scholarly journals Variation inequalities for rough singular integrals and their commutators on Morrey spaces and Besov spaces

2021 ◽  
Vol 11 (1) ◽  
pp. 72-95
Author(s):  
Xiao Zhang ◽  
Feng Liu ◽  
Huiyun Zhang
Keyword(s):  
Hilbert Transform ◽  
Besov Spaces ◽  
Singular Integrals ◽  
Riesz Transform ◽  
Morrey Spaces ◽  
Riesz Transforms ◽  
Rough Kernels ◽  
Lebesgue Spaces ◽  
Variation Operators

Abstract This paper is devoted to investigating the boundedness, continuity and compactness for variation operators of singular integrals and their commutators on Morrey spaces and Besov spaces. More precisely, we establish the boundedness for the variation operators of singular integrals with rough kernels Ω ∈ Lq (S n−1) (q > 1) and their commutators on Morrey spaces as well as the compactness for the above commutators on Lebesgue spaces and Morrey spaces. In addition, we present a criterion on the boundedness and continuity for a class of variation operators of singular integrals and their commutators on Besov spaces. As applications, we obtain the boundedness and continuity for the variation operators of Hilbert transform, Hermit Riesz transform, Riesz transforms and rough singular integrals as well as their commutators on Besov spaces.

scholarly journals A note on maximal singular integrals with rough kernels

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Xiao Zhang ◽  
Feng Liu
Keyword(s):  
Singular Integral ◽  
Singular Integrals ◽  
Integral Operators ◽  
Maximal Operators ◽  
Rough Kernels ◽  
Lebesgue Spaces ◽  
Homogeneous Mapping ◽  
Recent Developments ◽  
Maximal Singular Integrals ◽  
Maximal Singular Integral

Abstract In this note we study the maximal singular integral operators associated with a homogeneous mapping with rough kernels as well as the corresponding maximal operators. The boundedness and continuity on the Lebesgue spaces, Triebel–Lizorkin spaces, and Besov spaces are established for the above operators with rough kernels in $H^{1}({\mathrm{S}}^{n-1})$ H 1 ( S n − 1 ) , which complement some recent developments related to rough maximal singular integrals.

scholarly journals Weighted Morrey Spaces Related to Certain Nonnegative Potentials and Riesz Transforms

2019 ◽  
Vol 2019 ◽  
pp. 1-18
Author(s):  
Hua Wang
Keyword(s):  
Riesz Transform ◽  
Morrey Spaces ◽  
Riesz Transforms ◽  
Strong Type ◽  
Type Estimate ◽  
Reverse Hölder Class ◽  
Hölder Class ◽  
Weighted Morrey Spaces ◽  
Holder Class ◽  
Reverse Hölder

LetL=-Δ+Vbe a Schrödinger operator, whereΔis the Laplacian onRdand the nonnegative potentialVbelongs to the reverse Hölder classRHqforq≥d. The Riesz transform associated with the operatorL=-Δ+Vis denoted byR=∇(-Δ+V)-1/2and the dual Riesz transform is denoted byR⁎=(-Δ+V)-1/2∇. In this paper, we first introduce some kinds of weighted Morrey spaces related to certain nonnegative potentials belonging to the reverse Hölder classRHqforq≥d. Then we will establish the mapping properties of the operatorRand its adjointR⁎on these new spaces. Furthermore, the weighted strong-type estimate and weighted endpoint estimate for the corresponding commutators[b,R]and[b,R⁎]are also obtained. The classes of weights, classes of symbol functions, and weighted Morrey spaces discussed in this paper are larger thanAp,BMO(Rd), andLp,κ(w)corresponding to the classical Riesz transforms (V≡0).

scholarly journals Boundedness and Continuity of Several Integral Operators with Rough Kernels inWFβSn-1on Triebel-Lizorkin Spaces

2018 ◽  
Vol 2018 ◽  
pp. 1-18
Author(s):  
Feng Liu
Keyword(s):  
Besov Spaces ◽  
Singular Integrals ◽  
Integral Operators ◽  
Function Class ◽  
Function Spaces ◽  
Maximal Functions ◽  
Rough Kernels ◽  
Systematic Treatment ◽  
Marcinkiewicz Integrals

A systematic treatment is given of singular integrals and Marcinkiewicz integrals associated with surfaces generated by polynomial compound mappings as well as related maximal functions with rough kernels inWFβ(Sn-1), which relates to the Grafakos-Stefanov function class. Certain boundedness and continuity for these operators on Triebel-Lizorkin spaces and Besov spaces are proved by applying some criterions of bounds and continuity for several operators on the above function spaces.

