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Guozhen Lu

Publications and source records attributed to Guozhen Lu.

At least 73 records · Page 4Linked to original sources

arXiv2021

Sharp Hardy-Sobolev-Maz'ya, Adams and Hardy-Adams inequalities on the Siegel domains and complex hyperbolic spaces

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 invariant differential operators on the Heisenberg group and CR sphere. Secondly, by using, among other things, the Kunze-Stein phenomenon on a closed linear group $SU(1,n)$ and Helgason-Fourier analysis techniques on the complex hyperbolic spaces, we establish the Poincaré-Sobolev, Hardy-Sobolev-Maz'ya inequality on the Siegel domain $\mathcal{U}^{n}$ and the unit ball $\mathbb{B}_{\mathbb{C}}^{n}$. Finally, we establish the sharp Hardy-Adams inequalities and sharp Adams type inequalities on Sobolev spaces of any positive fractional order on the complex hyperbolic spaces. The factorization theorem we proved is of its independent interest in the Heisenberg group and CR sphere and CR invariant differential operators therein.

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arXiv2020

Bi-parameter trilinear Fourier multipliers and pseudo-differential operators with flag symbols

The main purpose of this paper is to study $L^r$ Hölder type estimates for a bi-parameter trilinear Fourier multiplier with flag singularity, and the analogous pseudo-differential operator, when the symbols are in a certain product form. More precisely, for $f,g,h\in \mathcal{S}(\mathbb{R}^{2})$, the bi-parameter trilinear flag Fourier multiplier operators we consider are defined by $$ T_{m_1,m_2}(f,g,h)(x):=\int_{\mathbb{R}^{6}}m_1(ξ,η,ζ)m_2(η,ζ)\hat f(ξ) \hat g(η)\hat h(ζ)e^{2πi(ξ+η+ζ)\cdot x}dξdηdζ, $$ when $m_1,m_2$ are two bi-parameter symbols. We will show that our problem can be reduced to establish the $L^r$ estimate for the special multiplier $m_1(ξ_1, η_1, ζ_1) m_2(η_2, ζ_2)$ (see Theorem 1.7). We also study these $L^r$ estimates for the corresponding bi-parameter trilinear pseudo-differential operators defined by $$ T_{ab}(f,g,h)(x):=\int_{\mathbb{R}^6}a(x,ξ,η,ζ)b(x,η,ζ)\hat f(ξ)\hat g(η)\hat h(ζ)e^{2πi x(ξ+η+ζ)}dξdηdζ, $$ where the smooth symbols $a,b$ satisfy certain bi-parameter Hörmander conditions. We will also show that the $L^r$ estimate holds for $T_{ab}$ as long as the $L^r$ estimate for the flag multiplier operator holds when the multiplier has the special form $m_1(ξ_1, η_1, ζ_1) m_2(η_2, ζ_2)$ (see Theorem 1.10). The bi-parameter and trilinear flag Fourier multipliers considered in this paper do not satisfy the conditions of the classical bi-parameter trilinear Fourier multipliers considered by Muscalu, Tao, Thiele and the second author [21, 22]. They may also be viewed as the bi-parameter trilinear variants of estimates obtained for the one-parameter flag paraproducts by Muscalu [18].

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arXiv2019

Ground states of bi-harmonic equations with critical exponential growth involving constant and trapping potentials

In this paper, we first give a necessary and sufficient condition for the boundedness and the compactness for a class of nonlinear functionals in $H^{2}(\mathbb{R}^4)$. Using this result and the principle of symmetric criticality, we can present a relationship between the existence of the nontrivial solutions to the semilinear bi-harmonic equation of the form \[ (-Δ)^{2}u+γu=f(u)\ \text{in}\ \mathbb{R}^4 \] and the range of $γ\in \mathbb{R}^{+}$, where $f(s)$ is the general nonlinear term having the critical exponential growth at infinity. Our next goal in this paper is to establish the existence of the ground-state solutions for the equation \begin{equation}\label{con} (-Δ)^{2}u+V(x)u=λs\exp(2|s|^{2}))\ \text{in}\ \mathbb{R}^{4}, \end{equation} when $V(x)$ is a positive constant using the Fourier rearrangement and the Pohozaev identity. Then we will explore the relationship between the Nehari manifold and the corresponding limiting Nehari manifold to derive the existence of the ground state solutions for the above equation when $V(x)$ is the Rabinowitz type trapping potential, namely it satisfies $$0<V_{0}=\underset{x\in\mathbb{R}^{4}}{\inf}V(x) <\underset{\ | x\ | \rightarrow\infty}{\lim}V(x) < +\infty. $$ The same result and proof applies to the harmonic equation with the critical exponential growth involving the Rabinowitz type trapping potential in $\mathbb{R}^2$.

