By Curtiss D.R.
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Additional resources for Analytic functions of a complex variable
X;r/ Our purpose here is to describe several results involving these operators that will be important later. But we begin with the simple assertion that M0 , the classical Hardy-Littlewood maximal function, is continuous from Lp; to Lp; ; 1 < p < 1; 0 < Ä n. Of course the case D n is the famous result of Hardy-Littlewood(Weiner). We give two proofs of this: the first based on the method of [A3], and the second based on our duality of Chapter 5. 1. 1) Proof. x/; for jx x0 j < 2r 0; otherwise. I1 C I2 /: © Springer International Publishing Switzerland 2015 D.
For g 2 H p; jjIn gjjLp; Ä c jjgjjH p; for some constant independent of g. Proof. 5 Notes 49 And now we need a lemma, whose proof we put in the Notes at the end of this chapter. 4. If ! ƒn /, then Z ! ÂZ n dy Ä c n ! dƒ n Ãn n for some constant independent of !. Thus continuing the argument above, we have that if ! p 1/n= by the standard Sobolev inequality. n 2 L ! x;r/ ÂZ ÄCr ! n n dy Ä C rn : And finally, the fact that In actually maps into VLp; is a consequence of the fact that C0 is dense in H p; .
Strong subadditivity 23 where the infimum is taken over all downward directed sequences fhi g of hi 2 USCC D non-negative upper semi-continuous functions on Rn . 7) and the fact that the sum of two compact simple functions is again a compact simple function. h/; for all h 2 USCC . 8). , ordering the sets where Ki 0 D [f\j2J Kj W J f1; 2; into an increasing sequence of compact sets K10 ; K20 ; ; Kn0 . 9) is the key to the proof. 9) is just strong subadditivity. K1 [ K2 / \ K3 . K3 /: This then shows how to go from two sets to three sets in the induction process.
Analytic functions of a complex variable by Curtiss D.R.