arXiv subjects
Publications and source records attributed to Óscar Ciaurri.
In this paper we present uniform weighted inequalities for the Hankel transform transplantation operator. A weighted vector-valued inequality is also obtained. As a consequence we deduce an extension of a transference theorem due to Rubio de Francia.
In this paper we analyze some classical operators in harmonic analysis associated to the multidimensional discrete Laplacian \[ Δ_N f(\mathbf{n})=\sum_{i=1}^{N}(f(\mathbf{n}+\mathbf{e}_i)-2f(\mathbf{n})+f(\mathbf{n}-\mathbf{e}_i)), \qquad \mathbf{n}\in \mathbb{Z}^N. \] We deal with the heat and Poisson semigroups, the fractional integrals, the Riesz transforms, the fractional powers of the Laplacian, and the $g_k$-square functions.
We present some simple proofs of the well-known expressions for \[ ζ(2k) = \sum_{m=1}^\infty \frac{1}{m^{2k}}, \qquad β(2k+1) = \sum_{m=0}^\infty \frac{(-1)^m}{(2m+1)^{2k+1}}, \] where $k = 1,2,3,\dots$, in terms of the Bernoulli and Euler polynomials. The computation is done using only the defining properties of these polynomials and employing telescoping series. The same method also yields integral formulas for $ζ(2k+1)$ and $β(2k)$. In addition, the method also applies to series of type \[ \sum_{m\in\mathbb{Z}} \frac{1}{(2m-μ)^s}, \qquad \sum_{m\in\mathbb{Z}} \frac{(-1)^m}{(2m+1-μ)^s}, \] in this case using Apostol-Bernoulli and Apostol-Euler polynomials.
We evaluate the moments of some functions composed with the fractional part of $1/x$. We name them fractional moments. In particular, we obtain expressions for the fractional moments of some trigonometric functions, the Bernoulli polynomials and the functions $x^m$ and $x^m(1-x)^m$.
We study a generalized spherical means operator, viz. generalized spherical mean Radon transform, acting on radial functions. As the main results, we find conditions for the associated maximal operator and its local variant to be bounded on power weighted Lebesgue spaces. This translates, in particular, into almost everywhere convergence to radial initial data results for solutions to certain Cauchy problems for classical Euler-Poisson-Darboux and wave equations. Moreover, our results shed some new light to the interesting and important question of optimality of the yet known $L^p$ boundedness results for the maximal operator in the general non-radial case. It appears that these could still be notably improved, as indicated by our conjecture of the ultimate sharp result.
The discrete counterpart of the problem related to the convergence of the Fourier-Jacobi series is studied. To this end, given a sequence, we construct the analogue of the partial sum operator related to Jacobi polynomials and characterize its convergence in the $\ell^p(\mathbb{N})$-norm.
The research about Harmonic Analysis associated with Jacobi expansions carried out in \cite{ACL-JacI} and \cite{ACL-JacII} is continued in this paper. Given the operator $\mathcal{J}^{(α,β)}=J^{(α,β)}-I$, where $J^{(α,β)}$ is the three-term recurrence relation for the normalized Jacobi polynomials and $I$ is the identity operator, we define the corresponding Littlewood-Paley-Stein $g_k^{(α,β)}$-functions associated with it and we prove an equivalence of norms with weights for them. As a consequence, we deduce a result for Laplace type multipliers.
This paper is the continuation of the study on discrete harmonic analysis related to Jacobi expansions initiated in [1]. Considering the operator $\mathcal{J}^{(α,β)}=J^{(α,β)}-I$, where $J^{(α,β)}$ is the three-term recurrence relation for the normalized Jacobi polynomials and $I$ is the identity operator, we focus on the study of weighted inequalities for the Riesz transform associated with it.