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arXiv2019

Existence and Nonexistence of Extremals for critical Adams inequalities in R4 and Trudinger-Moser inequalities in R2

Though much work has been done with respect to the existence of extremals of the critical first order Trudinger-Moser inequalities in $W^{1,n}(\mathbb{R}^n)$ and higher order Adams inequalities on finite domain $Ω\subset \mathbb{R}^n$, whether there exists an extremal function for the critical higher order Adams inequalities on the entire space $\mathbb{R}^n$ still remains open. The current paper represents the first attempt in this direction. The classical blow-up procedure cannot apply to solving the existence of critical Adams type inequality because of the absence of the Pólya-Szegö\ type inequality. In this paper, we develop some new ideas and approaches based on a sharp Fourier rearrangement principle (see \cite{Lenzmann}), sharp constants of the higher-order Gagliardo-Nirenberg inequalities and optimal poly-harmonic truncations to study the existence and nonexistence of the maximizers for the Adams inequalities in $\mathbb{R}^4$ of the form $$ S(α)=\sup_{\|u\|_{H^2}=1}\int_{\mathbb{R}^4}\big(\exp(32π^2|u|^2)-1-α|u|^2\big)dx,$$ where $α\in (-\infty, 32π^2)$. We establish the existence of the threshold $α^{\ast}$, where $α^{\ast}\geq \frac{(32π^{2})^2B_{2}}{2}$ and $B_2\geq \frac{1}{24π^2}$, such that $S\left( α\right) $ is attained if $32π^{2}-α<α^{\ast}$, and is not attained if $32π^{2}-α>α^{\ast}$. This phenomena has not been observed before even in the case of first order Trudinger-Moser inequality. Therefore, we also establish the existence and non-existence of an extremal function for the Trudinger-Moser inequality on $\mathbb{R}^2$. Furthermore, the symmetry of the extremal functions can also be deduced through the Fourier rearrangement principle.

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arXiv2019

Green's functions of Paneitz and GJMS operators on hyperbolic spaces and sharp Hardy-Sobolev-Maz'ya inequalities on half spaces

Using the Fourier analysis techniques on hyperbolic spaces and Green's function estimates, we confirm in this paper the conjecture given by the same authors in [43]. Namely, we prove that the sharp constant in the $\frac{n-1}{2}$-th order Hardy-Sobolev-Maz'ya inequality in the upper half space of dimension $n$ coincides with the best $\frac{n-1}{2}$-th order Sobolev constant when $n$ is odd and $n\geq9$ (See Theorem 1.6). We will also establish a lower bound of the coefficient of the Hardy term for the $k-$th order Hardy-Sobolev-Maz'ya inequality in upper half space in the remaining cases of dimension $n$ and $k$-th order derivatives (see Theorem 1.7). Precise expressions and optimal bounds for Green's functions of the operator $ -Δ_{\mathbb{H}}-\frac{(n-1)^{2}}{4}$ on the hyperbolic space $\mathbb{B}^n$ and operators of the product form are given, where $\frac{(n-1)^{2}}{4}$ is the spectral gap for the Laplacian $-Δ_{\mathbb{H}}$ on $\mathbb{B}^n$. Finally, we give the precise expression and optimal pointwise bound of Green's function of the Paneitz and GJMS operators on hyperbolic space, which are of their independent interest (see Theorem 1.10).

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arXiv2019

Existence of extremal functions for the Stein-Weiss inequalities on the Heisenberg group

In this paper, we establish the existence of extremals for two kinds of Stein-Weiss inequalities on the Heisenberg group. More precisely, we prove the existence of extremals for the Stein-Weiss inequalities with full weights in Theorem 1.1 and the Stein-Weiss inequalities with horizontal weights in Theorem 1.4. Different from the proof of the analogous inequality in Euclidean spaces given by Lieb [26] using Riesz rearrangement inequality which is not available on the Heisenberg group, we employ the concentration compactness principle to obtain the existence of the maximizers on the Heisenberg group. Our result is also new even in the Euclidean case because we don't assume that the exponents of the double weights in the Stein-Weiss inequality (1.1) are both nonnegative (see Theorem 1.3 and more generally Theorem 1.5). Therefore, we extend Lieb's celebrated result of the existence of extremal functions of the Stein-Weiss inequality in the Euclidean space to the case where the exponents are not necessarily both nonnegative (see Theorem 1.3). Furthermore, since the absence of translation invariance of the Stein-Weiss inequalities, additional difficulty presents and one cannot simply follow the same line of Lions' idea to obtain our desired result. Our methods can also be used to obtain the existence of optimizers for several other weighted integral inequalities (Theorem 1.5).

