Harmonic Analysishttp://hdl.handle.net/20.500.11824/32019-02-15T18:31:55Z2019-02-15T18:31:55ZTwo-weight mixed norm estimates for a generalized spherical mean Radon transform acting on radial functionsCiaurri Ó.Nowak, A.Roncal, L.http://hdl.handle.net/20.500.11824/9262019-02-13T02:00:11Z2018-01-01T00:00:00ZTwo-weight mixed norm estimates for a generalized spherical mean Radon transform acting on radial functions
Ciaurri Ó.; Nowak, A.; Roncal, L.
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
2018-01-01T00:00:00ZVector-valued extensions for fractional integrals of Laguerre expansionsCiaurri Ó.Roncal L.http://hdl.handle.net/20.500.11824/9252019-02-13T02:00:20Z2018-01-01T00:00:00ZVector-valued extensions for fractional integrals of Laguerre expansions
Ciaurri Ó.; Roncal L.
We prove some vector-valued inequalities for fractional integrals defined for several orthonormal systems of Laguerre functions. On the one hand, we obtain weighted $L^p-L^q$ vector-valued extensions, in a multidimensional setting, for negative powers of the operator related to so-called Laguerre expansions of Hermite type. On the other hand, we give necessary and sufficient conditions for vector-valued $L^p-L^q$ estimates related to negative powers of the Laguerre operator associated to expansions of convolution type, in a one-dimensional setting. Both types of vector-valued inequalities are based on estimates of the kernel with precise control of the parameters involved. As an application, mixed norm estimates for fractional integrals related to the harmonic oscillator are deduced.
2018-01-01T00:00:00ZHölder-Lebesgue regularity and almost periodicity for semidiscrete equations with a fractional LaplacianLizama C.Roncal L.http://hdl.handle.net/20.500.11824/9242019-02-13T02:00:17Z2018-01-01T00:00:00ZHölder-Lebesgue regularity and almost periodicity for semidiscrete equations with a fractional Laplacian
Lizama C.; Roncal L.
We study the equations
$
\partial_t u(t,n) = L u(t,n) + f(u(t,n),n); \partial_t u(t,n) = iL u(t,n) + f(u(t,n),n)$
and
$
\partial_{tt} u(t,n) =Lu(t,n) + f(u(t,n),n)$, where $n\in \mathbb{Z}$, $t\in (0,\infty)$, and $L$ is taken to be either the discrete Laplacian operator $\Delta_{\operatorname{dis}} f(n)=f(n+1)-2f(n)+f(n-1)$, or its fractional powers $-(-\Delta_{\operatorname{dis}})^{\sigma}$, $0<\sigma<1$. We combine operator theory techniques with the properties of the Bessel functions to develop a theory of analytic semigroups and cosine operators generated by $\Delta_{\operatorname{dis}}$ and $-(-\Delta_{\operatorname{dis}})^{\sigma}$. Such theory is then applied to prove existence and uniqueness of almost periodic solutions to the above-mentioned equations. Moreover, we show a new characterization of well-posedness on periodic H\"older spaces for linear heat equations involving discrete and fractional discrete Laplacians. The results obtained are applied to Nagumo and Fisher--KPP models with a discrete Laplacian. Further extensions to the multidimensional setting $\mathbb{Z}^N$ are also accomplished.
2018-01-01T00:00:00ZNonlocal discrete diffusion equations and the fractional discrete Laplacian, regularity and applicationsCiaurri Ó.Roncal L.Stinga P.R.Torrea J.L.Varona J.L.http://hdl.handle.net/20.500.11824/9222019-02-13T02:00:16Z2018-01-01T00:00:00ZNonlocal discrete diffusion equations and the fractional discrete Laplacian, regularity and applications
Ciaurri Ó.; Roncal L.; Stinga P.R.; Torrea J.L.; Varona J.L.
The analysis of nonlocal discrete equations driven by
fractional powers of the discrete Laplacian on a mesh of size $h>0$
\[
(-\Delta_h)^su=f,
\]
for $u,f:\Z_h\to\R$, $0<s<1$, is performed. The pointwise nonlocal formula for $(-\Delta_h)^su$ and
the nonlocal discrete mean value property for discrete $s$-harmonic functions are obtained.
We observe that a
characterization of $(-\Delta_h)^s$ as the Dirichlet-to-Neumann operator for a
semidiscrete degenerate elliptic local extension problem is valid.
Regularity properties and Schauder estimates in discrete H\"older spaces as well as
existence and uniqueness of solutions to the nonlocal Dirichlet problem are shown.
For the latter, the fractional discrete Sobolev embedding and the
fractional discrete Poincar\'e inequality are proved,
which are of independent interest. We introduce the negative power (fundamental solution)
\[
u=(-\Delta_h)^{-s}f,
\]
which can be seen as the Neumann-to-Dirichlet map
for the semidiscrete extension problem. We then prove
the discrete Hardy--Littlewood--Sobolev inequality for $(-\Delta_h)^{-s}$.
As applications, the convergence of our fractional discrete Laplacian to
the (continuous) fractional Laplacian as $h\to0$ in H\"older spaces
is analyzed. Indeed, uniform estimates for the error of the approximation
in terms of $h$ under minimal regularity assumptions are obtained. We finally prove
that solutions to the Poisson problem for the fractional Laplacian
\[
(-\Delta)^sU=F,
\]
in $\R$, can be approximated by solutions to the Dirichlet problem for our fractional
discrete Laplacian, with explicit uniform error estimates in terms of~$h$.
2018-01-01T00:00:00Z