zbMATH — the first resource for mathematics

Extrapolation theory: new results and applications. (English) Zbl 1081.46018
In [B. Jawerth and M. Milman, Mem. Am. Math. Soc. 440 (1991; Zbl 0733.46040)], the \(\Sigma^{(p)}\) and \(\Delta^{(p)}\) extrapolation methods were introduced and used to construct end point extrapolation spaces and to prove new extrapolation estimates. However, only the \(\Sigma^{(1)}\) and \(\Delta^{(\infty)}\) methods were studied in detail there. The aim of this paper is to give a more extensive study of the \(\Sigma^{(p)}\) and \(\Delta^{(p)}\) methods of extrapolation for \(p>0\) and to present new applications of these methods.
To illustrate results of the paper, we mention the following Yano type extrapolation theorems which are consequences of the \(\Sigma^{(p)}\) and \(\Delta^{(p)}\) methods of extrapolation.
(i) Let \(0<s\leq 1\). Put \(\| f\| _{L^{q,s}} = \{sq^{-1} \int^{\infty}_0 [t^{1/q} f^*(t)]^sdt/t\}^{1/s}\), and \(L^{q,\infty}:= (L^s,L^{\infty})_{\theta, \infty}\), \(1/q = (1-\theta)/s\). If \(T\) is a sublinear operator satisfying \[ \| Tf\| _{L^{q,\infty}} \leq c (q-s)^{-a} \| f\| _{L^{q,s}}, \quad 0 < s < q < p, \quad a > 0, \] then \[ T\: L^s (\log L)_a + L^{p,s} \rightarrow L^s + L^{p,s}. \] (ii) Let \(L^{q,\infty}:= (L^r,L^{\infty})_{\theta,\infty}\), \(1/q = (1-\theta)/r\), \(r= \min (p,s)\). If \(T\) is a sublinear operator satisfying \[ \| Tf\| _{L^{q,\infty}} \leq c q^a\| f\| _{L^{q,s}}, \quad 0 < p \leq q < \infty, \;s > 0, \;a > 0, \] then \[ T\: L^p\cap L^{\infty} \rightarrow L^r \cap L^{\infty} (\log L)_{-a}, \quad p \leq s, \] and \[ T\: L^{p,\infty} \cap L^{\infty} \rightarrow L^{p,\infty} \cap L^{\infty} (\log L)_{-a}, \quad p > s. \]
Two types of applications are given. First, the authors show that spaces currently appearing in analysis (e.g., Lorentz–Zygmund spaces, Donaldson–Sullivan spaces, logarithmic Sobolev spaces, etc.) are in fact extrapolation spaces for the \(\Sigma^{(p)}\) and \(\Delta^{(p)}\) methods. Second, the authors consider in detail certain classical operators (e.g., semigroups associated with the theory of logarithmic Sobolev inequalities) to give concrete applications of extrapolation theory to classical analysis.

46B70 Interpolation between normed linear spaces
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
Full Text: DOI
[1] D. Bakry, P.A. Meyer, Sur les inégalités de Sobolev logarithmiques I, Seminar on Probability, XVI, Lecture Notes in Mathematics, vol. 920, Springer, Berlin, New York, 1982, pp. 138-145. · Zbl 0489.60054
[2] D. Bakry, P.A. Meyer, Sur les inégalités de Sobolev logarithmiques II, Seminar on Probability, XVI, Lecture Notes in Mathematics, vol. 920, Springer, Berlin, New York, 1982, pp. 146-150. · Zbl 0489.60055
[3] Bastero, J.; Milman, M.; Ruiz, F., On the connection between weighted norm inequalities, commutators and real interpolation, Mem. amer. math. soc., 731, (2001) · Zbl 0992.46021
[4] Bennett, C.; Rudnick, K., On lorentz – zygmund spaces, Diss. math., 175, 1-72, (1980)
[5] Berg, J.; Löfström, J., Interpolation spaces, (1976), Springer Berlin
[6] Yu. Brudnyi, N. Kruglijak, Interpolation Functors and Interpolation Spaces, vol. 1, North-Holland, Amsterdam, 1991.
