On extrapolation blowups in the LP scale

Abstract

−α f p as p → r + (1 < r < ∞). The study has been motivated by current investigations of convolution maximal functions in stochastic analysis, where the problem occurs for r = 2. We also touch the problem of comparison of results in various scales of spaces.

Fundamental function for Grand Lebesgue Spaces

“…The detail investigation of this spaces (and more general spaces) see in [14], [19]. See also [5], [6], [8], [9], [10] etc.…”

“…This notion play a very important role in the functional analysis, [2], [22], [23]; in the theory of interpolation of operators, [2], [5], [7], in the theory of probability [13], [15], [16], [17]; in the theory of Partial Differential equations [7], [9]; in the theory of martingales [20]; in the theory of approximation, in the theory of random processes etc. E. Let g = g(p), p ∈ (a, b), 1 ≤ a < b ≤ ∞ be some numerical valued continuous strictly increasing (or decreasing) function.…”

Exact constant in Sobolev's and Sobolev's trace inequalities for Grand Lebesgue Spaces

“…Recently, see [16], [10], [11], [12], [13], [14], [22], [23], [24], [25], [26], [27] etc. appears the so-called Grand Lebesgue Spaces GLS = G(ψ) = G(ψ; A, B), A, B = const, A ≥ 1, A < B ≤ ∞, spaces consisting on all the measurable functions f :…”

A direct approach to the duality of grand and small Lebesgue spaces

“…Their role in Calculus of Variations (see [11]) stimulated the introduction of a more general class of spaces, the GΓ spaces (see [12]). Moreover, for a quite general class of operators they look as the appropriate spaces to be considered in the extrapolation process of families of inequalities, see [6].…”