Ali Baklouti, Aziz El Kacimi, Sadok Kallel, Nordine Mir's Analysis and Geometry: MIMS-GGTM, Tunis, Tunisia, March PDF

By Ali Baklouti, Aziz El Kacimi, Sadok Kallel, Nordine Mir

ISBN-10: 3319174428

ISBN-13: 9783319174426

ISBN-10: 3319174436

ISBN-13: 9783319174433

This ebook comprises chosen papers provided on the MIMS (Mediterranean Institute for the Mathematical Sciences) - GGTM (Geometry and Topology Grouping for the Maghreb) convention, held in reminiscence of Mohammed Salah Baouendi, a most famed determine within the box of a number of advanced variables, who passed on to the great beyond in 2011. All learn articles have been written by means of best specialists, a few of whom are prize winners within the fields of advanced geometry, algebraic geometry and research. The booklet bargains a precious source for all researchers attracted to contemporary advancements in research and geometry.

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Additional info for Analysis and Geometry: MIMS-GGTM, Tunis, Tunisia, March 2014. In Honour of Mohammed Salah Baouendi

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1) The space L(R2N ) it will be question in this paper is the Orlicz space associated to 2 the function φ(s) = es − 1. 2 u where β N = S2N −1 . 2) 2N π 2N 22N 2 πN , with ω2N −1 = the measure of the unit sphere ω2N −1 (N − 1)! 3) whose lack of compactness has been investigated by several authors (for further details, we refer to [7, 9, 12, 21, 22, 27]). -L. 3) in 2D is due to two reasons. The first reason is the lack of compactness at infinity that can be illustrated by the sequence u n (x) = ϕ(x + xn ), where 0 = ϕ ∈ D and |xn | → ∞, and the second reason is of concentration-type and can be highlighted by the example by Moser (see [21–23]) defined by: f αn (x) = ⎧ ⎪ ⎨ αn 2π log |x| √ − ⎪ 2αn π ⎩ 0 if |x| ≤ e−αn , if if e−αn ≤ |x| ≤ 1, |x| ≥ 1, Logarithmic Littlewood-Paley Decomposition and Applications to Orlicz Spaces 37 where α := (αn ) is a sequence of positive real numbers going to infinity.

12. Taking advantage of the fact that the spectrum of the function u is included in eλ C with λ ≥ 1, we find that u(ξ) = φ(λ−1 log |ξ|) u(ξ), where φ is a function of D(R) chosen as above. Therefore (log |D|)k u = (log |D|)k gλ u , where gλ (ξ) = φ(λ−1 log |ξ|). 7) Obviously F((log |D|)k gλ )(ξ) = λk (λ−1 log |ξ|)k φ(λ−1 log |ξ|) , thus in view of the relation 1 1 1 = − + 1 , our purpose is to establish that the r q p function gk,λ (x) := (2π)−2N R2N ei x·ξ φk (λ−1 log |ξ|) dξ 1 with φk (ρ) = ρk φ(ρ), satisfies gk,λ L r (R2N ) e2N λ b (1− r ) .

3, we state and establish some logarithmic Sobolev embeddings that occur in Orlicz spaces. We mention that the letter C will be used to denote an absolute constant which may vary from line to line. We also use A B to denote an estimate of the form A ≤ C B for some absolute constant C. 48 H. Bahouri 2 Proof of Bernstein Inequalities This section is devoted to the proof of Bernstein inequalities in the framework of the logarithmic Littlewood-Paley decomposition. Adapting these fundamental inequalities provides various functional inequalities such as Sobolev embeddings and their refined versions.

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Analysis and Geometry: MIMS-GGTM, Tunis, Tunisia, March 2014. In Honour of Mohammed Salah Baouendi by Ali Baklouti, Aziz El Kacimi, Sadok Kallel, Nordine Mir

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