ABSTRACT. If H > 0 is the generator of a hypercontractive semigroup (HCSG), it is known that (H + 1)~'''2 is a bounded operator from Lp to IP, 1 < p < °°. We prove that (H + 1)~'2 is bounded from L to the Orlicz space L ln"*"L, basing the proof on the uniform semiboundedness of the operator H + V, for suitable V. We also prove by an interpolation argument, that (H + l)-" is bounded from LP to iPXn^L, 2 < p < °°. Another interpolation argument shows that (H + 1)~'/2 is bounded from LP(ln+L)m to ¿P(ln+L)m + 1, 2 < p < °° and m a positive integer. Finally, we identify the topological duals of the spaces mentioned above.
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