By Laurent Saloff-Coste

ISBN-10: 0521006074

ISBN-13: 9780521006071

This publication 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 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 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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Ik ) and IIVkfllp = fn \ 1/p (lVkf(X)I'dX ) / with the obvious interpretation if p = oo. By induction based on the case k = 1 (which is treated in the previous sections), one easily obtains the following statement. 5 Fix two integers n, k, and 1 < p < oo. If 1 _< kp < n and q = np/(n - kp), there exists a constant C = C(n, k, p) such that V f E C (Rn), Ilf llq Vkf lip. If kp = n, there exist c = c(n, k) and C' = C(n, k) such that for all bounded subsets Il C Rn, b f E Co (SZ), neap ( If(x)l I p /(-1 ]dx

See [61]. We record it in the following theorem. 4 There exist two constants Cn, c,,. > 0 such that, for any 0 < c < c,,, for any bounded set 0 C Rn and any function f E Co (S2), JexpIc'' If(x)I IlpfIIn )n/(n-1) dx < Cnµn( l) CHAPTER 1. SOBOLEV INEQUALITIES IN RN 26 A more precise result is known [66]. Namely, let an = nwn(i -1) Then, for each bounded domain l C 1[l n, bf J exp a iflf (x)I E Co(1Z), )fl/(fl_1)) dx < Cµn(Q) st for all 0 < a < an whereas, if a > an, dx : f E Co (0), Iloflln = 1} _ 00.

Let also R be such that p ,,(Q) = µn(B(R)), that is R = (µn(S2)/SZn)'/n Then use the following computation. Q) (p-n)/n(p-1) . 2), 1 I I ! ll. 1. 1) depends on the set Q on which the function f is supported. As the volume of Il tends to infinity, the term µn(0)1/n-1/P also tends to infinity since n < p. In fact, when n < p, there is no way to control the size of f purely in terms Ilof IIP. To see this, consider the function fr : x --, (r - l xl)+. This function is supported in B(r) and Qn/Prf/P Ilofrllp = P-n(B( ))1/p = Also, µn(B(r/2))(r/2) = f2n/q(r/2)n/q+1.

### Aspects of Sobolev-Type Inequalities by Laurent Saloff-Coste

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