In this paper we commence the study of discrete harmonic analysis associated with Jacobi orthogonal polynomials of order $(α,β)$. Particularly, we give the solution $W^{(α,β)}_t$, $t\ge 0$, and some properties of the heat equation related to the operator $J^{(α,β)}-I$, where $J^{(α,β)}$ is the three-term recurrence relation for the normalized Jacobi polynomials and $I$ is the identity operator. These results will be a consequence of a much more general theorem concerning the solution of the heat equation for Jacobi matrices. In addition, we also prove the positivity of the operator $W^{(α,β)}_t$ under some suitable restrictions on the parameters $α$ and $β$. Finally, we investigate mapping properties of the maximal operators defined by the heat and Poisson semigroups in weighted $\ell^{p}$-spaces using discrete vector-valued local Calderón-Zygmund theory. For the Poisson semigroup, these properties follows readily from the control in terms of the heat one.
We present a transplantation theorem for Jacobi coefficients in weighted spaces. In fact, by using a discrete vector-valued local Calderón-Zygmund theory, which has recently been furnished, we prove the boundedness of transplantation operators from $\ell^p(\mathbb{N},w)$ into itself, where $w$ is a weight in the discrete Muckenhoupt class $A_{p}(\mathbb{N})$. Moreover, we obtain weighted weak $(1,1)$ estimates for those operators.
Let $\{q_n^{(α,β,m)}(x)\}_{n\ge 0}$ be the orthonormal polynomials respect to the Sobolev-type inner product \begin{equation*} \langle f,g\rangle_{α,β,m}=\sum_{k=0}^m \int_{-1}^{1}f^{(k)}(x)g^{(k)}(x)\, dw_{α+k,β+k}(x), \quad α,β>-1, \quad m\ge 1, \end{equation*} where $dw_{a,b}(x)=(1-x)^{a}(1+x)^b\, dx$. We obtain necessary and sufficient conditions for the uniform boundedness of the partial sum operators related to this sequence of polynomials in the Sobolev space $W_{α,β}^{p,m}$. As a consequence we deduce the convergence of such partial sums in the norm of $W_{α,β}^{p,m}$.
We study the partial sum operator for a Sobolev-type inner product related to the classical Gegenbauer polynomials. A complete characterization of the partial sum operator in an appropriate Sobolev space is given. Moreover, we analyze the convergence of the partial sum operators.
We prove a Hardy inequality for ultraspherical expansions by using a proper ground state representation. From this result we deduce some uncertainty principles for this kind of expansions. Our result also implies a Hardy inequality on spheres with a potential having a double singularity.
We investigate a generalized spherical means operator, viz. generalized spherical mean Radon transform, acting on radial functions. We establish an integral representation of this operator and find precise estimates of the corresponding kernel. As the main result, we prove two-weight mixed norm estimates for the integral operator, with general power weights involved. This leads to weighted Strichartz type estimates for solutions to certain Cauchy problems for classical Euler-Poisson-Darboux and wave equations with radial initial data.
We introduce Bernoulli-Dunkl and Apostol-Euler-Dunkl polynomials as generalizations of Bernoulli and Apostol-Euler polynomials, where the role of the derivative is now played by the Dunkl operator on the real line. We use them to sum a bunch of series involving the zeros of Bessel functions.
The conic sections, as well as the solids obtained by revolving these curves, and many of their surprising properties, were already studied by Greek mathematicians since at least the fourth century B.C. Some of these properties come to the light, or are rediscovered, from time to time. In this paper we characterize the conic sections as the plane curves whose tangent lines cut off from a certain similar curve segments of constant area. We also characterize some quadrics as the surfaces whose tangent planes cut off from a certain similar surface compact sets of constant volume. Our work is developed in the most general multidimensional case.
We show that the decreasing rearrangement of the Fourier series with respect to the Jacobi polynomials for functions in $L_p$ does not converge unless $p=2$. As a by-product of our work on quasi-greedy bases in $L_{p}(μ)$, we show that no normalized unconditional basis in $L_p$, $p\not=2$, can be semi-normalized in $L_q$ for $q\not=p$, thus extending a classical theorem of Kadets and Pełczy{ń}ski from 1968.
We obtain a $q$-linear analogue of Gegenbauer's expansion of the plane wave. It is expanded in terms of the little $q$-Gegenbauer polynomials and the \textit{third} Jackson $q$-Bessel function. The result is obtained by using a method based on bilinear biorthogonal expansions.