# On $$L^p$$ L p -Boundedness of Pseudo-Differential Operators of Sjöstrand’s Class

On $$L^p$$ L p -Boundedness of Pseudo-Differential Operators of Sjöstrand’s Class We extended the known result that symbols from modulation spaces $$M^{\infty ,1}(\mathbb {R}^{2n})$$ M ∞ , 1 ( R 2 n ) , also known as the Sjöstrand’s class, produce bounded operators in $$L^2(\mathbb {R}^n)$$ L 2 ( R n ) , to general $$L^p$$ L p boundedness at the cost of loss of derivatives. Indeed, we showed that pseudo-differential operators acting from $$L^p$$ L p -Sobolev spaces $$L^p_s(\mathbb {R}^n)$$ L s p ( R n ) to $$L^p(\mathbb {R}^n)$$ L p ( R n ) spaces with symbols from the modulation space $$M^{\infty ,1}(\mathbb {R}^{2n})$$ M ∞ , 1 ( R 2 n ) are bounded, whenever $$s\ge n|1/p-1/2|.$$ s ≥ n | 1 / p - 1 / 2 | . This estimate is sharp for all $$1< p<\infty$$ 1 < p < ∞ . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Fourier Analysis and Applications Springer Journals

# On $$L^p$$ L p -Boundedness of Pseudo-Differential Operators of Sjöstrand’s Class

, Volume 23 (4) – Jul 29, 2016
7 pages

/lp/springer_journal/on-l-p-l-p-boundedness-of-pseudo-differential-operators-of-sj-strand-s-UOeJUiSCW0
Publisher
Springer US
Subject
Mathematics; Fourier Analysis; Signal,Image and Speech Processing; Abstract Harmonic Analysis; Approximations and Expansions; Partial Differential Equations; Mathematical Methods in Physics
ISSN
1069-5869
eISSN
1531-5851
D.O.I.
10.1007/s00041-016-9490-x
Publisher site
See Article on Publisher Site

### Abstract

We extended the known result that symbols from modulation spaces $$M^{\infty ,1}(\mathbb {R}^{2n})$$ M ∞ , 1 ( R 2 n ) , also known as the Sjöstrand’s class, produce bounded operators in $$L^2(\mathbb {R}^n)$$ L 2 ( R n ) , to general $$L^p$$ L p boundedness at the cost of loss of derivatives. Indeed, we showed that pseudo-differential operators acting from $$L^p$$ L p -Sobolev spaces $$L^p_s(\mathbb {R}^n)$$ L s p ( R n ) to $$L^p(\mathbb {R}^n)$$ L p ( R n ) spaces with symbols from the modulation space $$M^{\infty ,1}(\mathbb {R}^{2n})$$ M ∞ , 1 ( R 2 n ) are bounded, whenever $$s\ge n|1/p-1/2|.$$ s ≥ n | 1 / p - 1 / 2 | . This estimate is sharp for all $$1< p<\infty$$ 1 < p < ∞ .

### Journal

Journal of Fourier Analysis and ApplicationsSpringer Journals

Published: Jul 29, 2016

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