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## Almost orthogonal operators on the bitorus II

### Mathematische Zeitschrift (2012-06-01) 271: 271-291 , June 01, 2012

We prove that the operator
$${Tf(x,y)=\int^\pi_{-\pi}\int_{|x^{\prime}|<|y^{\prime}|} \frac{e^{iN(x,y) x^{\prime}}}{x^{\prime}}\frac{e^{iN(x,y) y^{\prime}}}{y^{\prime}}f(x-x^{\prime}, y-y^{\prime}) dx^{\prime} dy^{\prime}}$$
, with
$${x,y \in[0,2\pi]}$$
and where the cut off
$${|x^{\prime}|<|y^{\prime}|}$$
is performed in a smooth and dyadic way, is bounded from *L*^{2} to weak-
$${L^{2-\epsilon}}$$
, any
$${\epsilon > 0 }$$
, under the basic assumption that the real-valued measurable function *N*(*x*, *y*) is “mainly” a function of *y* and the additional assumption that *N*(*x*, *y*) is non-decreasing in *x*, for every *y* fixed. This is an extension to 2D of C. Fefferman’s proof of a.e. convergence of Fourier series of *L*^{2} functions.

## An endpoint estimate for maximal multilinear singular integral operators

### Analysis in Theory and Applications (2007-12-01) 23: 307-314 , December 01, 2007

A weak type endpoint estimate for the maximal multilinear singular integral operator
$$T_A^* f(x) = \mathop {\sup }\limits_{\varepsilon > 0} \left| {\int_{\left| {x - y} \right| > \varepsilon } {\frac{{\Omega (x - y)}}{{\left| {x - y} \right|^{n + 1} }}(A(x) - A(y) - \nabla A(y)(x - y))f(y)dy} } \right|$$
is established, where Θ is homogeneous of degree zero, integrable on the unit sphere and has vanishing moment of order one, and *A* has derivatives of order one in BMO(ℝ^{n}). A regularity condition on Θ which implies an *L*log*L* type estimate of *T*_{A}^{*}
is given.

## Commutators of Littlewood-Paley operators

### Science in China Series A: Mathematics (2009-11-01) 52: 2493-2505 , November 01, 2009

Let *b* ∈ *L*_{loc}(ℝ^{n}) and *L* denote the Littlewood-Paley operators including the Littlewood-Paley *g* function, Lusin area integral and *g*_{λ}^{*}
function. In this paper, the authors prove that the *L*^{p} boundedness of commutators [*b*, *L*] implies that *b* ∈ BMO(ℝ^{n}). The authors therefore get a characterization of the *L*^{p}-boundedness of the commutators [*b*, *L*]. Notice that the condition of kernel function of *L* is weaker than the Lipshitz condition and the Littlewood-Paley operators *L* is only sublinear, so the results obtained in the present paper are essential improvement and extension of Uchiyama’s famous result.

## L p estimates for the Schrödinger type operators

### Applied Mathematics-A Journal of Chinese Universities (2011-12-01) 26: 412-424 , December 01, 2011

Let *L*_{k} = (−Δ)^{k} + *V*^{k} be a Schrödinger type operator, where *k* ≥ 1 is a positive integer and *V* is a nonnegative polynomial. We obtain the *L*^{p} estimates for the operators ∇^{2k}*L*_{k}^{−1}
and ∇^{k}*L*_{k}^{−1/2}
.

## Two weight norm inequalities for the bilinear fractional integrals

### Manuscripta Mathematica (2016-05-01) 150: 159-175 , May 01, 2016

In this paper, we give a characterization of the two weight strong and weak type norm inequalities for the bilinear fractional integrals in terms of Sawyer type testing conditions. Namely, we give the characterization of the following inequalities,
$$\|\mathcal{I}_\alpha (f_1\sigma_1, f_2\sigma_2)\|_{L^q(w)} \le \mathscr{N} \prod_{i=1}^2\|f_i\|_{L^{p_i}(\sigma_i)}$$
and
$$\|\mathcal{I}_\alpha (f_1\sigma_1, f_2\sigma_2)\|_{L^{q,\infty}(w)} \le \mathscr{N}_{\rm{weak}}\prod_{i=1}^2\|f_i\|_{L^{p_i}(\sigma_i)},$$
when *q* ≥ *p*_{1}, *p*_{2} > 1 and *p*_{1} + *p*_{2} ≥ *p*_{1}*p*_{2}.

