A sharp blow-up estimate for the Lebesgue norm

Extrapolation: Stories and Problems

“…which as we have seen is equivalent 29 So in this case there is no *gain* of logarithm, as indeed it should be, since the assumption that T : L 1 → L 1 and T : L ∞ → L ∞ is essentially stronger. We also note that once explicit inequalities are written down they can be proved by more direct methods.…”

“…These spaces have found many applications in analysis, including the study of maximal operators, PDEs, interpolation theory, etc (see [29,38] and the references therein). On the other hand, the expression (8.1) is somewhat difficult to work with.…”

Modulars from Nakano onwards

“…Moreover, we observe that one could introduce the grand Lebesgue spaces over a (Lebesgue) measurable set Ω ⊂ R n , n 1, 0 < |Ω| < ∞ (see F., Formica, Gogatishvili [53] and the more recent papers Farroni, F., Giova [50], Di Fratta, F., Slastikov [42], F., Formica [52]), as the set of the real valued, measurable functions such that…”

Grand and small norms in Lebesgue spaces

“…There are similar constructions of grand and small spaces for other classes of spaces [16]. However, even for the spaces L p) and L (p , the explicit formulas for the norms are quite complicated to handle (see, e.g., [17]). In fact, only for the set of Lorentz spaces Λ 𝛼 in [5,6] and for some cones in these spaces it was proposed an exact (with equality of norms) calculation of the norm in the space ∑ 0≤𝛽 0 <𝛼<𝛽 1 ≤1 𝜉(𝛼)Λ 𝛼 .…”

“…There are similar constructions of grand and small spaces for other classes of spaces [16]. However, even for the spaces $$$ {L}&amp;amp;#x0005E;{p\Big)} $$$ and $$$ {L}&amp;amp;#x0005E;{\Big(p} $$$, the explicit formulas for the norms are quite complicated to handle (see, e.g., [17]). In fact, only for the set of Lorentz spaces $$$ {\Lambda}&amp;amp;#x0005E;{\alpha } $$$ in [5, 6] and for some cones in these spaces it was proposed an exact (with equality of norms) calculation of the norm in the space $$$ {\sum}_{0\le {\beta}_0&amp;lt;\alpha &amp;lt;{\beta}_1\le 1}\xi \left(\alpha \right){\Lambda}&amp;amp;#x0005E;{\alpha }.$…”