MAXIMAL SEMIGROUP SYMMETRY AND DISCRETE RIESZ TRANSFORMS

2015 ◽  
Vol 100 (2) ◽  
pp. 216-240
Author(s):  
TOSHIYUKI KOBAYASHI ◽  
ANDREAS NILSSON ◽  
FUMIHIRO SATO
Keyword(s):  
Hilbert Transform ◽  
Riesz Transform ◽  
Riesz Transforms ◽  
Dimensional Case ◽  
One Dimensional ◽  
Linearly Independent ◽  
The One ◽  
The Hilbert Transform ◽  
Multiplier Operators

We raise a question of whether the Riesz transform on $\mathbb{T}^{n}$ or $\mathbb{Z}^{n}$ is characterized by the ‘maximal semigroup symmetry’ that the transform satisfies. We prove that this is the case if and only if the dimension is one, two or a multiple of four. This generalizes a theorem of Edwards and Gaudry for the Hilbert transform on $\mathbb{T}$ and $\mathbb{Z}$ in the one-dimensional case, and extends a theorem of Stein for the Riesz transform on $\mathbb{R}^{n}$. Unlike the $\mathbb{R}^{n}$ case, we show that there exist infinitely many linearly independent multiplier operators that enjoy the same maximal semigroup symmetry as the Riesz transforms on $\mathbb{T}^{n}$ and $\mathbb{Z}^{n}$ if the dimension $n$ is greater than or equal to three and is not a multiple of four.

scholarly journals Variation inequalities related to Schrödinger operators on weighted Morrey spaces

2019 ◽  
Vol 17 (1) ◽  
pp. 813-827
Author(s):  
Jing Zhang
Keyword(s):  
Schrödinger Operators ◽  
Morrey Spaces ◽  
Riesz Transforms ◽  
Schrodinger Operators ◽  
Reverse Hölder Class ◽  
Hölder Class ◽  
Weighted Morrey Spaces ◽  
Holder Class ◽  
Variation Operators ◽  
Reverse Hölder

Abstract This paper establishes the boundedness of the variation operators associated with Riesz transforms and commutators generated by the Riesz transforms and BMO-type functions in the Schrödinger setting on the weighted Morrey spaces related to certain nonnegative potentials belonging to the reverse Hölder class.

scholarly journals Variation Inequalities for One-Sided Singular Integrals and Related Commutators

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 876
Author(s):  
Feng Liu ◽  
Seongtae Jhang ◽  
Sung-Kwun Oh ◽  
Zunwei Fu
Keyword(s):  
Singular Integrals ◽  
Morrey Spaces ◽  
Weighted Morrey Spaces ◽  
Variation Operators

We establish one-sided weighted endpoint estimates for the ϱ -variation ( ϱ > 2 ) operators of one-sided singular integrals under certain priori assumption by applying one-sided Calderón–Zygmund argument. Using one-sided sharp maximal estimates, we further prove that the ϱ -variation operators of related commutators are bounded on one-sided weighted Lebesgue and Morrey spaces. In addition, we also show that these operators are bounded from one-sided weighted Morrey spaces to one-sided weighted Campanato spaces. As applications, we obtain some results for the λ -jump operators and the numbers of up-crossings. Our main results represent one-sided extensions of many previously known ones.

scholarly journals Boundedness of the Hilbert transform on Besov spaces

2020 ◽  
Vol 12 (2) ◽  
pp. 443-450
Author(s):  
A. Maatoug ◽  
S.E. Allaoui
Keyword(s):  
Integral Operator ◽  
Harmonic Analysis ◽  
Singular Integral Operator ◽  
Singular Integral ◽  
Hilbert Transform ◽  
Besov Spaces ◽  
Singular Integrals ◽  
Modern Theory ◽  
The Hilbert Transform ◽  
Positive Results

