Abstract
A generalization of the theory of Y. Brudnyi [7], and A. and Y. Brudnyi [5, 6], is presented. Our construction connects Brudnyi’s theory, which relies on local polynomial approximation, with new results on sparse domination. In particular, we find an analogue of the maximal theorem for the fractional maximal function, solving a problem proposed by Kruglyak–Kuznetsov. Our spaces shed light on the structure of the John–Nirenberg spaces. We show that SJN p (sparse John–Nirenberg space) coincides with L p ,1<p<∞. This characterization yields the John–Nirenberg inequality by extrapolation and is useful in the theory of commutators.
Highlights
The function spaces we use in Analysis can be described and characterized in different qualitative and quantitative ways
We find an analogue of the maximal theorem for the fractional maximal function, solving a problem proposed by Kruglyak–Kuznetsov
Yuri and Alexander Brudnyi have proposed the concept of best local polynomial approximation as a unifying characteristic to understand the structure of classical function spaces as diverse as, BMO, John–Nirenberg spaces J Np, Sobolev spaces, Besov spaces, Morrey spaces, Jordan–Wiener spaces, etc
Summary
The function spaces we use in Analysis can be described and characterized in different qualitative and quantitative ways (e.g. by duality, as being part of an interpolation scale, through the boundedness of suitable functionals, by the rate of approximation of their elements with respect to a fixed class of approximants, etc.). Yuri and Alexander Brudnyi (cf [5,6,7]) have proposed the concept of best local polynomial approximation as a unifying characteristic to understand the structure of classical function spaces as diverse as, BMO, John–Nirenberg spaces J Np , Sobolev spaces, Besov spaces, Morrey spaces, Jordan–Wiener spaces, etc. Their massive theory can be seen as a complement of the theories of function spaces that have evolved through the work of many authors, including names such as Coifman–Meyer, Frazier–Jawerth, Peetre, Triebel (cf [15,28,32] and the references therein), where the underlying unifying themes and tools are wavelet approximations, representation theorems, maximal inequalities, interpolation, etc.
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