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Hermite conjugate functions and rearrangement invariant spaces

1973
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Canadian mathematical bulletin
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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.

doi:10.4153/cmb-1973-059-5
fatcat:rjy7fxgzrnhyxet477khml4pxu