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摘要
We study multilinear operators T(f(1), f(2), ..., f(m)) that commutes with simultaneous translations and prove that if T is bounded from L-P1 X L-P2 X ... X L-Pm to L-P, then for any r >= p < p, q <= infinity and @@@ s > { n(1 - 1 boolean AND 1/q), (1/p, 1/q) is an element of D-1; n(1 boolean OR 1/p boolean OR 1/q - 1/q), (1/p, 1/q is an element of R-+(2) - D-1 (;) @@@ (D-1 = {(1/p, 1/p) is an element of : 1/q >= 2/p, 1/p <=})T is bounded from M-p1,q(s) X M-p2,q(s) X ... X M-p,q(s) to M-r,q(s) (which improves the results obtained by [5], [6].), where rn M-p,q(s) is the modulation spaces. Besides, we also obtain the similar results for Triebel-type spaces N-p,q(s) introduced by [21] (T is bounded from N-p,q(s) X N-p,q(s) X ... X N-p,q(s)). As applications, we obtain the boundedness on the modulation spaces for the bilinear Hilbert transform, bilinear fractional integral, the pointwise product of functions, and the bilinear oscillatory integral along parabolas. Also, in modulation spaces and N-p,q(s), we study the well-posedness of the Cauchy problem for the fractional heat and Schrodinger equations with some new nonlinear terms. Such nonlinear well-posedness problems are not studied in other function spaces.
- 出版日期2014-8
- 单位浙江大学
