We prove multiple vector-valued and mixed-norm estimates for multilinear operators in $\rr R^d$, more precisely for multilinear operators $T_k$ associated to a symbol singular along a $k$-dimensional space and for multilinear variants of the Hardy-Littlewood maximal function. When the dimension $d \geq 2$, the input functions are not necessarily in $L^p(\rr R^d)$ and can instead be elements of mixed-norm spaces $L^{p_1}_{x_1} \ldots L^{p_d}_{x_d}$.
Such a result has interesting consequences especially when $L^\infty$ spaces are involved. Among these, we mention mixed-norm Loomis-Whitney-type inequalities for singular integrals, as well as the boundedness of multilinear operators associated to certain rational symbols. We also present examples of operators that are not susceptible to isotropic rescaling, which only satisfy ``purely mixed-norm estimates" and no classical $L^p$ estimates.
Relying on previous estimates implied by the helicoidal method, we also prove (non-mixed-norm) estimates for generic singular Brascamp-Lieb-type inequalities.
We develop a new method of proving vector-valued estimates in harmonic analysis, which we like to call "the helicoidal method". As a consequence of it, we are able to give affirmative answers to some questions that have been circulating for some time. In particular, we show that the tensor product $BHT \otimes Π$ between the bilinear Hilbert transform $BHT$ and a paraproduct $Π$ satisfies the same $L^p$ estimates as the $BHT$ itself, solving completely a problem introduced in a paper of Muscalu, Pipher, Tao and Thiele. Then, we prove that for "locally $L^2$ exponents" the corresponding vector …
We extend the helicoidal method that we previously developed to the quasi-Banach context, proving in this way multiple Banach and quasi-Banach vector-valued inequalities for paraproducts $Π$ and for the bilinear Hilbert transform $BHT$. As an immediate application, we obtain mixed norm estimates for $Π\otimes Π$ in the whole range of Lebesgue exponents. One of the novelties in the quasi-Banach framework (that is, when $0<r<1$), which we expect to be useful in other contexts as well, is the "linearization" of the operator $ \left( \sum_k | T(f_k, g_k) |^r \right)^{1/r}$ by dualizing its weak-$L…
We prove that for any $L^Q$-valued Schwartz function $f$ defined on $\mathbb{R}^d$, one has the multiple vector-valued, mixed norm estimate $$ \| f \|_{L^P(L^Q)} \lesssim \| S f \|_{L^P(L^Q)} $$ valid for every $d$-tuple $P$ and every $n$-tuple $Q$ satisfying $0 < P, Q < \infty$ componentwise. Here $S:= S_{d_1}\otimes ... \otimes S_{d_N}$ is a tensor product of several Littlewood-Paley square functions $S_{d_j}$ defined on arbitrary Euclidean spaces $\mathbb{R}^{d_j}$ for $1\leq j\leq N$, with the property that $d_1 + ... + d_N = d$. This answers a question that came up implicitly in our recen…
In this article mixed norm estimates are obtained for some integral operators, from which those for the Hermite semigroup and the Bochner Riesz means associated with the Hermite expansions are deduced. Also, mixed norm estimates for the Littlewood Paley g functions and g* functions for the Hermite expansions are obtained, which lead to those for Hermite multipliers
We consider the Hardy-Littlewood-Sobolev inequality on mixed-norm Lebesgue spaces. We give a complete characterization of indices $\vec p$ and $\vec q$ such that the Riesz potential is bounded from $L^{\vec p}$ to $L^{\vec q}$, including all the endpoint cases. As a result, we get the mixed-norm Hardy-Littlewood-Sobolev inequality.
Starting from an adapted Whitney decomposition of tube domains in $\C^n$ over irreducible symmetric cones of $\R^n,$ we prove an atomic decomposition theorem in mixed norm weighted Bergman spaces on these domains. We also characterize the interpolation space via the complex method between two mixed norm weighted Bergman spaces.
In this paper we investigate the mapping properties in Lebesgue-type spaces of certain generalized Radon transforms defined by integration over curves.
