We develop a geometric invariant Littlewood-Paley theory for arbitrary tensors on a compact 2 dimensional manifold. We show that all the important features of the classical LP theory survive with estimates which depend only on very limited regularity assumptions on the metric. We give invariant descriptions of Sobolev and Besov spaces and prove some sharp product inequalities. This theory has being developed in connection to the work of the authors on the geometry of null hypersurfaces with a finite curvature flux condition, see \cite{KR1}, \cite{KR3}.
We are confident however that it can be applied, and extended, to many different situations.
For certain non compact Riemannian manifolds with ends, we obtain Littlewood-Paley type estimates on (weighted) Lp spaces, using the usual square function defined by a dyadic partition.
Using simple facts from harmonic analysis, namely Bernstein inequality and Plansherel isometry, we prove that the pseudodifferential equation $Δ^αu+Vu=0$ improves the Sobolev regularity of solutions provided the potential $V$ is integrable with the critical power $n/2α>1$.
We establish a general theorem improving regularity of solutions of elliptic pseudodifferential equations. It allows to resolve in a unified way the regularity issue for a broad class of nonlinear elliptic equations and systems appearing in different areas of geometry and analysis.
Let $H$ be a Schrödinger operator on $\R^n$. Under a polynomial decay condition for the kernel of its spectral operator, we show that the Besov spaces and Triebel-Lizorkin spaces associated with $H$ are well defined. We further give a Littlewood-Paley characterization of $L_p$ spaces as well as Sobolev spaces in terms of dyadic functions of $H$. This generalizes and strengthens the previous result when the heat kernel of $H$ satisfies certain upper Gaussian bound.
We obtain pointwise upper bounds on the derivatives of the heat kernel on Damek-Ricci spaces. Applying these estimates we prove the $L^p$-boundedness of Littlewood-Paley-Stein operators.
We derive time-averaged $L^1$ estimates on Littlewood--Paley decompositions for linear advection-diffusion equations. For wave numbers close to the dissipative cut-off, these estimates are consistent with Batchelor's predictions on the variance spectrum in passive scalar turbulent mixing.
Let $X$ be a ball quasi-Banach function space on ${\mathbb R}^n$. In this article, assuming that the powered Hardy--Littlewood maximal operator satisfies some Fefferman--Stein vector-valued maximal inequality on $X$ and is bounded on the associated space, the authors establish various Littlewood--Paley function characterizations of the Hardy space $H_X({\mathbb R}^n)$ associated with $X$, under some weak assumptions on the Littlewood--Paley functions. To this end, the authors also establish a useful estimate on the change of angles in tent spaces associated with $X$. All these results have wid…
Let $L:=-Δ+V$ be the Schrödinger operator on $\mathbb{R}^n$ with $n\geq 3$, where $V$ is a non-negative potential which belongs to certain reverse Hölder class $RH_q(\mathbb{R}^n)$ with $q\in (n/2,\,\infty)$. In this article, the authors obtain the quantitative weighted boundedness of Littlewood--Paley functions $g_L$, $S_L$ and $g_{L,\,λ}^\ast$, associated to $L$, on weighted Lebesgue spaces $L^p(w)$, where $w$ belongs to the class of Muckenhoupt $A_p$ weights adapted to $L$.
We study boundedness on $L^p(R^d)$ of vertical Littlewood-Paley-Stein functions for Schrödinger operators $-Δ+ V$ with nonnegative potential $V$. These functions are proved to be bounded on $L^p$ for all $p \in (1, 2]$. The situation for $p > 2$ is different. We prove for a class of potentials that the boundedness on $L^p$ for some $p > d$ holds if and only if $V= 0$.
We establish the kernel estimates for the Littlewood-Paley projections associated with a Schrödinger operator H=-Δ+V in \mathbb{R}^3 for a large class of short-range potentials V(x). As a corollary, we prove the homogeneous Sobolev inequality.
Let $m,n\ge 1$ and $g_{λ_1,λ_2}^*$ be the bi-parameter Littlewood-Paley $g_λ^{*}$-function defined by $$ g_{λ_1,λ_2}^*(f)(x)= \bigg(\iint_{\R^{m+1}_{+}} \big(\frac{t_2}{t_2 + |x_2 - y_2|}\big)^{m λ_2} \iint_{\R^{n+1}_{+}} \big(\frac{t_1}{t_1 + |x_1 - y_1|}\big)^{n λ_1}|θ_{t_1,t_2} f(y_1,y_2)|^2 \frac{dy_1 dt_1}{t_1^{n+1}} \frac{dy_2 dt_2}{t_2^{m+1}} \bigg)^{1/2}, λ_1>1,\quad λ_2>1 $$ where $θ_{t_1,t_2} f$ is a non-convolution kernel defined on $\mathbb{R}^{m+n}$. In this paper, we showed that the bi-parameter Littlewood-Paley function $g_{λ_1,λ_2}^*$ was bounded from $L^2(\R^{n+m})$ to $L^2(\R…
We study the Littlewood-Paley-Stein functions associated with Hodge-de Rham and Schr{ö}dinger operators on Riemannian manifolds. Under conditions on the Ricci curvature we prove their boundedness on L p for p in some interval (p 1 , 2] and make a link to the Riesz Transform. An important fact is that we do not make assumptions of doubling measure or estimates on the heat kernel in this case. For p > 2 we give a criterion to obtain the boundedness of the vertical Littlewood-Paley-Stein function associated with Schr{ö}dinger operators on L p .
The goal of this note is to prove a analogue of the Littewood-Paley decomposition for densities of operators and to use it in the context of Lieb-Thirring inequalities.
