Recent zbMATH articles in MSC 42B20https://zbmath.org/atom/cc/42B202021-06-15T18:09:00+00:00WerkzeugTwo weight commutators on spaces of homogeneous type and applications.https://zbmath.org/1460.420312021-06-15T18:09:00+00:00"Duong, Xuan Thinh"https://zbmath.org/authors/?q=ai:duong.xuan-thinh"Gong, Ruming"https://zbmath.org/authors/?q=ai:gong.ruming"Kuffner, Marie-Jose S."https://zbmath.org/authors/?q=ai:kuffner.marie-jose-s"Li, Ji"https://zbmath.org/authors/?q=ai:li.ji.1"Wick, Brett D."https://zbmath.org/authors/?q=ai:wick.brett-d"Yang, Dongyong"https://zbmath.org/authors/?q=ai:yang.dongyongSummary: In this paper, we establish the two weight commutator theorem of Calderón-Zygmund operators in the sense of Coifman-Weiss on spaces of homogeneous type, by studying the weighted Hardy and BMO space for \(A_2\) weights and by proving the sparse operator domination of commutators. The main tool here is the Haar basis, the adjacent dyadic systems on spaces of homogeneous type, and the construction of a suitable version of a sparse operator on spaces of homogeneous type. As applications, we provide a two weight commutator theorem (including the high order commutators) for the following Calderón-Zygmund operators: Cauchy integral operator on \(\mathbb{R}\), Cauchy-Szegö projection operator on Heisenberg groups, Szegö projection operators on a family of unbounded weakly pseudoconvex domains, the Riesz transform associated with the sub-Laplacian on stratified Lie groups, as well as the Bessel Riesz transforms (in one and several dimensions).Commutators of bilinear pseudo-differential operators on local Hardy spaces with variable exponents.https://zbmath.org/1460.420212021-06-15T18:09:00+00:00"Lu, Guanghui"https://zbmath.org/authors/?q=ai:lu.guanghuiSummary: The aim of this paper is to establish the boundedness of the commutator \([b_1, b_2,T_{\sigma}]\) generated by the bilinear pseudo-differential operator \(T_{\sigma}\) with smooth symbols and \(b_1,b_2\in \mathrm{BMO}(\mathbb{R}^n)\) on product of local Hardy spaces with variable exponents. By applying the refined atomic decomposition result, the authors prove that the bilinear pseudo-differential operator \(T_{\sigma}\) is bounded from the Lebesgue space \(L^p(\mathbb{R}^n)\) into \(h^{p_1(\cdot)}(\mathbb{R}^n)\times h^{p_2(\cdot)}(\mathbb{R}^n)\). Moreover, the boundedness of the commutator \([b_1, b_2, T_{\sigma}]\) on product of local Hardy spaces with variable exponents is also obtained.Weak Hardy-type spaces associated with ball quasi-Banach function spaces. II: Littlewood-Paley characterizations and real interpolation.https://zbmath.org/1460.420332021-06-15T18:09:00+00:00"Wang, Songbai"https://zbmath.org/authors/?q=ai:wang.songbai"Yang, Dachun"https://zbmath.org/authors/?q=ai:yang.dachun"Yuan, Wen"https://zbmath.org/authors/?q=ai:yuan.wen"Zhang, Yangyang"https://zbmath.org/authors/?q=ai:zhang.yangyangSummary: Let \(X\) be a ball quasi-Banach function space on \(\mathbb{R}^n\). In this article, assuming that the powered Hardy-Littlewood maximal operator satisfies some Fefferman-Stein vector-valued maximal inequality on \(X\) as well as it is bounded on both the weak ball quasi-Banach function space \(WX\) and the associated space, the authors establish various Littlewood-Paley function characterizations of \(WH_X(\mathbb{R}^n)\) under some weak assumptions on the Littlewood-Paley functions. The authors also prove that the real interpolation intermediate space \((H_X(\mathbb{R}^n),L^\infty(\mathbb{R}^n))_{\theta,\infty}\), between the Hardy space associated with \(X, H_X(\mathbb{R}^n)\), and the Lebesgue space \(L^\infty(\mathbb{R}^n)\), is \(WH_{X^{1/(1-\theta)}}(\mathbb{R}^n)\), where \(\theta\in (0,1)\). All these results are of wide applications. Particularly, when \(X:=M_q^p(\mathbb{R}^n)\) (the Morrey space), \(X:=L^{\vec{p}}(\mathbb{R}^n)\) (the mixed-norm Lebesgue space) and \(X:=(E_\Phi^q)_t(\mathbb{R}^n)\) (the Orlicz-slice space), all these results are even new; when \(X:=L_\omega^\Phi(\mathbb{R}^n)\) (the