Let $\mathcal L=-Δ+V$ be a Schrödinger operator on $\mathbb R^d$, $d\geq3$, where $Δ$ is the Laplacian operator on $\mathbb R^d$ and the nonnegative potential $V$ belongs to the reverse Hölder class $RH_s$ for $s\geq d/2$. For given $0<α<d$, the fractional integrals associated to the Schrödinger operator $\mathcal L$ is defined by $\mathcal I_α={\mathcal L}^{-α/2}$.Suppose that $b$ is a locally integrable function on $\mathbb R^d$, the commutator generated by $b$ and $\mathcal I_α$ is defined by $[b,\mathcal I_α]f(x)=b(x)\cdot \mathcal I_αf(x)-\mathcal I_α(bf)(x)$. In this paper, we first introduce some kinds of weighted Morrey spaces related to certain nonnegative potentials belonging to the reverse Hölder class $RH_s$ for $s\geq d/2$. Then we will establish the boundedness properties of the fractional integrals $\mathcal I_α$ on these new spaces. Furthermore, weighted strong-type estimate for the corresponding commutator $[b,\mathcal I_α]$ in the framework of Morrey spaces is also obtained. The classes of weights, the classes of symbol functions as well as weighted Morrey spaces discussed in this paper are larger than $A_{p,q}$, $\mathrm{BMO}(\mathbb R^d)$ and $L^{p,κ}(μ,ν)$ corresponding to the classical case (that is $V\equiv0$).
We prove well-posedness results for the Dirichlet problem in $\mathbb{R}^{n}_{+}$ for homogeneous, second order, constant complex coefficient elliptic systems with boundary data in generalized Hölder spaces $\mathscr{C}^ω(\mathbb{R}^{n-1},\mathbb{C}^M)$ and in generalized Morrey-Campanato spaces $\mathscr{E}^{ω,p}(\mathbb{R}^{n-1},\mathbb{C}^M)$ under certain assumptions on the growth function $ω$. We also identify a class of growth functions $ω$ for which $\mathscr{C}^ω(\mathbb{R}^{n-1},\mathbb{C}^M)=\mathscr{E}^{ω,p}(\mathbb{R}^{n-1},\mathbb{C}^M)$ and for which the aforementioned well-posed…
Let $\mathcal L=-Δ+V$ be a Schrödinger operator, where $Δ$ is the Laplacian on $\mathbb R^d$ and the nonnegative potential $V$ belongs to the reverse Hölder class $RH_q$ for $q\geq d$. The Riesz transform associated with the operator $\mathcal L=-Δ+V$ is denoted by $\mathcal R=\nabla{(-Δ+V)}^{-1/2}$ and the dual Riesz transform is denoted by $\mathcal R^{\ast}=(-Δ+V)^{-1/2}\nabla$. In this paper, we first introduce some kinds of weighted Morrey spaces related to certain nonnegative potentials belonging to the reverse Hölder class $RH_q$ for $q\geq d$. Then we will establish the boundedness pro…
Let L=-Δ+V(x) be a Schrodinger operator, where Δ is the Laplacian on while nonnegative potential V(x) belonging to the reverse Holder class. We establish the boundedness of the commutators of Marcinkiewicz integrals with rough kernel associated with Schrodinger operator on vanishing generalized Morrey spaces.
Rubio de Francia proved the one-sided Littlewood--Paley inequality for arbitrary intervals in $L^p$, $2 \le p < \infty$. In this article, his methods are developed and employed to prove an analogue of such an inequality "beyond the index $p=\infty$", i.e., for spaces of Hölder functions and BMO.
We prove an appropriate sharp quantitative reverse Hölder inequality for the $C_p$ class of weights from which we obtain as a limiting case the sharp reverse Hölder inequality for the $A_\infty$ class of weights. We use this result to provide a quantitative weighted norm inequality between Calderón-Zygmund operators and the Hardy-Littlewood maximal function, precisely \[ \|Tf\|_{L^p(w)} \lesssim_{T,n,p,q} [w]_{C_q}(1+\log^+[w]_{C_q})\|Mf\|_{L^p(w)}, \] for $w\in C_q$ and $q>p>1$, quantifying Sawyer's theorem.
