Hermite conjugate functions and rearrangement invariant spaces

Kenneth F. Andersen
1973 Canadian mathematical bulletin  
The Hermite conjugate Poisson integral f(x, y) of a given feL 1^) , d{i(y)= exp(-j 2 ) dy, was defined by Muckenhoupt [5, p. 247] as /*oo /(*> y) = Q( x > y> z )f( z ) d K z ) x > °> y e Q = (-°°> °°) J -00 where , /-i2 1/2 (z-rj;)exp(x 2 /21ogr) (-?y*+2ryz-r*z*\ e(x '* z) = Jo .(-log.ni-^ exp l-i^-) " r -If the Hermite conjugate function operator Tis defined by (Tf)(y)=lim x _ 0+ f(x, y) a.e., then one of the main results of [5] is that Tis of weak-type (1, 1) and strongtype (p,p) for all/7>l.
more » ... This result together with a theorem of Boyd [3, Theorem 1] shows that if L p (£l) is a rearrangement invariant space with upper and lower indices a and ft respectively (see [3] for definitions and notation) which satisfy 0<^\, and to reflexivity in the case of the Orlicz spaces. For the infinite non-atomic measure spaces this was proved by Boyd [2], and using similar methods Kerman [4] and the author [1] obtained the same results for the finite non-atomic and the purely atomic cases respectively. Thus, Kerman's result applies in the present situation, and in analogy with known results for the classical Hilbert transform [2], the classical conjugate function
doi:10.4153/cmb-1973-059-5 fatcat:rjy7fxgzrnhyxet477khml4pxu