Let $X$ be a ball quasi-Banach function space on ${\mathbb R}^n$ and assume that the Hardy--Littlewood maximal operator satisfies the Fefferman--Stein vector-valued maximal inequality on $X$, and let $q\in[1,\infty)$ and $d\in(0,\infty)$. In this article, the authors prove that, for any $f\in \mathcal{L}_{X,q,0,d}(\mathbb{R}^n)$ (the ball Campanato-type function space associated with $X$), the Littlewood--Paley $g$-function $g(f)$ is either infinite everywhere or finite almost everywhere and, in the latter case, $g(f)$ is bounded on $\mathcal{L}_{X,q,0,d}(\mathbb{R}^n)$. Similar results for bo…
Let $X$ be a ball quasi-Banach function space on ${\mathbb R}^n$. In this article, assuming that the powered Hardy--Littlewood maximal operator satisfies some Fefferman--Stein vector-valued maximal inequality on $X$ as well as it is bounded on both the weak ball quasi-Banach function space $WX$ and the associated space, the authors establish various Littlewood--Paley function characterizations of $WH_X({\mathbb R}^n)$ under some weak assumptions on the Littlewood--Paley functions. The authors also prove that the real interpolation intermediate space $(H_{X}({\mathbb R}^n),L^\infty({\mathbb R}^…
In this paper, we present a local $Tb$ theorem for the non-homogeneous Littlewood-Paley $g_λ^{*}$-function with non-convolution type kernels and upper power bound measure $μ$. We show that, under the assumptions $\supp b_Q \subset Q$, $|\int_Q b_Q dμ| \gtrsim μ(Q)$ and $||b_Q||^p_{L^p(μ)} \lesssim μ(Q)$, the norm inequality $\big\| g_λ^{*}(f) \big\|_{L^p(μ)} \lesssim \big\| f \big\|_{L^p(μ)}$ holds if and only if the following testing condition holds : $$\sup_{Q : cubes \ in \ \Rn} \frac{1}{μ(Q)}\int_Q \bigg(\int_{0}^{\ell(Q)} \int_{\Rn} \Big(\frac{t}{t+|x-y|}\Big)^{mλ}|θ_t(b_Q)(y,t)|^2 \frac{…
Let $Γ$ be a graph endowed with a reversible Markov kernel $p$, and $P$ the associated operator, defined by $Pf(x)=\sum_y p(x,y)f(y)$. Denote by $\nabla$ the discrete gradient. We give necessary and/or sufficient conditions on $Γ$ in order to compare $\Vert \nabla f \Vert_{p}$ and $\Vert (I-P)^{1/2}f \Vert_{p}$ uniformly in $f$ for $1<p<+\infty$. These conditions are different for $p<2$ and $p>2$. The proofs rely on recent techniques developed to handle operators beyond the class of Calderón-Zygmund operators. For our purpose, we also prove Littlewood-Paley inequalities and interpolation resul…
In this paper we investigate Besov spaces on graded Lie groups. We prove a Nikolskii type inequality (or the Reverse Hölder inequality) on graded Lie groups and as consequence we obtain embeddings of Besov spaces. We prove a version of the Littlewood-Paley theorem on graded Lie groups. The results are applied to obtain embedding properties of Besov spaces and multiplier theorems for both spectral and Fourier multipliers in Besov spaces on graded Lie groups. In particular, we give a number of sufficient conditions for the boundedness of Fourier multipliers in Besov spaces.
Let $\vec{a}:=(a_1,\ldots,a_n)\in[1,\infty)^n$, $\vec{p}:=(p_1,\ldots,p_n)\in(0,\infty)^n$ and $H_{\vec{a}}^{\vec{p}}(\mathbb{R}^n)$ be the anisotropic mixed-norm Hardy space associated with $\vec{a}$ defined via the non-tangential grand maximal function. In this article, via first establishing a Calderón-Zygmund decomposition and a discrete Calderón reproducing formula, the authors then characterize $H_{\vec{a}}^{\vec{p}}(\mathbb{R}^n)$, respectively, by means of atoms, the Lusin area function, the Littlewood-Paley $g$-function or $g_λ^\ast$-function. The obtained Littlewood-Paley $g$-functio…
Let $L = Δ+ V$ be a Schrödinger operator with a non-negative potential $V$ on a complete Riemannian manifold $M$. We prove that the vertical Littlewood-Paley-Stein functional associated with $L$ is bounded on $L^p(M)$ {\it if and only if} the set $\{\sqrt{t}\, \nabla e^{-tL}, \, t > 0\}$ is ${\mathcal R}$-bounded on $L^p(M)$. We also introduce and study more general functionals. For a sequence of functions $m_k : [0, \infty) \to \mathbb{C}$, we define $$H((f_k)) := \Big( \sum_k \int_0^\infty | \nabla m_k(tL) f_k |^2 dt \Big)^{1/2} + \Big( \sum_k \int_0^\infty | \sqrt{V} m_k(tL) f_k |^2 dt \Big…
In this article we give an overview on some recent development of Littlewood-Paley theory for Schrödinger operators. We extend the Littlewood-Paley theory for special potentials considered in the authors' previous work. We elaborate our approach by considering potential in $C^\infty_0$ or Schwartz class in one dimension. In particular the low energy estimates are treated by establishing some new and refined asymptotics for the eigenfunctions and their Fourier transforms. We give maximal function characterization of the Besov spaces and Triebel-Lizorkin spaces associated with $H$. Then we prove…
We use the Littlewood-Paley decomposition technique to obtain a $C^\infty$-well-posedness result for a weakly hyperbolic equation with a finite order of degeneration
Using Littlewood-Paley theory, one formulates the determining wavenumbers for the Hall-MHD system, defined for each individual solution $(u,b)$. It is shown that the long time behaviour of strong solutions is almost finite dimensional as the wavenumbers are bounded in certain average senses.