## Boundedness Properties of Pseudo-Differential and Calderón-Zygmund Operators on Modulation Spaces

### Journal of Fourier Analysis and Applications (2008-02-01) 14: 124-143 , February 01, 2008

In this article, we study the boundedness of pseudo-differential operators with symbols in *S*_{ρ,δ}^{m}
on the modulation spaces *M*^{p,q}. We discuss the order *m* for the boundedness Op(*S*_{ρ,δ}^{m}
)⊂ℒ(*M*^{p,q}) to be true. We also prove the existence of a Calderón-Zygmund operator which is not bounded on the modulation space *M*^{p,q} with *q*≠2. This unboundedness is still true even if we assume a generalized *T*(1) condition. These results are induced by the unboundedness of pseudo-differential operators on *M*^{p,q} whose symbols are of the class *S*_{1,δ}^{0}
with 0<*δ*<1.

## Morrey-Type Spaces on Gauss Measure Spaces and Boundedness of Singular Integrals

### The Journal of Geometric Analysis (2014-04-01) 24: 1007-1051 , April 01, 2014

In this paper, the authors introduce Morrey-type spaces on the locally doubling metric measure spaces, which means that the underlying measure enjoys the doubling and the reverse doubling properties only on a class of admissible balls, and then obtain the boundedness of the local Hardy–Littlewood maximal operator and the local fractional integral operator on such Morrey-type spaces. These Morrey-type spaces on the Gauss measure space are further proved to be naturally adapted to singular integrals associated with the Ornstein–Uhlenbeck operator. To be precise, by means of the locally doubling property and the geometric properties of the Gauss measure, the authors establish the equivalence between Morrey-type spaces and Campanato-type spaces on the Gauss measure space, and the boundedness for a class of singular integrals associated with the Ornstein–Uhlenbeck operator (including Riesz transforms of any order) on Morrey-type spaces over the Gauss measure space.

## Bounds for commutators of multilinear fractional integral operators with homogeneous kernels

### Analysis in Theory and Applications (2011-06-01) 27: 181-186 , June 01, 2011

We will show bounds for commutators of multilinear fractional integral operators with some homogeneous kernels.

## Boundedness for Marcinkiewicz integrals associated with Schrödinger operators

### Proceedings - Mathematical Sciences (2014-05-01) 124: 193-203 , May 01, 2014

Let *L* = −Δ + *V* be a Schrödinger operator, where Δ is the Laplacian on
$\mathbb {R}^{n}$
, while nonnegative potential *V* belongs to the reverse Hölder class. In this paper, we will show that Marcinkiewicz integral associated with Schrödinger operator is bounded on *BMO*_{L}, and from
$H^{1}_{L}(\mathbb {R}^{n})$
to
$L^{1}(\mathbb {R}^{n})$
.

## Bilinear Calderón-Zygmund operators on Sobolev, BMO and Lipschitz spaces

### Journal of Inequalities and Applications (2015-12-09) 2015: 1-12 , December 09, 2015

In this paper, the authors establish the necessary and sufficient condition such that the bilinear Calderón-Zygmund operators are bounded from $Lip_{\alpha}(\mathbb{R}^{n})\times L^{n/\alpha} (\mathbb{R}^{n})$ to $BMO(\mathbb{R}^{n})$ space and from $Lip_{\alpha}(\mathbb{R}^{n})\times L^{p}(\mathbb{R}^{n})$ to $Lip_{\alpha-n/p}(\mathbb{R}^{n})$ space. As an application, the bilinear Riesz transform is a good example which meets the related conditions.

Furthermore, the authors also establish another necessary and sufficient condition for the bilinear Calderón-Zygmund operators to be bounded from $Lip_{\alpha}(\mathbb{R}^{n})\times BMO(\mathbb{R}^{n})$ to $Lip_{\alpha}(\mathbb{R}^{n})$ space, from $Lip_{\alpha_{1}}(\mathbb{R}^{n})\times Lip_{\alpha_{2}}(\mathbb {R}^{n})$ to $Lip_{\alpha_{1}+\alpha_{2}}(\mathbb{R}^{n})$ space, and from $Lip_{\alpha}(\mathbb{R}^{n})\times \dot{B}^{s}(\mathbb{R}^{n})$ to $\dot{B}^{s-\alpha}(\mathbb{R}^{n})$ space.