The Lp-boundedness of pseudodifferential operators with estimates of parabolic type and product type
Masao Yamazaki · Journal of the Mathematical Society of Japan · 1986
Hence the condition in Theorem 1 is sharp.If $\omega_{1}(t)$ is of the form $(1-\log t)^{\delta}$ , then (1.3) holds if and only if $\delta<-N/2$ .On the other hand, by putting $N=1,$ $M^{(1)}=M$ and $[\cdot]_{1}=[\cdot ]$ , we have the following result on the symbols satisfying estimates of parabolic type, which is a modification of Theorem 7 in [14].COROLLARY.Let $\omega(t)$ be as above, and assume that $\int_{0}^{1}\frac{\omega(i)^{2}}{t}dt<\infty$ .If a symbol $P(x, \xi)$ sa fisfies the estimates $|\partial_{\xi_{l}}^{k}P(x, \xi)|\leqq C(1+[\xi])^{-m_{l}k}$ and