## Abstract

An extremal problem for integrals on a measure space with parameters y,y0∈Y is considered, where Y is a set of points. A class of integrands for which the integral attains its supremum over y∈ Y at a fixed point y0 is described. The integrands of such kind in R^{n} and on the unit sphere S^{n} ^{-} ^{1} in R^{n} with vector parameters are pointed out. As applications, we give a new simple proof of the sharp real-part estimate for analytic functions from the Hardy spaces in the upper half-plane as well as a solution of the extremal problem for some integrals on S^{n} ^{-} ^{1} with vector parameters. As consequence, we find the sharp constant in a pointwise estimate for solutions of the Lamé system in the upper half-space of R^{n} with boundary data from L^{p} for the case p= (n+ 2 m) / (2 m) , where m is a positive integer.

Original language | English |
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Pages (from-to) | 1477-1490 |

Number of pages | 14 |

Journal | Complex Analysis and Operator Theory |

Volume | 11 |

Issue number | 7 |

DOIs | |

State | Published - 1 Oct 2017 |

## Keywords

- Abstract parameters
- Analytic functions
- Extremal problem
- Integrals on a measure space
- Lamé system
- Sharp pointwise estimates