Ivo Klemes

A mean oscillation inequality

It is proved that lIf*IIBMO S Illf IBMO, where f* is the decreasing rearrangement of a function f E BMO([O, 1]). A generalization is given, as well as an example, showing the result fails for the symmetric decreasing rearrangement of a function on the circle. In this paper we prove that ( * ) ||f *E|BMO :I lIf IIBMO0 where f * is the decreasing rearrangement of the function f: [0, 1] -> R. The sharp constant one in this inequality refines a result contained in Theorem 3.1 of [1]. We also indicate generalizations, and finally mention symmetric decreasing rearrangement on the circle, for which (*) fails. The average of f over E c [0, 1], JEl > 0, will be denoted byfE = (l/IEI)JEf. The definition of the norm in (*) is lIf IIBMO = sup -J If-hiI JC[0,1] VIJ where J ranges over all subintervals of [0,1]. We need the following version of the rising sun lemma. See [3] or [2, p. 293] for the usual rising sun lemma. LEMMA. Let f E Ll([0, 1]) and suppose the average f [O 1] a. Then there is a finite or countable set 9 of pairwise disjoint subintervals of [0, 1] such that fL = a for each L E 9, andf 0 as large as possible so that F(to) = 0. Let ( = {t E [tog 1]: F(x) > F(t) for some x > t }. Then ( is open in [to, 1], so ( = UIJ for some disjoint intervals IJ open in [t 0 1]. Defining Y to be these intervals together with [0, t0) satisfies the lemma, as we now verify. On [0, to) the average of f is 0 by definition. Let IJ have endpoints t F(aj) then a1 E ( by definition, so aj = to. But, by hypothesis, F(1) = f[o l] < 0 = F(to) = F(aJ) < F(bj), so (by the intermediate value theorem) F(c) = 0 for some bj < c < 1, contradicting the choice of to. If F(bJ) < F(aJ) choose T E [aj, bj] as large as possible so that F(T) = 1 (F(aj) + F(b1)). Received by the editors March 25, 1984 and, in revised form, June 27, 1984. 1980 Mathemnatics Subject Classification. Primary 26A87. K1985 American Mathematical Society 0002-9939/85

Rank one transformations with singular spectral type

1.00 +

On permutations of lacunary intervals

.25 per page

PRODUCT AND MARKOV MEASURES OF TYPE III

AbstractWe show that a certain class of measures arising from generalized Riesz products is singular. In particular, cutting and stacking (i.e. rank one) transformations whose cuts do not grow too rapidly, have singular maximal spectral type. The precise condition is

A new type of Littlewood-Paley partition

\sum\nolimits_{n = 1}^\infty {(1/\omega _n^2 )}

Properties of Littlewood-Paley sets

, wherewh is the number of cuts at stagen.

A note on Hardy's inequality

Let {Ij} be an interval partition of the integers and consider the Littlewood-Paley type square function 5(f) ( If 12)1/2 where fj = fxi . We prove that if the lengths /(Ij) of the intervals IJ satisfy /(Ij+I )// (Ij) -* oo, then IIS(f)llp llfllp for 1 A > 1 , and we also conjecture a necessary and sufficient condition.

Symmetric polynomials and lp inequalities for certain intervals of p

We give some explicit constructions of type III product measures with various properties, resolving some conjectures of Brown, Dooley and Lake. We also define a family of Markov odometers of type III 0 and show that the associated flow is approximately transitive.