The Hilbert transform along curves is of a great importance in harmonic analysis. It is known that its boundedness on $L^p(\mathbb{R}^n)$ has been extensively studied by various authors in different contexts and the authors gave positive results for some or all $p,1<p<\infty$. Littlewood-Paley theory provides alternate methods for studying singular integrals. The Hilbert transform along curves, the classical example of a singular integral operator, led to the extensive modern theory of Calderón-Zygmund operators, mostly studied on the Lebesgue $L^p$ spaces. In this paper, we will use the Littlewood-Paley theory to prove that the boundedness of the Hilbert transform along curve $\Gamma$ on Besov spaces $ B^{s}_{p,q}(\mathbb{R}^n)$ can be obtained by its $L^p$-boundedness, where $ s\in \mathbb{R}, p,q \in ]1,+\infty[ $, and $\Gamma(t)$ is an appropriate curve in $\mathbb{R}^n$, also, it is known that the Besov spaces $ B^{s}_{p,q}(\mathbb{R}^n)$ are embedded into $L^p(\mathbb{R}^n)$ spaces for $s >0$ (i.e. $B^{s}_{p,q}(\mathbb{R}^n) \hookrightarrow L^p(\mathbb{R}^n), s>0)$. Thus, our result may be viewed as an extension of known results to the Besov spaces $ B^{s}_{p,q}(\mathbb{R}^n)$ for general values of $s$ in $\mathbb{R}$.

scholarly journals Transference on certain multilinear multiplier operators

2001 ◽  
Vol 70 (1) ◽  
pp. 37-55 ◽  
Author(s):  
Dashan Fan ◽  
Shuichi Sato
Keyword(s):  
Hilbert Transform ◽  
Hardy Spaces ◽  
Singular Integrals ◽  
Multilinear Operators ◽  
Lebesgue Spaces ◽  
Bilinear Hilbert Transform ◽  
Multiplier Operators ◽  
Multilinear Singular Integrals

AbstractWe study DeLeeuw type theorems for certain multilinear operators on the Lebesgue spaces and on the Hardy spaces. As applications, on the torus we obtain an analog of Lacey—Thiele's theorem on the bilinear Hilbert transform, as well as analogies of some recent theorems on multilinear singular integrals by Kenig—Stein and by Grafakos—Torres.

scholarly journals A note on singular integrals with angular integrability

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Feng Liu
Keyword(s):  
Singular Integral ◽  
Singular Integrals ◽  
Function Spaces ◽  
Riesz Transforms ◽  
Hilbert Transforms ◽  
Rotation Method ◽  
Vector Valued Function ◽  
Vector Valued ◽  
L Log L

Abstract In this note we study the rough singular integral $$ T_{\varOmega }f(x)=\mathrm{p.v.} \int _{\mathbb{R}^{n}}f(x-y)\frac{\varOmega (y/ \vert y \vert )}{ \vert y \vert ^{n}}\,dy, $$ T Ω f ( x ) = p . v . ∫ R n f ( x − y ) Ω ( y / | y | ) | y | n d y , where $n\geq 2$ n ≥ 2 and Ω is a function in $L\log L(\mathrm{S} ^{n-1})$ L log L ( S n − 1 ) with vanishing integral. We prove that $T_{\varOmega }$ T Ω is bounded on the mixed radial-angular spaces $L_{|x|}^{p}L_{\theta }^{\tilde{p}}( \mathbb{R}^{n})$ L | x | p L θ p ˜ ( R n ) , on the vector-valued mixed radial-angular spaces $L_{|x|}^{p}L_{\theta }^{\tilde{p}}(\mathbb{R}^{n},\ell ^{\tilde{p}})$ L | x | p L θ p ˜ ( R n , ℓ p ˜ ) and on the vector-valued function spaces $L^{p}(\mathbb{R}^{n}, \ell ^{\tilde{p}})$ L p ( R n , ℓ p ˜ ) if $1<\tilde{p}\leq p<\tilde{p}n/(n-1)$ 1 < p ˜ ≤ p < p ˜ n / ( n − 1 ) or $\tilde{p}n/(\tilde{p}+n-1)< p\leq \tilde{p}<\infty $ p ˜ n / ( p ˜ + n − 1 ) < p ≤ p ˜ < ∞ . The same conclusions hold for the well-known Riesz transforms and directional Hilbert transforms. It should be pointed out that our proof is based on the Calderón–Zygmund’s rotation method.

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