Heat flow, BMO, and the compactness of Toeplitz operators

In this paper, it is shown that the Berezin-Toeplitz operator Tg is compact or in the Schatten class Sp of the Segal-Bargmann space for 1<p<∞ whenever g̃^(s)∈C₀(C{double-struck}ⁿ) (vanishes at infinity) or g̃^(s)∈L^p(Cⁿ,dv), respectively, for some s with 0<s<14, where g̃(s) is the heat transform of g on C{double-struck}ⁿ. Moreover, we show that compactness of Tg implies that g̃^(s) is in C₀(Cⁿ) for all s>14 and use this to show that, for g∈BMO¹(C{double-struck}ⁿ), we have g̃^(s) is in C₀(C{double-struck}ⁿ) for some s>0 only if g̃^(s) is in C₀(Cⁿ) for all s>0. This " backwards heat flow" result seems to be unknown for g∈BMO¹ and even g∈L^∞. Finally, we show that our compactness and vanishing " backwards heat flow" results hold in the context of the weighted Bergman space L_a²(B{double-struck}_n,dv_α), where the "heat flow" g̃^(s) is replaced by the Berezin transform B_α(g) on L_a²(B{double-struck}_n,dv_α) for α>-1.