weighted Orlicz space), the result on the real interpolation is new and, when \(X:=L^{p(\cdot)}(\mathbb{R}^n)\) (the variable Lebesgue space) and \(X:=L_\omega^\Phi (\mathbb{R}^n)\), the Littlewood-Paley function characterizations of \(WH_X(\mathbb{R}^n)\) obtained in this article improves the existing results via weakening the assumptions on the Littlewood-Paley functions.Convergence of ergodic-martingale paraproducts.https://zbmath.org/1460.600302021-06-15T18:09:00+00:00"Kovač, Vjekoslav"https://zbmath.org/authors/?q=ai:kovac.vjekoslav"Stipčić, Mario"https://zbmath.org/authors/?q=ai:stipcic.marioSummary: In this note we introduce a sequence of bilinear operators that unify ergodic averages and backward martingales in a nontrivial way. We establish its convergence in a range of \(\text{L}^p\)-norms and leave its a.s.convergence as an open problem. This problem shares some similarities with a well-known unresolved conjecture on a.s. convergence of double ergodic averages with respect to two commuting transformations.Mapping properties of fundamental harmonic analysis operators in the exotic Bessel framework.https://zbmath.org/1460.420192021-06-15T18:09:00+00:00"Langowski, Bartosz"https://zbmath.org/authors/?q=ai:langowski.bartosz"Nowak, Adam"https://zbmath.org/authors/?q=ai:nowak.adamSummary: We prove sharp power-weighted \(L^p\), weak type and restricted weak type inequalities for the heat semigroup maximal operator and Riesz transforms associated with the Bessel operator \(B_\nu\) in the exotic range of the parameter \(- \infty < \nu < 1\). Moreover, in the same framework, we characterize basic mapping properties for other fundamental harmonic analysis operators, including the heat semigroup based vertical \(g\)-function and fractional integrals (Riesz potential operators).The \(A\)-integral and restricted complex Riesz transform.https://zbmath.org/1460.440052021-06-15T18:09:00+00:00"Aliev, R. A."https://zbmath.org/authors/?q=ai:aliev.rashid-a"Nebiyeva, Kh. I."https://zbmath.org/authors/?q=ai:nebiyeva.khanim-iSummary: In this paper, we prove that the restricted complex Riesz transform of a Lebesgue integrable function is \(A\)-integrable and we obtain an analogue of Riesz's equality.Monge-Ampère singular integral operators acting on Triebel-Lizorkin spaces.https://zbmath.org/1460.420142021-06-15T18:09:00+00:00"Cheng, Meifang"https://zbmath.org/authors/?q=ai:cheng.meifang"Lee, Ming-Yi"https://zbmath.org/authors/?q=ai:lee.ming-yi"Lin, Chin-Cheng"https://zbmath.org/authors/?q=ai:lin.chincheng"Qu, Meng"https://zbmath.org/authors/?q=ai:qu.mengSummary: We study the Triebel-Lizorkin spaces associated with sections which are closely related to the Monge-Ampère equations and show that Monge-Ampère singular integral operators are bounded on these Triebel-Lizorkin spaces.Restriction inequalities for the hyperbolic hyperboloid.https://zbmath.org/1460.420102021-06-15T18:09:00+00:00"Bruce, Benjamin Baker"https://zbmath.org/authors/?q=ai:bruce.benjamin-baker"Oliveira e. Silva, Diogo"https://zbmath.org/authors/?q=ai:oliveira-e-silva.diogo"Stovall, Betsy"https://zbmath.org/authors/?q=ai:stovall.betsySummary: In this article we establish new inequalities, both conditional and unconditional, for the restriction problem associated to the hyperbolic, or one-sheeted, hyperboloid in three dimensions, endowed with a Lorentz-invariant measure. These inequalities are unconditional (and optimal) in the bilinear range \(q>\frac{10}{3}\).Boundedness of differential transforms for one-sided fractional Poisson-type operator sequence.https://zbmath.org/1460.420132021-06-15T18:09:00+00:00"Chao, Zhang"https://zbmath.org/authors/?q=ai:chao.zhang"Ma, Tao"https://zbmath.org/authors/?q=ai:ma.tao"Torrea, José L."https://zbmath.org/authors/?q=ai:torrea.jose-luisSummary: Let \(\mathcal{P}_{\tau}^\alpha f\) be given by
\[
\mathcal{P}_{\tau}^\alpha f(t)=\frac{1}{4^\alpha\Gamma(\alpha)}\int_0^{+\infty}\frac{\tau^{2\alpha}e^{-{\tau^2}/(4s)}}{s^{1+\alpha}}f(t-s)\text{d}s,\,\tau >0,\, t\in\mathbb{R},\, 0<\alpha<1.
