By Lamb H.

Sir Horace Lamb (1849-1934) the British mathematician, wrote a few influential works in classical physics. A scholar of Stokes and Clerk Maxwell, he taught for ten years because the first professor of arithmetic on the collage of Adelaide sooner than returning to Britain to soak up the put up of professor of physics on the Victoria collage of Manchester (where he had first studied arithmetic at Owens College). As a instructor and author his said target used to be readability: 'somehow to make those dry bones live'. the 1st variation of this paintings was once released in 1897, the 3rd revised version in 1919, and another corrected model previous to his loss of life. This variation, reissued the following, remained in print until eventually the Nineteen Fifties. As with Lamb's different textbooks, every one part is through examples.

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JD1 0 ˇ ˇ Z 0Z ˇ ˇ@ Œdiam. jD1 @r Z Œdiam. 1(i). t u We refer to [48], Sect. 5, for an interesting connection between the Poincaré inequality and the measure of non-compactness of the embedding W 1;p . / ,! Lp . /. 7. n p/: Then D0 . / is continuously embedded in Ls . / for any s 2 Œp; p and there is a constant C, depending only 1;p on p and n; such that for all u 2 D0 . 6) 1;p is bounded, then D0 . / is compactly embedded in Lq . 6) is the Sobolev inequality and p is the Sobolev conjugate 1;p of p: Part (i) of the theorem asserts that any u 2 D0 .

29) P 2 the last inequality follows since j. x/j2 D j;k xj xk j k D jxj . 28), Z Z Z 1 j. h h0 jxj/j' j2 dx Ä gj. x/ D 1Cjxj . 30), Z Z Z 1 1 2 j'C j2 dx Ä j. 31) and Z R3 1 j' j2 dx Ä jxj Z R3 1 1 C jxj j. 32) Since ' D 'C C ' and the subspaces XC and X are orthogonal, it follows that Z R3 j'j2 dx D Z R3 j'C j2 dx C Z R3 j' j2 dx: The proof will be completed by Lemma 5 in [47], which asserts that P . r/2 PC D PC . 7 Relativistic Hardy-Type Inequalities 47 To verify this, the crucial point is that .

Which we now define. First, we define Ck . / to be the vector space of all bounded functions u 2 Ck . / such that u and all its derivatives D˛ u with j˛j Ä k can be extended so as to be bounded and continuous on . Then Ck; . / is the space of functions u 2 Ck . y0 /; y0 2 G: We shall write Ck in place of Ck;0 and C for C0;0 : 20 1 Hardy, Sobolev, and CLR Inequalities We refer to [48], Sect. 4 for a comparison of various smoothness criteria on @ . 2, that the boundary of a convex open set is in the Lipschitz class C0;1 .