Download e-book for kindle: Aspects of Sobolev-Type Inequalities by Laurent Saloff-Coste
By Laurent Saloff-Coste
This booklet makes a speciality of Poincaré, Nash and different Sobolev-type inequalities and their purposes 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 position 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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Extra info for Aspects of Sobolev-Type Inequalities
Then 1/pt - 1/po < (1 + 9)(1/p - 1/po). 2. 3 An abstract lemma This section presents an elementary but subtle lemma due to Bombieri and Giusti [9) which simplifies considerably Moser's original proof of the Harnack inequality. e, the exponential integrability of BMO functions). This simplification is even more significant when dealing with parabolic equations. See . Consider a collection of measurable subsets UQ, 0 < o < 1, of a fixed measure space endowed with a measure µ, such that U,, C Uo if a' < a.
A scalar product If f c C°°(M), its gradient is defined as the unique vector field V f such that d x E M, V E TM, (V f (x), (x))x = df (e)(x). There is a canonical distance function associated to the Riemannian structure of M. We will denote it by (x, y) '--+ d(x, y). It can be defined as the shortest length of all piecewise Cl curves from x to y. The topology of (M, d) as a metric space is the same as that of M as a manifold. 6]. There is also a canonical Riemannian measure on M which wy denote by either dx or p.
AB We assume that B is the unit ball. 6 to a-lu and ecu-i where c is a well-chosen constant. Pick p so that 6 < p < 1 (for instance p = 6 + (1 - 6)/2). Set U = B' = pB, V' = p(B'), and set U, = aB' = opB. 6 is satisfied by any constant multiple of u with ao = p < 0. 6, we apply the L2-Poincar6 inequality in the ball U = B' to the function log u. This yields f I log u- cI2dµ < C. J I V log ul2dp BI 1 where C,i depends only on n and c = (log U)BI _' JB log u dp is the mean of log u over B'. Now, as u is a positive supersolution, ai,,aiuO B >0 =,i 0 where for all non-negative test functions ¢ E W01'2(B).
Aspects of Sobolev-Type Inequalities by Laurent Saloff-Coste