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This paper is devoted to the weak and strong estimates for the linear and multilinear fractional Hausdorff operators on the Heisenberg group H n {{\mathbb{H}}}^{n} . A sharp strong estimate for T Φ m {T}_{\Phi }^{m} is obtained. As an application, we derive the sharp constant for the product Hardy operator on H n {{\mathbb{H}}}^{n} . Some weak-type ( p , q ) \left(p,q) ( 1 ≤ p ≤ ∞ ) \left(1\le p\le \infty ) estimates for T Φ , β {T}_{\Phi ,\beta } are also obtained. As applications, we calculate some sharp weak constants for the fractional Hausdorff operator on the Heisenberg group. Besides, we give an explicit weak estimate for T Φ , β → m {T}_{\Phi ,\overrightarrow{\beta }}^{m} under some mild assumptions on Φ \Phi . We extend the results of Guo et al. [Hausdorff operators on the Heisenberg group, Acta Math. Sin. (Engl. Ser.) 31 (2015), no. 11, 1703–1714] to the fractional setting.

A necessary and sufficient condition on the precompactness of extremal sequences for one dimensional $\alpha$-Strichartz inequalities, equivalently $\alpha$-Fourier extension estimates, is established based on the profile decomposition arguments. One of our main tools is an operator-convergence dislocation property consequence which comes from the van der Corput Lemma. Our result is valid in asymmetric cases as well. In addition, we obtain the existence of extremals for non-endpoint $\alpha$-Strichartz inequalities., Comment: 31 pages. Some small typos fixed

We prove a plethora of the boundedness property of Adams type for bilinear fractional integral operators of the form $$\begin{aligned} B_{\alpha }(f,g)(x)=\int _{{\mathbb {R}}^{n}}\frac{f(x-y)g(x+y)}{|y|^{n-\alpha }}dy,\quad 0

We will prove that for 1 < p < ∞ and 0 < λ < n, the central Morrey norm of the truncated centered Hardy-Littlewood maximal operator Mγc equals that of the centered Hardy-Littlewood maximal operator for all 0 < γ < +∞. When p = 1 and 0 < λ < n, it turns out that the weak central Morrey norm of the truncated centered Hardy-Littlewood maximal operator Mγc equals that of the centered Hardy-Littlewood maximal operator for all 0 < γ < +∞. Moreover, the same results are true for the truncated uncentered Hardy-Littlewood maximal operator. Our work extends the previous results of Lebesgue spaces to Morrey spaces.

The goal of this note is to establish non-tangential convergence results for Schrodinger operators along restricted curves. We consider the relationship between the dimension of this kind of approach region and the regularity for the initial data which implies convergence. As a consequence, we obtain an upper bound for p such that the Schrodinger maximal function is bounded from $$H^{s}(\mathbb {R}^{n})$$ to $$L^{p}(\mathbb {R}^{n})$$ for any $$s > \frac{n}{2(n+1)}$$ .

In this paper, we investigate some necessary and sufficient conditions which ensure validity of a k-fold Beta integral formula for any $$x^{i}\in \mathbb {R}^n$$ and some nonzero real numbers $$d_i$$ with $$i=1,2,\ldots ,k$$ . We establish that Eq. (0.1) holds if and only if and This yields a complete answer to the question raised by Grafakos and Morpurgo in [4]. In addition, it turns out that formula (0.1) does not hold if $$k\ge 4$$ . For those $$k, d_1, d_2, \ldots , d_k$$ not satisfying (0.2) or (0.3), we prove that the real-valued integral $$\displaystyle {\int _{\mathbb {R}^n}\prod \limits ^k_{i=1}|x^{i}-t|^{-d_i}dt}$$ can be represented as a function of distances of consecutive differences of the sequence $$x^1, x^2, \ldots , x^k$$ .

In this paper, we derive an incompressible Oldroyd-B model with hybrid dissipation and partial damping on stress tensor $$\tau $$ via the velocity equations and the generalized constitutive law so that global well-posedness of the model is established in the Sobolev space framework. Precisely speaking, the proof is based on the curl-div free property of $$\tau - {\nabla }\frac{1}{\Delta } \mathbb {P}\mathrm{div}\tau - ( {\nabla } \frac{1}{\Delta } \mathbb {P}\mathrm{div}\tau )^T $$ , the low frequency dissipation and high frequency damping of $$\tau $$ and the dissipation of u.

This paper is devoted to $L^2$ estimates for trilinear oscillatory integrals of convolution type on $\mathbb{R}^2$. The phases in the oscillatory factors include smooth functions and polynomials. We shall establish sharp $L^2$ decay estimates of trilinear oscillatory integrals with smooth phases, and then give $L^2$ uniform estimates for these integrals with polynomial phases., Comment: 26 pages

In this paper, we obtain the weighted boundedness for the local multi(sub)linear Hardy-Littlewood maximal operators and local multilinear fractional integral operators associated with the local Muckenhoupt weights on Gaussian measure spaces. We deal with these problems by introducing a new pointwise equivalent "radial" definitions of these local operators. Moreover using a similar approach, we also get the weighted boundedness for the local fractional maximal operators with rough kernel and local fractional integral operators with rough kernel on Gaussian measure spaces., Comment: 17 pages

In this paper, we establish the boundedness of a class of oscillatory singular integral operators with rough kernel on central Morrey spaces. Moreover, the boundedness for each of their commutators on weighted central Morrey spaces was also obtained. We generalized some existing results.

