Nuclear embeddings in general vector-valued sequence spaces with an application to Sobolev embeddings of function spaces on quasi-bounded domains. (English) Zbl 1493.46055
The authors study nuclear operators on Besov and Triebel-Lizorkin spaces on quasi-bounded domains. Nuclear operators have recently been characterized for these spaces on bounded domains. The authors prove two results on nuclearity, one for embeddings of Besov and Triebel-Lizorkin spaces on unbounded domains and one for sequence spaces.
If \(\Omega\) is a bounded Lipschitz domain in \(\mathbb R^d\), the identity map \(i:B^{s_1}_{p_1, q_1} (\Omega) \mapsto B^{s_2}_{p_2, q_2}(\Omega)\) is nuclear if and only if \[ s_1 - s_2 > d - d \max \left( \frac{1}{p_2} - \frac{1}{p_1} , 0 \right), \] where \(1 \leq p_i\), \(q_i \leq \infty\), \(s_i \in \mathbb R\), \(i = 1,2\).
The authors wish to consider whether this might hold for unbounded domains, perhaps with nuclear replaced by compactness of the embedding. For unbounded domains, they show that if \(\Omega\) is unbounded and \(b(\Omega) = \infty\), then the embedding is nuclear if and only if \(p_1 = 1\), \(p_2 = \infty\), \(s_1 - s_2 > d\). If \(b(\Omega) < \infty\), the results look more like the bounded case. The embedding is nuclear if \[ s_1 - s_2 - d \left(\frac{1}{p_1} - \frac{1}{p_2} \right) > \frac{ b(\Omega) }{\mathbf{t}(p_1, p_2) } \] where \(1 \leq p_i, q_i \leq \infty\), \(s_i \in \mathbb R\), \(i = 1,2\), \(s_1 > s_2\), and \[\frac{1} {\mathbf{t}(p_1, p_2)} = 1 - \max \left(\frac{1}{p_1} - \frac{1}{p_2}, 0 \right) = 1 - \left(\frac{1}{p_1} - \frac{1}{p_2}\right)_+. \]
\(b(\Omega)\) has a complicated technical definition, with \(b_j(\Omega)\) being the largest number of dyadic cubes of side length \(2^{-j}\) that are in \(\Omega\), and then \(b(\Omega) =\sup \{ t \in \mathbb R_+ : \limsup_{j \to \infty} b_j(\Omega) 2^{-jt} = \infty \}\). Note with the authors that for any non-empty open set \(\Omega \subset \mathbb R^d\), we have \(d \leq b(\Omega) \leq \infty\). If \(\Omega\) is unbounded and not quasi-bounded, then \(b(\Omega) = \infty\). But there are also quasi-bounded domains such that \(b(\Omega) = \infty\). Moreover, if the measure \(|\Omega |\) is finite, then \(b(\Omega) =d\) and one can check that when \(\Omega\) is bounded, the above result agrees with the first formula.
The proof is based on a compactness result of A. Tong [Trans. Am. Math. Soc. 143, 235–247 (1969; Zbl 0186.45602)] for embeddings of \(\ell_p\), extended by H.-G. Leopold [Georgian Math. J. 7, No. 4, 731–743 (2000; Zbl 0976.46025); in: Function spaces, differential operators and nonlinear analysis. Proceedings of the conference, FSDONA-99, Syöte, Finland, June 10–16, 1999. Prague: Mathematical Institute of the Academy of Sciences of the Czech Republic. 170–186 (2000; Zbl 0964.46011)] to the sequence space \(\ell_q ( \beta_j \ell_p^{M_j} )\). Leopold showed that if, with \(\beta_j >0\), \(M_j \in \mathbb N_0\), \(j \in \mathbb N_0\), \(M_j \geq 1\), \[ \ell_q ( \beta_j \ell_p^{M_j} )=\{ x = (x_{j,k})_{n \in \mathbb N_0, k = 1, \ldots, M_j}; x_{j,k} \in \mathbb C, \| x | \ell_q ( \beta_j \ell_p^{M_j} ) \| <\infty\} \] where \[ \| x | \ell_q ( \beta_j \ell_p^{M_j} )\| = \left( \sum_{j = 0}^{\infty} \beta_j^q \left( \sum_{k = 1}^{M_j} |x_{j,k}|^p \right)^{\frac{q}{p}} \right)^{\frac{1}{q}} , \] then the embedding of \(\ell_{p_1} ( \beta_j \ell_{p_2}^{M_j} )\) into \(\ell_{q_1} ( \ell_{q_2}^{M_j} )\) is compact if and only if \(( \beta_j^{-1} M_j^{\frac{1}{p^*}})_{j \in \mathbb N_0} \in \ell_{q^*}\), with \(\frac{1}{p^*} = \left(\frac{1}{p_2} - \frac{1}{p_1} \right)_+\), \(\frac{1}{q^*} = \left(\frac{1}{q_2} - \frac{1}{q_1} \right)_+\), modified if \(q^* = \infty\). Their result is based on their proof that the embedding of \(\ell_{p_1} ( \beta_j \ell_{p_2}^{M_j} )\) into \(\ell_{q_1} ( \ell_{q_2}^{M_j} )\) is nuclear if and only if \(( \beta_j^{-1} M_j^{\frac{1}{\mathbf{t}(p_1, p_2)}})_{j \in \mathbb N_0} \in \ell_{\mathbf{t}(q_1, q_2)}\), with \(\mathbf{t}(p,q)\) defined as above and modified if \(\mathbf{t}(q_1,q_2)= \infty\).
