By G. Jameson

This publication can be of curiosity to undergraduates taking classes on advanced capabilities.

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**Additional resources for A First Course on Complex Functions**

**Example text**

12) for some constant Cn . 2, which use very different techniques. Later proofs were obtained by Li and Yau in [109] and Conlon [38]. 13) is not known; the best known estimate is that obtained by Lieb in [110]. 12) is sharp in the following sense. 1 C n=2/ 1 I see [112], Sect. 1. 13) is sharp both in the power of ˛ and in the function class of V. 3, as converse. 2 is satisfied. 12) is satisfied. Rn / ˇZ ˇ ˇ ˇ Z ˇ ˇ Wjuj dxˇˇ n 2 R jruj2 dx: Rn Since Cen > Cn is arbitrary, the theorem follows. 3 imply that the Sobolev and CLR inequalities in Rn are equivalent in view of the positivity of the heat operator.

E. T C k/1=2 /. The space Q is called the form domain of T. T0 u; v/ has a closure tŒ and in this case the self-adjoint operator T in Kato’s theorem is the Friedrichs extension of T0 . Rn / ,! Rn /. Rn / and Q are isomorphic. V 1=2 u; V 1=2 u/ is a bounded quadratic form on Q Q, and so there exists a bounded linear operator VO W Q ! Vu; 30 1 Hardy, Sobolev, and CLR Inequalities Multiplication by V is said to be compact relative to the form t0 Œu D kruk2 if VO W Q ! Q is compact, where the norm of Q is now given by kukQ D 1=2 t0 Œu C kuk2 D kukH 1 .

Y/ if jxj Ä jyj: (ii) Let f ; g be Lebesgue measurable on . 23. 5 for details. Ff /. Rn / I it is not invertible. 13) (v) Plancherel’s theorem The map f 7! p/dp: It is standard for the Sobolev spaces W01;2 . /; W 1;2 . / to be denoted by /; H 1 . /, respectively; this is compatible with the comment in Sect. 1 that W k;p . / coincides with the completion H k;p . / in the W k;p . / norm, of C1 . / \ W k;p . /. 13) that H01 . 6 The Dirichlet and Neumann Laplacians We denote by D; u, the Dirichlet Laplacian of u 2 H01 .