Boundary oscillations of harmonic functions in Lipschitz domains

Standard

Boundary oscillations of harmonic functions in Lipschitz domains. / Mozolyako, P.

In: Collectanea Mathematica, Vol. 68, No. 3, 01.09.2017, p. 359-376.

Research output: Contribution to journal › Article › peer-review

BibTeX

@article{a578e9f1700e4601bd16ec4d9888b9c2,

title = "Boundary oscillations of harmonic functions in Lipschitz domains",

abstract = "Let u(x, y) be a harmonic function in the halfspace Rn× R+ that grows near the boundary not faster than some fixed majorant w(y). Recently it was proven that an appropriate weighted average along the vertical lines of such a function satisfies the law of iterated logarithm (LIL). We extend this result to a class of Lipschitz domains in Rn+1. In particular, we obtain the local version of this LIL for the upper halfspace. The proof is based on approximation of the weighted averages by a Bloch function, satisfying some additional condition determined by the weight w. The growth rate of such Bloch function depends on w and, for slowly increasing w, turns out to be slower than the one provided by LILs of Makarov and Llorente. We discuss the necessary condition for an arbitrary Bloch function to exhibit this type of behaviour.",

keywords = "Bloch functions, Growth classes, Harmonic functions, Radial weights",

author = "P. Mozolyako",

year = "2017",

month = sep,

day = "1",

doi = "10.1007/s13348-016-0177-z",

language = "English",

volume = "68",

pages = "359--376",

journal = "Collectanea Mathematica",

issn = "0010-0757",

publisher = "Springer Nature",

number = "3",

}

RIS

TY - JOUR

T1 - Boundary oscillations of harmonic functions in Lipschitz domains

AU - Mozolyako, P.

PY - 2017/9/1

Y1 - 2017/9/1

N2 - Let u(x, y) be a harmonic function in the halfspace Rn× R+ that grows near the boundary not faster than some fixed majorant w(y). Recently it was proven that an appropriate weighted average along the vertical lines of such a function satisfies the law of iterated logarithm (LIL). We extend this result to a class of Lipschitz domains in Rn+1. In particular, we obtain the local version of this LIL for the upper halfspace. The proof is based on approximation of the weighted averages by a Bloch function, satisfying some additional condition determined by the weight w. The growth rate of such Bloch function depends on w and, for slowly increasing w, turns out to be slower than the one provided by LILs of Makarov and Llorente. We discuss the necessary condition for an arbitrary Bloch function to exhibit this type of behaviour.

AB - Let u(x, y) be a harmonic function in the halfspace Rn× R+ that grows near the boundary not faster than some fixed majorant w(y). Recently it was proven that an appropriate weighted average along the vertical lines of such a function satisfies the law of iterated logarithm (LIL). We extend this result to a class of Lipschitz domains in Rn+1. In particular, we obtain the local version of this LIL for the upper halfspace. The proof is based on approximation of the weighted averages by a Bloch function, satisfying some additional condition determined by the weight w. The growth rate of such Bloch function depends on w and, for slowly increasing w, turns out to be slower than the one provided by LILs of Makarov and Llorente. We discuss the necessary condition for an arbitrary Bloch function to exhibit this type of behaviour.

KW - Bloch functions

KW - Growth classes

KW - Harmonic functions

KW - Radial weights

UR - http://www.scopus.com/inward/record.url?scp=85026912723&partnerID=8YFLogxK

U2 - 10.1007/s13348-016-0177-z

DO - 10.1007/s13348-016-0177-z

M3 - Article

AN - SCOPUS:85026912723

VL - 68

SP - 359

EP - 376

JO - Collectanea Mathematica

JF - Collectanea Mathematica

SN - 0010-0757

IS - 3

ER -

ID: 119109226