If \(\Omega\) is a bounded Lipschitz domain in \(\mathbb R^d\), the identity map \(i:B^{s_1}_{p_1, q_1} (\Omega) \mapsto B^{s_2}_{p_2, q_2}(\Omega)\) is nuclear if and only if \[ s_1 - s_2 > d - d \max \left( \frac{1}{p_2} - \frac{1}{p_1} , 0 \right), \] where \(1 \leq p_i\), \(q_i \leq \infty\), \(s_i \in \mathbb R\), \(i = 1,2\).
The authors wish to consider whether this might hold for unbounded domains, perhaps with nuclear replaced by compactness of the embedding. For unbounded domains, they show that if \(\Omega\) is unbounded and \(b(\Omega) = \infty\), then the embedding is nuclear if and only if \(p_1 = 1\), \(p_2 = \infty\), \(s_1 - s_2 > d\). If \(b(\Omega) < \infty\), the results look more like the bounded case. The embedding is nuclear if \[ s_1 - s_2 - d \left(\frac{1}{p_1} - \frac{1}{p_2} \right) > \frac{ b(\Omega) }{\mathbf{t}(p_1, p_2) } \] where \(1 \leq p_i, q_i \leq \infty\), \(s_i \in \mathbb R\), \(i = 1,2\), \(s_1 > s_2\), and \[\frac{1} {\mathbf{t}(p_1, p_2)} = 1 - \max \left(\frac{1}{p_1} - \frac{1}{p_2}, 0 \right) = 1 - \left(\frac{1}{p_1} - \frac{1}{p_2}\right)_+. \]
\(b(\Omega)\) has a complicated technical definition, with \(b_j(\Omega)\) being the largest number of dyadic cubes of side length \(2^{-j}\) that are in \(\Omega\), and then \(b(\Omega) =\sup \{ t \in \mathbb R_+ : \limsup_{j \to \infty} b_j(\Omega) 2^{-jt} = \infty \}\). Note with the authors that for any non-empty open set \(\Omega \subset \mathbb R^d\), we have \(d \leq b(\Omega) \leq \infty\). If \(\Omega\) is unbounded and not quasi-bounded, then \(b(\Omega) = \infty\). But there are also quasi-bounded domains such that \(b(\Omega) = \infty\). Moreover, if the measure \(|\Omega |\) is finite, then \(b(\Omega) =d\) and one can check that when \(\Omega\) is bounded, the above result agrees with the first formula.
