×

Poincaré and Opial inequalities for vector-valued convolution products. (English) Zbl 1263.26030

Let \(\delta : [0,\infty) \rightarrow X\) (or \((\delta(u))_{u\geq 0} \subset X\)) be a continuous function. Then the function \(\delta_\alpha\), (\(\alpha >0\)), is defined as \[ \delta_\alpha (t) = \frac{1}{\Gamma(\alpha)} \int_0^t (t-u)^{\alpha -1}\delta(u) du, \;\;\;t>0. \] Also, we have \( \| \| \delta_\alpha (t) \|_X \|_{L^{\nu}([0,a])} = \left( \int_0^a \| \delta_\alpha (t)\|_X^{\nu} dt \right)^{1/{\nu}}\).
The following result is the Poincaré type inequality for certain convolution products.
Theorem. Let \((\delta(u))_{u\geq 0} \subset X\) be a continuous map and \(\alpha >0\). Then \[ \| \| \delta_\alpha (t) \|_X \|_{L^{\nu}([0,a])} \leq \frac{a^{\alpha -1+\frac 1p +\frac{1}{\nu}}}{\Gamma(\alpha) (1-(1-\alpha)p)^{\frac 1p} (\nu(\frac 1p +\alpha -1)+1)^{\frac{1}{\nu}}} \| \| \delta (t) \|_X \|_{L^{q}([0,a])} \] for \(a>0\), in the case \(0<\alpha \leq 1\) with \(1<p<\frac{1}{1-\alpha}\); and in the case \(\alpha \geq 1\) with \(1<p<\infty\); \(\frac 1p +\frac 1q =1\), and \(\nu \geq 1\).
Further, an Opial type inequality for vector-valued convolution products is given.
Theorem. Let \((\delta(u))_{u\geq 0} \subset X\) be a continuous map and \(\alpha >0\). Then \[ \begin{split} \int_0^t \| \delta_\alpha (s) \|_X \| \delta (s) \|_X ds \\ \leq \frac{t^{\alpha-1+\frac 2p}}{2^{\frac 1q} \Gamma(\alpha) (p(\alpha -1)+1)^{\frac 1p} (p(\alpha -1)+2)^{\frac 1p}} \left( \int_0^t \| \delta (s) \|_X^q ds\right)^{\frac 2q}\end{split} \] for \(t>0, 1<p\), \(p(\alpha -1)>-1\) and \(\frac 1p +\frac 1q =1\).
These results are applied to infinitesimal generators of \(C_0\)-semigroups and cosine functions.

MSC:

26D10 Inequalities involving derivatives and differential and integral operators
26D15 Inequalities for sums, series and integrals
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Chua, S. K.; Wheeden, R. L., A note on sharp 1-dimensional Poincaré inequalities, Proc. Amer. Math. Soc., 134, 2309-2316 (2006) · Zbl 1116.26012
[2] Anastassiou, G. A., Poincaré like inequalities for semigroups, cosine and sine operator functions, Semigroup Forum, 78, 54-67 (2009) · Zbl 1166.47039
[3] Opial, Z., Sur une inégalité, Ann. Polon. Math., 8, 29-32 (1960) · Zbl 0089.27403
[4] Anastassiou, G. A., Opial type inequalities for semigroups, Semigroup Forum, 75, 624-633 (2007) · Zbl 1133.26028
[5] Anastassiou, G. A., Opial type inequalities for cosine and sine operator functions, Semigroup Forum, 76, 149-158 (2008) · Zbl 1137.47033
[6] Martínez, C.; Sanz, M., (The Theory of Fractional Powers of Operators. The Theory of Fractional Powers of Operators, North-Holland Mathematics Studies, vol. 187 (2001), Elsevier Science, B.V.) · Zbl 0971.47011
[7] Yosida, K., (Functional Analysis. Functional Analysis, Grundlehren der mathematischen Wissenschaften, vol. 123 (1965), Springer: Springer Berlin)
[8] Goldstein, J. A., Semigroups of Linear Operators and Applications (1985), Oxford Univ. Press: Oxford Univ. Press Oxford · Zbl 0592.47034
[9] Krägeloh, A. M., Two families of functions related to the fractional powers of generators of strongly continuous contraction semigroups, J. Math. Anal. Appl., 283, 459-467 (2003) · Zbl 1061.47038
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.