arXiv subjects

José L. Torrea

Publications and source records attributed to José L. Torrea.

7 recordsLinked to original sources

arXiv2020

Lipschitz spaces adapted to Schrödinger operators and regularity properties

Consider the Schrödinger operator $\mathcal{L}=-Δ+V$ in $\mathbb{R}^n, n\ge 3,$ where $V$ is a nonnegative potential satisfying a reverse Hölder condition of the type \begin{equation*} \left( \frac{1}{|B|}\int_B V(y)^qdy\right)^{1/q}\le \frac{C}{|B|}\int_B V(y)dy, \, \text{ for some }q>n/2. \end{equation*} We define $Λ^α_{\mathcal{L}},\, 0<α<2,$ the class of measurable functions such that $$ \|ρ(\cdot)^{-α}f(\cdot)\|_\infty<\infty \quad \, \, \text{and}\:\: \quad \sup_{|z|>0}\frac{\|f(\cdot+z)+f(\cdot-z)-2f(\cdot)\|_\infty}{|z|^α}<\infty, $$ where $ρ$ is the critical radius function associated to $\mathcal{L}$. Let $W_y f = e^{-y\mathcal{L}}f$ be the heat semigroup of $\mathcal{L}$. Given $α>0,$ we denote by $Λ_{α/2}^{W}$ the set of functions $f$ which satisfy \begin{equation*} \|ρ(\cdot)^{-α}f(\cdot)\|_\infty<\infty \hbox{ and } \Big\|\partial_y^k{W}_y f \Big\|_{L^\infty(\mathbb{R}^{n})}\leq C_αy^{-k+α/2},\;\: \, {\rm with }\, k=[α/2]+1, y>0. \end{equation*} We prove that for $0<α\le 2-n/q$, $Λ^α_{\mathcal{L}} = Λ_{α/2}^{W}.$ As application, we obtain regularity properties of fractional powers (positive and negative) of the operator $\mathcal{L}$, Schrödinger Riesz transforms, Bessel potentials and multipliers of Laplace transforms type. The proofs of these results need in an essential way the language of semigroups. Parallel results are obtained for the classes defined through the Poisson semigroup, $P_yf= e^{-y\sqrt{\mathcal{L}}}f.$

math.FA↗

arXiv2019

Boundedness of differential transforms for one-sided fractional Poisson-type operator sequence

In this paper, we analyze the convergence speed of a series related with $\mathcal{P}_τ^αf$ by discussing the behavior of the family of operators \begin{equation*} T_N^αf(t) = \sum_{j=N_1}^{N_2} v_j(\mathcal{P}_{a_{j+1}}^αf(t)-\mathcal{P}_{a_j}^αf(t)),\quad ~N=(N_1,N_2)\in \mathbb{Z}^2\quad \hbox{with} \quad N_1<N_2, \end{equation*} where $\{v_j\}_{j\in \mathbb Z}$ is a bounded number sequence, and $\{a_j\}_{j\in \mathbb{Z}}$ is a $ρ$-lacunary sequence of positive numbers, that is, $1<ρ\leq a_{j+1}/a_j, \text{for all}\ j\in \mathbb{Z}.$ We shall show the boundedness of the maximal operator \begin{equation*}T^*f(t)=\sup_N |T_N^αf(t)|, \quad t\in\mathbb{R}, \end{equation*} in the one-sided weighted Lebesgue spaces $L^p(\mathbb{R},ω)(ω\in A_p^-$), $1< p < \infty$. As a consequence we infer the existence of the limit, in norm and almost everywhere, of the family $T_N^αf$ for functions in $L^p(\mathbb{R},ω)$. Results for $L^1(\mathbb{R},ω)(ω\in A_1^-)$, $L^\infty(\mathbb{R})$ and $BMO(\mathbb{R})$ are also obtained. It is also shown that the local size of $T^*f$, for functions $f$ having local support, is the same with the order of a singular integral. Moreover, if $\{v_j\}_{j\in \mathbb Z}\in \ell^p(\mathbb Z)$, we get an intermediate size between the local size of singular integrals and Hardy-Littlewood maximal operator.

math.CA↗

arXiv2017

Schauder estimates for discrete fractional integrals

In this note we focus on the discrete fractional integrals as a natural continuation of our previous work about nonlocal fractional derivatives, discrete and continuous. We define the discrete fractional integrals by using the semigroup theory and we study the regularity of {discrete fractional integrals} on the discrete Hölder spaces, which it is known in the differential equations field as the discrete Schauder estimates.

