In this paper we establish decay estimates for Fourier transform on Hardy-Morrey spaces and its localizable version. Our work include some aspects to these spaces linked up with pointwise Fourier estimates, in particular a natural approach on cancellation moment conditions. As application, we discuss the optimality for continuity of Fourier multipliers and pseudodifferential operators in Hardy-Morrey spaces.
We use the Morrey norm estimate for the imaginary power of the Laplacian to prove an interpolation inequality for the fractional power of the Laplacian on Morrey spaces. We then prove a Hardy-type inequality and use it together with the interpolation inequality to obtain a Heisenberg-type inequality in Morrey spaces.
In this work we improve the sharp Hardy inequality in the case $p>n$ by adding an optimal weighted Hoelder semi-norm. To achieve this we first obtain a local improvement. We also obtain a refinement of both the Sobolev inequality for $p>n$ and the Hardy inequality, the latter having the best constant.
In this note we address the continuity of strongly singular Calderón-Zygmund operators on Hardy-Morrey spaces $\mathcal{HM}_{q}^λ(\mathbb{R}^n)$, assuming weaker integral conditions on the associated kernel. Important examples that falls into this scope are pseudodifferential operators on the Hörmander classes $OpS^{m}_{σ,μ}(\mathbb{R}^n)$ with $0<σ\leq 1$, $0 \leq μ<1$, $μ\leq σ$ and $m\leq -n(1-σ)/2$.
When the growth at infinity of a function $u$ on $\Bbb{R}^{N}$ is compared with the growth of $|x|^{s}$ for some $s\in \Bbb{R},$ this comparison is invariably made pointwise. This paper argues that the comparison can also be made in a suitably defined $L^{p}$ sense for every $1\leq p<\infty $ and that, in this perspective, inequalities of Hardy, Sobolev or Morrey type account for the fact that sub $|x|^{-N/p}$ growth of $\nabla u$ in the $% L^{p} $ sense implies sub $|x|^{1-N/p}$ growth of $u$ in the $L^{q}$ sense for well chosen values of $q.$ By investigating how sub $|x|^{s}$ growth of $\na…
Let $L$ be a homogeneous divergence form higher order elliptic operator with complex bounded measurable coefficients on $\mathbb{R}^n$ and $X$ a ball quasi-Banach function space on $\mathbb{R}^n$ satisfying some mild assumptions. Denote by $H_{X,\, L}(\mathbb{R}^n)$ the Hardy space, associated with both $L$ and $X$, which is defined via the Lusin area function related to the semigroup generated by $L$. In this article, the authors establish both the maximal function and the Riesz transform characterizations of $H_{X,\, L}(\mathbb{R}^n)$. The results obtained in this article have a wide range o…
In this paper, we prove the boundedness of Bessel-Riesz operators on generalized Morrey spaces. The proof uses the usual dyadic decomposition, a Hedberg-type inequality for the operators, and the boundedness of Hardy-Littlewood maximal operator. Our results reveal that the norm of the operators is dominated by the norm of the kernels.
We investigate some smoothness properties for a transport-diffusion equation involving a class of non-degerate L{é}vy type operators with singular drift. Our main argument is based on a duality method using the molecular decomposition of Hardy spaces through which we derive some H{ö}lder continuity for the associated parabolic PDE. This property will be fulfilled as far as the singular drift belongs to a suitable Morrey-Campanato space for which the regularizing properties of the L{é}vy operator suffice to obtain global H{ö}lder continuity.
We are concerned with a family of dissipative active scalar equation with velocity fields coupled via multiplier operators that can be of high-order. We consider sub-critical values for the fractional diffusion and prove global well-posedness of solutions with small initial data belonging to a framework based on Fourier transform, namely Fourier-Besov-Morrey spaces. Since the smallness condition is with respect to the weak norm of this space, some initial data with large $L^{2}$-norm can be considered. Self-similar solutions are obtained depending on the homogeneity of the initial data and cou…
Let $C_Γ$ be the Cauchy integral operator on a Lipschitz curve $Γ$. In this article, the authors show that the commutator $[b,C_Γ]$ is bounded (resp., compact) on the Morrey space $L^{p,\,λ}(\mathbb R)$ for any (or some) $p\in(1, \infty)$ and $λ\in(0, 1)$ if and only if $b\in {\rm BMO}(\mathbb R)$ (resp., ${\rm CMO}(\mathbb R)$). As an application, a factorization of the classical Hardy space $H^1(\mathbb R)$ in terms of $C_Γ$ and its adjoint operator is obtained.
