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4. The measure m satisfies the doubling condition o(A(x, 2r))<~Co(A(x, r)). 1 and its corollaries can be strengthened. 5 (Fabes, Kenig, Neri [161]). For any P0 > I there exists an ~ > 0 such that in the domain D with Lipschitz constant M < e, k and k -z satisfy (Ap0(O)). 6. p > I. 7. in BMO(~). In a domain of class C z, k and k -z satisfy all the conditions Ap(O), In a domain of class C I, the function logk is in the closure of L~(aD) Remark. , has vanishing mean oscillation [175], is unproved.

Are Riesz transforms of the function f:f~ = R~f. It turns out that the s p a c e H I c L 1 ( R ") ca~ be defined equivalently by the condition n I/Iq- ~ IRff lED(R"). 1--I Finally, the Poisson integral Pf can be replaced by the convolution with a sufficiently smooth function of quite general form [165]. Let w be a weight in R n. Is the equivalence of the definitions listed preserved if LI(R n) is replaced by L1(w)? Wheeden [312] proved the following fact. 4. I. are equivalent : If the weight w~(A1), then the following conditions on a function [~Lt(w) N (Pf)s (w); (ii) RffELJ(w), j = l , (i) 2 .

2 . L e t T = M, 1 < p < ~. F o r a g i v e n w e i g h t w, t h e r e everywhere finite weight v, satisfying ( 7 . 3 ) i f and o n l y i f I(1 C-F~")p-~ < + Rn [285]. exists an a l m o s t co. in the direction o f Theorem 7 . 3 . 2 ; i t was Integrals. THEOREM 7 . 4 . 1 (Rubio de F r a n c i a [ 2 7 9 , 2 8 0 ] , C a r l e s o n and J o n e s [ 1 2 2 ] ) . Let 1 < p < and v be a weight in R n. 3) is valid for any Calderon--Zygmund operator. (ii) The Riesz transforms Rj a c t from LP(w) to the spaces L~ n) of almost everywhere finite measurable functions.

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Survey of Text Mining, Clustering, Classification and Retrieval 2 by Berry M.W.,Castellanos M.


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