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arXiv2019

Stein-Weiss inequalities with the fractional Poisson kernel

In this paper, we establish the following Stein-Weiss inequality with the fractional Poisson kernel (see Theorem 1.1): \begin{equation}\label{int1} \int_{\mathbb{R}^n_{+}}\int_{\partial\mathbb{R}^n_{+}}|ξ|^{-α}f(ξ)P(x,ξ,γ)g(x)|x|^{-β}dξdx\leq C_{n,α,β,p,q'}\|g\|_{L^{q'}(\mathbb{R}^n_{+})}\|f\|_{L^p(\partial \mathbb{R}^{n}_{+})}, \end{equation} where $P(x,ξ,γ)=\frac{x_n}{(|x'-ξ|^2+x_n^2)^{\frac{n+2-γ}{2}}}$, $2\le γ<n$, $f\in L^{p}(\partial\mathbb{R}^n_{+})$, $g\in L^{q'}(\mathbb{R}^n_{+})$ and $p,\ q'\in (1,\infty)$ and satisfy $\frac{n-1}{n}\frac{1}{p}+\frac{1}{q'}+\frac{α+β+2-γ}{n}=1$. Then we prove that there exist extremals for the Stein-Weiss inequality (0.1) and the extremals must be radially decreasing about the origin (see Theorem 1.5). We also provide the regularity and asymptotic estimates of positive solutions to the integral systems which are the Euler-Lagrange equations of the extremals to the Stein-Weiss inequality (0.1) with the fractional Poisson kernel (see Theorems 1.7 and 1.8). Our result is inspired by the work of Hang, Wang and Yan [29] where the Hardy-Littlewood-Sobolev type inequality was first establishedmwhen $γ=2$ and $α=β=0$ (see (1.5)). The proof of the Stein-Weiss inequality (0.1) with the fractional Poisson kernel in this paper uses our recent work on the Hardy-Littlewood-Sobolev inequality with the fractional Poisson kernel [18] and the present paper is a further study in this direction.

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arXiv2017

Sharp Hardy-Adams inequalities for bi-Laplacian on hyperbolic space of dimension four

We establish sharp Hardy-Adams inequalities on hyperbolic space $\mathbb{B}^{4}$ of dimension four. Namely, we will show that for any $α>0$ there exists a constant $C_α>0$ such that \[ \int_{\mathbb{B}^{4}}(e^{32π^{2} u^{2}}-1-32π^{2} u^{2})dV=16\int_{\mathbb{B}^{4}}\frac{e^{32π^{2} u^{2}}-1-32π^{2} u^{2}}{(1-|x|^{2})^{4}}dx\leq C_α. \] for any $u\in C^{\infty}_{0}(\mathbb{B}^{4})$ with \[ \int_{\mathbb{B}^{4}}\left(-Δ_{\mathbb{H}}-\frac{9}{4}\right)(-Δ_{\mathbb{H}}+α)u\cdot udV\leq1. \] As applications, we obtain a sharpened Adams inequality on hyperbolic space $\mathbb{B}^{4}$ and an inequality which improves the classical Adams' inequality and the Hardy inequality simultaneously. The later inequality is in the spirit of the Hardy-Trudinger-Moser inequality on a disk in dimension two given by Wang and Ye [37] and on any convex planar domain by the authors [26]. The tools of fractional Laplacian, Fourier transform and the Plancherel formula on hyperbolic spaces and symmetric spaces play an important role in our work.

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arXiv2017

Hardy-Sobolev-Maz'ya inequalities for higher order derivatives on half spaces

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 space of dimension five coincides with the Sobolev constant.