Let $p \in (1,\infty)$ and $Ω\subset \mathbb{R}^N$ be a domain. Let $ A: =(a_{ij}) \in L^{\infty}_{\text{loc}}(Ω; \mathbb{R}^{N\times N})$ be a symmetric and locally uniformly positive definite matrix. Set $|ξ|_A^2:= \displaystyle \sum_{i,j=1}^N a_{ij}(x) ξ_i ξ_j$, $ξ\in \mathbb{R}^N$, and let $V$ be a given potential in a certain local Morrey space. We assume that the energy functional $$Q_{p,A,V}(ϕ):=\displaystyle \int_Ω [|\nabla ϕ|_A^p + V|ϕ|^p] {\rm dx} $$ is nonnegative in $W^{1,p}(Ω)\cap C_c(Ω)$. We introduce a generalized notion of $Q_{p,A,V}$-capacity and characterize the space of all …
The targets of this article are threefold. The first one is to give a survey on the recent developments of function spaces with mixed norms, including mixed Lebesgue spaces, iterated weak Lebesgue spaces, weak mixed-norm Lebesgue spaces and mixed Morrey spaces as well as anisotropic mixed-norm Hardy spaces. The second one is to provide a detailed proof for a useful inequality about mixed Lebesgue norms and the Hardy--Littlewood maximal operator and also to improve some known results on the maximal function characterizations of anisotropic mixed-norm Hardy spaces and the boundedness of Calderón…
Let $X$ be a ball quasi-Banach function space satisfying some minor assumptions. In this article, the authors establish the characterizations of $H_X(\mathbb{R}^n)$, the Hardy space associated with $X$, via the Littlewood--Paley $g$-functions and $g_λ^\ast$-functions. Moreover, the authors obtain the boundedness of Calderón--Zygmund operators on $H_X(\mathbb{R}^n)$. For the local Hardy-type space $h_X(\mathbb{R}^n)$ associated with $X$, the authors also obtain the boundedness of $S^0_{1,0}(\mathbb{R}^n)$ pseudo-differential operators on $h_X(\mathbb{R}^n)$ via first establishing the atomic cha…
The Hardy spaces for Fourier integral operators $\mathcal{H}_{FIO}^{p}(\mathbb{R}^{n})$, for $1\leq p\leq \infty$, were introduced by Smith in [Smith,1998] and Hassell et al. in [Hassell-Portal-Rozendaal,2020]. In this article, we give several equivalent characterizations of $\mathcal{H}_{FIO}^{1}(\mathbb{R}^{n})$, for example in terms of Littlewood--Paley $g$ functions and maximal functions. This answers a question from [Rozendaal,2021]. We also give several applications of the characterizations.
Let $X$ be a ball quasi-Banach function space on ${\mathbb R}^n$. In this article, the authors introduce the weak Hardy-type space $WH_X({\mathbb R}^n)$, associated with $X$, via the radial maximal function. 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 then establish several real-variable characterizations of $WH_X({\mathbb R}^n)$, respectively, in terms of various maximal functions, atoms a…
We define a scale of Hardy spaces $\mathcal{H}^{p}_{FIO}(\mathbb{R}^{n})$, $p\in[1,\infty]$, that are invariant under suitable Fourier integral operators of order zero. This builds on work by Smith for $p=1$. We also introduce a notion of off-singularity decay for kernels on the cosphere bundle of $\mathbb{R}^{n}$, and we combine this with wave packet transforms and tent spaces over the cosphere bundle to develop a full Hardy space theory for oscillatory integral operators. In the process we extend the known results about $L^{p}$-boundedness of Fourier integral operators, from local boundednes…
We show that the Hardy spaces for Fourier integral operators form natural spaces of initial data when applying $\ell^{p}$-decoupling inequalities to local smoothing for the wave equation. This yields new local smoothing estimates which, in a quantified manner, improve the bounds in the local smoothing conjecture on $\mathbb{R}^{n}$ for $p\geq 2(n+1)/(n-1)$, and complement them for $2<p<2(n+1)/(n-1)$. These estimates are invariant under application of Fourier integral operators, and they are essentially sharp.
We prove several characterizations of the Hardy spaces for Fourier integral operators $\mathcal{H}^{p}_{FIO}(\mathbb{R}^{n})$, for $1<p<\infty$. First we characterize $\mathcal{H}^{p}_{FIO}(\mathbb{R}^{n})$ in terms of $L^{p}(\mathbb{R}^{n})$-norms of parabolic frequency localizations. As a corollary, any characterization of $L^{p}(\mathbb{R}^{n})$ yields a corresponding version for $\mathcal{H}^{p}_{FIO}(\mathbb{R}^{n})$. In particular, we obtain a maximal function characterization and a characterization in terms of vertical square functions.
