We prove that, under certain conditions on the function pair arphi_1 and arphi_1 , bilinear average arphi_1 along curve arphi_1 satisfies certain decay estimate. As a consequence, Roth type theorems hold in the setting of finite fields. In particular, if arphi_1 with arphi_1 are linearly independent polynomials, then for any arphi_1 with arphi_1 , there are arphi_1 triplets arphi_1 . This extends a recent result of Bourgain and Chang who initiated this type of problems, and strengthens the bound in a result of Peluse, who generalized Bourgain and Chang's work. The proof uses discrete Fourier analysis and algebraic geometry.

To characterize the complete structure of the Fucik spectrum of the p-Laplacian on higher dimensional domains is a long-standing problem. In this paper, we study the p-Laplacian with integrable potentials on an interval under the Dirichlet or the Neumann boundary conditions. Based on the strong continuity and continuous differentiability of solutions in potentials, we will give a comprehensive characterization of the corresponding Fucik spectra: each of them is composed of two trivial lines and a double-sequence of differentiable, strictly decreasing, hyperbolic-like curves; all asymptotic lines of these spectral curves are precisely described by using eigenvalues of the p-Laplacian with potentials; and moreover, all these spectral curves have strong continuity in potentials, i.e. as potentials vary in the weak topology, these spectral curves are continuously dependent on potentials in a certain sense.

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We prove that for a large class of functions P and Q, the discrete bilinear operator T P, Q (f, g)(n)= m Z{0} f (n P (m)) g (n Q (m)) 1 m is bounded from l 2 l 2 into l 1+ , for any (0, 1].

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