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In this paper, we will establish the existence and the Hölder regularity of the local time of the Brownian bridge. Our results are obtained by using a result on Malliavin calculus in [K. Es-Sebaiy, D. Nualart, Y. Ouknine and C. A. Tudor, Occupation densities for certain processes related to fractional Brownian motion, Stochastics 82 2010, 1–3, 133–147] for a Gaussian process with an absolutely continuous random drift, jointly with the classical approach based on the concept of local nondeterminism for Gaussian processes introduced by Berman [S. M. Berman, Local nondeterminism and local times of Gaussian processes, Indiana Univ. Math. J. 23 1973/74, 69–94]. [ABSTRACT FROM AUTHOR]

In this paper, we establish the endpoint estimate ( 0 < p ≤ 1) for a trilinear pseudo-differential operator, where the symbol involved is given by the product of two standard symbols from the bilinear Hörmander class BS1,00. The study of this operator is motivated from the Lp (1 < p < ∞) estimates for the trilinear Fourier multiplier operator with flag singularities considered in [C. Muscalu, Paraproducts with flag singularities. I. A case study, Rev. Mat. Iberoam. 23 2007, 2, 705–742] and Hardy space estimates in [A. Miyachi and N. Tomita, Estimates for trilinear flag paraproducts on L∞ and Hardy spaces, Math. Z. 282 2016, 1–2, 577–613], and the Lp ( 1 < p < ∞) estimates for the trilinear pseudo-differential operator with flag symbols in [G. Lu and L. Zhang, Lp-estimates for a trilinear pseudo-differential operator with flag symbols, Indiana Univ. Math. J. 66 2017, 3, 877–900]. More precisely, we will show that the trilinear pseudo-differential operator with flag symbols defined in (1.3) maps from the product of local Hardy spaces to the Lebesgue space, i.e., hp1 × hp2 × hp3 → Lp with 1/p1 + 1/p2 + 1/p3 = 1/p with 0 < p < ∞ (see Theorem 1.6 and Theorem 1.7). [ABSTRACT FROM AUTHOR]