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Aspects of Sobolev-Type Inequalities by Laurent Saloff-Coste

By Laurent Saloff-Coste

This booklet specializes in Poincaré, Nash and different Sobolev-type inequalities and their purposes to the Laplace and warmth diffusion equations on Riemannian manifolds. functions coated contain 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 function of households of neighborhood Poincaré and Sobolev inequalities. The textual content presents 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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T where v = log u. Hence f P2 aija=vajV5dp E a=,AO,vO,vdp < 2 IV) ij ij CHAPTER 2. (1 - p)-2)\-4 = C(n, A, 6). Thus tp(B' n {I logu - cl > t}) < j I logu - cIdp 1/2 < (/ 1 log u - cl2dp) \ B' < C(n,A,6). 6 is satisfied by f = e-cu (and also by ecu 1, for that matter). 1) JI Ujjp,SB < Ae` with A = A(n, A, 6). 6 applies to ecu-1 with the same c = (log u)5 as above. In this case, ao = oo. Hence sup{u-' } < Ae-°. 2) together yields 'IUI+p,6B < A2 6inBf{u} which is the desired inequality. u(x) - u(y) I x,yIz - y'a JJ1 B for positive solutions of Lu = 0 in the ball B of radius r.

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.

Ok flipl C s x) I (IA11 For the proof, recall the representation formula Ax) = J(Pn,k(x - y), Vkf (y))d(y) where Pn,k is homogeneous of degree -n + k, 0 < k < n. Ix - ylnF' dy ^ Ial=k where 6 = (99)i = (x - y)/Ix - yI. 6. The case p = 1, k = n is a very special case. Indeed, we obviously have I f (x) I < f +°° ... 00 f +00 Ial ... 5 IIVnfll1. 1). , a = 0) is excluded. For instance the case p = 2, k = 2, n = 2 is not treated. When n/p = Q is an integer, the optimal result is as follows. CHAPTER 1.

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