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In this paper, assume that   L  =  −    Δℍ    n+  V   is a Schrödinger operator on the Heisenberg groupℍ    n    , where the nonnegative potential   V   belongs to the reverse Hölder classB    Q  /  2    . By the aid of the subordinate formula, we investigate the regularity properties of the time-fractional derivatives of semigroupse    −  t  Lt  &gt;  0ande    −  t    Lt  &gt;  0    , respectively. As applications, using fractional square functions, we characterize the Hardy-Sobolev type spaceH    L    1  ,  αℍ    nassociated with   L  . Moreover, the fractional square function characterizations indicate an equivalence relation of two classes of Hardy-Sobolev spaces related with   L  .

journal of function spaces

Regularities of Time-Fractional Derivatives of Semigroups Related to Schrodinger Operators with Application to Hardy-Sobolev Spaces on Heisenberg Groups