Multilinear Commutators Research Articles

On groups with BFC-covered word values

For a group G and a positive integer n write Bn(G)={x∈G:|xG|≤n}. If s≥1 and w is a group word, say that G satisfies the (n,s)-covering condition with respect to the word w if there exists a subset S⊆G such that |S|≤s and all w-values of G are contained in Bn(G)S. In a natural way, this condition emerged in the study of probabilistically nilpotent groups of class two. In this paper we obtain the following results.Let w be a multilinear commutator word on k variables and let G be a group satisfying the(n,s)-covering condition with respect to the word w. Then G has a soluble subgroup T such that[G:T]and the derived length of T are both(k,n,s)-bounded. (Theorem 1.1.)Letk≥1and G be a group satisfying the(n,s)-covering condition with respect to the wordγk. Then (1)γ2k−1(G)has a subgroup T such that[γ2k−1(G):T]and|T′|are both(k,n,s)-bounded; and (2) G has a nilpotent subgroup U such that[G:U]and the nilpotency class of U are both(k,n,s)-bounded. (Theorem 1.2.)

Some extensions of classes involving pair of weights related to the boundedness of multilinear commutators associated to generalized fractional integral operators

Some extensions of classes involving pair of weights related to the boundedness of multilinear commutators associated to generalized fractional integral operators

Multilinear Commutators of Fractional Integral Operator on Central Morrey Spaces with Variable Exponent

Multilinear Commutators of Fractional Integral Operator on Central Morrey Spaces with Variable Exponent

Weighted norm inequalities for multilinear strongly singular Calderón–Zygmund operators on RD‐spaces

AbstractLet be an RD‐space, namely, a space of homogeneous type in the sense of Coifman and Weiss with the Borel measure μ satisfying the reverse doubling condition on . Based on this space, the authors define a multilinear strongly singular Calderón–Zygmund operator whose kernel does not need any size condition and has more singularities near the diagonal than that of a standard multilinear Calderón–Zygmund operator. For such an operator, we establish its boundedness on product of weighted Lebesgue spaces by means of the pointwise estimate for the sharp maximal function. In addition, the endpoint estimates of the type are also obtained. Moreover, we prove weighted boundedness results for multilinear commutators generated by multilinear strongly singular Calderón–Zygmund operators and BMO functions. These results contribute to the extension of multilinear strongly singular Calderón–Zygmund operator theory in the Euclidean case to the context of space of homogeneous type.

The commutators of multilinear Calderón–Zygmund operators on weighted Hardy spaces

In this paper, we study the behaviours of the commutators $[\vec b,\,T]$ generated by multilinear Calderón–Zygmund operators $T$ with $\vec b=(b_1,\,\ldots,\,b_m)\in L_{\rm loc}(\mathbb {R}^n)$ on weighted Hardy spaces. We show that for some $p_i\in (0,\,1]$ with $1/p=1/p_1+\cdots +1/p_m$ , $\omega \in A_\infty$ and $b_i\in \mathcal {BMO}_{\omega,p_i}$ ( $1\le i\le m$ ), which are a class of non-trivial subspaces of ${\rm BMO}$ , the commutators $[\vec b,\,T]$ are bounded from $H^{p_1}(\omega )\times \cdots \times H^{p_m}(\omega )$ to $L^p(\omega )$ . Meanwhile, we also establish the corresponding results for a class of maximal truncated multilinear commutators $T_{\vec b}^*$ .

Multilinear commutators of multilinear strongly singular integral operators with generalized kernels

Abstract In [S. Yang and Y. Lin, Multilinear strongly singular integral operators with generalized kernels and applications, AIMS Math. 6 2021, 12, 13533–13551], the authors of the present paper further weaken the smoothness condition of kernel functions with multilinear strongly singular Calderón–Zygmund operators of [Y. Lin, Multilinear theory of strongly singular Calderón–Zygmund operators and applications, Nonlinear Anal. 192 2020, Article ID 111699]. They defined a new class of multilinear strongly singular integral operators, and studied its weighted L p {L^{p}} boundedness, variable exponent L p ⁢ ( ⋅ ) {L^{p(\cdot)}} boundedness and endpoint estimates. In this paper, we naturally consider the boundedness of multilinear commutators and multilinear iterated commutators, which are generated by multilinear strongly singular integral operators with generalized kernels and Lipschitz functions. Our results include the corresponding results of multilinear strongly singular Calderón–Zygmund operators and classical multilinear Calderón–Zygmund operators, respectively.

Strong conciseness of Engel words in profinite groups

AbstractA group wordwis said to be strongly concise in a class of profinite groups if, for any groupGin , eitherwtakes at least continuum many values inGor the verbal subgroup is finite. It is conjectured that all words are strongly concise in the class of all profinite groups. Earlier Detomi, Klopsch, and Shumyatsky proved this conjecture for multilinear commutator words, as well as for some other particular words. They also proved that every group word is strongly concise in the class of nilpotent profinite groups, as well as that 2‐Engel words are strongly concise (but their approach does not seem to generalize ton‐Engel words for ). In this paper, we prove that for anyn, then‐Engel word (whereyis repeatedntimes) is strongly concise in the class of finitely generated profinite groups.

