arXiv subjects
Publications and source records attributed to Gustavo Garrigos.
When $L$ is the Hermite or the Ornstein-Uhlenbeck operator, we find minimal integrability and smoothness conditions on a function $f$ so that the fractional power $L^σf(x_0)$ is well-defined at a given point $x_0$. We illustrate the optimality of the conditions with various examples. Finally, we obtain similar results for the fractional operators $(-Δ+R)^σ$, with $R>0$.
This survey is a slightly extended version of the lecture given by the author at the \emph{VI International Course of Mathematical Analysis in Andaluc\'ıa} (CIDAMA), in September 2014. Most results are contained (in a slightly less general setting) in the earlier paper [3] (Garrigós, Hartzstein, Signes, Torrea and Viviani, Pointwise convergence to initial data of heat and Laplace equations, Trans. Amer. Math. Soc. 368 (9) (2016), 6575-6600).
We present new results regarding Lebesgue-type inequalities for the Weak Chebyshev Greedy Algorithm (WCGA) in uniformly smooth Banach spaces. We improve earlier bounds in Temlyakov (Forum Math Sigma 2014), for dictionaries satisfying a new property introduced here. We apply these results to derive optimal bounds in two natural examples of sequence spaces. In particular, optimality is obtained in the case of the multivariate Haar system in Lp with 1<p<2, under the Littlewood-Paley norm.
We find optimal decay estimates for the Poisson kernels associated with various Laguerre-type operators L. From these, we solve two problems about the Poisson semigroup $e^{-t\sqrt{L}}$. First, we find the largest space of initial data $f$ so that $e^{-t\sqrt{L}}f(x)\to f(x)$ at a.e. $x$. Secondly, we characterize the largest class of weights $w$ which admit 2-weight inequalities of the form $\|\sup_{0<t\leq t_0}|e^{-t\sqrt{L}}f|\,\|_{L^p(v)}\lesssim \|f\|_{L^p(w)}$, for some other weight $v$.
An important inequality due to Wolff on plate decompositions of cone multipliers is known to have consequences for a variety of problems in harmonic analysis. We observe that the range in Wolff's inequality, for the conic and the spherical versions, can be improved by using bilinear restriction results. We also use this inequality to give some improved estimates on square functions associated to decompositions of cone multipliers in low dimensions. This gives a new L^4 bound for the cone multiplier operator in three dimensions.
We give characterizations of radial Fourier multipliers as acting on radial L^p-functions, 1<p<2d/(d+1), in terms of Lebesgue space norms for Fourier localized pieces of the convolution kernel. This is a special case of corresponding results for general Hankel multipliers. Besides L^p-L^q bounds we also characterize weak type inequalities and intermediate inequalities involving Lorentz spaces. Applications include results on interpolation of multiplier spaces.
In this paper, we pursue the study of the radar ambiguity problem started in \cite{Ja,GJP}. More precisely, for a given function $u$ we ask for all functions $v$ (called \emph{ambiguity partners}) such that the ambiguity functions of $u$ and $v$ have same modulus. In some cases, $v$ may be given by some elementary transformation of $u$ and is then called a \emph{trivial partner} of $u$ otherwise we call it a \emph{strange partner}. Our focus here is on two discrete versions of the problem. For the first one, we restrict the problem to functions $u$ of the Hermite class, $u=P(x)e^{-x^2/2}$, thus reducing it to an algebraic problem on polynomials. Up to some mild restriction satisfied by quasi-all and almost-all polynomials, we show that such a function has only trivial partners. The second discretization, restricting the problem to pulse type signals, reduces to a combinatorial problem on matrices of a special form. We then exploit this to obtain new examples of functions that have only trivial partners. In particular, we show that most pulse type signals have only trivial partners. Finally, we clarify the notion of \emph{trivial partner}, showing that most previous counterexamples are still trivial in some restricted sense.