By Lam T.Y., Magid A.R. (eds.)

ISBN-10: 0821810871

ISBN-13: 9780821810873

**Read Online or Download Algebra, K-Theory, Groups, and Education: On the Occasion of Hyman Bass's 65th Birthday PDF**

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4). If n = 1, then R1 f (x) is the Hilbert transform H. 2. Let 1 < p < ∞. 6) where the supremum is taken over all balls B in Rn and |B| is a volume of B. 3 ([224], p. 205, p. 210). 3) are satisfied and w ∈ A p (Rn ), then the operator K is bounded in Lwp (Rn ). 4) hold and K is bounded in Lwp (Rn ), then w ∈ A p (Rn ). The following statements will be useful in Chapter 6. 6). 3) be satisfied and let u0 be the unit vector in Rn . Then by choosing u = tu0 , with t fixed sufficiently large we can guarantee that 1 |k(r(u + v)) − k(ru)| ≤ |k(ru)| 2 whenever r ∈ R \ {0} and |v| ≤ 2.

Muckenhoupt and R. L. Wheeden [171]. 2 for Euclidean spaces was derived in [171] and for homogeneous groups and quasimetric measure spaces with doubling condition, for instance, in [70], Ch. 6; [76], Ch. 4. 36 Alexander Meskhi Two-weight criteria involving the operator itself for the Hardy-Littlewood maximal operator have been established in [212] (see [172], [191] for another type of sufficient conditions). The two-weight problem for the fractional maximal functions has been solved in [212], [214], [217], [236] (see also [78], [76], Ch.

10 of [12]) according to which a measurable function f belong to the dual space X ′ of a Banach function space X if and only if f g is integrable for every g in X. , [138], [204]). 7([204]). For all f ∈ L p(·) (Ω) the inequality f L p(·) (Ω) ≤ sup f (y)g(y)dy Ω ≤1 ′ L p (·) (Ω) g holds. In the sequel we will assume that Ek := [2k , 2k+1 ); Ik := [2k−1 , 2k+1 ), k ∈ Z. The next statements will be useful in Chapter 5. 8. Let 1 < p− (R+ ) ≤ p(x) ≤ q(x) ≤ q+ (R+ ) < ∞ and let p, q ∈ W L(R+ ). Suppose that p(x) ≡ pc = const, q(x) ≡ qc = const when x > a for some positive number a.

### Algebra, K-Theory, Groups, and Education: On the Occasion of Hyman Bass's 65th Birthday by Lam T.Y., Magid A.R. (eds.)

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