MULTILINEAR ESTIMATES ON FREQUENCY-UNIFORM DECOMPOSITION SPACES AND APPLICATIONS

Abstract

We study multilinear operators $T(f_{1},f_{2},...,f_{m})$ that commutes with simultaneous translations and prove that if T is bounded from $L^{p_{1}} \times L^{p_{2}} \times ... \times L^{p_{m}}$ to $L^{p}$, then for any $r \geqslant p$, $0 \lt p,q \leqslant \infty$ and \[s \gt \left\{ \begin{array}{lll} n(1-1\wedge\frac{1}{q}), &(\frac{1}{p},\frac{1}{q})\in D_{1};\\ n(1\vee\frac{1}{p}\vee\frac{1}{q}-\frac{1}{q}), &(\frac{1}{p},\frac{1}{q})\in \mathbb{R}_{+}^{2}-D_{1}, \end{array} \right.\] ($D_{1}=\{(\frac{1}{p},\frac{1}{q})\in\mathbb{R}_{+}^{2}:\frac{1}{q}\geqslant\frac{2}{p},\frac{1}{p}\leqslant\frac{1}{2}\}$)T is bounded from $M_{p_{1},q}^{s}\times M_{p_{2},q}^{s}\times...\times M_{p_{m},q}^{s}$ to $M_{r,q}^{s}$ (which improves the results obtained by [5], [6].), where $M_{p,q}^{s}$ is the modulation spaces. Besides, we alsoobtain the similar results for Triebel-type spaces $N_{p,q}^{s}$ introduced by [21] (T is bounded from $N_{p,q}^{s} \times N_{p,q}^{s} \times ... \times N_{p,q}^{s}$ to $N_{p,q}^{s}$). 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 Schrödinger equations with some new nonlinear terms. Such nonlinear well-posedness problems are not studied in other function spaces.