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PreprintCC BY 4.0arXiv

A Morrey-to-Lebesgue Equivalence for Convolution Calderón--Zygmund Operators

  • Rishad Shahmurov
  • Veli Shahmurov
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Abstract

We prove that, for vector-valued convolution Calderón--Zygmund operators, boundedness on a single nontrivial Morrey space is equivalent to the corresponding global Lp boundedness. Thus one Morrey scale already contains the full finite-p boundedness information. The implication from Morrey to Lp is obtained by a separated-copy amplification argument that reconstructs the global norm from a single scale-local estimate; the converse is proved in the same vector-valued framework by a local/far-field decomposition. As a consequence, boundedness of the vector-valued Hilbert transform on one nontrivial Morrey space is equivalent to the UMD property. The result shows that Morrey estimates do not bypass the classical Banach-space obstruction: they detect it exactly.

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Cite

BibTeXRIS
BibTeX
@misc{shahmurov2026morreytolebesgue,
  title = {{A Morrey-to-Lebesgue Equivalence for Convolution Calderón--Zygmund Operators}},
  author = {Rishad Shahmurov and Veli Shahmurov},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.29158},
  eprint = {2609.29158},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.29158},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - A Morrey-to-Lebesgue Equivalence for Convolution Calderón--Zygmund Operators
AU  - Rishad Shahmurov
AU  - Veli Shahmurov
PY  - 2026
DA  - 2026/09/24
PB  - arXiv
DO  - 10.48550/arxiv.2609.29158
UR  - https://arxiv.org/abs/2609.29158
AB  - We prove that, for vector-valued convolution Calderón--Zygmund operators, boundedness on a single nontrivial Morrey space is equivalent to the corresponding global L^p boundedness. Thus one Morrey scale already contains the full finite-p boundedness information. The implication from Morrey to L^p is obtained by a separated-copy amplification argument that reconstructs the global norm from a single scale-local estimate; the converse is proved in the same vector-valued framework by a local/far-field decomposition. As a consequence, boundedness of the vector-valued Hilbert transform on one nontrivial Morrey space is equivalent to the UMD property. The result shows that Morrey estimates do not bypass the classical Banach-space obstruction: they detect it exactly.
ER  -