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## Studia Mathematica

1993 | 107 | 1 | 61-100
Tytuł artykułu

### Molecular decompositions and embedding theorems for vector-valued Sobolev spaces with gradient norm

Autorzy
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Języki publikacji
EN
Abstrakty
EN
Let E be a Banach space. Let $L¹_{(1)}(ℝ^d,E)$ be the Sobolev space of E-valued functions on $ℝ^d$ with the norm $ʃ_{ℝ^d} ∥f∥_E dx + ʃ_{ℝ^d} ∥∇f∥_E dx = ∥f∥₁ + ∥∇f∥₁$. It is proved that if $f ∈ L¹_{(1)}(ℝ^d,E)$ then there exists a sequence $(g_m) ⊂ L_{(1)}¹(ℝ^d,E)$ such that $f = ∑_m g_m$; $∑_m (∥g_m∥₁ + ∥∇g_m ∥₁) < ∞$; and $∥g_m∥_∞^{1/d} ∥g_m∥₁^{(d-1)/d} ≤ b∥∇g_m∥₁$ for m = 1, 2,..., where b is an absolute constant independent of f and E. The result is applied to prove various refinements of the Sobolev type embedding $L_{(1)}¹(ℝ^d,E) ↪ L²(ℝ^d,E)$. In particular, the embedding into Besov spaces $L¹_{(1)} (ℝ^d,E) ↪ B_{p,1}^{θ(p,d)}(ℝ^d,E)$ is proved, where $θ(p,d) = d(p^{-1} + d^{-1} -1)$ for 1 < p ≤ d/(d-1), d=1,2,... The latter embedding in the scalar case is due to Bourgain and Kolyada.
Słowa kluczowe
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
61-100
Opis fizyczny
Daty
wydano
1993
otrzymano
1993-02-24
Twórcy
autor
• Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-950 Warszawa, Poland
autor
• Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-950 Warszawa, Poland
Bibliografia
• [A] T. Aubin, Problèmes isopérimétriques et espaces de Sobolev, C. R. Acad. Sci. Paris 280 (1975), 279-281.
• [BS] C. Bennett and R. Sharpley, Interpolation of Operators, Academic Press, London, 1988.
• [Br1] J. Bourgain, A Hardy inequality in Sobolev spaces, Vrije University, Brussels, 1981.
• [Br2] J. Bourgain, Some examples of multipliers in Sobolev spaces, IHES, 1985.
• [BZ] Yu. D. Burago and V. A. Zalgaller, Geometric Inequalities, Nauka, Leningrad, 1980 (in Russian); English transl.: Springer, 1988.
• [CSV] T. Coulhon, L. Saloff-Coste and N. Varopoulos, Analysis and Geometry on Groups, Cambridge University Press, Cambridge, 1992.
• [DS] N. Dunford and J. T. Schwartz, Linear Operators I, Interscience, New York, 1958.
• [F] H. Federer, Geometric Measure Theory, Springer, Berlin, 1969.
• [FF] H. Federer and W. H. Fleming, Normal and integral currents, Ann. of Math. 72 (1960), 458-520.
• [G] E. Gagliardo, Proprietà di alcune classi di funzioni in più variabili, Ricerche Mat. 7 (1958), 102-137.
• [Hö] L. Hörmander, The Analysis of Linear Partial Differential Operators I, Springer, Berlin, 1983.
• [H] R. Hunt, On L(p,q) spaces, Enseign. Math. (2) 12 (1966), 249-275.
• [Jo] P. W. Jones, Quasiconformal mappings and extendability of functions in Sobolev spaces, Acta Math. 147 (1981), 71-88.
• [K] V. I. Kolyada, On relations between moduli of continuity in different metrics, Trudy Mat. Inst. Steklov. 181 (1988), 117-136 (in Russian); English transl.: Proc. Steklov Inst. Math. 4 (1989), 127-148.
• [Kr] A. S. Kronrod, On functions of two variables, Uspekhi Mat. Nauk 5 (1) (1950), 24-134 (in Russian).
• [Le] M. Ledoux, Semigroup proofs of the isoperimetric inequality in euclidean and Gauss space, Bull. Sci. Math., to appear.
• [LW] L. H. Loomis and H. Whitney, An inequality related to the isoperimetric inequality, Bull. Amer. Math. Soc. 55 (1949), 961-962.
• [M1] V. G. Maz'ya, Classes of sets and embedding theorems for function spaces, Dokl. Akad. Nauk SSSR 133 (1960), 527-530 (in Russian).
• [M2] V. G. Maz'ya, S. L. Sobolev's Spaces, Leningrad University Publishing House, Leningrad, 1985 (in Russian).
• [N] L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa (3) 13 (1959), 116-162.
• [Pee] J. Peetre, New Thoughts on Besov Spaces, Duke Univ. Math. Ser. 1, Durham, N.C., 1976.
• [Po] S. Poornima, An embedding theorem for the Sobolev space $W^{1,1}$, Bull. Sci. Math. (2) 107 (1983), 253-259.
• [St] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, N.J., 1970.
• [T] G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl. (4) 110 (1976), 353-372.
• [Tr] H. Triebel, Theory of Function Spaces, Birkhäuser, Basel 1983.
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Bibliografia
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