TY - GEN

T1 - Maximally Modulated Singular Integral Operators and their Applications to Pseudodifferential Operators on Banach Function Spaces

AU - Karlovich, Alexei Yu.

PY - 2015

Y1 - 2015

N2 - We prove that if the Hardy-Littlewood maximal operator is bounded on a separable Banach function space X(R-n) and on its associate space X'(R-n) and a maximally modulated Calderon-Zygmund singular integral operator T-Phi is of weak type (r, r) for all r is an element of E (1, infinity), then T-Phi extends to a bounded operator on X(R-n). This theorem implies the boundedness of the maximally modulated Hilbert transform on variable Lebesgue spaces L-p((.)) (H) under natural assumptions on the variable exponent p : R -> (1, infinity). Applications of the above result to the boundedness and compactness of pseudodifferential operators with L-infinity, V(R))-symbols on variable Lebesgue spaces L-p((.)) (H) are considered. Here the Banach algebra L-infinity(R, V(R)) consists of all bounded measurable V(R)-valued functions on R where V(R) is the Banach algebra of all functions of bounded total variation.

AB - We prove that if the Hardy-Littlewood maximal operator is bounded on a separable Banach function space X(R-n) and on its associate space X'(R-n) and a maximally modulated Calderon-Zygmund singular integral operator T-Phi is of weak type (r, r) for all r is an element of E (1, infinity), then T-Phi extends to a bounded operator on X(R-n). This theorem implies the boundedness of the maximally modulated Hilbert transform on variable Lebesgue spaces L-p((.)) (H) under natural assumptions on the variable exponent p : R -> (1, infinity). Applications of the above result to the boundedness and compactness of pseudodifferential operators with L-infinity, V(R))-symbols on variable Lebesgue spaces L-p((.)) (H) are considered. Here the Banach algebra L-infinity(R, V(R)) consists of all bounded measurable V(R)-valued functions on R where V(R) is the Banach algebra of all functions of bounded total variation.

KW - Maximally modulated singular integral operator

KW - Calderon-Zygmund operator

KW - Hilbert transform

KW - pseudodifferential operator with non-regular symbol

KW - Banach function space

KW - variable Lebesgue space

KW - LEBESGUE SPACES

KW - BOUNDEDNESS

KW - COMPACTNESS

KW - SYMBOLS

UR - https://arxiv.org/pdf/1408.4400.pdf

U2 - 10.1090/conm/645/1.2908

DO - 10.1090/conm/645/1.2908

M3 - Conference contribution

SN - 978-1-4704-1694-2

T3 - Contemporary Mathematics

SP - 165

EP - 178

BT - FUNCTION SPACES IN ANALYSIS

A2 - Jarosz, K

PB - AMER MATHEMATICAL SOC

T2 - 7th Conference on Function Spaces

Y2 - 20 May 2014 through 24 May 2014

ER -