The conjugation operator on \$A\sb q(G)\$

Sanjiv Kumar Gupta, Shobha Madan, U. B. Tewari
1994 Proceedings of the American Mathematical Society
Let G be a compact abelian group and r its dual. For 1 < q < oo , the space Aq(G) is defined as Aq(G) = {f\feLx(G), felq(T)} with the norm \\f\\Aq = ll/ll¿i + \\f\\tq ■ We prove: Let G be a compact, connected abelian group and P any fixed order on T. If q > 2 and is a Young's function, then the conjugation operator H does not extend to a bounded operator from Aq(G) to the Orlicz space L't'(G). Let G be a compact abelian group and Y its dual. For 1 < q < oo, the space Aq(G) is defined as with
more » ... defined as with the norm \\f\\Aq = II/IIl' + II/II/ • Then Aq(G) is a commutative semisimple Banach algebra with maximal ideal space Y, in which the set of trigonometric polynomials 7T is dense [4] . If G is, in addition, a connected group, then its dual can be ordered; i.e., there exists a semigroup P ç Y such that P n -P = {0} , Pu-P = Y [5], and we say that y e Y is positive if y e P. If / = 52y€Ff(y)y is a trigonometric polynomial, the conjugation operator is defined as 77/=5Z-z'sgn()')/(y))> y€F where sgn(y) = +1 if y e P, -1 if y e -P, and 0 if y = 0. If 1 < q < 2, then Aq(G) ç L2(G), and it is easy to see that 77 extends to a bounded operator on Aq(G). The corresponding result for q > 2 is not known. Clearly 77 extends to a bounded operator on Aq if and only if 77 extends to a bounded operator from Aq(G) to LX(G). In this paper we use Rudin-Shapiro polynomials (see [1] or [2]) to study the conjugation operator from Aq(G) to Orlicz spaces L't'(G). We recall the definition of Orlicz spaces below.