We prove mapping properties of the form T : Ḃ11 p1 × L p2 → Ḃ22 p3 and T : Ḃ11 p1 × Ḃ α2,q2 p2 → L p3 , for certain related indices p1, p2, p3, q1, q2, α1, α2 ∈ R, where T is a bilinear Hörmander-Mihlin multiplier or a molecular paraproduct. Applications to bilinear Littlewood-Paley theory are discussed.

The term paraproduct is nowadays used rather loosely in the literature to indicate a bilinear operator that, although noncommutative, is somehow better behaved than the usual product of functions. Paraproducts emerged in J.-M. Bony’s theory of paradifferential operators [1], which stands as a milestone on the road beyond pseudodifferential operators pioneered by R. R. Coifman and Y. Meyer in [3...

We first prove an adiabatic limit formula for the rf-invariant of aDirac operator, generalizing the recent work of J.-M. Bismut and J. Cheeger.An essential part of the proof is the study of the spectrum of the Dirac op-erator in the adiabatic limit. A new contribution arises in the adiabatic limitformula, in the form of a global term coming from the (asymptotically) verysmal...

We consider saddle-point problems that typically arise from the mixed finite element discretization of second order elliptic problems. By proper equivalent algebraic operations the considered saddle-point problem is transformed to another saddle-point problem. The resulting problem can then be efficiently preconditioned by a block-diagonal matrix or by a factored block-matrix (the blocks corres...

We consider the integral domain restriction operator TΩ for certain bilinear operator T . We obtain that if (s, p1, p2) satisfies 1 p1 + 1 p2 ≥ 2 min{1,s} and ‖T‖Lp1×Lp2→Ls < ∞, then ‖TΩ‖Lp1×Lp2→Ls < ∞. For some special domain Ω, this property holds for triplets (s, p1, p2) satisfying 1 p1 + 1 p2 > 1 min{1,s} . 2010 Mathematics Subject Classification: 42B25.

Laplace–Beltrami operator and its discretization play a central role in the fields of image processing, computer graphics, computer aided geometric design and so on. In this paper, a discrete scheme for Laplace–Beltrami operator over quadrilateral meshes is constructed based on a bilinear interpolation of the quadrilateral. Convergence results for the proposed discrete scheme are established un...

Results analogous to those proved by Rubio de Francia [28] are obtained for a class of maximal functions formed dilations bilinear multiplier operators limited decay. We focus our attention on L2 × → L1 estimates. discuss two applications: the boundedness Bochner-Riesz operator and spherical operator. For latter we improve known results in [1] reducing dimension restriction from n ≥ 8 4.

A necessary and sufficient condition for a central simple algebra with involution over a field of characteristic two to be decomposable as a tensor product of quaternion algebras with involution, in terms of its Frobenius subalgebras, is given. It is also proved that a bilinear Pfister form, recently introduced by A. Dolphin, can classify totally decomposable central simple algebras of orthogon...

We study the regularity of the bilinear maximal operator when applied to Sobolev functions, proving that it maps W (R) × W (R) → W (R) with 1 < p, q < ∞ and r ≥ 1, boundedly and continuously. The same result holds on R when r > 1. We also investigate the almost everywhere and weak convergence under the action of the classical Hardy-Littlewood maximal operator, both in its global and local versi...

The optimal robust disturbance rejection problem plays an important role in feedback control theory. Here its time-varying version is solved explicitly in terms of duality and operator theory. In particular, the optimum is shown to satisfy a time-varying allpass property. Moreover, optimal performance is given in terms of the norm of a bilinear form. The latter depends on a lower triangular pro...