𝑝} of the polydisc
for 0<𝑝<1, Duke Math. J. 39 (1972),
369–379. MR 0293119
(45 #2198)
 [4]
G.
H. Hardy and J.
E. Littlewood, Some properties of fractional integrals. II,
Math. Z. 34 (1932), no. 1, 403–439. MR
1545260, http://dx.doi.org/10.1007/BF01180596
 [5]
G.
H. Hardy and J.
E. Littlewood, Theorems concerning mean values of analytic or
harmonic functions, Quart. J. Math., Oxford Ser. 12
(1941), 221–256. MR 0006581
(4,8d)
 [6]
B. Jawerth and A. Torchinsky, On a Hardy and Littlewood imbedding theorem, preprint.
 [7]
Hong
Oh Kim, Derivatives of Blaschke products, Pacific J. Math.
114 (1984), no. 1, 175–190. MR 755488
(85h:30045)
 [8]
Walter
Rudin, Function theory in polydiscs, W. A. Benjamin, Inc., New
YorkAmsterdam, 1969. MR 0255841
(41 #501)
 [9]
, Function theory in the unit ball of , SpringerVerlag, New York, 1980.
 [10]
A.
Zygmund, On the boundary values of functions of several complex
variables. I, Fund. Math. 36 (1949), 207–235.
MR
0035832 (12,18b)
 [1]
 P. L. Duren, Theory of spaces, Academic Press, New York, 1970. MR 0268655 (42:3552)
 [2]
 T. M. Flett, The dual of an inequality of Hardy and Littlewood and some related inequalities, J. Math. Anal. Appl. 38 (1972), 746765. MR 0304667 (46:3799)
 [3]
 A. Frazier, The dual space of of the polydisc for , Duke Math. J. 39 (1972), 369379. MR 0293119 (45:2198)
 [4]
 G. H. Hardy and J. E. Littlewood, Some properties of fractional integrals. II, Math. Z. 34 (1932), 403439. MR 1545260
 [5]
 , Theorems concerning mean values of analytic or harmonic functions, Quart. J. Math. 12 (1941), 221256. MR 0006581 (4:8d)
 [6]
 B. Jawerth and A. Torchinsky, On a Hardy and Littlewood imbedding theorem, preprint.
 [7]
 H. O. Kim. Derivatives of Blaschke products, Pacific J. Math. 114 (1984), 175191. MR 755488 (85h:30045)
 [8]
 W. Rudin, Function theory in polydiscs, Benjamin, New York, 1969. MR 0255841 (41:501)
 [9]
 , Function theory in the unit ball of , SpringerVerlag, New York, 1980.
 [10]
 A. Zygmund, On the boundary value of functions of several complex variables. I, Fund. Math. 36 (1949), 207235. MR 0035832 (12:18b)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC:
32A35,
30D55
Retrieve articles in all journals
with MSC:
32A35,
30D55
Additional Information
DOI:
http://dx.doi.org/10.1090/S0002993919860840619X
PII:
S 00029939(1986)0840619X
Keywords:
Fractional integral,
maximal theorem,
Hardy space
Article copyright:
© Copyright 1986
American Mathematical Society
