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Boundedness of some sublinear operators on Herz spaces. (English) Zbl 0956.46025

From the introduction: It is well known that Beurling and Herz introduced some new spaces that characterize certain properties of functions. These new spaces are called the Herz spaces. Many studies involving these spaces can be found in the literature. One of the main reasons is that Hardy space theory associated with Herz spaces is very rich. Actually, these new Hardy spaces are a sort of local version of the ordinary Hardy spaces; the former, sometimes, are good substitutes of the latter when considering, for example, the boundedness of non-translation invariant singular integral operators. This paper is motivated by previous work of E. Hernández, S. Lu and D. Yang [Sci. China, Ser. A 38, No. 2, 147-158 (1995; Zbl 0830.42014) and Math. Nachr. 205, 69-87 (1999)], and also by more applications, such as the boundedness of bilinear operators and the regularity of solutions of the Laplacian and the wave equations on Herz-type spaces. Our main interest is to study the boundedness of some sublinear operators on these spaces under certain weak size conditions. These conditions are similar to those introduced by F. Soria and G. Weiss in [Indiana Univ. Math. J. 43, No. 1, 187-204 (1994; Zbl 0803.42004)], and are satisfied by most of the operators in harmonic analysis.

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
42B30 \(H^p\)-spaces
47A30 Norms (inequalities, more than one norm, etc.) of linear operators
46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces
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