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arXiv2017

Concentration-compactness principle for Trudinger-Moser inequalities on Heisenberg Groups and existence of ground state solutions

Let $\mathbb{H}^{n}=\mathbb{C}^{n}\times\mathbb{R}$ be the $n$-dimensional Heisenberg group, $Q=2n+2$ be the homogeneous dimension of $\mathbb{H}^{n}$. We extend the well-known concentration-compactness principle on finite domains in the Euclidean spaces of \ P. L. Lions to the setting of the Heisenberg group $\mathbb{H}^{n}$. Furthermore, we also obtain the corresponding concentration-compactness principle for the Sobolev space $HW^{1,Q}\left( \mathbb{H}^{n}\right) $ on the entire Heisenberg group $\mathbb{H}^{n}$. Our results improve the sharp Trudinger-Moser inequality on domains of finite measure in $\mathbb{H}^{n}$ by Cohn and the second author [8] and the corresponding one on the whole space $\mathbb{H}^n$ by Lam and the second author [21]. All the proofs of the concentration-compactness principles in the literature even in the Euclidean spaces use the rearrangement argument and the Polyá-Szegö inequality. Due to the absence of the Polyá-Szegö inequality on the Heisenberg group, we will develop a different argument. Our approach is surprisingly simple and general and can be easily applied to other settings where symmetrization argument does not work. As an application of the concentration-compactness principle, we establish the existence of ground state solutions for a class of $Q$- Laplacian subelliptic equations on $\mathbb{H}^{n}$ with nonlinear terms $f$ of maximal exponential growth $\exp\left( αt^{\frac{Q}{Q-1}}\right) $ as $t\rightarrow+\infty$.

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arXiv2017

A sharp Trudinger-Moser type inequality involving $L^{n}$ norm in the entire space $\mathbb{R}^{n}$

Let $W^{1,n} ( \mathbb{R}^{n} $ be the standard Sobolev space and $\left\Vert \cdot\right\Vert _{n}$ be the $L^{n}$ norm on $\mathbb{R}^n$. We establish a sharp form of the following Trudinger-Moser inequality involving the $L^{n}$ norm \[ \underset{\left\Vert u\right\Vert _{W^{1,n}\left(\mathbb{R} ^{n}\right) }=1}{\sup}\int_{ \mathbb{R}^{n}}Φ\left( α_{n}\left\vert u\right\vert ^{\frac{n}{n-1}}\left( 1+α\left\Vert u\right\Vert _{n}^{n}\right) ^{\frac{1}{n-1}}\right) dx<+\infty \]in the entire space $\mathbb{R}^n$ for any $0\leqα<1$, where $Φ\left( t\right) =e^{t}-\underset{j=0}{\overset{n-2}{\sum}}% \frac{t^{j}}{j!}$, $α_{n}=nω_{n-1}^{\frac{1}{n-1}}$ and $ω_{n-1}$ is the $n-1$ dimensional surface measure of the unit ball in $\mathbb{R}^n$. We also show that the above supremum is infinity for all $α\geq1$. Moreover, we prove the supremum is attained, namely, there exists a maximizer for the above supremum when $α>0$ is sufficiently small. The proof is based on the method of blow-up analysis of the nonlinear Euler-Lagrange equations of the Trudinger-Moser functionals. Our result sharpens the recent work \cite{J. M. do1} in which they show that the above inequality holds in a weaker form when $Φ(t)$ is replaced by a strictly smaller $Φ^*(t)=e^{t}-\underset{j=0}{\overset{n-1}{\sum}}% \frac{t^{j}}{j!}$. (Note that $Φ(t)=Φ^*(t)+\frac{t^{n-1}}{(n-1)!}$).

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arXiv2015

A sharp Trudinger-Moser inequality on any bounded and convex planar domain

Wang and Ye conjectured in [22]: Let $Ω$ be a regular, bounded and convex domain in $\mathbb{R}^{2}$. There exists a finite constant $C(Ω)>0$ such that \[ \int_Ωe^{\frac{4πu^{2}}{H_{d}(u)}}dxdy\le C(Ω),\;\;\forall u\in C^{\infty}_{0}(Ω), \] where $H_{d}=\int_Ω|\nabla u|^{2}dxdy-\frac{1}{4}\int_Ω\frac{u^{2}}{d(z,\partialΩ)^{2}}dxdy$ and $d(z,\partialΩ)=\min\limits_{z_{1}\in\partialΩ}|z-z_{1}|$.} The main purpose of this paper is to confirm that this conjecture indeed holds for any bounded and convex domain in $\mathbb{R}^{2}$ via the Riemann mapping theorem (the smoothness of the boundary of the domain is thus irrelevant). We also give a rearrangement-free argument for the following Trudinger-Moser inequality on the hyperbolic space $\mathbb{B}=\{z=x+iy:|z|=\sqrt{x^{2}+y^{2}}<1\}$: \[ \sup_{\|u\|_{\mathcal{H}}\leq 1} \int_{\mathbb{B}}(e^{4πu^{2}}-1-4πu^{2})dV=\sup_{\|u\|_{\mathcal{H}}\leq 1}\int_{\mathbb{B}}\frac{(e^{4πu^{2}}-1-4πu^{2})}{(1-|z|^{2})^{2}}dxdy< \infty, \] by using the method employed earlier by Lam and the first author [9, 10], where $\mathcal{H}$ denotes the closure of $C^{\infty}_{0}(\mathbb{B})$ with respect to the norm $$\|u\|_{\mathcal{H}}=\int_{\mathbb{B}}|\nabla u|^{2}dxdy-\int_{\mathbb{B}}\frac{u^{2}}{(1-|z|^{2})^{2}}dxdy.$$ Using this strengthened Trudinger-Moser inequality, we also give a simpler proof of the Hardy-Moser-Trudinger inequality obtained by Wang and Ye [22].

