Extremal Extrapolation Spaces
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tremal Extrapolation Spaces E. I. Berezhnoi Received January 17, 2019; in final form, May 30, 2019; accepted June 5, 2019
Abstract. We show that the extremal extrapolation spaces for an operator with any function ξ characterizing the growth of the operator norms are the sum and the intersection of spaces whose norms depend on ξ and are written out explicitly. Key words: extremal spaces, sum and intersection of spaces, extrapolation theorems. DOI: 10.1134/S0016266320010013
It is well known ([1], [2]) that many operators occurring in analysis, say, the Hilbert operator, are bounded in the Lebesgue spaces Lp for p ∈ (1, ∞) but are not bounded in L1 and L∞ . One way for determining the “extremal” space in which the operator is bounded in the limit case is extrapolation theorems. The first extrapolation theorem for estimating operators in a neighborhood of the space L∞ was given in implicit form by Titchmarsh and Zygmund (see [1; Vol. 2, Ch. XII, p. 179]), and the corresponding theorem for estimating operators in a neighborhood of the space L1 was proposed by Yano [3]. Later, a lot of papers appeared that generalized the Yano or the Titchmarsh–Zygmund theorem for Lebesgue spaces and close spaces (e.g., see [4]–[7]). We also note three large papers by Jawerth and Milman [8], Milman [9], and Karadzhov and Milman [10], which analyze the extrapolation theorem from the viewpoint of general ideas of interpolation theory of linear operators (K- and J-methods of real interpolation). In the present paper, we write out exact extremal extrapolation spaces for the extrapolation problem arising from the Yano theorem and the dual problem arising from the Titchmarsh–Zygmund theorem. It turns out that the extremal spaces in extrapolation theorems are the sum and intersection of spaces. These theorems permit one to take a fresh look at the problems of calculating sums and intersections for specific spaces, for example, the grand Lebesgue spaces Lp) and the small Lebesgue spaces L(p [11], which have found wide applications in partial differential equations and in the study of maximal and other typical operators in harmonic analysis [12]. Similar constructions have appeared for other sets of spaces [13]. Unfortunately, there does not exist an explicit description in conventional terms even for the spaces Lp) and L(p , which, in the light of the present paper, are simply extrapolation spaces for a given operator growth function. In fact, it is only for the set of Lorentz spaces Λα that an exact (with the equality of norms) calculation was proposed in [14] and [15] for the space 0β0
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