TY - GEN

T1 - Pseudodifferential operators on variable lebesgue spaces

AU - Karlovich, Alexei Yu

AU - Spitkovsky, Ilya M.

N1 - Funding Information:
Accepted September 2/, /992. The authors arefrom the Departments ofPsychiatry and Pediatrics, Columbia University College of Physicians and Surgeons, the New York State Psychiatric Institute, and the Columbia University School of Public Health. This research was supported by N/MH/NIDA grant 5-P50MH43520, Heather J. Walter, M.D., M.P.H., Principal Investigator, AIDS Prevention for Adolescents in School, Anke A. Ehrhardt, Principal Investigator, HIV Center for Clinical and Behavioral Studies. The authors acknowledge Deborah Fish Ragin, Ph.D., and Stephanie Kasen, Ph.D., for their contributions to this research. Reprint requests to Dr. Walter, Center for Population and Family Health, 60 Haven Avenue, New York, NY 10032. 0890-8567/93/3205-0975$03.00/0©1 993 by the American Academy of Child and Adolescent Psychiatry.
Publisher Copyright:
© 2013 Springer Basel.

PY - 2013

Y1 - 2013

N2 - Let M(ℝn) be the class of bounded away from one and infinity functions p: ℝn → [1,∞] such that the Hardy-Littlewood maximal operator is bounded on the variable Lebesgue space Lp(⋅)(ℝn). We show that if a belongs to the Hörmander class Sn(ρ-1) ρ, δ with 0 < ρ ≤ 1, 0 ≤ δ < 1, then the pseudodifferential operator Op(a) is bounded on Lp(⋅)(ℝn) provided that p∈M(ℝn). Let M* (ℝn) be the class of variable exponents p∈M(ℝn) represented as 1/p(x) = θ/p0 + (1 – θ)/p1(x) where p0 ∈ (1,∞), θ ∈(0, 1), and p1 ∈M(ℝn). We prove that if a ∈ S0 1,0 slowly oscillates at infinity in the first variable, then the condition (Formula presented.) is sufficient for the Fredholmness of Op(a) on Lp(⋅)(ℝn) whenever p∈M* (ℝn). Both theorems generalize pioneering results by Rabinovich and Samko [24] obtained for globally log-Hölder continuous exponents p, constituting a proper subset of M* (ℝn).

AB - Let M(ℝn) be the class of bounded away from one and infinity functions p: ℝn → [1,∞] such that the Hardy-Littlewood maximal operator is bounded on the variable Lebesgue space Lp(⋅)(ℝn). We show that if a belongs to the Hörmander class Sn(ρ-1) ρ, δ with 0 < ρ ≤ 1, 0 ≤ δ < 1, then the pseudodifferential operator Op(a) is bounded on Lp(⋅)(ℝn) provided that p∈M(ℝn). Let M* (ℝn) be the class of variable exponents p∈M(ℝn) represented as 1/p(x) = θ/p0 + (1 – θ)/p1(x) where p0 ∈ (1,∞), θ ∈(0, 1), and p1 ∈M(ℝn). We prove that if a ∈ S0 1,0 slowly oscillates at infinity in the first variable, then the condition (Formula presented.) is sufficient for the Fredholmness of Op(a) on Lp(⋅)(ℝn) whenever p∈M* (ℝn). Both theorems generalize pioneering results by Rabinovich and Samko [24] obtained for globally log-Hölder continuous exponents p, constituting a proper subset of M* (ℝn).

KW - Fefferman-Stein sharp maximal operator

KW - Fredholmness

KW - Hardy-Littlewood maximal operator

KW - Hörmander symbol

KW - Pseudodifferential operator

KW - Slowly oscillating symbol

KW - Variable Lebesgue space

UR - http://www.scopus.com/inward/record.url?scp=84946043539&partnerID=8YFLogxK

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U2 - 10.1007/978-3-0348-0537-7_9

DO - 10.1007/978-3-0348-0537-7_9

M3 - Conference contribution

AN - SCOPUS:84946043539

SN - 9783034805360

T3 - Operator Theory: Advances and Applications

SP - 173

EP - 183

BT - Operator Theory, Pseudo-Differential Equations, and Mathematical Physics

A2 - Karlovich, Yuri I.

A2 - Rodino, Luigi

A2 - Silbermann, Bernd

A2 - Spitkovsky, Ilya M.

PB - Springer International Publishing

T2 - International workshop on Analysis, Operator Theory, and Mathematical Physics, 2012

Y2 - 23 January 2012 through 27 January 2012

ER -