By Yoav Benyamini and Joram Lindenstrauss

The publication provides a scientific and unified learn of geometric nonlinear useful research. This sector has its classical roots initially of the 20th century and is now a really energetic examine zone, having shut connections to geometric degree idea, likelihood, classical research, combinatorics, and Banach house thought. the most topic of the booklet is the research of uniformly non-stop and Lipschitz capabilities among Banach areas (e.g., differentiability, balance, approximation, lifestyles of extensions, fastened issues, etc.). This research leads certainly additionally to the type of Banach areas and in their vital subsets (mainly spheres) within the uniform and Lipschitz different types. Many fresh really deep theorems and gentle examples are incorporated with entire and distinctive proofs. difficult open difficulties are defined and defined, and promising new study instructions are indicated.

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2 involve evolution equations (in the form of partial diﬀerential equations), while the example in Sect. 3 involves a discrete-time dynamical system. In Sect. 2 we shall discuss certain applications of the inequalities (1)–(4) to the study of the evolution equations introduced there, while in Sect. 3 we shall apply the discrete-time dynamics introduced there to prove the inequalities themselves. The remaining sections further develop the themes introduce in the ﬁrst three, but in the context of other examples.

1 19 The Sharp Hardy-Littlewood-Sobolev Inequality For two non-negative measurable functions f and g on Rd , d ≥ 1, and a number λ with 0 ≤ λ < d, deﬁne the functional Iλ (f, g) by Iλ (f, g) = and for 1 ≤ p < ∞, let f p Rd Rd f (x) 1 g(y)dxdy, |x − y|λ (1) denote the Lp norm of f deﬁned by 1/p f p = Rd f p dx . (2) The HLS inequality provides an upper bound on Iλ (f, f ) in terms of p(λ) where 2d p(λ) = . (3) 2d − λ 2d is determined by the scale invariance of the The value p(λ) = 2d − λ functional Iλ (f, f ): For s > 0, deﬁne f (s) by f f (s) (x) = f (x/s).

The bacteria themselves also diﬀuse across the surface, but with a drift: The bacteria tend to move towards higher concentrations of the attractant, and this induces a drift term tending to concentrate the population, and countering the spreading eﬀects of the diﬀusion. • The key point of mathematical interest in this model is the competition between the concentrating eﬀects of the drift induced by the chemical attractant and the spreading eﬀects of the diﬀusion. If ρ(x) denotes the population density on R2 , and c(x) denotes the concentration of the chemical attractant, also on R2 of course, the system of equations is Functional Inequalities and Dynamics 41 ⎧ ∂ρ ⎪ ⎪ (t, x) = div [∇ρ(t, x) − ρ(t, x)∇c(t, x)] ⎨ ∂t −Δc(t, x) = ρ(t, x) , ⎪ ⎪ ⎩ ρ(0, x) = ρ0 (x) ≥ 0 t > 0 , x ∈ R2 , t > 0 , x ∈ R2 , x ∈ R2 .