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Distribution and rearrangement estimates of the maximal function and interpolation

100%
U. Asekritova I., Krugljak N. Y., Maligranda L., Persson L.-E.
Studia Mathematica
|
1997
|
tom 124
|
nr 2
107-132
EN
There are given necessary and sufficient conditions on a measure dμ(x)=w(x)dx under which the key estimates for the distribution and rearrangement of the maximal function due to Riesz, Wiener, Herz and Stein are valid. As a consequence, we obtain the equivalence of the Riesz and Wiener inequalities which seems to be new even for the Lebesgue measure. Our main tools are estimates of the distribution of the averaging function f** and a modified version of the Calderón-Zygmund decomposition. Analogous methods allow us to obtain K-functional formulas in terms of the maximal function for couples of weighted $L_p$-spaces.
2
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A study of some constants characterizing the weighted Hardy inequality

100%
Kufner A., Persson L.-E., Wedestig A.
Banach Center Publications
|
2004
|
tom 64
|
nr 1
135-146
EN
The modern form of Hardy's inequality means that we have a necessary and sufficient condition on the weights u and v on [0,b] so that the mapping
$H: L^{p}(0,b;v) → L^{q}(0,b;u)$
is continuous, where $Hf(x) = ∫_{0}^{x} f(t)dt$ is the Hardy operator. We consider the case 1 < p ≤ q < ∞ and then this condition is usually written in the Muckenhoupt form
(*) $A₁: = sup_{0 In this paper we discuss and compare some old and new other constants $A_{i}$ of the form (*), which also characterize Hardy's inequality. We also point out some dual forms of these characterizations, prove some new compactness results and state some open problems.
3
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Lions-Peetre reiteration formulas for triples and their applications

88%
Asekritova I., Krugljak N., Maligranda L., Nikolova L., Persson L.-E.
Studia Mathematica
|
2001
|
tom 145
|
nr 3
219-254
EN
We present, discuss and apply two reiteration theorems for triples of quasi-Banach function lattices. Some interpolation results for block-Lorentz spaces and triples of weighted $L_{p}$-spaces are proved. By using these results and a wavelet theory approach we calculate (θ,q)-spaces for triples of smooth function spaces (such as Besov spaces, Sobolev spaces, etc.). In contrast to the case of couples, for which even the scale of Besov spaces is not stable under interpolation, for triples we obtain stability in the frame of Besov spaces based on Lorentz spaces. Moreover, by using the results and ideas of this paper, we can extend the Stein-Weiss interpolation theorem known for $L_{p}(μ)$-spaces with change of measures to Lorentz spaces with change of measures. In particular, the results obtained show that for some problems in analysis the three-space real interpolation approach is really more useful than the usual real interpolation between couples.
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