The natural maximal and minimal functions commute pointwise with the logarithm on $A_\infty$. We use this observation to characterize the spaces $A_1$ and $RH_\infty$ on metric measure spaces with a doubling measure. As the limiting cases of Muckenhoupt $A_p$ and reverse Hölder classes, respectively, their behavior is remarkably symmetric. On general metric measure spaces, an additional geometric assumption is needed in order to pass between $A_p$ and reverse Hölder descriptions. Finally, we apply the characterization to give simple proofs of several known properties of $A_1$ and $RH_\infty$, …
In the Euclidean setting, the Fujii-Wilson-type $A_\infty$ weights satisfy a Reverse Hölder Inequality (RHI) but in spaces of homogeneous type the best known result has been that $A_\infty$ weights satisfy only a weak Reverse Hölder Inequality. In this paper, we compliment the results of Hytönen, Pérez and Rela and show that there exist both $A_\infty$ weights that do not satisfy an RHI and a genuinely weaker weight class that still satisfies a weak RHI. We also show that all the weights that satisfy a weak RHI have a self-improving property but the self-improving property of the strong Revers…
In this note we prove a multilinear version of the reverse Hölder inequality in the theory of Muckenhoupt $A_p$ weights. We give two applications of this inequality to the study of multilinear weighted norm inequalities. First, we prove a structure theorem for multilinear $A_{\vec{p}}$ weights; second, we give a new sufficient condition for multilinear, two-weight norm inequalities for the maximal operator.
We develop a theory of extrapolation for weights that satisfy a generalized reverse Hölder inequality in the scale of Orlicz spaces. This extends previous results by Auscher and Martell [2] on limited range extrapolation. As an application, we show that a number of weighted norm inequalities for linear and bilinear operators can be proved using our extrapolation theorem.
Let $\mathcal L=-Δ_{\mathbb H^n}+V$ be a Schrödinger operator on the Heisenberg group $\mathbb H^n$, where $Δ_{\mathbb H^n}$ is the sub-Laplacian on $\mathbb H^n$ and the nonnegative potential $V$ belongs to the reverse Hölder class $RH_s$ with $s\geq Q/2$. Here $Q=2n+2$ is the homogeneous dimension of $\mathbb H^n$. For given $α\in(0,Q)$, the fractional integrals associated to the Schrödinger operator $\mathcal L$ is defined by $\mathcal I_α={\mathcal L}^{-α/2}$. In this article, we first introduce the Morrey space $L^{p,κ}_{ρ,\infty}(\mathbb H^n)$ and weak Morrey space $WL^{p,κ}_{ρ,\infty}(\…
We examine the problem of the Fourier transform mapping one weighted Lebesgue space into another, by studying necessary conditions and sufficient conditions which expose an underlying geometry. In the necessary conditions, this geometry is connected to an old result of Mahler concerning the the measure of a convex set and its geometric polar being essentially reciprocal. An additional assumption, that the weights must belong to a reverse Hölder class, is used to formulate the sufficient condition.
We present ten different characterizations of functions satisfying a weak reverse Hölder inequality on an open subset of a metric measure space with a doubling measure. Among others, we describe these functions as a class of weak $A_\infty$ weights, which is a generalization of Muckenhoupt weights that allows for nondoubling weights. Although our main results are modeled after conditions that hold true for Muckenhoupt weights, we also discuss two conditions for Muckenhoupt $A_\infty$ weights that fail to hold for weak $A_\infty$ weights.
