# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Some new classes of Hardy spaces. (English) Zbl 0677.30030
The authors develop a Hardy space theory for certain function spaces, among them the spaces $$B\sp p=\{f\in L\sp 1\sb{loc}({\bbfR}):\quad \Vert f\Vert =\sup\sb{T\ge 1}[(2T)\sp{-1}\int\sp{T}\sb{-T}\vert f(t)\vert\sp p dt]\sp{1/p}<\infty \},$$ 1$<p<\infty$ (formerly considered by {\it A. Beurling} [Ann. Inst. Fourier 14, No.2, 1-32 (1964; Zbl 0133.075)]), and the harmonic extension of their elements to the upper half-plane. Their results include a Burkholder-Gundy-Silverstein maximal function characterization of spaces related to the spaces $B\sp p$ above. Also considered are duality relations; for example, an analogue to the Fefferman-Stein theorem [{\it C. Fefferman} and {\it E. M. Stein}, Acta Math. 129, 137-193 (1972; Zbl 0257.46078)] on the duality between the classical Hardy space $H\sp 1$ and BMO is proved.
Reviewer: R.Mortini

##### MSC:
 30H05 Bounded analytic functions 30D55 H (sup p)-classes (MSC2000) 46J15 Banach algebras of differentiable or analytic functions, $H^p$-spaces
##### Keywords:
generalized Hardy spaces
Full Text:
##### References:
 [1] Bass, J.: Fonctions de correlation, fonctions pseudo-aleatoires et applications. (1984) · Zbl 0557.76053 [2] Beurling, A.: Construction and analysis of some convolution algebra. Ann. inst. Fourier (Grenoble) 14, 1-32 (1964) · Zbl 0133.07501 [3] Besicovitch, A.: Almost periodic functions. (1954) · Zbl 0065.07102 [4] Bertrandias, J.: Operateurs subordinatifs sur des fonctions bornees en moyenne quadratique. J. math. Pures appl. (9) 52, 27-63 (1973) · Zbl 0266.46020 [5] Burkholder, D.; Gundy, R.; Silverstein, M.: A maximal function characterization of the class hp. Trans. amer. Math. soc. 157, 27-53 (1971) · Zbl 0223.30048 [6] Calderón, A.: An atomic decomposition of distributions in parabolic hp spaces. Adv. in math. 25, 216-225 (1977) [7] Y. Chen and K. Lau, Wiener transformation on functions with bounded averages, Proc. Amer. Math. Soc. (to appear). · Zbl 0683.42015 [8] Chen, Y.; Lau, K.: On an equivalent class of norms for BMO. J. aust. Math. soc. A 46, 289-295 (1989) · Zbl 0675.42018 [9] Coifman, R.; Weiss, G.: Extensions of Hardy spaces and their use in analysis. Bull. amer. Math. soc. (N. S.) 83, 569-645 (1977) · Zbl 0358.30023 [10] Fefferman, C.; Stein, E.: Some maximal inequalities. Amer. J. Math. 93, 107-115 (1971) · Zbl 0222.26019 [11] Fefferman, C.; Stein, E.: Hp spaces of several variables. Acta math. 129, 137-193 (1972) · Zbl 0257.46078 [12] Feichtinger, H.: Sympos. math.. An elementrary approach to Wiener’s third Tauberian theorem on the Euclidean n-space, proc. Conf. catona (1987) [13] Garnett, J.: Bounded analytic functions. (1981) · Zbl 0469.30024 [14] Lau, K.: On the Banach spaces of functions with bounded upper means. Pacific J. Math. 91, 153-173 (1980) · Zbl 0492.46027 [15] Lau, K.: Extensions of Wiener’s Tauberian identity and multipliers on the Marcinkiewicz space. Trans. amer. Math. soc. 277, 489-506 (1983) · Zbl 0521.42011 [16] Lau, K.; Lee, J.: On generalized harmonic analysis. Trans. amer. Math. soc. 259, 75-97 (1980) · Zbl 0441.42007 [17] Torchinsky, A.: Real variable methods in harmonic analysis. (1986) · Zbl 0621.42001 [18] Wiener, N.: Generalized harmonic analysis. Acta math. 55, 117-258 (1930) · Zbl 56.0954.02 [19] Wiener, N.: Extrapolation, interpolation, and smoothing of stationary time series, with engineering applications. (1949) · Zbl 0036.09705 [20] Wilson, J.: On the atomic decomposition for Hardy spaces. Pacific J. Math. 116, 201-207 (1985) · Zbl 0563.42012