In this article we revisit some classical conjectures in harmonic analysis in the setting of mixed norm spaces $L^p_{rad} L^2_{ang} (\mathbb{R}^n)$. We produce sharp bounds for the restriction of the Fourier transform to compact hypersurfaces of revolution in the mixed norm setting and study an extension of the disc multiplier. We also present some results for the discrete restriction conjecture and state an intriguing open problem.
We present a transference principle of Lebesgue mixed norm estimates for Bergman projectors from tube domains over homogeneous cones to homogeneous Siegel domains of type II associated to the same cones. This principle implies improvements of these estimates for homogeneous Siegel domains of type II associated with Lorentz cones, e.g. the Pyateckii-Shapiro Siegel domain of type II.
We prove multi-parameter Leibniz rules corresponding to flag paraproducts of arbitrary complexity in mixed-norm spaces, including endpoint estimates. The proof relies on multi-linear harmonic analysis techniques and a quantitative treatment of the commutators introduced by Bourgain and Li. The argument is robust and applicable to a generic class of multipliers, including (symmetric) Mikhlin multipliers of positive order and asymmetric variants of partial differential operators and Mikhlin multipliers of positive order.
We prove an $L^2 \times L^2 \rightarrow L_t^qL_x^p $ bilinear Fourier extension estimate for the cone when $p,q$ are on the critical line $1/q=(\frac{n+1}{2})(1-1/p)$. This extends previous results by Wolff, Tao and Lee-Vargas.
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.
It is known that that the endpoint mixed norm estimate $|| \, ||Tf(x,y)||_{L_{x}^{p}}||_{L_{y}^{\infty}} \lesssim || \, ||f(x,y)||_{L_{x}^{p}}||_{L_{y}^{\infty}}$ in general does not hold for Calderón-Zygmund operator $T$. In this article, we show that when $p=2$, even if we make the right hand side of the above estimate larger by replacing it with $ || \, ||e^{x^2+y^2}f(x,y)||_{L_{y}^{\infty}}||_{L_{x}^{\infty}} $, the estimate does not hold for the double Riesz transform given by the kernel $K(x,y)=\frac{xy}{2π(x^2+y^2)^{2}}$. As a consequence we will show that the mixed norm estimate $|| \,…
In this paper, the $L^2 \times L^{\infty} \to L^2$ and $L^2 \times L^2 \to L^1$ boundedness of bilinear Fourier multiplier operators is discussed under weak smoothness conditions on multipliers. As an application, we prove the $L^2 \times BMO \to L^2$ and $L^2 \times L^2 \to H^1$ boundedness of bilinear operators with multipliers of limited smoothness satisfying vanishing conditions.
We study two classes of bounded operators on mixed norm Lebesgue spaces, namely composition operators and product operators. A complete description of bounded composition operators on mixed norm Lebesgue spaces are given. For a certain class of integral operators, we provide sufficient conditions for boundedness. We conclude by applying the developed technique to the investigation of Hardy-Steklov type operators.
In the last decade, the problem of characterizing the normability of the weighted Lorentz spaces has been completely solved (\cite{Sa}, \cite{CaSoA}). However, the question for multidimensional Lorentz spaces is still open. In this paper, we consider weights of product type, and give necessary and sufficient conditions for the Lorentz spaces, defined with respect to the two-dimensional decreasing rearrangement, to be normable. To this end, it is also useful to study the mixed norm Lorentz spaces. Finally, we prove embeddings between all the classical, multidimensional, and mixed norm Lorentz s…
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…
Our main goal in this article is to study mixed norm estimates for the Cesàro means associated with Dunkl--Hermite expansions on $\mathbb{R}^d$. These expansions arise when one consider the Dunkl--Hermite operator (or Dunkl harmonic oscillator) $H_κ:=-Δ_κ+|x|^2$, where $Δ_κ$ stands for the Dunkl--Laplacian. It is shown that the desired mixed norm estimates are equivalent to vector-valued inequalities for a sequence of Cesàro means for Laguerre expansions with shifted parameter. In order to obtain the latter, we develop an argument to extend these operators for complex values of the parameters …
In their work [IM16] I.A. Ikromov and D. Müller proved the full range $L^p-L^2$ Fourier restriction estimates for a very general class of hypersurfaces in $\R^3$ which includes the class of real analytic hypersurfaces. In this article we partly extend their results to the mixed norm case where the coordinates are split in two directions, one tangential and the other normal to the surface at a fixed given point. In particular, we resolve completely the adapted case and partly the non-adapted case. In the non-adapted case the case when the linear height $h_\text{lin}(ϕ)$ is below two is settled …
Novel global weighted parabolic Sobolev estimates, weighted mixed-norm estimates and a.e. convergence results of singular integrals for evolution equations are obtained. Our results include the classical heat equation, the harmonic oscillator evolution equation $$\partial_tu=Δu-|x|^2u+f,$$ and their corresponding Cauchy problems. We also show weighted mixed-norm estimates for solutions to degenerate parabolic extension problems arising in connection with the fractional space-time nonlocal equations $(\partial_t-Δ)^su=f$ and $(\partial_t-Δ+|x|^2)^su=f$, for $0<s<1$.