\]
It is known that the function \(U^\alpha(t,\tau)=\mathcal{P}^\alpha_\tau f(t)\) is a classical solution to the extension problem
\[
-D_{\text{left}}U^\alpha+\frac{1-2\alpha}{\tau}\,U^\alpha_\tau+U^\alpha_{\tau\tau}=0,\text{ in }\mathbb{R}\times (0,\infty)
\]
and
\[
\lim\limits_{\tau\rightarrow 0^+}\mathcal{P}_\tau^\alpha f(t)=f(t),\quad a.e. \text{ and in }L^p(\mathbb{R},w)\text{-norm},\,w\in A_p^-.
\]
In this paper, we analyze the convergence speed of a series related with \(\mathcal{P}_\tau^\alpha f\) by discussing the behavior of the family of operators
\[
T_N^\alpha f(t)=\sum\limits_{j=N_1}^{N_2} v_j(\mathcal{P}_{a_{j+1}}^\alpha f(t)-\mathcal{P}_{a_j}^\alpha f(t)),\quad N=(N_1,N_2)\in\mathbb{Z}^2\text{ with }N_1<N_2,
\]
where \(\{v_j\}_{j\in\mathbb{Z}}\) is a bounded number sequence, and \(\{a_j\}_{j\in\mathbb{Z}}\) is a \(\rho\)-lacunary sequence of positive numbers, that is, \(1<\rho\leq a_{j+1}/a_j\), for all \(j\in\mathbb{Z}\). We shall show the boundedness of the maximal operator
\[
T^*f(t)=\sup\limits_N\left|T_N^\alpha f(t)\right|,\quad t\in\mathbb{R},
\]
in the one-sided weighted Lebesgue spaces \(L^p(\mathbb{R},\omega)(\omega\in A_p^-)\), \(1<p<\infty\). As a consequence we infer the existence of the limit, in norm and almost everywhere, of the family \(T_N^\alpha f\) for functions in \(L^p(\mathbb{R},\omega)\). Results for \(L^1(\mathbb{R},\omega)(\omega\in A_1^-)\), \(L^\infty (\mathbb{R})\) and BMO\((\mathbb{R})\) are also obtained. It is also shown that the local size of \(T^*f\), for functions \(f\) having local support, is the same with the order of a singular integral. Moreover, if \(\{v_j\}_{j\in\mathbb{Z}}\in \ell^p(\mathbb{Z})\), we get an intermediate size between the local size of singular integrals and Hardy-Littlewood maximal operator.Sublinear operators on mixed-norm Hardy spaces with variable exponents.https://zbmath.org/1460.420372021-06-15T18:09:00+00:00"Ho, Kwok-Pun"https://zbmath.org/authors/?q=ai:ho.kwok-punSummary: In this paper, we define and study the mixed-norm Hardy spaces with variable exponents. We establish some general principles for the mapping properties of sublinear operators on the mixed-norm Hardy spaces with variable exponents. By using these principles, we obtain the mapping properties of the Calderón-Zygmund operators, the oscillatory singular integral operators, the multiplier operators, the Littlewood-Paley functions, the intrinsic square functions, the parametric Marcinkiewicz integrals and the maximal Bochner-Riesz means on the mixed-norm Hardy spaces with variable exponents.\(L^p\)-estimates for the heat semigroup on differential forms, and related problems.https://zbmath.org/1460.580152021-06-15T18:09:00+00:00"Magniez, Jocelyn"https://zbmath.org/authors/?q=ai:magniez.jocelyn"Ouhabaz, El Maati"https://zbmath.org/authors/?q=ai:ouhabaz.el-maatiFor a \(M^n\) complete Riemannian manifold and \(\Delta\) the (non-negative) Laplace-Beltrami operator, consider \((e^{-t\Delta})_{t\geq 0}\) the associated heat semigroup acting as a contraction semigroup on \(L^p(M)\) for all \(1\leq p \leq \infty\), and the semigroup is strongly continuous on \(L^p(M)\) for \(1\leq p < \infty\). Now instead of \(\Delta\), one may consider the Hodge-de Rham Laplacian \(\overrightarrow{\Delta}_k=d_k^*d_k+d_{k-1}d_{k-1}^*\) and the associated contraction semigroup \((e^{-t\Delta_k})_{t\geq 0}\) on \(L^2(\Lambda^kT^*M)\). Note that \(\overrightarrow{\Delta}_k\) is non-negative.