In this paper, we obtain the existence and boundedness of Marcinkiewicz integrals with homogeneous kernels on central Campanato spaces. Moreover, the existence and boundedness of multilinear Marcinkiewicz integrals on central Campanato spaces are also deduced. We extend various previous results to central Campanato spaces which can be seen as the local version of Campanato spaces.

In this paper, we obtain the sharp bound for fractional conjugate Hardy operator on higher-dimensional product spaces from L1ℝn1×⋯×ℝnm to the space wLQℝn1×⋯×ℝnm and Lpℝn1×⋯×ℝnm to the space Lqℝn1×⋯×ℝnm. More generally, the operator norm of the fractional Hardy operator on higher-dimensional product spaces from LPℝn1×⋯×ℝnm to LQIℝn1×⋯×ℝnm is obtained.

We consider certain kinds of weighted multi-linear fractional integral inequalities which can be regarded as extensions of the Hardy-Littlewood-Sobolev inequality. For a particular case, we characterize the sufficient and necessary conditions which ensure that the corresponding inequality holds. For the general case, we give some sufficient conditions and necessary conditions, respectively.

LetΩ∈Ls(Sn-1)fors⩾1be a homogeneous function of degree zero andbbe BMO functions. In this paper, we obtain some boundedness of the parametric Marcinkiewicz integral operatorμΩρand its higher-order commutator[bm,μΩρ]on Herz spaces with variable exponent.

In this article, we obtain the sharp bounds from LP ( G n ) to the space wLP ( G n ) for Hardy operators on product spaces. More generally, the precise norms of Hardy operators on product spaces from LP ( G n ) to the space LPI ( G n ) are obtained.

We investigate the boundedness of the strongly singular convolution operators on Herz-type Hardy spaces with variable exponent.

We investigate a class of multilinear integral operators with the nonnegative kernels, and prove that the norms of the operators can be obtained by integral of the product of the kernel function and finitely many basic functions. Using the integral, we can easily calculate the sharp constants for the multilinear Hilbert inequality, the generalized Hardy-Littlewood-Sobolev inequality and the multilinear Hardy operator.

In this paper, we use the constructive methods to show that every Schwartz function can be decom-posed as a product of two Schwartz functions. The same is true for D (R n ) function with the same compact set. As their applications, it immediately follows that g (R n )= g (R n ) g (R n ) and D (R n )= D (R n ) D (R n ). It should be pointed out that we use the adaptive idea and method to give the decompositions.

In this paper, we investigate sharp damping estimates for a class of one dimensional oscillatory integral operators with real-analytic phases. By establishing endpoint estimates for suitably damped oscillatory integral operators, we are able to give a new proof of the sharp $L^p$ estimates which have been proved by Xiao in Endpoint estimates for one-dimensional oscillatory integral operators, \emph{Advances in Mathematics}, \textbf{316}, 255-291 (2017). The damping estimates obtained in this paper are of independent interest., Comment: 30 pages

We consider the following two functions, (1-| x | ρ 1)+ α and (1-| x | ρ 2)+ α + which are called the generalized Bochner-Riesz multipliers, where ρ 1, ρ 2 and α are positive real numbers. We shall prove that, as the multiplier operators with respect to the two functions, their L p -boundedness and H p -boundedness are equivalent for any ρ 1, ρ 2 and α .

We investigate the bilinear Hilbert transform with oscillatory factors and the truncated bilinear Hilbert transform. The main result is that theLp1×Lp2→Lq-boundedness of the two operators is equivalent with1≤p1,p2<∞, and1/q=1/p1+1/p2. In addition, we also discuss the boundedness of a variant operator of bilinear Hilbert transform with a nontrivial polynomial phase.

It is well known that the doubly weighted Hardy-Littlewood-Sobolev inequality is as follows

We characterize a class of piecewise linear spectral sequences. Associated with the spectral sequence, we construct an orthonormal exponential bases for L2([0, 1)d), which are called generalized Fourier bases. Moreover, we investigate the convergence of Bochner-Riesz means of the generalized Fourier series.

We introduce a space Open image in new window , where Open image in new window is the testing function space whose functions are infinitely differentiable and have bounded support, and Open image in new window is the space the double Hilbert transform acting on the testing function space. We prove that the double Hilbert transform is a homeomorphism from Open image in new window onto itself.

In this paper, we study a class of multilinear fractional integral operators which have correlation kernels $\prod_{1\leq i, Comment: 42 pages. A new proof by Brascamp-Lieb inequality is given in this version. V1 submitted 19 Feb 2016, originally announced 19 Feb 2016

In this paper, we shall prove the $L^{p}$ endpoint decay estimates of oscillatory integral operators with homogeneous polynomial phases $S$ in $\mathbb{R} \times \mathbb{R}$. As a consequence, sharp $L^{p}$ decay estimates are also obtained when polynomial phases have the form $S(x^{m_{1}},y^{m_{2}})$ with $m_1$ and $m_2$ being positive integers., 27 pages