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Eric Amar

Publications and source records attributed to Eric Amar.

28 records · Page 2Linked to original sources
arXiv2014

Estimates $ L^{r}-L^{s}$ for solutions of the $\bar \partial $ equation in strictly pseudo convex domains in ${\mathbb{C}}^{n}.$

We prove estimates for solutions of the $\bar \partial u=ω$ equation in a strictly pseudo convex domain $ Ω$ in ${\mathbb{C}}^{n}.$ For instance if the $ (p,q)$ current $ω$ has its coefficients in $L^{r}(Ω)$ with $1\leq r<2(n+1)$ then there is a solution $u$ in $L^{s}(Ω)$ with $\ \frac{1}{s}=\frac{1}{r}-\frac{1}{2(n+1)}.$ We also have $BMO$ and Lipschitz estimates for $r\geq 2(n+1).$ These results were already done by S. Krantz in the case of $(0,1)$ forms and just for the $L^{r}-L^{s}$ part by L. Ma and S. Vassiliadou for general $(p,q)$ forms. To get the complete result we propose another approach, based on Carleson measures of order $α$ and on the subordination lemma.

math.CV↗
arXiv2014

A subordination Principle. Applications

The subordination principle states roughly : if a property is true for Hardy spaces in some kind of domains in $C^n$ then it is also true for the Bergman spaces of the same kind of domains in $C^{n-1}$. We give applications of this principle to Bergman-Carleson measures, interpolating sequences for Bergman spaces, $A^p$ Corona theorem and characterization of the zeros set of Bergman-Nevanlinna class.

math.CV↗
arXiv2014

The raising steps method. Applications to the $\bar \partial $ equation in Stein manifolds

In order to get estimates on the solutions of the equation $\bar \partial u=ω$ on Stein manifold, we introduce a new method the "raising steps method", to get global results from local ones. In particular it allows us to transfer results form open sets in ${\mathbb{C}}^{n}$ to open sets in a Stein manifold.\ \par Using it we get $\displaystyle L^{r}-L^{s}$ results for solutions of equation $\bar \partial u=ω$ with a gain, $\displaystyle s>r,$ in strictly pseudo convex domains in Stein manifolds.\ \par We also get $\displaystyle L^{r}-L^{s}$ results for domains in ${\mathbb{C}}^{n}$ locally biholomorphic to convex domains of finite type.

math.CV↗
arXiv2012

On separated Carleson sequences in the unit disc of ${\mathbb{C}}.$

The interpolating sequences for $H^{\infty}({\mathbb{D}}),$ the bounded holomorphic function in the unit disc ${\mathbb{D}}$ of the complex plane ${\mathbb{C}},$ {\small where characterised by L. Carleson by metric conditions on the points. They are also characterised by "dual boundedness" conditions which imply an infinity of functions. A. Hartmann proved recently that just one function in $H^{\infty}({\mathbb{D}})$ was enough to characterize interpolating sequences for $H^{\infty}({\mathbb{D}}).$ In this work we use the "hard" part of the proof of Carleson for the Corona theorem, to extend Hartman's result and answer a question he asked in his paper.}\ \par

math.CV↗
arXiv2011

On $L^r$ hypoellipticity of solutions with compact support of the Cauchy-Riemann equation

In one complex variable, the existence of a compactly supported solution to the Cauchy-Riemann equation is related to the vanishing of certain integrals of the data; trying to generalize this approach, we find an explicit construction, via convolution, for a compactly supported solution in $\mathbb{C}^n$, which allows to estimate the $L^p$ norm of the solution. We also investigate the possible generalizations of this method to domains of the form $P\setminus Z$, where $P$ is a polydisc and $Z$ is the zero locus of some holomorphic function.

math.CV↗
arXiv2009

Uniform minimality, unconditionality and interpolation in backward shift invariant spaces

We discuss relations between uniform minimality, unconditionality and interpolation for families of reproducing kernels in backward shift invariant subspaces. This class of spaces contains as prominent examples the Paley-Wiener spaces for which it is known that uniform minimality does in general neither imply interpolation nor unconditionality. Hence, contrarily to the situation of standard Hardy spaces (and other scales of spaces), changing the size of the space seems %in this context necessary to deduce unconditionality or interpolation from uniform minimality. Such a change can take two directions: lowering the power of integration, or "increasing" the defining inner function (e.g. increasing the type in the case of Paley-Wiener space). Khinchin's inequalities play a substantial rôle in the proofs of our main results.

math.CV↗
arXiv2006

On linear extension for interpolating sequences

Title: On linear extension for interpolating sequences. Author: Eric Amar Abstract: Let A be a uniform algebra on the compact space X and $σ$ a probability measure on X. We define the Hardy spaces $H^{p}(σ)$ and the $H^{p}(σ)$ interpolating sequences S in the p-spectrum ${\mathcal{M}}_{p}$ of $σ$. We prove, under some structural hypotheses on $σ$ that "Carleson type" conditions on S imply that S is interpolating with a linear extension operator in $\displaystyle H^{s}(σ), s<p$ provided that either $p=\infty $ or $p\leq 2$. This gives new results on interpolating sequences for Hardy spaces of the ball and the polydisc. In particular in the case of the unit ball of ${\mathbb{C}}^{n}$ we get that if there is a sequence $\{ρ_{a}\}_{a\in S}$ bounded in $H^{\infty}({\mathbb{B}})$ such that $\forall a,b\in S, ρ_{a}(b)=δ_{ab}$, then S is $H^{p}({\mathbb{B}})$-interpolating with a linear extension operator for any $1\leq p<\infty $.

math.CV↗
arXiv1997

Finite interpolation with minimum uniform norm in C^n

Given a finite sequence $a:={a_1, ..., a_N}$ in a domain $Ω\subset C^n$, and complex scalars $v:={v_1, ..., v_N}$, consider the classical extremal problem of finding the smallest uniform norm of a holomorphic function verifying $f(a_j)=v_j$ for all $j$. We show that the modulus of the solutions to this problem must approach its least upper bound along a subset of the boundary of the domain large enough to contain the support of a measure whose hull contains a subset of the original $a$ large enough to force the same minimum norm. Furthermore, all the solutions must agree on a variety which also contains this hull. An example is given to show that the inclusions can be strict.

math.CV↗