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On the trace problem for Lizorkin–Triebel spaces with mixed norms. (English) Zbl 1154.46017

For \(\bar{p} = (p_1, \dots, p_n)\) with \(0<p_l\leq\infty\), let
\[ \| u | L_{\bar{p}} (\mathbb R^n) \|=\Biggl(\dots\biggl(\int_R| u(x_1,\dots)|^{p_1} \biggr)^{p_2/p_1}\dots\Biggr)^{1/p_n} \]
be mixed \(L_p\)-spaces. Let \(a= (a_1, \dots, a_n)\), \(0<a_l<\infty\), be an anisotropy. Then \(F^{s,a}_{\bar{p},q}(\mathbb R^n)\) collects all \(f\in S'(\mathbb R^n)\) such that
\[ \Biggl\|\Biggl(\sum^\infty_{j=0} 2^{jsq}\bigl|(\Phi_j\hat{f})^\vee(\cdot)\bigr|^q \Biggr)^{1/q} \Biggl| L_{\bar{p}} (\mathbb R^n) \Biggr\| \]
is finite. Here, \(s \in\mathbb R\), \(0<q\leq \infty\) and \(\{\Phi_j \}\) is a related anisotropic resolution of unity. The authors develop a theory of these spaces and their \(B\)-counterparts. The main point is the study of traces \(f(x) \to f(0,x')\) from \(\mathbb R^n\) to \(\mathbb R^{n-1}\). It may happen that this trace is a map of \(F^{s,a}_{\bar{p},q}(\mathbb R^n)\) onto a corresponding \(F\)-space on \(\mathbb R^{n-1}\), in self-explaining notation \(F^{s- \frac{a_{1}}{p_1}, a''}_{p''. p_{1}}(\mathbb R^{n-1})\) (Theorem 2.2). This is a surprising effect which stands in sharp contrast to the usual isotropic and also anistropic spaces based on unmixed \(L_p\)-spaces. The motivation comes from parabolic boundary value problems with boundary data in terms of mixed \(L_p\)-spaces.

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
42B35 Function spaces arising in harmonic analysis
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[1] Linear and Quasilinear Parabolic Problems. Vol. I: Abstract Linear Theory, Monographs in Mathematics Vol. 89 (Birkhäuser Boston Inc., Boston, MA, 1995).
[2] Bagby, Proc. Amer. Math. Soc. 48 pp 419– (1975)
[3] Benedek, Duke Math. J. 28 pp 301– (1961)
[4] Berkola??ko, Dokl. Akad. Nauk SSSR 27730 pp 270– (1984)
[5] Berkola??ko, Dokl. Akad. Nauk SSSR 28231 pp 1042– (1985)
[6] Berkola??ko, Trudy Inst. Mat. 7 pp 30– (1987)
[7] Berkola??ko, Trudy Inst. Mat. 9 pp 34– (1987)
[8] , and , Integral Representations of Functions and Embedding Theorems. Vol. I, translated from the Russian, edited by Mitchell H. Taibleson, Scripta Series in Mathematics (V. H. Winston & Sons, Washington, D. C., 1978).
[9] , and , Integral Representations of Functions and Embedding Theorems. Vol. II, translated from the Russian, edited by Mitchell H. Taibleson, Scripta Series in Mathematics (V. H. Winston & Sons, Washington, D. C., 1979).
[10] , and , Integralnye predstavleniya funktsii i teoremy vlozheniya, 2nd ed. (Fizmatlit ”Nauka”, Moscow, 1996).
[11] Bugrov, Izv. Akad. Nauk. SSSR Ser. Mat. 355 pp 1137– (1971)
[12] Denk, Math. Z. 257 pp 193– (2007)
[13] Farkas, Math. Bohem. 125 pp 1– (2000)
[14] and , An Axiomatic Approach to Function Spaces, Spectral Synthesis, and Luzin Approximation, Memoirs of the American Mathematical Society Vol. 188, No. 882 (Amer. Math. Soc., Providence, RI, 2007).
[15] Johnsen, Math. Scand. 79 pp 25– (1996) · Zbl 0873.35023 · doi:10.7146/math.scand.a-12593
[16] Johnsen, Z. Anal. Anwendungen 19 pp 763– (2000) · Zbl 0972.46023 · doi:10.4171/ZAA/979
[17] Johnsen, J. Funct. Spaces Appl. 5 pp 183– (2007) · Zbl 1140.46014 · doi:10.1155/2007/714905
[18] Krée, Boll. Un. Mat. Ital. (3) 22 pp 330– (1967)
[19] Lizorkin, Izv. Akad. Nauk SSSR Ser. Mat. 344 pp 225– (1970)
[20] Marschall, Z. Anal. Anwendungen 15 pp 109– (1996) · Zbl 0840.47040 · doi:10.4171/ZAA/691
[21] and , Ondelettes et Opérateurs. III (Hermann, Paris, 1991); English version by Cambridge Univ. Press, Cambridge (1997).
[22] Approximation of Functions of Several Variables and Embedding Theorems (Springer-Verlag, New York, 1975).
[23] Schmeisser, Z. Anal. Anwendungen 3 pp 153– (1984)
[24] and , Topics in Fourier Analysis and Function Spaces (Akademische Verlagsgesellschaft Geest & Portig K.-G., Leipzig, 1987). Published also by John Wiley, Chichester (1987). · Zbl 0661.46025
[25] Theory of Function Spaces (Birkhäuser Verlag, Basel, 1983).
[26] Existence results in Lp -Lq spaces for second order parabolic equations with inhomogeneous Dirichlet boundary conditions, in: Progress in Partial Differential Equations Vol. 2, Pitman Research Notes Mathematics Series Vol. 384 (Longman, Harlow, 1998), pp. 189–200.
[27] Weidemaier, Electron. Res. Announc. Amer. Math. Soc. 8 pp 47– (2002)
[28] Weidemaier, Mat. Sb. 196 pp 3– (2005)
[29] Yamazaki, J. Math. Soc. Japan 38 pp 199– (1986)
[30] Yamazaki, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 33 pp 131– (1986)
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