| 1. |
- Kruglyak, Natan, et al.
(författare)
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Sharp estimates for the identity minus Hardy operator on the cone of decreasing functions
- 2008
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Ingår i: Proceedings of the American Mathematical Society. - : American Mathematical Society (AMS). - 0002-9939 .- 1088-6826. ; 136:7, s. 2505-2513
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Tidskriftsartikel (refereegranskat)abstract
- It is shown that if we restrict the identity minus Hardy operator on the cone of nonnegative decreasing functions f in L-p, then we have the sharp estimate parallel to(I - H)f parallel to(Lp) <= 1/(p - 1)(1/p) parallel to f parallel to(Lp) for p = 2, 3, 4, .... In other words,parallel to f** - f*parallel to(Lp) <= 1/(p - 1)(1/p) parallel to f parallel to(Lp) for each f is an element of L-p and each integer p >= 2.It is also shown, via a connection between the operator I - H and Laguerre functions, thatparallel to(1 - alpha)I + Phi(I - H)parallel to(L2 -> L2) = parallel to I - alpha H parallel to(L2 -> L2) = 1 for all a is an element of [ 0, 1].
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| 2. |
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| 3. |
- Kruglyak, Natan, et al.
(författare)
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The limiting case of the Marcinkiewicz integral : growth for convex sets
- 2007
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Ingår i: Proceedings of the American Mathematical Society. - : American Mathematical Society (AMS). - 0002-9939 .- 1088-6826. ; 135:10, s. 3283-3293
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Tidskriftsartikel (refereegranskat)abstract
- The Marcinkiewicz integral Iλ (x)= ∫ (dist (y, ℝn\Ω))λ/Ω| x - y | n+λ dy, where λ > 0, plays a well-known and prominent role in harmonic analysis. In this paper, we estimate the growth of it in the limiting case λ → 0. Throughout, we assume that Ω is convex; it is interesting that this condition cannot be dropped
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| 4. |
- 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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| 7. |
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| 8. |
- Asekritova, Irina, et al.
(författare)
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Interpolation of Besov spaces in the nondiagonal case
- 2006
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Ingår i: St. Petersburg Mathematical Journal. - 1061-0022 .- 1547-7371. ; 18:4, s. 1-9
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Tidskriftsartikel (refereegranskat)abstract
- In the nondiagonal case, interpolation spaces for a collection of Besov spaces are described. The results are consequences of the fact that, whenever the convex hull of points (formula presented) includes a ball of ℝm+1, we have (formula presented).
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| 9. |
- Asekritova, Irina, et al.
(författare)
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Invertibility of operators in spaces of real interpolation
- 2008
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Ingår i: Revista Matematica Complutense. - 1139-1138. ; 21:1, s. 207-217
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Tidskriftsartikel (refereegranskat)abstract
- Let A be a linear bounded operator from a couple X→ = (X0, X1) to a couple Y→ = (Y0, Y1) such that the restrictions of A on the spaces X0 and X1 have bounded inverses. This condition does not imply that the restriction of A on the real interpolation space (X0, X 1)θ, q has a bounded inverse for all values of the parameters θ and q. In this paper under some conditions on the kernel of A we describe all spaces (X0, X1)θ,q such that the operator A : (x0,X1)θ,q → (Y0, Y1) has a bounded inverse.
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| 10. |
- 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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