TY - JOUR
T1 - Banach algebra of the Fourier multipliers on weighted Banach function spaces
AU - Karlovich, Alexei
N1 - Funding Information:
This work was partially supported by the Funda??o para a Ci?ncia e a Tecnologia (Portuguese Foundation for Science and Technology) through the projects UID/MAT/00297/2013 (Centro de Matem?tica e Aplica??es) and EXPL/MAT-CAL/0840/2013 (Problemas Variacionais em Espa?os de Sobolev de Expoente Vari?vel).
Publisher Copyright:
© 2015 Alexei Karlovich.
PY - 2015/3/10
Y1 - 2015/3/10
N2 - Let MX,w(ℝ) denote the algebra of the Fourier multipliers on a separable weighted Banach function space X(ℝ, w). We prove that if the Cauchy singular integral operator S is bounded on X(ℝ, w), thenMX,w(ℝ) is continuously embedded into L∞(ℝ). An important consequence of the continuous embedding MX,w(ℝ) ⊂ L∞(ℝ) is that MX,w(ℝ) is a Banach algebra.
AB - Let MX,w(ℝ) denote the algebra of the Fourier multipliers on a separable weighted Banach function space X(ℝ, w). We prove that if the Cauchy singular integral operator S is bounded on X(ℝ, w), thenMX,w(ℝ) is continuously embedded into L∞(ℝ). An important consequence of the continuous embedding MX,w(ℝ) ⊂ L∞(ℝ) is that MX,w(ℝ) is a Banach algebra.
KW - Banach function space
KW - Cauchy singular integral operator
KW - Fourier convolution operator
KW - Fourier multiplier
KW - Muckenhoupt-type weight
KW - rearrangement-invariant space
KW - variable Lebesgue space
UR - http://www.scopus.com/inward/record.url?scp=84946405449&partnerID=8YFLogxK
U2 - 10.1515/conop-2015-0001
DO - 10.1515/conop-2015-0001
M3 - Article
AN - SCOPUS:84946405449
SN - 2299-3282
VL - 2
SP - 27
EP - 36
JO - Concrete Operators
JF - Concrete Operators
IS - 1
ER -