$H^\infty$谓词演算与平方函数估计的一个等价性
A sharp equivalence between $H^\infty$ functional calculus and square function estimates
摘要：
Let T_t = e^{-tA} be a bounded analytic semigroup on Lp, with 1<p< \infty . It is known that if A and its adjoint A^* both satisfy square function estimates \bignorm { \bigl ( \int_{0}^{\infty} | A^{1/2} T_t(x)|^2\, dt\, \bigr )^{1/2}_{Lp} \lesssim \norm{x} and \bignorm { \bigl ( \int_{0}^{\infty} |A^{*}^{1/2} T_t^*(y)|^2\, dt\, \bigr )^{1/2}_{Lp'} \lesssim \norm{y} for x in Lp and y in Lp', then A admits a bounded H^{ \infty }(Σ_θ) functional calculus for any θ> \frac π{2}. We show that this actually holds true for some θ< \frac π{2}.
英文摘要：
Let T_t = e^{-tA} be a bounded analytic semigroup on Lp, with 1<p<\infty. It is known that if A and its adjoint A^* both satisfy square function estimates \bignorm{\bigl(\int_{0}^{\infty}| A^{1/2} T_t(x)|^2\, dt\,\bigr)^{1/2}_{Lp} \lesssim \norm{x} and \bignorm{\bigl(\int_{0}^{\infty}|A^{*}^{1/2} T_t^*(y)|^2\, dt\,\bigr)^{1/2}_{Lp'} \lesssim \norm{y} for x in Lp and y in Lp', then A admits a bounded H^{\infty}(Σ_θ) functional calculus for any θ>\fracπ{2}. We show that this actually holds true for some θ<\fracπ{2}.
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