Estimation optimale du gradient du semi-groupe de la chaleur sur le ...

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510 J. Van Schaftingen / Journal of Functional Analysis 236 (2006) 490–516 5.4. Integrability of functions in Dk(Rn ) If u ∈ BMO(Rn ), uBMO(Rn ) 1 and B(0,1) udx = 0, the John–Nirenberg theorem states that exp μ|u| dx c, (5.2) B(0,1) where μ>0 and c>0 can be chosen independently of u. Since the spaces Dk(Rn ) are embedded in BMO(Rn ), this might be improved on Dk(Rn ). On the other hand, if sp = n,0
J. Van Schaftingen / Journal of Functional Analysis 236 (2006) 490–516 511 The improvement of Dk(R n ) on BMO(R n ) should thus not be seen as an improvement of the integrand, but as an improvement on the domains of integration: by the trace Theorem 3.4, the embedding Theorem 5.3, and the John–Nirenberg inequality, functions in Vk(R n ) are exponentially integrable on (n − k + 1)-dimensional subspaces. 6. Further problems 6.1. Traces of V1(R n ) on VMO(R n−1 ) By Theorems 3.4 and 5.3, functions in Vk(R n ) have VMO traces on (n − k + 1)-dimensional spaces. The dimension n − k seems more natural: functions in Vn(R n ) are continuous, and hence have traces on 0-dimensional spaces, i.e. points. If there was such a trace inequality, one could define D0(R n ) = BMO(R n ). This notation would be consistent with the mutual injection Theorem 3.1, the extension Theorem 3.2 and the examples of Proposition 4.6. It would then be nice to have a definition of Dk(R n ) which encompasses the case k = 0. The two-dimensional case would already solve the problem of traces of Vn−1(R n ) on lines. 6.2. Geometric characterizations By Propositions 2.10 and 2.11, the spaces V1(R n ) and Vn−1(R n ) can be defined by oscillations respectively along boundaries of bounded domains and along closed curves. Further refinements would restrict the set of domains and of curves. The most striking result would be if the oscillation could be simply evaluated respectively on spheres and on circles. The spaces Vk(R n ) for 1
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Journal of Functional Analysis 236
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H.-Q. Li / Journal of Functional An
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x(s) = H.-Q. Li / Journal of Functi
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et et Donc, H.-Q. Li / Journal of F
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et Posons H.-Q. Li / Journal of Fun
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⎛ ⎜ ⎝ − H.-Q. Li / Journal
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Donc, H.-Q. Li / Journal of Functio
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5. Quelques problèmes ouverts H.-Q
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The assumption tells us that the fo
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3.4. The zeroes of ( √ z, η) D.
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4.1. Persistence of surface waves D
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When G = η ⊗ r, we obtain Becaus
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This polynomial satisfies D. Serre
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A. Wi´snicki / Journal of Function
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Therefore, A. Wi´snicki / Journal
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Its limit when t → 0is K. Koufany
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and K. Koufany, G. Zhang / Journal
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A. Porretta, L. Véron / Journal of
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1.2. Spectral and growth bounds L.
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By duality, Kp,T is also defined by
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E. Kissin / Journal of Functional A
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Using it, we obtain as in (3.2) tha
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5. Structure of the group ZS E. Kis
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M.J. Crabb / Journal of Functional
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S. Albeverio, A. Kosyak / Journal o
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The minimum is realized for S. Albe
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For m = 3wehave S. Albeverio, A. Ko
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hence S. Albeverio, A. Kosyak / Jou
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φpq(t; tqq) = Since = S. Albeveri
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hence C = C3 = S. Albeverio, A. Kos
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and S. Albeverio, A. Kosyak / Journ
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Furthermore, D. Brydges, A. Talarcz
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lim n→∞ Ω lim sup n→∞
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