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**Extra info for Encyclopedia Of Mathematical Physics. I-O**

**Sample text**

The sharp second constants depend on the manifold Àp=n and are given by B0 = Bp = Vg , where Vg is the volume of (M, g). A typical question in the AB program is to know whether or not we can take A or B to be the sharp constants in [11] and, similarly, whether or not we can take A0 or B0 to be the sharp constants in [12]. Another typical question in the AB program is whether or not there are nonzero extremal functions for the saturated form of the sharp inequalities when they are valid. Concerning the B-part of the program, the sharp inequality [11] À1=n is true on any manifold, and constant with B = Vg functions are extremal functions.

Ni= 1 dpi ^ dqi = d, where = ni= 1 pi ^ dqi . The same formula for ! holds locally in T Ã Q for any finite-dimensional Q (Darboux’s lemma). For the infinite-dimensional example P = V Â V Ã , the symplectic form ! ((’1 , 1 ), (’2 , 2 )) = h’1 , 2 i À h’2 , 1 i. Again, these two formulas for ! are identical if V = Rn . ½7 where the prime denotes differentiation with respect to ’, which imply the wave equation @ ’ ¼ r2 ’ À F0 ð’Þ @t2 bundle (phase space) of a manifold Q (configuration space).

Springer Monographs in Mathematics. Berlin: Springer. Druet O and Hebey E (2002) The AB Program in Geometric Analysis: Sharp Sobolev Inequalities and Related Problems. Memoirs of the American Mathematical Society, 160, 761. American Mathematical Society. Evans LC and Gariepy RF (1992) Measure Theory and Fine Properties of Functions. CRC Press. Hebey E (2000) Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities. Courant Lecture Notes in Mathematics, vol. 5. American Mathematical Society.