Download Blaschke products in B by Sarason D. PDF
By Sarason D.
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Example text
44. 21 Proof. THEOREM L p (f2) is separable if 1 < p < ~ . For m -- 1, 2 . . let ~'2m - - {X E ~ " Ixl ~ m and dist(x,bdry(f2)) > 1/m]. Then Qm is a compact subset of g2. Let P be the set of all polynomials on I~n having rational-complex coefficients, and let Pm - - { X m f " f ~ P} where Xm is the characteristic function of ~2m. 32, Pm is dense in C(~"2m). Moreover, U m % l Pm is countable. If u e: LP (~ ) and E > 0, there exists 4~ 6 C0(f2) such that I l u - Clip < E/2. If 1/m < dist(supp (40, bdry(f2)), then there exists f in the set Pm such that 114~- f l l ~ < (E/2)(VOI(~'2m)) -lIp" It follows that I1r flip S I 1 r fll~ (vol(~m)) lip < e/2 and so Ilu - flip < E.
The best (smallest) constant is pl/pql/qrl/r K ( p , q , r, n ) -- ( p t ) l~ ' F-~ , i1i-~,' -(r' ) l /r, )n/2 34 The Lebesgue Spaces L P(~2) See [LL] for a proof of this. 25 COROLLARY If ( l / p ) + ( l / q ) -- 1 + ( l / r ) , and if u ~ L p ( ~ n ) and v E L q (~n), then u 9 v E L r (]~n), and Ilu * vllr ~ K ( p , q, r', n)Ilullp Ilvllq ~ Ilullp Ilvllq 9 This is known as Young's inequality f o r convolution. It also implies Young's Theorem. 7 and the case of inequality (17) proved above, with r' in place of r.
J~(x) dx = 1. J, is called a mollifier and the convolution J6 :~ U(X) = fRn J6(x -- y)u(y) dy, (18) defined for functions u for which the right side of (18) makes sense, is called a mollification or regularization of u. The following theorem summarizes some properties of mollification. 29 T H E O R E M (Properties of Mollification) is defined on I~n and vanishes identically outside f~. Let u be a function which (a) If u E L~oc (~n), then J~ 9 u ~ C ~ (R n ). (b) If u ~ L~oc(f2) and supp (u) ~ ~ , then J~ 9 u ~ C ~ ( f 2 ) provided E < dist(supp (u), bdry (fl)).