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arXiv2015

Sharp constants and optimizers for a class of the Caffarelli-Kohn-Nirenberg inequalities

In this paper, we will use a suitable tranform to investigate the sharp constants and optimizers for the following Caffarelli-Kohn-Nirenberg inequalities for a wide range of parameters $(r,p,q,s,μ,σ)$ and $0\leq a\leq1$: \begin{equation} \left({\displaystyle\int} \left\vert u\right\vert ^{r}\frac{dx}{\left\vert x\right\vert ^{s}}\right)^{1/r}\leq C\left( {\displaystyle\int} \left\vert \nabla u\right\vert ^{p}\frac{dx}{\left\vert x\right\vert ^μ% }\right) ^{a/p}\left({\displaystyle\int} \left\vert u\right\vert ^{q}\frac{dx}{\left\vert x\right\vert ^σ% }\right) ^{\left( 1-a\right) /q}. \end{equation} We are able to compute the best constants and the explicit forms of the extremal functions in numerous cases. When $0<a<1$, we can deduce the existence and symmetry of optimizers for a wide range of parameters. Moreover, in the particular classes $r=p\frac{q-1}{p-1}$ and $q=p\frac{r-1}{p-1}$, the forms of maximizers will also be provided in the spirit of Del Pino and Dolbeault ([12], 13]). In the case $a=1$, that is the Caffarelli-Kohn-Nirenberg inequality without the interpolation term, we will provide the exact maximizers for all the range of $μ\geq0$. The Caffarelli-Kohn-Nirenberg inequalities with arbitrary norms on the Euclidean spaces will also be considered in the spirit of Cordero-Erausquin, Nazaret and Villani [10].

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arXiv2015

$L^p$ Boundedness of rough Bi-parameter Fourier Integral Operators

In this paper, we will investigate the boundedness of the bi-parameter Fourier integral operators (or FIOs for short) of the following form: $$T(f)(x)=\frac{1}{(2π)^{2n}}\int_{\mathbb{R}^{2n}}e^{iφ(x,ξ,η)}\cdot a(x,ξ,η)\cdot\widehat{f}(ξ,η)dξdη,$$ where for $x=(x_1,x_2)\in \mathbb{R}^{n}\times \mathbb{R}^{n}$ and $ξ,η\in \mathbb{R}^{n}\setminus\{0\}$, the amplitude $a(x,ξ,η)\in L^\infty BS^m_ρ$ and the phase function is of the form $ φ(x,ξ,η)=φ_1(x_1,ξ)+φ_2(x_2,η)$ with $\quad φ_1,φ_2 \in L^\infty Φ^2 (\mathbb{R}^{n}\times\mathbb{R}^{n}\setminus\{0\})$ and $φ(x, ξ, η)$ satisfies a certain rough non-degeneracy condition. The study of these operators are motivated by the $L^p$ estimates for one-parameter FIOs and bi-parameter Fourier multipliers and pseudo-differential operators. We will first define the bi-parameter FIOs and then study the $L^p$ boundedness of such operators when their phase functions have compact support in frequency variables with certain necessary non-degeneracy conditions. We will then establish the $L^p$ boundedness of the more general FIOs with amplitude $a(x,ξ,η)\in L^\infty BS^m_ρ$ and non-smooth phase function $φ(x,ξ,η)$ on $x$ satisfying a rough non-degeneracy condition.

