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

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702 D. Brydges, A. Ta<strong>la</strong>rczyk / Journal of Functional Analysis 236 (2006) 682–711 Proposition 4.4. Let d = 1, B = d/dx, Λ = (a, b) and L>0, then the operator AL of Theorem 2.8 is of the form ALϕ(y) = (a,b) aL(y, x)ϕ(x) dx, where ⎧ 0 for (y, x) /∈ (a, b) × (a, b), ⎪⎨ y−a 2L(x−a) for a
D. Brydges, A. Ta<strong>la</strong>rczyk / Journal of Functional Analysis 236 (2006) 682–711 703 ALϕ(y) = 1 2L a+L a−L + 1 2L y − a ϕ(x + L) x + L − a 1|x−y|L dx b+L b−L + 1 2L b−L a+L + 1 2L = 1 2L b−L a+L a+2L a + 1 2L + 1 2L b − y ϕ(x − L) b − (x − L) 1|x−y|L dx ϕ(x − L) ϕ(x + L) x + L − y 1|x−y|L dx 2L y − (x − L) 1|x−y|L dx 2L y − a ϕ(x) x − a 1yx dx + 1 2L b−2L a b a+2L ϕ(x) ϕ(x) which gives (4.10). If b − a
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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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D. Kucerovsky / Journal of Function
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D. Serre / Journal of Functional An
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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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H. Schultz / Journal of Functional
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we now have (cf. (7.8)) that H. Sch
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J. Van Schaftingen / Journal of Fun
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2.2. Definition J. Van Schaftingen
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Since ϕ = ϕ0 + ϕ1 ∧ ωN, one h
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3. The Poisson transform K. Koufany
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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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S. Albeverio, A. Kosyak / Journal o
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The minimum is realized for S. Albe
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