arXiv subjects

Serban Costea

Publications and source records attributed to Serban Costea.

7 recordsLinked to original sources

arXiv2018

Sobolev-Lorentz capacity and its regularity in the Euclidean setting

This paper studies the Sobolev-Lorentz capacity and its regularity in the Euclidean setting for $n \ge 1$ integer. We extend here our previous results on the Sobolev-Lorentz capacity obtained for $n \ge 2.$ Moreover, for $n \ge 2$ integer we obtain a few new results concerning the $n,1$ relative and global capacities. We obtain sharp estimates for the $n,1$ relative capacity of the concentric condensers $(\overline{B}(0,r), B(0,1))$ for all $r$ in $[0,1).$ As a consequence we obtain the exact value of the $n,1$ capacity of a point relative to all its bounded open neighborhoods from ${\mathbf{R}}^n$ when $n \ge 2.$ We also show that this aforementioned constant is the value of the $n,1$ global capacity of any point from ${\mathbf{R}}^n,$ where $n \ge 2$ is integer. This allows us to give a new proof of the embedding $H_{0}^{1,(n,1)}(Ω) \hookrightarrow C(\overlineΩ) \cap L^{\infty}(Ω),$ where $Ω\subset {\mathbf{R}}^n$ is open and $n \ge 2$ is an integer. In the penultimate section of our paper we prove a new weak convergence result for bounded sequences in the non-reflexive spaces $H^{1,(p,1)}(Ω)$ and $H_{0}^{1,(p,1)}(Ω).$ The weak convergence result concerning the spaces $H^{1,(p,1)}(Ω)$ is valid whenever $1<p<\infty,$ while the weak convergence result concerning the spaces $H_{0}^{1,(p,1)}(Ω)$ is valid whenever $1 \le n<p<\infty$ or $1<n=p<\infty.$ As a consequence of the weak convergence result concerning the spaces $H_{0}^{1,(p,1)}(Ω),$ in the last section of our paper we show that the relative and the global $(p,1)$ and $p,1$ capacities are Choquet whenever $1 \le n<p<\infty$ or $1<n=p<\infty.$

math.AP↗

arXiv2017

Sobolev-Lorentz spaces in the Euclidean setting and counterexamples

This paper studies the inclusions between different Sobolev-Lorentz spaces $W^{1,(p,q)}(Ω)$ defined on open sets $Ω\subset {\mathbf{R}^n},$ where $n \ge 1$ is an integer, $1<p<\infty$ and $1 \le q \le \infty.$ We prove that if $1 \le q<r \le \infty,$ then $W^{1,(p,q)}(Ω)$ is strictly included in $W^{1,(p,r)}(Ω).$ We show that although $H^{1,(p,\infty)}(Ω) \subsetneq W^{1,(p,\infty)}(Ω)$ where $Ω\subset {\mathbf{R}}^n$ is open and $n \ge 1,$ there exists a partial converse. Namely, we show that if a function $u$ in $W^{1,(p,\infty)}(Ω), n \ge 1$ is such that $u$ and its distributional gradient $\nabla u$ have absolutely continuous $(p,\infty)$-norm, then $u$ belongs to $H^{1,(p,\infty)}(Ω)$ as well. We also extend the Morrey embedding theorem to the Sobolev-Lorentz spaces $H_{0}^{1,(p,q)}(Ω)$ with $1 \le n<p<\infty$ and $1 \le q \le \infty.$ Namely, we prove that the Sobolev-Lorentz spaces $H_{0}^{1,(p,q)}(Ω)$ embed into the space of Hölder continuous functions on $\overlineΩ$ with exponent $1-\frac{n}{p}$ whenever $Ω\subset {\mathbf{R}}^n$ is open, $1 \le n<p<\infty,$ and $1 \le q \le \infty.$

math.AP↗

arXiv2012

Newtonian Lorentz Metric Spaces

This paper studies Newtonian Sobolev-Lorentz spaces. We prove that these spaces are Banach. We also study the global p,q-capacity and the p,q-modulus of families of rectifiable curves. Under some additional assumptions (that is, the space carries a doubling measure and a weak Poincare inequality) and some restrictions on q, we show that the Lipschitz functions are dense in those spaces. Moreover, in the same setting we show that the p,q-capacity is Choquet provided that q is strictly greater than 1. We also provide a counterexample to the density result of Lipschitz functions in the Euclidean setting when q is infinite.

math.MG↗

arXiv2009

BMO Estimates for the $H^{\infty}(\mathbb{B}_n)$ Corona Problem

We study the $H^{\infty}(\mathbb{B}_{n})$ Corona problem $\sum_{j=1}^{N}f_{j}g_{j}=h$ and show it is always possible to find solutions $f$ that belong to $BMOA(\mathbb{B}_{n})$ for any $n>1$, including infinitely many generators $N$. This theorem improves upon both a 2000 result of Andersson and Carlsson and the classical 1977 result of Varopoulos. The former result obtains solutions for strictly pseudoconvex domains in the larger space $H^{\infty}\cdot BMOA$ with $N=\infty $, while the latter result obtains $BMOA(\mathbb{B}_{n})$ solutions for just N=2 generators with $h=1$. Our method of proof is to solve $\overline{\partial}$-problems and to exploit the connection between $BMO$ functions and Carleson measures for $H^{2}(\mathbb{B}_{n})$. Key to this is the exact structure of the kernels that solve the $\overline{\partial}$ equation for $(0,q)$ forms, as well as new estimates for iterates of these operators. A generalization to multiplier algebras of Besov-Sobolev spaces is also given.

math.CA↗

arXiv2009

The Corona Theorem for the Drury-Arveson Hardy space and other holomorphic Besov-Sobolev spaces on the unit ball in $\mathbb{C}^{n}$

We prove that the multiplier algebra of the Drury-Arveson Hardy space $H_{n}^{2}$ on the unit ball in $\mathbb{C}^{n}$ has no corona in its maximal ideal space, thus generalizing the famous Corona Theorem of L. Carleson to higher dimensions. This result is obtained as a corollary of the Toeplitz corona theorem and a new Banach space result: the Besov-Sobolev space $B_{p}^σ$ has the "baby corona property" for all $σ\geq 0$ and $1<p<\infty $. In addition we obtain infinite generator and semi-infinite matrix versions of these theorems.

math.CV↗

arXiv2008

Strong A-infinity weights, Besov and Sobolev capacities in metric measure spaces

This article studies strong A-infinity weights in Ahlfors Q-regular and geodesic metric spaces satisfying a weak (1,s)-Poincare inequality for some 1<s<=Q, where Q is finite. It is shown that whenever max(1,Q-1)<s<=Q, a function u yields a strong A-infinity weight of the form w=exp(Qu) if u has a minimal s-weak upper gradient with sufficiently small Morrey norm. Similarly, it is proved that if 1<Q<p for some finite p, then w=exp(Qu) is a strong A-infinity weight whenever u has sufficiently small Besov p-seminorm.

math.AP↗

arXiv2008

Conductor inequalities and criteria for Sobolev-Lorentz two-weight inequalities

In this paper we present integral conductor inequalities connecting the Lorentz p,q-(quasi)norm of a gradient of a function to a one-dimensional integral of the p,q-capacitance of the conductor between two level surfaces of the same function. These inequalities generalize an inequality obtained by the second author in the case of the Sobolev norm. Such conductor inequalities lead to necessary and sufficient conditions for Sobolev-Lorentz type inequalities involving two arbitrary measures.

math.AP↗