Let $\mathcal{L}$ be the Schrödinger operator with potential $V$, that is, $\mathcal L=-Δ+V$, where it is assumed that $V$ satisfies a reverse Hölder inequality. We consider weighted Morrey-Campanato spaces $BMO_{\mathcal L,w}^α(\mathbb R^d)$ and $BLO_{L,w}^α(\mathbb R^d)$ in the Schrödinger setting. We prove that the variation operator $V_σ(\{T_t\}_{t>0})$, $σ>2$, and the oscillation operator $O(\{T_t\}_{t>0}, \{t_j\}_{j\in \mathbb Z})$, where $t_j<t_{j+1}$, $j\in \mathbb Z$, $\lim_{j\rightarrow +\infty}t_j=+\infty$ and $\lim_{j\rightarrow -\infty} t_j=0$, being $T_t=t^k\partial_t^k e^{-t\mat…
In this article, the authors introduce the Newton-Morrey-Sobolev space on a metric measure space $(\mathscr{X},d,μ)$. The embedding of the Newton-Morrey-Sobolev space into the Hölder space is obtained if $\mathscr{X}$ supports a weak Poincaré inequality and the measure $μ$ is doubling and satisfies a lower bounded condition. Moreover, in the Ahlfors $Q$-regular case, a Rellich-Kondrachov type embedding theorem is also obtained. Using the Hajłasz gradient, the authors also introduce the Hajłasz-Morrey-Sobolev spaces, and prove that the Newton-Morrey-Sobolev space coincides with the Hajłasz-Morr…
We consider the class of all non-negative on $\mathbb{R_+}$ functions such that each of them satisfies the Reverse Hölder Inequality uniformly over all intervals with some constant the minimum value of which can be regarded as the corresponding "norm" of a function. We compare this "norm" with the "norm" of an even extension of a function from $\mathbb{R_+}$ on $\mathbb{R}.$ In this paper the upper estimate for the ratio of such "norms" has been obtained. In the particular case of power functions on $\mathbb{R_+}$ the precise value of the increase of the "norm" of its even extension is given. …
In this expository article we collect and discuss some recent results on different consequences of a Sharp Reverse Hölder Inequality for $A_{\infty}$ weights. For two given operators $T$ and $S$, we study $L^p(w)$ bounds of Coifman-Fefferman type. $ \|Tf\|_{L^p(w)}\le c_{n,w,p} \|Sf\|_{L^p(w)}, $ that can be understood as a way to control $T$ by $S$. We will focus on a \emph{quantitative} analysis of the constants involved and show that we can improve classical results regarding the dependence on the weight $w$ in terms of Wilson's $A_{\infty}$ constant $ [w]_{A_{\infty}}:=\sup_Q\frac{1}{w(Q)}…
In this article we present a new proof of a sharp Reverse Hölder Inequality for $A_\infty$ weights that is valid in the context of spaces of homogeneous type. Then we derive two applications: a precise open property of Muckenhoupt classes and, as a consequence of this last result, we obtain a simple proof of a sharp weighted bound for the Hardy-Littlewood maximal function involving $A_\infty$ constants: |M|_{L^p(w)} \leq c (\frac{1}{p-1} [w]_{A_p}[σ]_{A_\infty})^{1/p}, where $1<p<\infty$, $σ=w^{\frac{1}{1-p}}$ and $c$ depends only on the doubling constant of the measure $μ$ and the geometric c…
The aim of this paper is to introduce and investigative some new function classes of Morrey-Campanato type. Let $0<p<\infty$ and $0\leq λ<n+p$. We say that $f\in \mathcal{\bar{L}}^{p,λ}(Ω)$ if $$\sup_{x_{0}\in Ω,ρ>0}ρ^{-λ}\int_{Ω(x_{0},ρ)}\big|f(x)-|f|_{Ω(x_{0},ρ)}\big|^pdx<\infty,$$ where $Ω(x_{0},ρ)=Q(x_{0},ρ)\cap Ω$ and $Q(x,ρ)$ is denote the cube of $\mathbb{R}^n$. Some basic properties and characterizations of these classes are presented. If $0\leq λ<n$, the space is equivalent to related Morrey space. If $λ=n$, then $f \in \mathcal{\bar{L}}^{p,n}(Ω)$ if and only if $f\in BMO(Ω)$ with $f^…
