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- Asekritova, Irina, et al.
(författare)
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Distribution and Rearranement Estimates of the Maximal Functions and Interpolation
- 1997
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Ingår i: Studia Mathematica. - 0039-3223 .- 1730-6337. ; 124:2, s. 107-132
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 3. |
- Asekritova, Irina, et al.
(författare)
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Lions-Peetre Reiteration Formulas for Triples and Their Application
- 2001
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Ingår i: Studia Mathematica. - : Institute of Mathematics, Polish Academy of Sciences. - 0039-3223 .- 1730-6337. ; 145:3, s. 219-254
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Tidskriftsartikel (refereegranskat)abstract
- 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 Lp-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 Lp(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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| 6. |
- Kruglyak, Natan, et al.
(författare)
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On an elementary approach to the fractional Hardy inequality
- 2000
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Ingår i: Proceedings of the American Mathematical Society. - 0002-9939 .- 1088-6826. ; 128:3, s. 727-734
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Tidskriftsartikel (refereegranskat)abstract
- Let be the usual Hardy operator, i.e., . We prove that the operator is bounded and has a bounded inverse on the weighted spaces for -1$" type="#_x005F_x0000_t75">and . Moreover, by using these inequalities we derive a somewhat generalized form of some well-known fractional Hardy type inequalities and also of a result due to Bennett-DeVore-Sharpley, where the usual Lorentz norm is replaced by an equivalent expression. Examples show that the restrictions in the theorems are essential.
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| 7. |
- Kruglyak, Natan, et al.
(författare)
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Structure of the Hardy operator related to Laguerre polynomials and Euler differential equation
- 2006
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Ingår i: Revista Matematica Complutense. - 1139-1138. ; 19:2, s. 467-476
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Tidskriftsartikel (refereegranskat)abstract
- We present a direct proof of a known result that the Hardy operator Hf (x) =1xR x0 f (t) dt in the space L2 = L2(0, ∞) can be written as H = I − U , whereU is a shift operator (U en = en+1, n ∈ Z) for some orthonormal basis {en}.The basis {en} is constructed by using classical Laguerre polynomials. Wealso explain connections with the Euler differential equation of the first ordery′ − 1x y = g and point out some generalizations to the case with weighted L2w (a, b)spaces.
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| 8. |
- Kruglyak, Natan, et al.
(författare)
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The failure of the Hardy inequality and interpolation of intersections
- 1999
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Ingår i: Arkiv för matematik. - 0004-2080 .- 1871-2487. ; 37:2, s. 323-244
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Tidskriftsartikel (refereegranskat)abstract
- The main idea of this paper is to clarify why it is sometimes incorrect to interpolate inequalities in a “formal” way. For this we consider two Hardy type inequalities, which are true for each parameter α≠0 but which fail for the “critical” point α=0. This means that we cannot interpolate these inequalities between the noncritical points α=1 and α=−1 and conclude that it is also true at the critical point α=0. Why? An accurate analysis shows that this problem is connected with the investigation of the interpolation of intersections (N∩L p(w0), N∩Lp(w1)), whereN is the linear space which consists of all functions with the integral equal to 0. We calculate theK-functional for the couple (N∩L p(w0),N∩L p (w1)), which turns out to be essentially different from theK-functional for (L p(w0), Lp(w1)), even for the case whenN∩L p(wi) is dense inL p(wi) (i=0,1). This essential difference is the reason why the “naive” interpolation above gives an incorrect result.
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