We obtain improved bounds for pseudodifferential operators with rough symbols on Hardy spaces for Fourier integral operators. The symbols $a(x,η)$ are elements of $C^{r}_{*}S^{m}_{1,δ}$ classes that have limited regularity in the $x$ variable. We show that the associated pseudodifferential operator $a(x,D)$ maps between Sobolev spaces $\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n})$ and $\mathcal{H}^{t,p}_{FIO}(\mathbb{R}^{n})$ over the Hardy space for Fourier integral operators $\mathcal{H}^{p}_{FIO}(\mathbb{R}^{n})$. Our main result is that for all $r>0$, $m=0$ and $δ=1/2$, there exists an interval…
We introduce the Hardy spaces $\mathcal{H}^{p}_{FIO}(\mathbb{R}^{n})$ for Fourier integral operators for $0<p<1$, thereby extending earlier constructions for $1\leq p\leq \infty$. We then establish various properties of these spaces, including their behavior under complex interpolation and duality, and their invariance under Fourier integral operators. We also obtain Sobolev embeddings, equivalent characterizations, and a molecular decomposition. These spaces are used in the companion article arXiv:2502.02511 to determine the sharp $\mathcal{H}^{1}(\mathbb{R}^{n})$ and $\mathrm{bmo}(\mathbb{R}…
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 $X$ be a ball quasi-Banach function space on ${\mathbb R}^n$ and $H_X({\mathbb R}^n)$ the Hardy space associated with $X$, and let $α\in(0,n)$ and $β\in(1,\infty)$. In this article, assuming that the (powered) Hardy--Littlewood maximal operator satisfies the Fefferman--Stein vector-valued maximal inequality on $X$ and is bounded on the associate space of $X$, the authors prove that the fractional integral $I_α$ can be extended to a bounded linear operator from $H_X({\mathbb R}^n)$ to $H_{X^β}({\mathbb R}^n)$ if and only if there exists a positive constant $C$ such that, for any ball $B\sub…
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}^…
Assuming that both a function and its Fourier transform are dominated by a Gaussian of large variance, it is shown that the Hermite coefficients of the function decay exponentially. A sharp estimate for the rate of exponential decay is obtained in terms of the variance, and in the limiting case (when the variance becomes so small that the Gaussian is its own Fourier transform), Hardy's theorem on Fourier transform pairs is obtained. A quantitative result on the confinement of particle-like states of a quantum harmonic oscillator is obtained. A stronger form of the result is conjectured. Furthe…
We give a real-variable proof of the Hardy uncertainty principle. The method is based on energy estimates for evolutions with positive viscosity, convexity properties of free waves with Gaussian decay at two different times, elliptic $L^2$-estimates and the invertibility of the Fourier transform on $L^2(\Rn)$ and $\mathcal S'(\Rn)$.
We provide characterizations for boundedness of multilinear Fourier operators on Hardy-Lebesgue spaces with symbols locally in Sobolev spaces. Let $H^q(\mathbb R^n)$ denote the Hardy space when $0<q\le 1$ and the Lebesgue space $L^q(\mathbb R^n)$ when $1<q\le \infty$. We find optimal conditions on $m$-linear Fourier multiplier operators to be bounded from $H^{p_1}\times \cdots \times H^{p_m}$ to $L^p$ when $1/p=1/p_1+\cdots +1/p_m$ in terms of local $L^2$-Sobolev space estimates for the symbol of the operator. Our conditions provide multilinear analogues of the linear results of Calderón and T…
By using, among other things, the Fourier analysis techniques on hyperbolic and symmetric spaces, we establish the Hardy-Sobolev-Maz'ya inequalities for higher order derivatives on half spaces. The proof relies on a Hardy-Littlewood-Sobolev inequality on hyperbolic spaces which is of its independent interest. We also give an alternative proof of Benguria, Frank and Loss' work concerning the sharp constant in the Hardy-Sobolev-Maz'ya inequality in the three dimensional upper half space. Finally, we show the sharp constant in the Hardy-Sobolev-Maz'ya inequality for bi-Laplacian in the upper half…
This paper continues the program initiated in the works by the authors [60], [61] and [62] and by the authors with Li [51] and [52] to establish higher order Poincaré-Sobolev, Hardy-Sobolev-Maz'ya, Adams and Hardy-Adams inequalities on real hyperbolic spaces using the method of Helgason-Fourier analysis on the hyperbolic spaces. The aim of this paper is to establish such inequalities on the Siegel domains and complex hyperbolic spaces. Firstly, we prove a factorization theorem for the operators on the complex hyperbolic space which is closely related to Geller' operator, as well as the CR inva…
