arXiv subjects
Publications and source records attributed to Salvador Rodríguez-López.
In this paper we establish some endpoint estimates for bilinearpseudodifferential operators with symbol in the class BS^m_{1,1}, involving the space of functions with local bounded mean oscillation bmo. As a consequence we also obtain an endpoint estimate of Kato-Ponce type.
This article is devoted to a study of the Hardy space $H^{\log} (\mathbb{R}^d)$ introduced by Bonami, Grellier, and Ky. We present an alternative approach to their result relating the product of a function in the real Hardy space $H^1$ and a function in $BMO$ to distributions that belong to $H^{\log}$ based on dyadic paraproducts. We also point out analogues of classical results of Hardy-Littlewood, Zygmund, and Stein for $H^{\log}$ and related Musielak-Orlicz spaces.
In this paper we establish mapping properties of bilinear Coifman-Meyer multipliers acting on the product spaces $H^1(\mathbb{R}^n)\times\mathrm{bmo}(\mathbb{R}^n)$ and $L^p(\mathbb{R}^n)\times\mathrm{bmo}(\mathbb{R}^n)$, with $1<p<\infty$. As application of these results, we obtain some related Kato-Ponce-type inequalities involving the endpoint space $\mathrm{bmo}(\mathbb{R}^n)$, and we also study the pointwise product of a function in $\mathrm{bmo}(\mathbb{R}^n)$ with functions in $H^1(\mathbb{R}^n)$, $h^1(\mathbb{R}^n)$ and $L^p(\mathbb{R}^n)$, with $1<p<\infty$.
We obtain versions of some classical results of Zygmund and Stein for functions belonging to the Hardy space $H^{\log} (\mathbb{R}^d)$ introduced by Bonami, Grellier, and Ky. We present further applications in the context of more general Orlicz spaces.
In this paper we give an elementary proof for transference of local to global maximal estimates for dispersive PDEs. This is done by transferring local $L^2$ estimates for certain oscillatory integrals with rough phase functions, to the corresponding global estimates. The elementary feature of our approach is that it entirely avoids the use of the wave packet techniques which are quite common in this context, and instead is based on scalings and classical oscillatory integral estimates.
We establish the $L^2$-solvability of Dirichlet, Neumann and regularity problems for divergence-form heat (or diffusion) equations with Hölder-continuous diffusion coefficients, on bounded Lipschitz domains in $\mathbb{R}^n$. This is achieved through the demonstration of invertibility of the relevant layer-potentials which is in turn based on Fredholm theory and a new systematic approach which yields suitable parabolic Rellich-type estimates.
We study the boundedness of rough Fourier integral and pseudodifferential operators, defined by general rough Hörmander class amplitudes, on Banach and quasi-Banach $L^p$ spaces. Thereafter we apply the aforementioned boundedness in order to improve on some of the existing boundedness results for Hörmander class bilinear pseudodifferential operators and certain classes of bilinear (as well as multilinear) Fourier integral operators. For these classes of amplitudes, the boundedness of the aforementioned Fourier integral operators turn out to be sharp. Furthermore we also obtain results for rough multilinear operators.
In this paper we establish the boundedness of bilinear paraproducts on local BMO spaces. As applications, we also investigate the boundedness of bilinear Fourier integral operators and bilinear Coifman-Meyer multipliers on these spaces and also obtain a certain end-point result concerning Kato-Ponce type estimates.
We establish the regularity of bilinear Fourier integral operators with bilinear amplitudes in $S^m_{1,0} (n,2)$ and non-degenerate phase functions, from $L^p \times L^q \to L^r$ under the assumptions that $m\leq -(n-1)(|\frac{1}{p}-\frac{1}{2}|+|\frac{1}{q}-\frac{1}{2}|)$ and $\frac{1}{p}+\frac{1}{q}=\frac{1}{r}$. This is a bilinear version of the classical theorem of Seeger-Sogge-Stein concerning the $L^p$ boundedness of linear Fourier integral operators. Moreover, our result goes beyond the aforementioned theorem in that it also includes the case of non-Banach target spaces.
We obtain restriction results of K. de Leeuw's type for maximal operators defined through multilinear Fourier multipliers of either strong or weak type acting on weighted Lebesgue spaces. We give some application of our development. In particular we obtain periodic weighted results for Coifman-Meyer, Hörmander and Hörmander-Mihlin type multilinear multipliers.
In this paper we prove an abstract homomorphism theorem for bilinear multipliers in the setting of locally compact Abelian (LCA) groups. We also provide some applications. In particular, we obtain a bilinear abstract version of K. de Leeuw's theorem for bilinear multipliers of strong and weak type. We also obtain necessary conditions on bilinear multipliers on non-compact LCA groups, yielding boundedness for the corresponding operators on products of rearrangement invariant spaces. Our investigations extend some existing results in Euclidean spaces to the framework of general LCA groups, and yield new boundedness results for bilinear multipliers in quasi Banach spaces.