The proof is based on a compactness result of A. Tong [Trans. Am. Math. Soc. 143, 235–247 (1969; Zbl 0186.45602)] for embeddings of \(\ell_p\), extended by H.-G. Leopold [Georgian Math. J. 7, No. 4, 731–743 (2000; Zbl 0976.46025); in: Function spaces, differential operators and nonlinear analysis. Proceedings of the conference, FSDONA-99, Syöte, Finland, June 10–16, 1999. Prague: Mathematical Institute of the Academy of Sciences of the Czech Republic. 170–186 (2000; Zbl 0964.46011)] to the sequence space \(\ell_q ( \beta_j \ell_p^{M_j} )\). Leopold showed that if, with \(\beta_j >0\), \(M_j \in \mathbb N_0\), \(j \in \mathbb N_0\), \(M_j \geq 1\), \[ \ell_q ( \beta_j \ell_p^{M_j} )=\{ x = (x_{j,k})_{n \in \mathbb N_0, k = 1, \ldots, M_j}; x_{j,k} \in \mathbb C, \| x | \ell_q ( \beta_j \ell_p^{M_j} ) \| <\infty\} \] where \[ \| x | \ell_q ( \beta_j \ell_p^{M_j} )\| = \left( \sum_{j = 0}^{\infty} \beta_j^q \left( \sum_{k = 1}^{M_j} |x_{j,k}|^p \right)^{\frac{q}{p}} \right)^{\frac{1}{q}} , \] then the embedding of \(\ell_{p_1} ( \beta_j \ell_{p_2}^{M_j} )\) into \(\ell_{q_1} ( \ell_{q_2}^{M_j} )\) is compact if and only if \(( \beta_j^{-1} M_j^{\frac{1}{p^*}})_{j \in \mathbb N_0} \in \ell_{q^*}\), with \(\frac{1}{p^*} = \left(\frac{1}{p_2} - \frac{1}{p_1} \right)_+\), \(\frac{1}{q^*} = \left(\frac{1}{q_2} - \frac{1}{q_1} \right)_+\), modified if \(q^* = \infty\). Their result is based on their proof that the embedding of \(\ell_{p_1} ( \beta_j \ell_{p_2}^{M_j} )\) into \(\ell_{q_1} ( \ell_{q_2}^{M_j} )\) is nuclear if and only if \(( \beta_j^{-1} M_j^{\frac{1}{\mathbf{t}(p_1, p_2)}})_{j \in \mathbb N_0} \in \ell_{\mathbf{t}(q_1, q_2)}\), with \(\mathbf{t}(p,q)\) defined as above and modified if \(\mathbf{t}(q_1,q_2)= \infty\).
Reviewer: Raymond Johnson (Columbia)
MSC:
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
| 47B10 | Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.) |
Keywords:
nuclear embeddings; function spaces on quasi-bounded domains; vector-valued sequence spacesReferences:
| [1] | Adams, R. A.; Fournier, J. J.F., Sobolev Spaces, Pure and Applied Mathematics, vol. 140 (2003), Elsevier, Academic Press: Elsevier, Academic Press New York · Zbl 1098.46001 |
| [2] | Bhatia, R., Pinching, trimming, truncating, and averaging of matrices, Am. Math. Mon., 107, 7, 602-608 (2000) · Zbl 0984.15024 |
| [3] | Blasco, O.; Zaragoza-Berzosa, C., Multipliers on generalized mixed norm sequence spaces, Abstr. Appl. Anal., Article 983273 pp. (2014) · Zbl 1474.42028 |
| [4] | Cobos, F.; Domínguez, Ó.; Kühn, Th., On nuclearity of embeddings between Besov spaces, J. Approx. Theory, 225, 209-223 (2018) · Zbl 1386.46034 |
| [5] | Cobos, F.; Edmunds, D. E.; Kühn, Th., Nuclear embeddings of Besov spaces into Zygmund spaces, J. Fourier Anal. Appl., 26, 1, Article 9 pp. (2020) · Zbl 1445.46035 |
| [6] | Dota, A.; Zhang, S., s-numbers of compact embeddings of function spaces on quasi-bounded domains, J. Complex., 30, 495-513 (2014) · Zbl 1298.47030 |
| [7] | Edmunds, D. E.; Lang, J., Non-nuclearity of a Sobolev embedding on an interval, J. Approx. Theory, 178, 22-29 (2014) · Zbl 1301.46012 |
| [8] | Edmunds, D. E.; Gurka, P.; Lang, J., Nuclearity and non-nuclearity of some Sobolev embeddings on domains, J. Approx. Theory, 211, 94-103 (2016) · Zbl 1361.46028 |
| [9] | Enflo, P., A counterexample to the approximation problem in Banach spaces, Acta Math., 130, 309-317 (1973) · Zbl 0267.46012 |
| [10] | Grothendieck, A., Produits tensoriels topologiques et espaces nucléaires, Mem. Am. Math. Soc., 16, 140 (1955) · Zbl 0064.35501 |
| [11] | Haroske, D. D.; Skrzypczak, L., Nuclear embeddings in weighted function spaces, Integral Equ. Oper. Theory, 92, 37 (2020) · Zbl 1480.46040 |
