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Georgios Psaromiligkos

Bio: Georgios Psaromiligkos is an academic researcher. The author has contributed to research in topics: Embedding & Counterexample. The author has an hindex of 4, co-authored 8 publications receiving 34 citations.

Papers
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TL;DR: In this paper, it was shown that a Carleson embedding for a scale of Dirichlet spaces from the bi-torus to the Bi-disc is equivalent to a simple ''box'' condition, for product weights on the bidisc and arbitrary weights in the bitorus.
Abstract: Coifman--Meyer multipliers represent a very important class of bi-linear singular operators, which were extensively studied and generalized. They have a natural multi-parameter counterpart. Decomposition of those operators into paraproducts, and, more generally to multi-parameter paraproducts is a staple of the theory. In this paper we consider weighted estimates for bi-parameter paraproducts that appear from such multipliers. Then we apply our harmonic analysis results to several complex variables. Namely, we show that a (weighted) Carleson embedding for a scale of Dirichlet spaces from the bi-torus to the bi-disc is equivalent to a simple ``box'' condition, for product weights on the bi-disc and arbitrary weights on the bi-torus. This gives a new simple necessary and sufficient condition for the embedding of the whole scale of weighted Dirichlet spaces of holomorphic functions on the bi-disc. This scale of Dirichlet spaces includes the classical Dirichlet space on the bi-disc. Our result is in contrast to the classical situation on the bi-disc considered by Chang and Fefferman, when a counterexample due to Carleson shows that the ``box'' condition does not suffice for the embedding to hold. But this was the embedding of bi-harmonic functions in bi-harmonic Hardy class. Our result can be viewed as a new and unexpected combinatorial property of all positive finite planar measures.

11 citations

Journal ArticleDOI
TL;DR: In this paper, Mozolyako et al. build a plethora of counterexamples to bi-parameter two-parameters embedding theorems and show that without tensor structure requirement all results break down.

7 citations

Journal ArticleDOI
TL;DR: In this paper, Mozolyako, Psaromiligkos, Volberg and Kranich proved the embedding theorem of holomorphic spaces on bi-disc and tri-disc.

6 citations

Posted Content
TL;DR: In this article, a simple tree is considered as a counterexample for two weight bi-parameter embedding of Carleson type, and it is shown that the box condition, Carlesone condition and two weight embedding are all equivalent.
Abstract: In this note we give an example of measure satisfying the box condition on certain sub-bi-trees (see below) but not satisfying Carleson condition on those sub-bi-trees. This can be considered as a certain counterexample for two weight bi-parameter embedding of Carleson type. Our type of counterexample is impossible for a simple tree. In the case of a simple tree, the box condition, Carleson condition and two weight embedding are all equivalent. In the last section we show that bi-parameter box condition implies the bi-parameter capacitary estimate for dyadic rectangles.

6 citations

Posted Content
TL;DR: This work proves multi-parameter dyadic embedding theorem for Hardy operator on the multi-tree and shows that for a large class of Dirichlet spaces in bi-disc and tri-disc this proves theembedding theorem of those Dirichlets spaces of holomorphic function on bi- andTri-disc.
Abstract: We prove multi-parameter dyadic embedding theorem for Hardy operator on the multi-tree. We also show that for a large class of Dirichlet spaces in bi-disc and tri-disc this proves the embedding theorem of those Dirichlet spaces of holomorphic function on bi- and tri-disc. We completely describe the Carleson measures for such embeddings. The result below generalizes embedding result of \cite{AMPVZ} from bi-tree to tri-tree. One of our embedding description is similar to Carleson--Chang--Fefferman condition and involves dyadic open sets. On the other hand, the unusual feature of \cite{AMPVZ} was that embedding on bi-tree turned out to be equivalent to one box Carleson condition. This is in striking difference to works of Chang--Fefferman and well known Carleson quilt counterexample. We prove here the same unexpected result for the tri-tree. Finally, we explain the obstacle that prevents us from proving our results on polydiscs of dimension four and higher.

4 citations


Cited by
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TL;DR: In this paper, the authors give another proof of a bi-parameter Carleson embedding theorem that avoids the use of bi-tree capacity, based on some rather subtle comparisons of energies of measures on the Bi-tree.
Abstract: Nicola Arcozzi, Pavel Mozolyako, Karl-Mikael Perfekt, and Giulia Sarfatti recently gave the proof of a bi-parameter Carleson embedding theorem. Their proof uses heavily the notion of capacity on the bi-tree. In this note we give another proof of a bi-parameter Carleson embedding theorem that avoids the use of bi-tree capacity. Unlike the proof on a simple tree in a previous paper of the authors (Arcozzi et al. in Bellman function sitting on a tree, arXiv:1809.03397, 2018), which used the Bellman function technique, the proof here is based on some rather subtle comparisons of energies of measures on the bi-tree.

14 citations

Posted Content
TL;DR: In this paper, it was shown that a Carleson embedding for a scale of Dirichlet spaces from the bi-torus to the Bi-disc is equivalent to a simple ''box'' condition, for product weights on the bidisc and arbitrary weights in the bitorus.
Abstract: Coifman--Meyer multipliers represent a very important class of bi-linear singular operators, which were extensively studied and generalized. They have a natural multi-parameter counterpart. Decomposition of those operators into paraproducts, and, more generally to multi-parameter paraproducts is a staple of the theory. In this paper we consider weighted estimates for bi-parameter paraproducts that appear from such multipliers. Then we apply our harmonic analysis results to several complex variables. Namely, we show that a (weighted) Carleson embedding for a scale of Dirichlet spaces from the bi-torus to the bi-disc is equivalent to a simple ``box'' condition, for product weights on the bi-disc and arbitrary weights on the bi-torus. This gives a new simple necessary and sufficient condition for the embedding of the whole scale of weighted Dirichlet spaces of holomorphic functions on the bi-disc. This scale of Dirichlet spaces includes the classical Dirichlet space on the bi-disc. Our result is in contrast to the classical situation on the bi-disc considered by Chang and Fefferman, when a counterexample due to Carleson shows that the ``box'' condition does not suffice for the embedding to hold. But this was the embedding of bi-harmonic functions in bi-harmonic Hardy class. Our result can be viewed as a new and unexpected combinatorial property of all positive finite planar measures.

11 citations

Posted Content
TL;DR: In this article, the authors give another proof of a bi-parameter Carleson embedding theorem that avoids the use of bi-tree capacity, based on some rather subtle comparison of energies of measures on Bi-tree.
Abstract: Nicola Arcozzi, Pavel Mozolyako, Karl-Mikael Perfekt, and Giulia Sarfatti recently gave the proof of a bi-parameter Carleson embedding theorem. Their proof uses heavily the notion of capacity on bi-tree. In this note we give one more proof of a bi-parameter Carleson embedding theorem that avoids the use of bi-tree capacity. Unlike the proof on a simple tree (in a pervious paper of the authors) that used the Bellman function technique, the proof here is based on some rather subtle comparison of energies of measures on bi-tree.

9 citations

Journal ArticleDOI
TL;DR: In this paper, Mozolyako et al. build a plethora of counterexamples to bi-parameter two-parameters embedding theorems and show that without tensor structure requirement all results break down.

7 citations