Rough fractional integral and its multilinear commutators on $ p $-adic generalized Morrey spaces

<abstract><p>In this paper, we establish the boundedness of rough $ p $-adic fractional integral operators on $ p $-adic generalized Morrey spaces, as well as the boundedness of multilinear commutators generated by rough $ p $-adic fractional integral operator and $ p $-adic generalized Campanato functions. Moreover, the boundedness in classical Morrey is given as corollaries.</p></abstract>

Boundedness of sublinear operators on weighted grand Herz-Morrey spaces

<abstract><p>In this paper, we introduce weighted grand Herz-Morrey type spaces and prove the boundedness of sublinear operators and their multilinear commutators on these spaces. The results are still new even in the unweighted setting.</p></abstract>

Two-Weight Inequalities for Multilinear Commutators in Product Spaces

This article is devoted to establishing two-weight estimates for commutators of singular integrals. We combine multilinearity with product spaces. A new type of two-weight extrapolation result is used to yield the quasi-Banach range of estimates.

p -Adic Riesz Potential and Its Commutators on Morrey-Herz Spaces

In this paper, we establish the boundedness of p -adic Riesz potential on Morrey-Herz spaces, as well as the λ -central BMO estimates for multilinear commutators of p -adic Riesz potential on Morrey-Herz spaces.

Multilinear commutators in the two‐weight setting

We extend the recently much-studied two-weight commutator estimates to the multilinear setting. In contrast to previous results, our result respects the multilinear nature of the problem fully and is formulated with the genuinely multilinear weights.

Multilinear commutators related to maximal function on Morrey-Banach space and its application

Multilinear commutators related to maximal function on Morrey-Banach space and its application

Calderón-Zygmund operators with kernels of Dini's type and their multilinear commutators on generalized variable exponent Morrey spaces

Calderón-Zygmund operators with kernels of Dini's type and their multilinear commutators on generalized variable exponent Morrey spaces

ON THE RANK OF A VERBAL SUBGROUP OF A FINITE GROUP

AbstractWe show that if w is a multilinear commutator word and G a finite group in which every metanilpotent subgroup generated by w-values is of rank at most r, then the rank of the verbal subgroup $w(G)$ is bounded in terms of r and w only. In the case where G is soluble, we obtain a better result: if G is a finite soluble group in which every nilpotent subgroup generated by w-values is of rank at most r, then the rank of $w(G)$ is at most $r+1$ .

Words of Engel type are concise in residually finite groups. Part II

This work is a natural follow-up of the article [5]. Given a group-word w and a group G , the verbal subgroup w(G) is the one generated by all w -values in G . The word w is called concise if w(G) is finite whenever the set of w -values in G is finite. It is an open question whether every word is concise in residually finite groups. Let w=w(x_1,\ldots,x_k) be a multilinear commutator word, n a positive integer and q a prime power. In the present article we show that the word [w^q,_ny] is concise in residually finite groups (Theorem 1.2) while the word [w,_ny] is boundedly concise in residually finite groups (Theorem 1.1).

Strong conciseness in profinite groups

A group word $w$ is said to be strongly concise in a class $\mathcal{C}$ of profinite groups if, for every group $G$ in $\mathcal{C}$ such that $w$ takes less than $2^{\aleph_0}$ values in $G$, the verbal subgroup $w(G)$ is finite. Detomi, Morigi and Shumyatsky established that multilinear commutator words -- and the particular words $x^2$ and $[x^2,y]$ -- have the property that the corresponding verbal subgroup is finite in a profinite group $G$ whenever the word takes at most countably many values in $G$. They conjectured that, in fact, this should be true for every word. In particular, their conjecture included as open cases power words and Engel words. In the present paper, we take a new approach via parametrised words that leads to stronger results. First we prove that multilinear commutator words are strongly concise in the class of all profinite groups. Then we establish that every group word is strongly concise in the class of nilpotent profinite groups. From this we deduce, for instance, that, if $w$ is one of the group words $x^2$, $x^3$, $x^6$, $[x^3,y]$ or $[x,y,y]$, then $w$ is strongly concise in the class of all profinite groups. Indeed, the same conclusion can be reached for all words of the infinite families $[x^m,z_1,\ldots,z_r]$ and $[x,y,y,z_1,\ldots,z_r]$, where $m \in \{2,3\}$ and $r \ge 1$.

A note for some parabolic multilinear commutators generated by a class of parabolic maximal and linear operators with rough kernel on the parabolic generalized local Morrey spaces

In this paper, we give the boundedness of some parabolic multilinear commutators generated by a class of parabolic maximal and linear operators with rough kernel and parabolic local Campanato functions on the parabolic generalized local Morrey spaces, respectively. Indeed, the results in this paper are extensions of some known results.

On finite groups in which commutators are covered by Engel subgroups

Abstract Let m , n {m,n} be positive integers and w a multilinear commutator word. Assume that G is a finite group having subgroups G 1 , … , G m {G_{1},\ldots,G_{m}} whose union contains all w-values in G. Assume further that all elements of the subgroups G 1 , … , G m {G_{1},\ldots,G_{m}} are n-Engel in G. It is shown that the verbal subgroup w ⁢ ( G ) {w(G)} is s-Engel for some { m , n , w } {\{m,n,w\}} -bounded number s.

Boundedness results for commutators with BMO functions via weighted estimates: a comprehensive approach

We present a unified method to obtain weighted estimates of linear and multilinear commutators with BMO functions, that is amenable to a plethora of operators and functional settings. Our approach elaborates on a commonly used Cauchy integral trick, recovering many known results but yielding also numerous new ones.