A discrete framework for the interpolation of Banach spaces. (English) Zbl 07811947
The authors develop a discrete framework for interpolation of Banach spaces which contain most of the well-known methods of interpolation such as the real interpolation method introduced by J. L. Lions and J. Peetre [Publ. Math., Inst. Hautes Étud. Sci. 19, 5–68 (1964; Zbl 0148.11403)] and complex interpolation method introduced by A. P. Calderón [Stud. Math. 24, 113–190 (1964; Zbl 0204.13703)]. Also some others such as the Rademacher interpolation method [N. Kalton et al., Math. Ann. 336, No. 4, 747–801 (2006; Zbl 1111.47020)], \(\gamma\)-interpolation method [J. Suárez and L. Weis, Lond. Math. Soc. Lect. Note Ser. 337, 293–306 (2006; Zbl 1127.46014)] and \(\ell^q\)-interpolation method [P. C. Kunstmann, Ill. J. Math. 59, No. 1, 1–19 (2015; Zbl 1346.46018)].
The starting point is the discrete mean method of real interpolation of Lions and Peetre. Recall that given a compatible pair of Banach spaces \((X_0,X_1)\) and a parameter \(0<\theta<1\) the real interpolation space \((X_0,X_1)_{\theta,p_0,p_1}\) is the space of elements \(x\in X_0+X_1\) such that \[\|x\|_{(X_0,X_1)_{\theta,p_0,p_1}}= \inf \max_{j=0,1}\|(e^{k(j-\theta)}x_k)_{k\in \mathbb Z}\|_{\ell^{p_j}(\mathbb Z, X_j)}<\infty\] where the infimum is taken over all sequences \((x_k)\in X_0\cap X_1\) such that \(\sum_{k\in \mathbb Z} x_k=x\). In the paper, the authors replace the spaces \(\ell^{p_j}(\mathbb Z, X_j)\) by sequence structures on the spaces \(X_j\). A sequence structure \(\mathfrak G\) on a space \(X\) is a translation invariant Banach space of \(X\)-valued sequences such that \(\ell^1(\mathbb Z, X)\subset \mathfrak G\subset \ell^\infty(\mathbb Z, X)\) contractively. Now given \(0<\theta<1\) and sequence structures \(\mathfrak G_j\) on \(X_j\) for \(j=0,1\), writing \(\mathcal X_j=[X_j,\mathfrak G_j]\), the authors introduce the norm \[\|x\|_{(\mathcal X_0,\mathcal X_1)_\theta}= \inf \max_{j=0,1}\|(e^{k(j-\theta)}x_k)_{k\in \mathbb Z}\|_{\mathfrak G_j}<\infty\] where the infimum is taken over all sequences \((x_k)\in X_0\cap X_1\) such that \(\sum_{k\in \mathbb Z} x_k=x\). For specific choices of \(\mathfrak G_j\) the procedure includes many known interpolation methods: the real interpolation method (using \(\ell^{p_j }(\mathbb Z, X_j)\)), the lower and upper complex interpolation method (using the space of Fourier coefficients of functions in \(C(\mathbb T,X_j)\) and measures in \(\Lambda^\infty(\mathbb T,X_j)\) respectively), the Rademacher and \(\gamma\)-interpolation methods (using certain random sequence spaces), the \(\ell^q\)-interpolation method (using \(X_j(\ell^q(\mathbb Z))\)) and others. This approach enables the authors to extend some previously known results for either real or complex interpolation methods to all interpolation methods that fit in their framework. As applications, they prove some interpolation results for analytic operator families or for intersections of spaces.
The starting point is the discrete mean method of real interpolation of Lions and Peetre. Recall that given a compatible pair of Banach spaces \((X_0,X_1)\) and a parameter \(0<\theta<1\) the real interpolation space \((X_0,X_1)_{\theta,p_0,p_1}\) is the space of elements \(x\in X_0+X_1\) such that \[\|x\|_{(X_0,X_1)_{\theta,p_0,p_1}}= \inf \max_{j=0,1}\|(e^{k(j-\theta)}x_k)_{k\in \mathbb Z}\|_{\ell^{p_j}(\mathbb Z, X_j)}<\infty\] where the infimum is taken over all sequences \((x_k)\in X_0\cap X_1\) such that \(\sum_{k\in \mathbb Z} x_k=x\). In the paper, the authors replace the spaces \(\ell^{p_j}(\mathbb Z, X_j)\) by sequence structures on the spaces \(X_j\). A sequence structure \(\mathfrak G\) on a space \(X\) is a translation invariant Banach space of \(X\)-valued sequences such that \(\ell^1(\mathbb Z, X)\subset \mathfrak G\subset \ell^\infty(\mathbb Z, X)\) contractively. Now given \(0<\theta<1\) and sequence structures \(\mathfrak G_j\) on \(X_j\) for \(j=0,1\), writing \(\mathcal X_j=[X_j,\mathfrak G_j]\), the authors introduce the norm \[\|x\|_{(\mathcal X_0,\mathcal X_1)_\theta}= \inf \max_{j=0,1}\|(e^{k(j-\theta)}x_k)_{k\in \mathbb Z}\|_{\mathfrak G_j}<\infty\] where the infimum is taken over all sequences \((x_k)\in X_0\cap X_1\) such that \(\sum_{k\in \mathbb Z} x_k=x\). For specific choices of \(\mathfrak G_j\) the procedure includes many known interpolation methods: the real interpolation method (using \(\ell^{p_j }(\mathbb Z, X_j)\)), the lower and upper complex interpolation method (using the space of Fourier coefficients of functions in \(C(\mathbb T,X_j)\) and measures in \(\Lambda^\infty(\mathbb T,X_j)\) respectively), the Rademacher and \(\gamma\)-interpolation methods (using certain random sequence spaces), the \(\ell^q\)-interpolation method (using \(X_j(\ell^q(\mathbb Z))\)) and others. This approach enables the authors to extend some previously known results for either real or complex interpolation methods to all interpolation methods that fit in their framework. As applications, they prove some interpolation results for analytic operator families or for intersections of spaces.
Reviewer: Oscar Blasco (València)
MSC:
| 46B70 | Interpolation between normed linear spaces |
| 46M35 | Abstract interpolation of topological vector spaces |
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