Let $\mathcal L=-Δ_{\mathbb H^n}+V$ be a Schrödinger operator on the Heisenberg group $\mathbb H^n$, where $Δ_{\mathbb H^n}$ is the sublaplacian on $\mathbb H^n$ and the nonnegative potential $V$ belongs to the reverse Hölder class $RH_s$ with $s\in[Q/2,\infty)$. Here $Q=2n+2$ is the homogeneous dimension of $\mathbb H^n$. For given $α\in(0,Q)$, the fractional integral operator associated with the Schrödinger operator $\mathcal L$ is defined by $\mathcal I_α={\mathcal L}^{-α/2}$. In this article, the author introduces the Morrey space $L^{p,κ}_{ρ,\infty}(\mathbb H^n)$ and weak Morrey space $WL…
This paper determines the sharp asymptotic order of the following reverse Hölder inequality for spherical harmonics $Y_n$ of degree $n$ on the unit sphere $\mathbb{S}^{d-1}$ of $\mathbb{R}^d$ as $n\to \infty$: \[\|Y_n\|_{L^q(\mathbb{S}^{d-1})}\leq C n^{α(p,q)}\|Y_n\|_{L^p(\mathbb{S}^{d-1})},\quad 0<p<q\leq \infty.\] In many cases, these sharp estimates turn out to be significantly better than the corresponding estimates in the Nilkolskii inequality for spherical polynomials. Furthermore, they allow us to improve two recent results on the restriction conjecture and the sharp Pitt inequalities f…
Let $\mathcal{L}$ be a Schrödinger operator of the form $\mathcal{L} = -Δ+V$ acting on $L^2(\mathbb R^n)$ where the nonnegative potential $V$ belongs to the reverse Hölder class $B_q$ for some $q\geq n.$ Let $L^{p,λ}(\mathbb{R}^{n})$, $0\le λ<n$ denote the Morrey space on $\mathbb{R}^{n}$. In this paper, we will show that a function $f\in L^{2,λ}(\mathbb{R}^{n})$ is the trace of the solution of ${\mathbb L}u=u_{t}+{\mathcal{L}}u=0, u(x,0)= f(x),$ where $u$ satisfies a Carleson-type condition \begin{eqnarray*} \sup_{x_B, r_B} r_B^{-λ}\int_0^{r_B^2}\int_{B(x_B, r_B)} |\nabla u(x,t)|^2 {dx dt} \l…
We prove a two dimensional Holder and reverse-Holder inequality on time scales via the diamond-alpha integral. Other integral inequalities are established as well, which have as corollaries some recent proved Hardy-type inequalities on time scales.
We study the boundedness of Riesz transforms in $L^p$ for $p>2$ on a doubling metric measure space endowed with a gradient operator and an injective, $ω$-accretive operator $L$ satisfying Davies-Gaffney estimates. If $L$ is non-negative self-adjoint, we show that under a reverse Hölder inequality, the Riesz transform is always bounded on $L^p$ for $p$ in some interval $[2,2+\varepsilon)$, and that $L^p$ gradient estimates for the semigroup imply boundedness of the Riesz transform in $L^q$ for $q \in [2,p)$. This improves results of \cite{ACDH} and \cite{AC}, where the stronger assumption of a …
Let $\mathcal L=-Δ_{\mathbb H^n}+V$ be a Schrödinger operator on the Heisenberg group $\mathbb H^n$, where $Δ_{\mathbb H^n}$ is the sublaplacian on $\mathbb H^n$ and the nonnegative potential $V$ belongs to the reverse Hölder class $RH_q$ with $q\geq Q/2$. Here $Q=2n+2$ is the homogeneous dimension of $\mathbb H^n$. Assume that $\{e^{-s\mathcal L}\}_{s>0}$ is the heat semigroup generated by $\mathcal L$. The Littlewood-Paley function $\mathfrak{g}_{\mathcal L}$ and the Lusin area integral $\mathcal{S}_{\mathcal L}$ associated with the Schrödinger operator $\mathcal L$ are defined, respectively…
We show various $L^p$ estimates for Schrödinger operators $-Δ+V$ on $\RR^n$ and their square roots. We assume reverse Hölder estimates on the potential, and improve some results of Shen \cite{Sh1}. Our main tools are improved Fefferman-Phong inequalities and reverse Hölder estimates for weak solutions of $-Δ+V$ and their gradients.
We state a version of Gehring's self improvement Theorem for reverse \Holder weights which is valid for dyadic cubes over spaces of homogeneous type and explore some of the consequences and applications.
