One-parameter families of operators in ℂ
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We introduce classes of one-parameter families (OPF) of operators on C c t8 (ℂ) which characterize the behavior of operators associated to the\(\bar \partial - problem\) problem in the weighted space L2 (ℂ, e−2p) where p is a subharmonic, nonharmonic polynomial. We prove that an order 0 OPF operator extends to a bounded operator from Lq (ℂ) to itself, 1 < q < ∞, with a bound that depends on q and the degree of p but not on the parameter τ or the coefficients of p. Last, we show that there is a one-to-one correspondence given by the partial Fourier transform in τ between OPF operators of order m ≤ 2 and nonisotropic smoothing (NIS) operators of order m ≤ 2 on polynomial models in ℂ2.
Math Subject Classifications32W50 32W30 32T25
Key Words and PhrasesFinite type NIS operator one-parameter families weakly pseudoconvex domain
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