Мб Some open problems concerning H and BMO

2.9.

SOME OPEN PROBLEMS CONCERNING H ~ AND BMO*

I. I. An interpolating Blaschke product is a Blaschke product having distinct zeros which lie on an H ~ interpolating sequence. Is H ~176the uniformly closed linear span of the interpolating Blaschke products (see [1, 2])? It is known that the interpolating Blaschke products separate the points of the maximal ideal space (Peter Jones, thesis, University of California, Los Angeles 1978). 2.

where ~I

Let ~

be a real locally integrable function on ~

is the mean value of ~

~=~+Hd , where i ~ . ~

and |~B~$~

Assume that for every interval

over I, and where C is a constant. ?

Does it follow that

(H denotes the Hilbert transform.)

This-is the limiting

case of the equivalence of the Muckenhoupt (A2) condition with the condition of Helson and Szeg~ (see [3, 4]). This question is due to Peter Jones. A positive solution should have several applications. 3.

Let f be a function of bounded mean oscillation on

~

Construct L ~ functions u

and v so that ~=~+H~, ll1~11~+gl~|~'~I~B~0 , with C a constant not depending on f (see [5, 6]). 4.

Let TI, T2,...,T n be singular

and sufficient conditions

integral operators

on IT1, T2,...,Tn}

such that

on

~H(#)

~m (see [7]).

Find necessary

if and only if I ~ I + ~ [ ~ [ ~ -

j._~

-

-

~

-

~.'C~") (see [5, 8]). LITERATURE CITED I 9

2. 3. 4. 5. 6. 7. 8.

D. Marshall, "Blaschke products generate H~, '' Bull. Am. Math. Soc., 82, 494-496 (1976). D. Marshall, "Subalgebras of L ~ containing H~, '' Acta Math., 137, 91-98 (1976). R. A. Hunt, B. Muckenhoupt, and R. L. Wheeden, "Weighted norm inequalities for the conjugate function and Hilbert transform," Trans. Am. Math. Soc., 176, 227-251 (1973). H. Helson and G. SzegS, "A problem in prediction theory," Ann. Math. Pure Appl., 51, 107-138 (1960). C. Fefferman and E. M. Stein, "HP spaces of several variables," Acta Math., 129, 137193 (1972). L. Carleson, "Two remarks on H l and BMO," Adv. Math., 22, 269-277 (1976). E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton N.J. (1970). S. Janson, "Characterizations of H I by singular integral transforms on martingales and ~ ," Math. Scand., 4__I, 140-152 (1977).

*JOHN GARNETT.

2254

University

of California,

Los Angeles,

California 90024.