Interpolation of Nonlinear Maps

T. Kappeler ; A. M. Savchuk ; P. Topalov ; A. A. Shkalikov

Matematičeskie zametki, Tome 96 (2014) no. 6, pp. 896-904.

Résumé

Let $(X_0, X_1)$ and $(Y_0, Y_1)$ be complex Banach couples and assume that $X_1\subseteq X_0$ with norms satisfying $\|x\|_{X_0} \le c\|x\|_{X_1}$ for some $c >\nobreak 0$. For any $0\theta 1$, denote by $X_\theta = [X_0, X_1]_\theta$ and $Y_\theta = [Y_0, Y_1]_\theta$ the complex interpolation spaces and by $B(r, X_\theta)$, $0 \le \theta \le 1$, the open ball of radius $r>0$ in $X_\theta$ centered at zero. Then, for any analytic map $\Phi\colon B(r, X_0) \to Y_0+ Y_1$ such that $\Phi\colon B(r, X_0)\to Y_0$ and $\Phi\colon B(c^{-1}r, X_1)\to Y_1$ are continuous and bounded by constants $M_0$ and $M_1$, respectively, the restriction of $\Phi$ to $B(c^{-\theta}r, X_\theta)$, $0 \theta \nobreak 1$, is shown to be a map with values in $Y_\theta$ which is analytic and bounded by $ M_0^{1-\theta} M_1^\theta$.

Détail

Mots-clés : interpolation, Hausdorff space.
Keywords: nonlinear maps, Banach couple

Bibliographie

[1] Kh. Tribel, Teoriya interpolyatsii, funktsionalnye prostranstva, differentsialnye operatory, Mir, M., 1980 | MR | Zbl

[2] I. Berg, I. Lefstrem, Interpolyatsionnye prostranstva. Vvedenie, Mir, M., 1980 | MR | Zbl

[3] J. Dieudonné, Foundations of Modern Analysis, Pure Appl. Math., 10, Academic Press, New York, 1960 | MR | Zbl

[4] J. Pöschel, E. Trubowitz, Inverse Spectral Theory, Pure Appl. Math., 130, Academic Press, New York, 1987 | MR | Zbl

[5] J. Bergh, “A nonlinear complex interpolation result”, Interpolation Spaces and Apllied Topics in Analysis, Lecture Notes in Math., 1070, Springer, Berlin, 1984, 45–47 | DOI | MR | Zbl

[6] A. Calderón, A. Zygmund, “A note on the interpolation of sublinear operations”, Amer. J. Math., 78 (1956), 282–288 | DOI | MR | Zbl

[7] E. Gagliardo, “Interpolation d'espaces de Banach et applications. III”, C. R. Acad. Sci. Paris, 248 (1959), 3517–3518 | MR | Zbl

[8] J. Lions, “Interpolation linéaire et non linéaire et regularité”, Symposia Math., 7 (1971), 443–458 | MR | Zbl

[9] L. Maligranda, “On interpolation of nonlinear operators”, Comment. Math. Prace Mat., 28:2 (1989), 253–275 | MR | Zbl

[10] L. Maligranda, A Bibliography on “Interpolation of Operators and Applications” (1926–{\allowbreak}1990), Högskolan i Luleå, Luleå, 1990

[11] L. Maligranda, L. Persson, J. Wyller, “Interpolation and partial differential equations”, J. Math. Phys., 35:9 (1994), 5035–5046 | DOI | MR | Zbl

[12] J. Peetre, “Interpolation of Lipschitz operators and metric spaces”, Mathematica (Cluj), 12 (1970), 325–334 | MR | Zbl

[13] L. Tartar, “Interpolation non linéaire et régularité”, J. Funct. Anal., 9 (1972), 469–489 | DOI | MR | Zbl

[14] T. Kappeler, P. Topalov, “On nonlinear interpolation”, Proc. Amer. Math. Soc. (to appear) , arXiv: 1306.5721

[15] A. M. Savchuk, A. A. Shkalikov, “O sobstvennykh znacheniyakh operatora Shturma–Liuvillya s potentsialami iz prostranstv Soboleva”, Matem. zametki, 80:6 (2006), 864–884 | DOI | MR | Zbl

[16] A. M. Savchuk, A. A. Shkalikov, “Ob interpolyatsii analiticheskikh otobrazhenii”, Matem. zametki, 94:4 (2013), 578–581, arXiv: 1307.0623 | DOI | Zbl

[17] M. Böhm, “Global interpolation estimates for nonlinear operators based on a global and a local estimate in the corner spaces”, C. R. Acad. Sci. Paris Sér. I Math., 318:9 (1994), 821–826 | MR | Zbl

[18] M. Cwikel, “Complex interpolation of nonlinear operators and Lebedev-Milin inequalities for Taylor coefficients of composed analytic functions”, Math. Nachr., 132 (1987), 153–161 | DOI | MR | Zbl