By Laurent Saloff-Coste

ISBN-10: 0521006074

ISBN-13: 9780521006071

This booklet makes a speciality of Poincaré, Nash and different Sobolev-type inequalities and their functions to the Laplace and warmth diffusion equations on Riemannian manifolds. functions lined comprise the ultracontractivity of the warmth diffusion semigroup, Gaussian warmth kernel bounds, the Rozenblum-Lieb-Cwikel inequality and elliptic and parabolic Harnack inequalities. Emphasis is put on the position of households of neighborhood Poincaré and Sobolev inequalities. The textual content offers the 1st self contained account of the equivalence among the uniform parabolic Harnack inequality, at the one hand, and the conjunction of the doubling quantity estate and Poincaré's inequality at the different.

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**Sample text**

3). Assume that B is the unit ball. Fix 6 E (0, 1). For x, y E 6B, consider two cases. If Ix - yI > (1 - 6), then Iu(x) - u(y)I < 2sup{u} < 2(1 - h)-"Ix - yl'sup{u}. B B If Ix - yI < (1 - 6) then the ball B' of radius (1 - 6) and center (x + y)/2 contains both x and y and is contained in B. Moreover, x, y are contained in (Ix - yI/(1 - b))B'. 2 in B' yields I u(x) - u(y)l 5 21 «(1 - b)-'fix - yI° Sup{u}. 1. iemannian manifolds We want now to replace the Euclidean space lilt" by a Riemannian manifold M and consider the possibility of having some kind of Sobolev inequalities.

Let -y, C be positive constants and 0 < ao < oo. , for all or, o', a such that 0 < b < o' < or < 1 and 0 < a < min{1, ao/2}. Assume further that f satisfies /'(log f > \) <- CA(U)a-1 for all A > 0. Then Ill IIQ0,u6 < Aj(U)'/Qo where A depends only on b, ry, C, and a lower bound on ao. P =Ri(o)=log(IIfi(00,uo), for0** < e''(2C/')11Q-1100 + e'P/2. 9) Here, we have used successively the Holder inequality and the second hypothesis of the lemma. 9) are equal, and 0 < a < min{1, ao/2}. **

2. 3 applies here and yields the following inequalities. 1 and only the case s = q needs to be proved. Note also that it suffices to treat the case where B is the unit ball. 2 Fix 1 < p < n and set q = np/(n - p). 3) 32 CHAPTER 1. SOBOLEV INEQUALITIES IN RN for all 1 < s < q. 4) II p,B for all 1 < s < q. Here, fB is the mean off over the ball B. It is natural to wonder whether the ball B can be replaced by some more general bounded domain. Let S2 be a bounded domain in Rn. On the one hand, there is no difficulty with the case of functions with compact support in f because any f E Co (1) can be extended to a function in C°(R) by setting f = 0 outside Q.

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