Banach函数空间上的Bessel(Riesz)位势及其应用(I)理论

本文首先引入Ｂｅｓｅｌ（Ｒｉｅｓｚ）位势Ｋ¨ｏｔｈｅ函数空间Ｘｓ（Ｘｓ）的概念，然后讨论一类算子在Ｌｅｂｅｓｇｕｅ－位势Ｋ¨ｏｔｈｅ函数空间Ｌｑ（－Ｔ，Ｔ；Ｘｓ）上的对偶估计．由此我们得到半群ｅｘｐ（ｉｔ（－Δ）ｍ／２）和算子Ａ：＝∫ｔ０ｅｘｐ（ｉ（ｔ－τ）（－Δ）ｍ／２）·ｄτ在Ｌｅｂｅｓｇｕｅ－Ｂｅｓｏｖ空间Ｌｑ－Ｔ，Ｔ；·Ｂｓｐ，２中的一些时间－－空间Ｌｐ－Ｌｐ′估计．本文的系列文将给出这些估计的应用

Bessel(Riesz)Potentials on Banach Function Spacesand Their Applications I, Theory

In this paper, we shall introduce the notion of the Bessel(Riesz)potential Kthe function spaces X s( s) and give some dual estimates for a class of operators determined by a semi-group in the spaces L q(-T, T; X s)(L q(-T, T; s)).Moreover, some time-space L p-L p′  estimates for the semi-groupexp ( i t(-Δ) m/2 ) and the operator A:=∫ t 0exp( i (t-τ)(-Δ) m/2 )·dτ in the Lebesgue-Besov spacesL q-T, T; AB·U s p,2  are given. On the basis of these results, in a sequel paper we shall present some further applications to a class of nonlinear evolution equation.