Let $\mu$ be a positive Borel measure on the interval $[0,1)$. The Hankel matrix $\mathcal{H}_{\mu}=(\mu_{n,k})_{n,k\geq 0}$ with entries $\mu_{n,k}=\mu_{n+k}$, where $\mu_{n}=\int_{[0,1)}t^n{\rm d}\mu(t)$, induces formally the operator as $\mathcal{DH}_\mu(f)(z)=\sum\limits_{n=0}^\infty\Big(\sum\limits_{k=0}^\infty \mu_{n,k}a_k\Big)(n+1)z^n , z\in \mathbb{D},$where $f(z)=\sum\limits_{n=0}^{\infty}a_nz^n$ is an analytic function in $\mathbb{D}$.We characterize the positive Borel measures on $[0,1)$ such that $\mathcal{DH}_\mu(f)(z)= \int_{[0,1)} \frac{f(t)}{{(1-tz)^2}} {\rm d}\mu(t)$ for all $f$ in the Hardy spaces $H^p(0<p<\infty)$, and among these we describe those for which $\mathcal{DH}_\mu$ is a bounded (resp., compact) operator from $H^p(0<p <\infty)$ into $H^q(q > p$ and $q\geq 1$). We also study the analogous problem in the Hardy spaces $H^p(1\leq p\leq 2)$.
Shanli YE
,
Guanghao FENG
. A DERIVATIVE-HILBERT OPERATOR ACTING ON HARDY SPACES*[J]. Acta mathematica scientia, Series B, 2023
, 43(6)
: 2398
-2412
.
DOI: 10.1007/s10473-023-0605-6
[1] Anderson J, Pommerenke C, Clunie J. On Bloch functions and normal functions. J Reine Angew Math, 1974, 270(1): 12-37
[2] Chatzifountas C, Girela D, Peláez J Á. A generalized Hilbert matrix acting on Hardy spaces. J Math Anal Appl, 2014, 413(1): 154-168
[3] Carleson L. Interpolations by bounded analytic functions and the corona problem. Ann of Math, 1962, 76(3): 547-559
[4] Cowen C C, MacCluer B D. Composition Operators on Spaces of Analytic Functions. Boca Raton: CRC Press, 1995
[5] Cwikel M, Kalton N J. Interpolation of compact operators by the methods of Calderón and Gustavsson-Peetre. Proc Edinburgh Math Soc, 1995, 38(2): 261-276
[6] Deng F, Ouyan C, Deng G. Some questions regarding verification of Carleson measure. Acta Math Sci, 2021, 41B(6): 2136-2148
[7] Diamantopoulos E, Siskakis A G. Composition operators and the Hilbert matrix. Studia Math, 2000, 140(2): 191-198
[8] Duren P L. Extension of a theorem of Carleson. Bull Amer Math Soc, 1969, 75: 143-146
[9] Duren P L.Theory of $H^p$ Spaces. New York: Academic Press, 1970
[10] Duren P L, Romberg B W, Shields A L. Linear functionals on $H^p$ spaces with $0< p< 1$. Reine Angew Math, 1969, 238(1): 32-60
[11] Galanopoulos P, Girela D, Peláez J Á, Siskakis A G. Generalized Hilbert operators. Ann Acad Sci Fenn Math, 2014, 39: 231-258
[12] Galanopoulos P, Peláez J Á. A Hankel matrix acting on Hardy and Bergman spaces. Studia Math, 2010, 200(3): 201-220
[13] Girela D.Analytic functions of bounded mean oscillation//Complex Function Spaces, University Joensuu Dept Math Rep Ser, vol 4. Joensuu: University Joensuu, 2001: 61-170
[14] Girela D, Merchán N. A generalized Hilbert operator acting on conformally invariant spaces. Banach J Math Anal, 2016, 12(2): 374-398
[15] Gnuschke-Hauschild D, Pommerenke C. On Bloch functions and gap series. J Reine Angew Math, 1986, 367: 172-186
[16] Li S, Zhou J. Essential norm of generalized Hilbert matrix from Bloch type spaces to BMOA and Bloch space. AIMS Math, 2021, 6: 3305-3318
[17] Maccluer B, Zhao R. Vanishing logarithmic Carleson measures. Illinois J Math, 2002, 46(2): 507-518
[18] Ye S, Zhou Z. A derivative-Hilbert operator acting on the Bloch space. Complex Anal Oper Theory, 2021, 15(5): 88
[19] Ye S, Zhou Z. A derivative-Hilbert operator acting on Bergman spaces. J Math Anal Appl, 2022, 506(1): 125553
[20] Zhao R. On logarithmic Carleson measures. Acta Sci Math, 2003, 69: 605-618
[21] Zhu K. Bloch type spaces of analytic functions. Rocky Mountain J Math, 1993, 23(3): 1143-1177
[22] Zhu K.Operator Theory in Function Spaces. Surveys Monoge 138. 2nd ed. Providence RI: Amer Math Soc, 2007