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arXiv2015

Hardy Space Estimates for Littlewood-Paley-Stein Square Functions and Calderón-Zygmund Operators

In this work, we give new sufficient conditions for a Littlewood-Paley-Stein square function and necessary and sufficient conditions for a Calderón-Zygmund operator to be bounded on Hardy spaces $H^p$ with indices smaller than $1$. New Carleson measure type conditions are defined for Littlewood-Paley-Stein operators, and we show that they are sufficient for the associated square function to be bounded from $H^p$ into $L^p$. New polynomial growth $BMO$ conditions are also introduced for Calderón-Zygmund operators. These results are applied to prove that Bony paraproducts can be constructed such that they are bounded on Hardy spaces with exponents ranging all the way down to zero.

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arXiv2015

Best constants and existence of maximizers for weighted Moser-Trudinger inequalities

In this paper, we will establish the best constants for certain classes of weighted Moser-Trudinger inequalities on the entire Euclidean spaces $\mathbb{R}^N$. We will also prove the existence of maximizers of these sharp weighted inequalities. The class of functions considered here are not necessarily spherically symmetric. Our inequality in Theorem 1.1 improves the earlier one where such type of inequality was only considered for spherically symmetric functions by M. Ishiwata, M. Nakamura, H. Wadade in \cite{INW} (except in the case $s\not=0$). Since $\int_{\mathbb{R}^N}Φ_N(α|u|^{N/(N-1)})\frac{dx}{|x|^t}\le \int_{\mathbb{R}^N}e^{α|u|^{N/(N-1)}}|u|^N\frac{dx}{|x|^t}$, our inequality in Theorem 1.2 is stronger than the inequality in Theorem 1.1. We note that it suffices for us to prove the above inequalities for all functions not necessarily radially symmetric when $s=t$ by the well-known Caffareli-Kokn-Nirenberg inequalities \cite{CKN}.

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arXiv2015

Equivalence of critical and subcritical sharp Trudinger-Moser-Adams inequalities

Sharp Trudinger-Moser inequalities on the first order Sobolev spaces and their analogous Adams inequalities on high order Sobolev spaces play an important role in geometric analysis, partial differential equations and other branches of modern mathematics. Such geometric inequalities have been studied extensively by many authors in recent years and there is a vast literature. There are two types of such optimal inequalities: critical and subcritical sharp inequalities, both are with best constants. Critical sharp inequalities are under the restriction of the full Sobolev norms for the functions under consideration, while the subcritical inequalities are under the restriction of the partial Sobolev norms for the functions under consideration. There are subtle differences between these two type of inequalities. Surprisingly, we prove in this paper that these critical and subcritical Trudinger-Moser and Adams inequalities are actually equivalent. Moreover, we also establish the asymptotic behavior of the supremum for the subcritical Trudinger-Moser and Adams inequalities on the entire Euclidean spaces (Theorem 1.1 and Theorem 1.3) and provide a precise relationship between the supremums for the critical and subcritical Trudinger-Moser and Adams inequalities (Theorem 1.2 and Theorem 1.4).

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arXiv2014

$L^{p}$ estimates for the bilinear Hilbert transform for $1/2<p\leq2/3$: A counterexample and generalizations to non-smooth symbols

M. Lacey and C. Thiele proved in [27] (Annals of Math. (1997)) and [28] (Annals of Math. (1999)) that the bilinear Hilbert transform maps $L^{p_1}\times L^{p_2}\rightarrow L^{p}$ boundedly when $\frac{1}{p_1}+\frac{1}{p_2}=\frac{1}{p}$ with $1<p_{1}, \, p_{2}\leq\infty$ and $\frac{2}{3}<p<\infty$. Whether the $L^p$ estimates hold in the range $p\in (1/2,2/3]$ has remained an open problem since then. In this paper, we prove that the bilinear Hilbert transform does not map $\mathcal{F}L^{p'_{1}}\times L^{p_{2}}\rightarrow L^{p}$ for $p_1<2$ and $L^{p_{1}}\times \mathcal{F}L^{p'_{2}}\rightarrow L^{p}$ for $p_2<2$ boundedly (Theorem 1.2). In particular, this shows that the bilinear Hilbert transform neither maps $\mathcal{F}L^{p'_{1}}\times L^{p_{2}}\rightarrow L^{p}$ nor $L^{p_{1}}\times \mathcal{F}L^{p'_{2}}\rightarrow L^{p}$ for $\frac{1}{2}<p<\frac{2}{3}$. Nevertheless, we can establish $L^p$ estimates for the bilinear Fourier multipliers whose symbols are not identical to but arbitrarily close to that of the bilinear Hilbert transform in the full range $p\in(1/2,\infty)$ (Theorem 1.3).

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