# Nonconvolution transforms with oscillating kernels that map ${Ḃ}_{1}^{0,1}$ into itself

Studia Mathematica (1993)

• Volume: 106, Issue: 1, page 1-44
• ISSN: 0039-3223

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## Abstract

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We consider operators of the form $\left(\Omega f\right)\left(y\right)={ʃ}_{-\infty }^{\infty }\Omega \left(y,u\right)f\left(u\right)du$ with Ω(y,u) = K(y,u)h(y-u), where K is a Calderón-Zygmund kernel and $h\in {L}^{\infty }$ (see (0.1) and (0.2)). We give necessary and sufficient conditions for such operators to map the Besov space ${Ḃ}_{1}^{0,1}$ (= B) into itself. In particular, all operators with $h\left(y\right)={e}^{{i|y|}^{a}}$, a > 0, a ≠ 1, map B into itself.

## How to cite

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Sampson, G.. "Nonconvolution transforms with oscillating kernels that map $Ḃ_{1}^{0,1}$ into itself." Studia Mathematica 106.1 (1993): 1-44. <http://eudml.org/doc/216001>.

@article{Sampson1993,
abstract = {We consider operators of the form $(Ωf)(y) = ʃ_\{-∞\}^∞ Ω(y,u)f(u)du$ with Ω(y,u) = K(y,u)h(y-u), where K is a Calderón-Zygmund kernel and $h ∈ L^∞$ (see (0.1) and (0.2)). We give necessary and sufficient conditions for such operators to map the Besov space $Ḃ^\{0,1\}_1$ (= B) into itself. In particular, all operators with $h(y) = e^\{i|y|^a\}$, a > 0, a ≠ 1, map B into itself.},
author = {Sampson, G.},
journal = {Studia Mathematica},
keywords = {Calderón-Zygmund kernel},
language = {eng},
number = {1},
pages = {1-44},
title = {Nonconvolution transforms with oscillating kernels that map $Ḃ_\{1\}^\{0,1\}$ into itself},
url = {http://eudml.org/doc/216001},
volume = {106},
year = {1993},
}

TY - JOUR
AU - Sampson, G.
TI - Nonconvolution transforms with oscillating kernels that map $Ḃ_{1}^{0,1}$ into itself
JO - Studia Mathematica
PY - 1993
VL - 106
IS - 1
SP - 1
EP - 44
AB - We consider operators of the form $(Ωf)(y) = ʃ_{-∞}^∞ Ω(y,u)f(u)du$ with Ω(y,u) = K(y,u)h(y-u), where K is a Calderón-Zygmund kernel and $h ∈ L^∞$ (see (0.1) and (0.2)). We give necessary and sufficient conditions for such operators to map the Besov space $Ḃ^{0,1}_1$ (= B) into itself. In particular, all operators with $h(y) = e^{i|y|^a}$, a > 0, a ≠ 1, map B into itself.
LA - eng
KW - Calderón-Zygmund kernel
UR - http://eudml.org/doc/216001
ER -

## References

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