The derivative of the atomic function is not in 𝐵^{2/3}

Belna, Charles L.; Muckenhoupt, Benjamin

doi:10.1090/S0002-9939-1977-0586555-2

The derivative of the atomic function is not in $B^{2/3}$
HTML articles powered by AMS MathViewer

by Charles L. Belna and Benjamin Muckenhoupt PDF

Proc. Amer. Math. Soc. 63 (1977), 129-130 Request permission

Abstract:

H. A. Allen and C. L. Belna have shown that the derivative of the atomic function $A(z) = \exp [(z + 1)/(z - 1)]$ is in ${B^p}$ for $0 < p < 2/3$, where ${B^p}$ is the containing Banach space for the Hardy class ${H^p}(0 < p < 1)$. Here we show that $A’(z)$ does not belong to any of the other ${B^p}$ spaces.

References

P. R. Ahern and D. N. Clark, On inner functions with $B^{p}$ derivative, Michigan Math. J. 23 (1976), no. 2, 107–118. MR 414884, DOI 10.1307/mmj/1029001659
H. A. Allen and C. L. Belna, Singular inner functions with derivative in $B^{p}$, Michigan Math. J. 19 (1972), 185–188. MR 299796, DOI 10.1307/mmj/1029000852
P. L. Duren, B. W. Romberg, and A. L. Shields, Linear functionals on $H^{p}$ spaces with $0<p<1$, J. Reine Angew. Math. 238 (1969), 32–60. MR 259579
D. J. Newman and Harold S. Shapiro, The Taylor coefficients of inner functions, Michigan Math. J. 9 (1962), 249–255. MR 148874, DOI 10.1307/mmj/1028998724