Abstract
We introduce a new type of variable exponent function spaces Ḣ p(·),q(·),α(·)(\({\mathbb{R}^n}\)) and H p(·),q(·),α(·)(\({\mathbb{R}^n}\)) of Herz type, homogeneous and non-homogeneous versions, where all the three parameters are variable, and give comparison of continual and discrete approaches to their definition. Under the only assumption that the exponents p, q and α are subject to the log-decay condition at infinity, we prove that sublinear operators, satisfying the size condition known for singular integrals and bounded in L p(·)(\({\mathbb{R}^n}\)), are also bounded in the nonhomogeneous version of the introduced spaces, which includes the case maximal and Calderón-Zygmund singular operators.
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Samko, S. Variable Exponent Herz Spaces. Mediterr. J. Math. 10, 2007–2025 (2013). https://doi.org/10.1007/s00009-013-0285-x
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DOI: https://doi.org/10.1007/s00009-013-0285-x