TY - GEN

T1 - Pseudodifferential operators on variable lebesgue spaces

AU - Karlovich, Alexei Yu

AU - Spitkovsky, Ilya M.

N1 - 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

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

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 -