[7] Cao, W.; Sagher, Y., Stability of Fredholm properties on interpolation scales, Ark. mat., 28, 249-258, (1990) · Zbl 0753.46036
[8] Carro, M., New extrapolation estimates, J. funct. anal., 174, 155-166, (2000) · Zbl 0962.42014
[9] Carro, M.; Martín, J., Extrapolation theory for the real interpolation method, Collect. math., 53, 165-186, (2002) · Zbl 1034.46077
[10] Cruz-Uribe, D.; SFO; Krbec, M., Localization and extrapolation in orlicz – lorentz spaces, (), 273-284 · Zbl 1128.46302
[11] Cwikel, M.; Jawerth, B.; Milman, M., On the fundamental lemma of interpolation theory, J. approx. theory, 60, 70-82, (1990) · Zbl 0694.41041
[12] Donaldson, S.K.; Sullivan, D.P., Quasiconformal 4-manifolds, Acta math., 163, 181-252, (1989) · Zbl 0704.57008
[13] Durán, R.G., Error estimates for 3-d narrow finite elements, Math. comp., 68, 187-199, (1999) · Zbl 0910.65078
[14] Edmunds, D.E.; Triebel, H., Function spaces, entropy numbers and differential operators, (1996), Cambridge University Press Cambridge · Zbl 0629.46034
[15] Fiorenza, A.; Krbec, M., On an optimal decomposition in Zygmund spaces, Georgian math. J., 9, 271-286, (2002) · Zbl 1015.46018
[16] Iwaniec, T.; Koskela, P.; Onninen, J., Mappings of finite distortionmonotonicity and continuity, Invent. math., 144, 507-531, (2001) · Zbl 1006.30016
[17] Jawerth, B.; Milman, M., Extrapolation theory with applications, Mem. amer. math. soc., 440, (1991) · Zbl 0733.46040
[18] G.E. Karadzhov, M. Milman, Extrapolation Theory: New Results and Applications, Institute of Mathematics and Informatics, Bulgarian Academy of Science, preprint no. 1, 2002. · Zbl 1081.46018
[19] M. Krbec, H.J. Schmeisser, On Extrapolation of Sobolev and Morrey type Imbeddings, Function Spaces (Poznan, 1998), Lecture Notes in Pure and Applied Mathematics, vol. 213, Dekker, New York, 2000, pp. 297-321. · Zbl 0966.46016
[20] P.A. Meyer, Interpolation entre espaces d’Orlicz, Seminar on Probability, XVI, Lecture Notes in Mathematics, vol. 920, Springer, Berlin, New York, 1982, pp. 153-158.
[21] M. Milman, Extrapolation and Optimal Decompositions with Applications to Analysis, Lecture Notes in Mathematics, vol. 1580, Springer, New York, 1994. · Zbl 0852.46059
[22] M. Milman, A note on extrapolation theory, J. Math. Anal. Appl. 282 (2003) 26-47. · Zbl 1031.46085
[23] Neves, J., On decompositions in generalised lorentz – zygmund spaces, Boll. un. mat. ital. sez. B artic. ric. mat., 4, 8, 239-267, (2001) · Zbl 1178.46029
[24] Neves, J., Extrapolation results on general besov-Hölder-Lipschitz spaces, Math. nachr., 230, 117-141, (2001) · Zbl 1033.26004
[25] Sagher, Y., Real interpolation with weights, Indiana univ. math. J., 30, 113-121, (1981) · Zbl 0421.47023
[26] Simon, B., Schroedinger semigroups, Bull. amer. math. soc., 7, 447-526, (1982) · Zbl 0524.35002
[27] T. Sobukawa, Extrapolation theorem on Lorentz spaces, preprint. · Zbl 0846.46048
[28] Tao, T., A converse extrapolation theorem for translation-invariant operators, J. funct. anal., 180, 1-10, (2001) · Zbl 0987.47021
[29] Yano, S., An extrapolation theorem, J. math. soc. Japan, 3, 296-305, (1951) · Zbl 0045.17901
[30] Zafran, M., Spectral theory and interpolation of operators, J. funct. anal., 36, 185-204, (1980) · Zbl 0429.47002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.