In this article, we investigate systems of generalized Schrödinger operators and their fundamental matrices. More specifically, we establish the existence of such fundamental matrices and then prove sharp upper and lower exponential decay estimates for them. The Schrödinger operators that we consider have leading coefficients that are bounded and uniformly elliptic, while the zeroth-order terms are assumed to be nondegenerate and belong to a reverse Hölder class of matrices. In particular, our operators need not be self-adjoint. The exponential bounds are governed by the so-called upper and lo…
We prove that, for totally irregular measures $μ$ on $\mathbb{R}^{d}$ with $d\geq3$, the $(d-1)$-dimensional Riesz transform $$ T_{A,μ}^{V}f(x) = \int_{\mathbb{R}^d} \nabla_{1}\mathcal{E}_{A}^{V}(x,y) f(y) \, d μ(y) $$ adapted to the Schrödinger operator $L_{A}^{V} = -\mathrm{div} A \nabla + V$ with fundamental solution $\mathcal{E}_{A}^{V}$ is not bounded on $L^{2}(μ)$. This generalises recent results obtained by Conde-Alonso, Mourgoglou and Tolsa for free-space elliptic operators with Hölder continuous coefficients $A$ since it allows for the presence of potentials $V$ in the reverse Hölder …
Let $\mathcal{L}=-Δ+\mathit{V}(x)$ be a Schrödinger operator, where $Δ$ is the Laplacian operator on $\mathbb{R}^{d}$ $(d\geq 3)$, while the nonnegative potential $\mathit{V}(x)$ belongs to the reverse Hölder class $B_{q}, q>d/2$. In this paper, we study weighted compactness of commutators of some Schrödinger operators, which include Riesz transforms, standard Calderón-Zygmund operatos and Littlewood-Paley functions. These results generalize substantially some well-know results.
Let $L=-Δ+V$ with non-negative potential $V$ satisfying some appropriate reverse Hölder inequality. In this paper, we study the boundedness of the commutators of some singular integrals associated to $L$ such as Riesz transforms and fractional integrals with the new BMO functions introduced in \cite{BHS1} on the weighted spaces $L^p(w)$ where $w$ belongs to the new classes of weights introduced by \cite{BHS2}.
Let L be a Schrödinger operator of the form L=-Δ+V, where the nonnegative potential V satisfies a reverse Hölder inequality. Using the method of L-harmonic extensions we study regularity estimates at the scale of adapted Hölder spaces. We give a pointwise description of L-Hölder spaces and provide some characterizations in terms of the growth of fractional derivatives of any order and Carleson measures. Applications to fractional powers of L and multipliers of Laplace transform type developed.
The aim of this article is to deepen the understanding of the derivation of $\mathrm{L}^p$-estimates of non-local operators. We review the $\mathrm{L}^p$-extrapolation theorem of Shen which builds on a real variable argument of Caffarelli and Peral and adapt this theorem to account for non-local weak reverse Hölder estimates. These non-local weak reverse Hölder estimates appear for example in the investigation of non-local elliptic integrodifferential operators. This originates from the fact that here only a non-local Caccioppoli inequality is valid, see Kuusi, Mingione, and Sire. As an applic…
In this article, the authors consider the Schrödinger type operator $L:=-{\rm div}(A\nabla)+V$ on $\mathbb{R}^n$ with $n\geq 3$, where the matrix $A$ is symmetric and satisfies uniformly elliptic condition and the nonnegative potential $V$ belongs to the reverse Hölder class $RH_q(\mathbb{R}^n)$ with $q\in(n/2,\,\infty)$. Let $p(\cdot):\ \mathbb{R}^n\to(0,\,1]$ be a variable exponent function satisfying the globally $\log$-Hölder continuous condition. The authors introduce the variable Hardy space $H_L^{p(\cdot)}(\mathbb{R}^n)$ associated to $L$ and establish its atomic characterization. The a…
The paper concerns the magnetic Schrödinger operator on $R^n$. Under certain conditions, given in terms of the reverse Hölder inequality on the magnetic field and the electric potential, we prove some $L^p$ estimates on the Riesz transforms and we establish some related maximal inequalities.
Let $L= -Δ+ V$ be a Schrödinger operator on $\mathbb R^d$, $d\geq 3$, where $V$ is a nonnegative function, $V\ne 0$, and belongs to the reverse Hölder class $RH_{d/2}$. In this paper, we prove a version of the classical theorem of Jones and Journé on weak$^*$-convergence in the Hardy space $H^1_L(\mathbb R^d)$.
