Sharp bounds for Hardy operators on product spaces

In this article, we obtain the sharp bounds from L^P$\left({\mathbb{G}}^n\right)$ to the space wL^P$\left({\mathbb{G}}^n\right)$ for Hardy operators on product spaces. More generally, the precise norms of Hardy operators on product spaces from L^P$\left({\mathbb{G}}^n\right)$ to the space L^P_I$\left({\mathbb{G}}^n\right)$ are obtained.     

Sharp estimates for the non-centered maximal operator associated to Gaussian and other radial measures

In this work we investigate the integrability properties of the maximal operator M_μ, associated with a non-doubling measure μ defined on the Euclidean space , with special emphasis on the Gaussian and similar measures. Among other results we show for a wide class of radial and decreasing measures μ, that M_μ satisfies the modular inequality

     

Solution of two-weight problems for integral transforms with positive kernels

Criteria for weak and strong two-weighted inequalities are obtained for integral transforms with positive kernels.     

Marcinkiewicz integral on hardy spaces

In this paper we prove that the Marcinkiewicz integral is an operator of type (H ¹,L ¹) and of type (H ^1,,L ^1,). As a corollary of the results above, we obtain again the the weak type (1,1) boundedness of , but the smoothness condition assumed on is weaker than Stein's condition.The research was supported partly by Doctoral Programme Foundation of Institution of Higher Education (Grant No. 98002703) of China.The author was supported partly by NSF of China (Grant No. 19971010).The author was supported partly by NSF of China (Grant No. 19131080).     

Hypersingular integral operators along surfaces

In this note, we estimate the boundedness for singular integral operators along curves and surfaces with highly singular kernels.     

Marcinkiewicz integral on weighted Hardy spaces

In this paper we give sufficient conditions to imply the $H^{1}_{w}-L^{1}_{w}$ boundedness of the Marcinkiewicz integral operator $\mu_\Omega$, where w is a Muckenhoupt weight. We also prove that, under the stronger condition $\Omega \in {\rm Lip}_\alpha$, the operator $\mu_\Omega$ is bounded from $H^{p}_{w}$ to $L^{p}_{w}$ for $\max\{n/(n+1/2), n/(n+\alpha)\}$ < p < 1.     

Two-weightedL1 p -inequalities for singular integral operators on Heisenberg groups

Some sufficient conditions are found for a pair of weight functions, providing the validity of two-weighted inequalities for singular integrals defined on Heisenberg groups.     

Sharp two-weight inequalities for singular integrals, with applications to the Hilbert transform and the Sarason conjecture

We prove two-weight norm inequalities for Calderón-Zygmund singular integrals that are sharp for the Hilbert transform and for the Riesz transforms. In addition, we give results for the dyadic square function and for commutators of singular integrals. As an application we give new results for the Sarason conjecture on the product of unbounded Toeplitz operators on Hardy spaces.     

Boundedness of generalized marcinkiewicz integral on the <Emphasis Type="Italic">Hp</Emphasis>-sobolev spaces

In this paper, the general Marcinkiewicz integral operator μΩ,α on the Hp Sobolev spaces under the proper condition of kernel Ω(x′) is considered. It is obtained that μΩ,α is bounded from H{ } to Lp for some 0<p≤1. Jiang and Jia were supported in part by Education Department of Zhejiang province.     

Sharp weighted norm inequalities for Littlewood–Paley operators and singular integrals

We prove sharp Lp(w) norm inequalities for the intrinsic square function (introduced recently by M. Wilson) in terms of the Ap characteristic of w for all 1<p<∞. This implies the same sharp inequalities for the classical Lusin area integral S(f), the Littlewood–Paley g-function, and their continuous analogs Sψ and gψ. Also, as a corollary, we obtain sharp weighted inequalities for any convolution Calderón–Zygmund operator for all 1<p?3/2 and 3?p<∞, and for its maximal truncations for 3?p<∞.     

Sharp weighted bounds for fractional integral operators

The relationship between the operator norms of fractional integral operators acting on weighted Lebesgue spaces and the constant of the weights is investigated. Sharp bounds are obtained for both the fractional integral operators and the associated fractional maximal functions. As an application improved Sobolev inequalities are obtained. Some of the techniques used include a sharp off-diagonal version of the extrapolation theorem of Rubio de Francia and characterizations of two-weight norm inequalities.     

Weighted estimates for bilinear fractional integral operators

Moen (2016) proved weighted estimates for the bilinear fractional integrals where . We improve his results when and consider the case . As a corollary we obtain a bilinear Stein–Weiss inequality where .     

Characterizations of Besov-type and Triebel-Lizorkin-type spaces via maximal functions and local means

Let s∈R. In this paper, the authors first establish the maximal function characterizations of the Besov-type space with and τ∈[0,∞), the Triebel-Lizorkin-type space with p∈(0,∞), q∈(0,∞] and τ∈[0,∞), the Besov-Hausdorff space with p∈(1,∞), q∈[1,∞) and and the Triebel-Lizorkin-Hausdorff space with and , where t^′ denotes the conjugate index of t∈[1,∞]. Using this characterization, the authors further obtain the local mean characterizations of these function spaces via functions satisfying the Tauberian condition and establish a Fourier multiplier theorem on these spaces. All these results generalize the existing classical results on Besov and Triebel-Lizorkin spaces by taking τ=0 and are also new even for Q spaces and Hardy-Hausdorff spaces.     

Two-weighted estimates for some integral transforms in Lebesgue spaces with mixed norm and imbedding theorems

Two-weighted inequalities are proved for anisotropic potentials. These estimates are used to obtain refinements of the well-known imbedding theorems in the scale of weighted Lebesgue spaces.     

Weighted estimates for integral operators on local type spaces

We prove the weighted boundedness for a family of integral operators on Lebesgue spaces and local type spaces. To this end we show that can be controlled by the Calderón operator and a local maximal operator. This approach allows us to characterize the power weighted boundedness on Lebesgue spaces.     

Sharp weighted inequalities for multilinear fractional maximal operators and fractional integrals

In this paper, we study weighted inequalities for multilinear fractional maximal operators and fractional integrals. We prove sharp weighted Lebesgue space estimates for both operators when the vector of weights belongs to . In addition we prove sharp two weight mixed estimates for multilinear operators in the spirit of the linear estimates given in 3 .     

Sharp bounds for singular values of fractional integral operators

From the results of Dostanic [M.R. Dostanic, Asymptotic behavior of the singular values of fractional integral operators, J. Math. Anal. Appl. 175 (1993) 380-391] and V? and Gorenflo [Kim Tuan V?, R. Gorenflo, Singular values of fractional and Volterra integral operators, in: Inverse Problems and Applications to Geophysics, Industry, Medicine and Technology, Ho Chi Minh City, 1995, Ho Chi Minh City Math. Soc., Ho Chi Minh City, 1995, pp. 174-185] it is known that the jth singular value of the fractional integral operator of order α>0 is approximately (πj)^−α for all large j. In this note we refine this result by obtaining sharp bounds for the singular values and use these bounds to show that the jth singular value is (πj)^−α[1+O(j⁻¹)].     

Weighted bounds for the carleson maximal operator inR n

The Carleson maximal operator is shown to be bounded inL ^p (w) for certain values ofp and certain radial weightsw when acting on products of radial functions and homogeneous harmonic polynomials. Partially supported by DGICYT (MEC Spain). PB 92/187.     

Criteria of strong type two-weighted inequalities for fractional maximal functions

A strong type two-weight problem is solved for fractional maximal functions defined in homogeneous type general spaces. A similar problem is also solved for one-sided fractional maximal functions.