Taiwanese Journal of Mathematics

VECTOR-VALUED FUNCTIONS INTEGRABLE WITH RESPECT TO BILINEAR MAPS

O. Blasco and J. M. Calabuig

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Abstract

Let $(\Omega, \Sigma, \mu)$ be a $\sigma-$finite measure space, $1\le p \lt \infty$, $X$ be a Banach space $X$ and ${\cal B} :X\times Y \to Z$ be a bounded bilinear map. We say that an $X$-valued function $f$ is $p-$integrable with respect to ${\cal B}$ whenever $\sup\{\int_\Omega\|{\cal B}(f(w),y)\|^pd\mu: \|y\|=1\}$ is finite. We identify the spaces of functions integrable with respect to the bilinear maps arising from H\"older's and Young's inequalities. We apply the theory to give conditions on $X$-valued kernels for the boundedness of integral operators $T_{{\cal B}}(f) (w)=\int_{\Omega'}{{\cal B}}(k(w,w'),$ $f(w'))d\mu'(w')$ from ${\mathrm L}^p(Y)$ into ${\mathrm L}^p(Z)$, extending the results known in the operator-valued case, corresponding to ${\cal B}:{\mathrm L}(X,Y)\times X\to Y$ given by ${\cal B}(T,x)=Tx$.

Article information

Source
Taiwanese J. Math., Volume 12, Number 9 (2008), 2387-2403.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500405186

Digital Object Identifier
doi:10.11650/twjm/1500405186

Mathematical Reviews number (MathSciNet)
MR2479062

Zentralblatt MATH identifier
1171.42010

Subjects
Primary: 42B30: $H^p$-spaces 47B35: Toeplitz operators, Hankel operators, Wiener-Hopf operators [See also 45P05, 47G10 for other integral operators; see also 32A25, 32M15]

Keywords
vector-valued functions bilinear map

Citation

Blasco, O.; Calabuig, J. M. VECTOR-VALUED FUNCTIONS INTEGRABLE WITH RESPECT TO BILINEAR MAPS. Taiwanese J. Math. 12 (2008), no. 9, 2387--2403. doi:10.11650/twjm/1500405186. https://projecteuclid.org/euclid.twjm/1500405186


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References

  • H. Amann, Operator-valued Fourier multipliers,vector-vaued Besov spaces and applications, Math. Nachr., 186 (1997), 15-56.
  • J. L. Arregui and O. Blasco, On the Bloch space and convolutions of functions in the $\L^p$-valued case, Collect. Math., 48 (1997), 363-373.
  • J. L. Arregui and O. Blasco, Convolutions of three functions by means of bilinear maps and applications, Illinois J. Math., 43 (1999), 264-280.
  • O. Blasco, Convolutions by means of bilinear maps, Contemp. Math., 232 (1999), 85-103.
  • O. Blasco, Bilinear maps and convolutions, Research and Expositions in Math., 24 (2000), 45-55.
  • J. Diestel and J. J. Uhl Vector measures, American Mathematical Society Mathematical Surveys, No. 15, 1977.
  • J. Garc\'\tiny la-Cuerva and J. L. Rubio de Francia, Weighted norm inequalities and relatedtopics, North-Holland, Amsterdam, 1985.
  • M. Girardi, and L. Weis, Integral operators with operator-valued kernels, J. Math. Anal. Appl., 290 (2004), 190-212.
  • R. A. Ryan, Introduction to tensor products of Banach spaces, Springer Monographs in Mathematics. Springer, 2002.