Boundedness Of Pseudo-differential Operators Research Articles

On the L 2 -Boundedness of Pseudo-Differential Operators

On the L 2 -Boundedness of Pseudo-Differential Operators

On the Boundedness of Pseudo Differential Operators in the Class L m ρ, 1

On the Boundedness of Pseudo Differential Operators in the Class L m ρ, 1

On the boundedness of pseudo differential operators in the class $L\sp{m}\sb{\rho }{}\sb{,1}.$

We prove that every pseudo differential operator in the class L™,, 0 < p < 1, is bounded in L2(R) if and only if m < n(p - l)/2.

On the Boundedness of Pseudo-Differential Operators

On the Boundedness of Pseudo-Differential Operators

On the boundedness of pseudo-differential operators

In this note, we use the sequence version of Cotlar's lemma and a partition of unity to give a proof of the L2-boundedness of a class of pseudo-differential operators. Introduction and results. Let p(x, 5) be a continuous function on RI x RI. Then, a pseudo-differential operator P with the symbol p(x, 5) is a linear map of CO (Rn) into CO(Rn), defined by p = ((27r)1/2)-n eixtp(x )2(4) d4 for u E CO (Rn), where u is the Fourier transform of u. In [1] A. Calderon and R. Vaillencourt prove the following THEOREM A. Let p(x, $) be a function defined on R' x R' such that 1(1 + aw)3 . . (I + ax1)3(I + a )3 . . (1 + at1)3p(x, t)I ? C for all (x, t) e Rn x Rn. Then the pseudo-diferential operator associated with the symbolp(x, $) can be extended to a bounded operator from L2(Rn) to L2(Rn). Using the sequence version of Cotlar's lemma (cf. [2]) and a partition of unity, we can prove the following result analogous to Theorem A. THEOREM. Let p(x, $) be a function defined on Rn x RL such that (1) J a,tp(x, dx _ C, for O _ oi < 2, 0 < f3j < 3, and all (x, t) E Q x Rt, where Q is any cube with edges of length two and parallel to the axes. Then the associated operator can be extended to a bounded operator from L2(Rn) to L2(R'). Received by the editors June 26, 1972 and, in revised form, December 30, 1972. AMS (MOS) subject classifications (1970). Primary 35S05.

On the boundedness of pseudo-differential operators of type $\rho ,\,\delta $ with $0łeq \rho =\delta &lt;1$

On the boundedness of pseudo-differential operators of type $\rho ,\,\delta $ with $0łeq \rho =\delta &lt;1$

On the boundedness of pseudo-differential operators

On the boundedness of pseudo-differential operators