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Eugenia Malinnikova

Publications and source records attributed to Eugenia Malinnikova.

27 records · Page 2Linked to original sources
arXiv2014

Coefficient multipliers of growth spaces of harmonic functions

Let $h_g^\infty$ be the space of harmonic functions in the unit ball that are bounded by some increasing radial function $g(r)$ with $\lim_{r\rightarrow 1} g(r)=+\infty$; these spaces are called growth spaces. We describe functions in growth spaces by the Cesàro means of their expansions in harmonic polynomials and apply this characterization to study coefficient multipliers between growth spaces. A series of examples of multipliers is given and some results by A. L. Shields and D. L. Williams and G. Bennett, D. A. Stegenga and R. M. Timoney are generalized to harmonic functions in higher dimensions.

math.CA↗
arXiv2014

On ratios of harmonic functions

Let $u$ and $v$ be harmonic in $ Ω\subset \mathbb{R}^n$ functions with the same zero set $Z$. We show that the ratio $f$ of such functions is always well-defined and is real analytic. Moreover it satisfies the maximum and minimum principles. For $n=3$ we also prove the Harnack inequality and the gradient estimate for the ratios of harmonic functions, namely ${ \sup\limits_{K} |f| \leq C \inf\limits_{K}| f| \quad \& \quad \sup\limits_{K} |\nabla f| \leq C \inf\limits_{K}| f| }$ for any compact subset $K$ of $Ω$, where the constant $C$ depends on $K$, $Z$, $Ω$ only. In dimension two the first inequality follows from the boundary Harnack principle and the second from the gradient estimate recently obtained by Mangoubi. It is an open question whether these inequalities remain true in higher dimensions ($n \geq 4$).

math.AP↗
arXiv2013

A note on unique continuation for discrete harmonic functions

In this note we give a quantitative version of the following simple observation: a discrete harmonic function on the lattice may vanish at each point of a large cube without being zero identically, at the same time there is a version of three balls theorem for discrete harmonic functions since they resemble continuous ones on large scales.

math.AP↗
arXiv2013

Wavelet decomposition of harmonic functions in growth spaces

Spaces of harmonic functions in upper half-space with controlled growth near the boundary are described in terms of multiresolution approximations. The results are applied to prove the law of the iterated logarithm for the oscillation of harmonic functions along vertical lines.

math.FA↗
arXiv2010

Radial growth of harmonic functions in the unit ball

We study harmonic functions which admit a certain majorant in the unit ball in $\R^m $. We prove that when the majorant fulfills a doubling condition, the extremal growth or decay may occur only along small sets of radii, and we give precise estimates of these exceptional sets.

math.CA↗
arXiv2009

Orthonormal sequences in $L^2(R^d)$ and time frequency localization

We study uncertainty principles for orthonormal bases and sequences in $L^2(\R^d)$. As in the classical Heisenberg inequality we focus on the product of the dispersions of a function and its Fourier transform. In particular we prove that there is no orthonormal basis for $L^2(\R)$ for which the time and frequency means as well as the product of dispersions are uniformly bounded. The problem is related to recent results of J. Benedetto, A. Powell, and Ph. Jaming. Our main tool is a time frequency localization inequality for orthonormal sequences in $L^2(\R^d)$. It has various other applications.

math.CA↗