math.AP↗

arXiv2017

Fractional powers of the parabolic Hermite operator. Regularity properties

Let $\mathcal{L}= \partial_t- Δ_x+|x|^2$. Consider its Poisson semigroup $e^{-y\sqrt{\mathcal{L}}}$. For $α>0$ define the Parabolic Hermite-Zygmund spaces $$ Λ^α_{\mathcal{L}}=\left\{f: \:f\in L^\infty(\mathbb{R}^{n+1})\:\; {\rm and} \:\; \left\|\partial_y^k e^{-y\sqrt{\mathcal{L}}} f \right\|_{L^\infty(\mathbb{R}^{n+1})}\leq C_k y^{-k+α},\;\: {\rm with }\, k=[α]+1, y>0. \right\}, $$ with the obvious norm. It is shown that these spaces have a pointwise description of Hölder type. The fractional powers $\mathcal{L}^{\pm β}$ are well defined in these spaces and the following regularity properties are proved: \begin{eqnarray*} α, β>0, \quad \|\mathcal{L}^{-β} f\|_{ Λ^{α+2β}_{\mathcal{L}}}\le C \|f\|_{ Λ^α_{\mathcal{L}}}. \end{eqnarray*} \begin{eqnarray*} 0< 2β< α, \quad \|\mathcal{L}^βf\|_{Λ_{\mathcal{L}}^{α-2β}}\le C \|f\|_{Λ^α_{\mathcal{L}}}. \end{eqnarray*} Parallel results are obtained for the Hermite operator $- Δ+|x|^2.$ The proofs use in a fundamental way the semigroup definition of the operators $\mathcal{L}^{\pm β}$ and $(-Δ+|x|^2)^{\pm β}$. The non-convolution structure of the operators produce an extra difficulty of the arguments.

math.FA↗

arXiv2016

Non-local fractional derivatives. Discrete and continuous

We prove maximum and comparison principles for fractional discrete derivatives in the integers. Regularity results when the space is a mesh of length $h$, and approximation theorems to the continuous fractional derivatives are shown. When the functions are good enough, these approximation procedures give a measure of the order of approximation. These results also allows us to prove the coincidence, for good enough functions, of the Marchaud and Grünwald-Letnikov derivatives in every point and the speed of convergence to the Grünwald-Letnikov derivative. The fractional discrete derivative will be also described as a Neumann-Dirichlet operator defined by a semi-discrete extension problem. Some operators related to the Harmonic Analysis associated to the discrete derivative will be also considered, in particular their behavior in the Lebesgue spaces $\ell^p(\mathbb{Z}).$

math.AP↗

arXiv2011

Fractional Vector-Valued Littlewood-Paley-Stein Theory for Semigroups

We consider the fractional derivative of a general Poisson semigroup. With this fractional derivative we define the generalized fractional Littlewood-Paley $g$-function for semigroups acting on $L^p$-spaces of functions with values in Banach spaces. We give a characterization of the classes of Banach spaces for which the fractional Litlewood-Paley $g$-function is bounded on $L^p$-spaces. We show that the class of Banach spaces is independent of the order of derivation and coincides with the classical (Lusin type/cotype) case. It is also shown that the same kind of results exist for the case of the fractional area function and the fractional $g^*_λ$-function on $\mathbb{R}^n$. At last, we consider the relationship of the almost sure finiteness of the fractional Littlewood-Paley $g$-function, area function and $g^*_λ$-function with the Lusin cotype property of the underlying Banach space. As a byproduct of the techniques developed, one can get some results of independent interest for vector-valued Calderón--Zygmund operators. For example, one can get the following characterization, a Banach space $\mathbb{B}$ is UMD if and only if for some (or, equivalently, for every) $p\in [1,\infty)$, $\displaystyle \lim_{ε\rightarrow 0} \int_{|x-y|> ε} \frac{f(y)}{x-y}dy $ exists \textup{a.e.} $x\in \mathbb{R}$ for every $f\in L^p_\mathbb{B}(\mathbb{R}).$

math.FA↗

arXiv2005

Vector-valued Littlewood-Paley-Stein theory for semigroups

We develop a generalized Littlewood-Paley theory for semigroups acting on $L^p$-spaces of functions with values in uniformly convex or smooth Banach spaces. We characterize, in the vector-valued setting, the validity of the one-sided inequalities concerning the generalized Littlewood-Paley-Stein $g$-function associated with a subordinated Poisson symmetric diffusion semigroup by the martingale cotype and type properties of the underlying Banach space. We show that in the case of the usual Poisson semigroup and the Poisson semigroup subordinated to the Ornstein-Uhlenbeck semigroup on ${\mathbb R}^n$, this general theory becomes more satisfactory (and easier to be handled) in virtue of the theory of vector-valued Calderón-Zygmund singular integral operators.

math.FA↗