On the Lp-boundedness of pseudo-differential operators with non-regular symbols

Abstract

In this paper, we consider the continuity property of pseudo-differential operators with symbols whose Fourier transforms have compact support. As applications, we obtain the L p -boundedness for symbols in Besov spaces and in modulation spaces.

Multilinear pseudo-differential operators with $S_{0,0}$ class symbols of limited smoothness

“…Moreover, it might be worth mentioning that these values are partially sharp (see [26,Section 5]). Some results on this direction can be also found in, for instance, Boulkhemair [3], Coifman and Meyer [5], Cordes [6], Hwang [20], Muramatu [29], and Sugimoto [32] for p = 2 and Tomita [33] for 0 < p < ∞. For the multilinear case, in [21,22], it was shown that, for the case 2/N ≤ p ≤ 2 and 2 ≤ p 1 , .…”

A Remark Related Pseudo-Differential Operators on Multi-Parameter Besov-Lipschitz and Triebel-Lizorkin Spaces

“…ðℝ n × ℝ n Þ implies the L 2 -boundedness. Furthermore, Tomita [28] generalized the results in [27] and studied the L p -boundedness of pseudodifferential operators with non-regular symbols, where 1 < p < ∞. As some applications, Tomita [28] also obtained the L p -boundedness for symbols in Besov spaces and modulation spaces, respectively.…”

“…Furthermore, Tomita [28] generalized the results in [27] and studied the L p -boundedness of pseudodifferential operators with non-regular symbols, where 1 < p < ∞. As some applications, Tomita [28] also obtained the L p -boundedness for symbols in Besov spaces and modulation spaces, respectively. Recently, the properties of multiparameter Besov-Lipschitz and Triebel-Lizorkin spaces and the boundedness of multi-parameter singular integral operators on them have been established in [29][30][31][32].…”

“…Here the parameter α ∈ ½0, 1Þ works as the "adjuster", modulation spaces M s p,q ðℝ n Þ are the special α-modulation spaces when α = 0 and Besov spaces B s p,q ðℝ n Þ can be treated as the limit case of α-modulation spaces when the parameter α tends to 1. The difference between modulation spaces M s p,q ðℝ n Þ and Besov spaces B s p,q ðℝ n Þ can be referred to Tomita [28]. We also remark that the boundedness of multi-parameter Fourier multiplier operators on Triebel-Lizorkin and Besov-Lipschitz spaces have been obtained in [34].…”