Let $\vec{p}\in(0,\,\infty)^n$, $A$ be an expansive dilation on $\mathbb{R}^n$,and $H^{\vec{p}}_A({\mathbb {R}}^n)$ be the anisotropic mixed-norm Hardy space defined via the non-tangential grand maximal function studied by \cite{hlyy20}. In this paper, the authors establish new atomic and molecular decompositions of $H^{\vec{p}}_A({\mathbb {R}}^n)$. As an application, the authors obtain a boundedness criterion for a class of linear operators from $H^{\vec{p}}_{A}(\mathbb{R}^n)$ to $H^{\vec{p}}_{A}(\mathbb{R}^n)$. Part of results are still new even in the classical isotropic setting (in the cas…
Let $\vec{p}\in(0,1]^n$ be a $n$-dimensional vector and $A$ a dilation. Let $H_A^{\vec{p}}(\mathbb{R}^n)$ denote the anisotropic mixed-norm Hardy space defined via the radial maximal function. Using the known atomic characterization of $H_{A}^{\vec{p}}(\mathbb{R}^n)$ and establishing a uniform estimate for corresponding atoms, the authors prove that the Fourier transform of $f\in H_A^{\vec{p}}(\mathbb{R}^n)$ coincides with a continuous function $F$ on $\mathbb{R}^n$ in the sense of tempered distributions. Moreover, the function $F$ can be controlled pointwisely by the product of the Hardy spac…
Let $\vec{p}\in(0,\infty)^n$ and $A$ be a general expansive matrix on $\mathbb{R}^n$. In this article, via the non-tangential grand maximal function, the authors first introduce the anisotropic mixed-norm Hardy spaces $H_A^{\vec{p}}(\mathbb{R}^n)$ associated with $A$ and then establish their radial or non-tangential maximal function characterizations. Moreover, the authors characterize $H_A^{\vec{p}}(\mathbb{R}^n)$, respectively, by means of atoms, finite atoms, Lusin area functions, Littlewood-Paley $g$-functions or $g_λ^\ast$-functions via first establishing an anisotropic Fefferman-Stein ve…
In this article, the authors introduce a class of mixed-norm Herz spaces, $\dot{E}^{\vecα,\vec{p}}_{\vec{q}}(\mathbb{R}^{n})$, which is a natural generalization of mixed Lebesgue spaces and some special cases of which naturally appear in the study of the summability of Fourier transforms on mixed-norm Lebesgue spaces. The authors also give their dual spaces and obtain the Riesz-Thorin interpolation theorem on $\dot{E}^{\vecα,\vec{p}}_{\vec{q}}(\mathbb{R}^{n})$. Applying these Riesz-Thorin interpolation theorem and using some ideas from the extrapolation theorem, the authors establish both the …
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, the authors give the dual space of $H_{\vec{a}}^{\vec{p}}(\mathbb{R}^n)$, which was asked by Cleanthous et al. in [J. Geom. Anal. 27 (2017), 2758-2787]. More precisely, via first introducing the anisotropic mixed-norm Campanato space $\mathcal{L}_{\vec{p},\,q,\,s}^{\vec{a}}(\mathbb{R}^n)$ with $q\in[1,\infty]$ and $s\in\mathb…