| [12] | Jameson, G. J.O., Summing and Nuclear Norms in Banach Space Theory, London Mathematical Society Student Texts, vol. 8 (1987), Cambridge University Press: Cambridge University Press Cambridge, ISBN: 0-521-34134-5, 0-521-34937-0-521-0 · Zbl 0634.46007 |
| [13] | Leopold, H.-G., Embeddings and entropy numbers for general weighted sequence spaces: the non-limiting case, Georgian Math. J., 7, 4, 731-743 (2000) · Zbl 0976.46025 |
| [14] | Leopold, H.-G., Embeddings for general weighted sequence spaces and entropy numbers, (Mustonen, V.; Rákosník, J., Function Spaces, Differential Operators and Nonlinear Analysis. Function Spaces, Differential Operators and Nonlinear Analysis, June 1999. Function Spaces, Differential Operators and Nonlinear Analysis. Function Spaces, Differential Operators and Nonlinear Analysis, June 1999, Proceedings of the Conference Held in Syöte (2000), Math. Inst. Acad. Sci.: Math. Inst. Acad. Sci. Czech Republic), 170-186 · Zbl 0964.46011 |
| [15] | Leopold, H.-G.; Skrzypczak, L., Compactness of embeddings of function spaces on quasi-bounded domains and the distribution of eigenvalues of related elliptic operators, Proc. Edinb. Math. Soc., 56, 3, 829-851 (2013) · Zbl 1291.46032 |
| [16] | Leopold, H.-G.; Skrzypczak, L., Compactness of embeddings of function spaces on quasi-bounded domains and the distribution of eigenvalues of related elliptic operators. Part II, J. Math. Anal. Appl., 429, 1, 439-460 (2015) · Zbl 1330.46035 |
| [17] | Parfenov, O. G., Nuclearity of embedding operators from Sobolev classes into weighted spaces, Zap. Nauč. Semin. POMI. Zap. Nauč. Semin. POMI, J. Math. Sci. (N.Y.), 101, 3, 3139-3145 (2000), 301-302, Russian; Engl. translation: |
| [18] | Pietsch, A., r-Nukleare Sobolevsche Einbettungsoperatoren, (Elliptische Differentialgleichungen, Band II. Elliptische Differentialgleichungen, Band II, Schriftenreihe Inst. Math. Deutsch. Akad. Wissensch. Berlin, Reihe A, vol. 8 (1971), Akademie-Verlag: Akademie-Verlag Berlin), 203-215 · Zbl 0219.46030 |
| [19] | Pietsch, A., Operator Ideals, North-Holland Mathematical Library, vol. 20 (1980), North-Holland: North-Holland Amsterdam · Zbl 0399.47039 |
| [20] | Pietsch, A., Grothendieck’s concept of a p-nuclear operator, Integral Equ. Oper. Theory, 7, 2, 282-284 (1984) · Zbl 0573.47014 |
| [21] | Pietsch, A., Eigenvalues and s-Numbers (1987), Akad. Verlagsgesellschaft Geest & Portig: Akad. Verlagsgesellschaft Geest & Portig Leipzig · Zbl 0615.47019 |
| [22] | Pietsch, A., History of Banach Spaces and Linear Operators (2007), Birkhäuser Boston Inc.: Birkhäuser Boston Inc. Boston, MA · Zbl 1121.46002 |
| [23] | Pietsch, A.; Triebel, H., Interpolationstheorie für Banachideale von beschränkten linearen Operatoren, Stud. Math., 31, 95-109 (1968) · Zbl 0182.17002 |
| [24] | Tong, A., Diagonal nuclear operators on \(l_p\) spaces, Trans. Am. Math. Soc., 143, 235-247 (1969) · Zbl 0186.45602 |
| [25] | Triebel, H., Theory of Function Spaces (1983), Birkhäuser: Birkhäuser Basel · Zbl 0546.46028 |
| [26] | Triebel, H., Theory of Function Spaces II (1992), Birkhäuser: Birkhäuser Basel · Zbl 0763.46025 |
| [27] | Triebel, H., Fractals and Spectra (1997), Birkhäuser: Birkhäuser Basel · Zbl 0898.46030 |
| [28] | Triebel, H., Theory of Function Spaces III (2006), Birkhäuser: Birkhäuser Basel · Zbl 1104.46001 |
| [29] | Triebel, H., Function Spaces and Wavelets on Domains, EMS Tracts in Mathematics (ETM), vol. 7 (2008), European Mathematical Society (EMS): European Mathematical Society (EMS) Zürich · Zbl 1158.46002 |
| [30] | Triebel, H., Nuclear embeddings in function spaces, Math. Nachr., 290, 17-18, 3038-3048 (2017) · Zbl 1391.46043 |
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