The article under review studies \(L^p\)-estimates of the semigroup \((e^{-t\Delta_k})_{t\geq 0}\). Apparently the precise estimate of the \(L^p\)-norm \(\|(e^{-t\Delta_k})_{t\geq 0}\|_{p-p}\) is not easy, and the article proves in Theorem 1.2 (i) such an estimate. To be more specific, consider the Bochner's formula \(\overrightarrow{\Delta}_k=\Delta^*\Delta+R_k\) where \(\Delta\) the Levi-Civita connection and \(R_k\) a symmetric section of \(\text{End}(\Lambda^k T^*M)\). Denote by \(R^\pm_k\) the positive and negative part of \(R_k\). The article, under an assumption of a volume-doubling property (Compare Equation (D) in p.3003), the Gaussian upper bound condition (Compare Equation (G) in p.3004), \(R_k^-\) in the enlarged Kato class \(\hat{K}\) (compare Definition 1.1), shows that \(\|(e^{-t\Delta_k})\|_{p-p}\leq C(t\log t)^{\frac{D}{4}(1-\frac{2}{p})}\) for large \(t\) where \(D\) is a homogeneous dimension appearing in the volume doubling property.
In addition, using an \(\epsilon\)-subcritical condition for \(R_k^-\) for \(\epsilon\in[0,1)\), the authors prove in Theorem 1.4 better estimates than that of Theorem 1.2. The article also provides in Proposition 1.5 a side result that under an additional assumption of uniform boundedness of \((e^{-t\Delta})_{t\geq 0}\) on \(L^p(\Lambda^1 T^*M)\) for some fixed \(p\), the Riesz transformation \(d\Delta^{-1/2}\) is bounded on \(L^r(M)\) for all \(r\in (1,\text{max}(p,p'))\).
Reviewer: Byungdo Park (Cheongju)Vector valued maximal Carleson type operators on the weighted Lorentz spaces.https://zbmath.org/1460.420162021-06-15T18:09:00+00:00"Duong, Dao Van"https://zbmath.org/authors/?q=ai:duong.dao-van"Dung, Kieu Huu"https://zbmath.org/authors/?q=ai:dung.kieu-huu"Nguyen Minh Chuong"https://zbmath.org/authors/?q=ai:nguyen-minh-chuong.Summary: In this paper, by using the idea of linearizing maximal operators originated by \textit{C. Fefferman} [Ann. Math. (2) 98, 551--571 (1973; Zbl 0268.42009)] and the \(TT^\ast\) method of Stein-Wainger [\textit{E. M. Stein} and \textit{S. Wainger}, Math. Res. Lett. 8, No. 5--6, 789--800 (2001; Zbl 0998.42007)], we establish a weighted inequality for vector valued maximal Carleson type operators with singular kernels proposed by \textit{K. F. Andersen} and \textit{R. T. John} [Stud. Math. 69, 19--31 (1980; Zbl 0448.42016)] on the weighted Lorentz spaces with vector-valued functions.Sparse bounds for maximally truncated oscillatory singular integrals.https://zbmath.org/1460.420182021-06-15T18:09:00+00:00"Krause, Ben"https://zbmath.org/authors/?q=ai:krause.ben"Lacey, Michael T."https://zbmath.org/authors/?q=ai:lacey.michael-tSummary: For a polynomial \(P(x,y)\), and any Calderón-Zygmund kernel \(K\), the operator below satisfies a \((1,{r})\) sparse bound, for \(1<r \leq 2:\)
\[
\sup_{\epsilon > 0} \left | \int_{|y|>\epsilon} f(x-y)e^{2\pi i P(x,y)}K(y)dy \right | .
\]
The implied bound depends upon \(P(x,y)\) only through the degree of \(P\). We derive from this a range of weighted inequalities, including weak type inequalities on \(L^1(w)\), which are new, even in the unweighted case. The unweighted weak type estimate, without maximal truncations, is due to \textit{S. Chanillo} and \textit{M. Christ} [Duke Math. J. 55, 141--155 (1987; Zbl 0667.42007)].Calderón-Zygmund operators on homogeneous product Lipschitz spaces.https://zbmath.org/1460.420232021-06-15T18:09:00+00:00"Zheng, Taotao"https://zbmath.org/authors/?q=ai:zheng.taotao"Chen, Jiecheng"https://zbmath.org/authors/?q=ai:chen.jiecheng"Dai, Jiawei"https://zbmath.org/authors/?q=ai:dai.jiawei"He, Shaoyong"https://zbmath.org/authors/?q=ai:he.shaoyong"Tao, Xiangxing"https://zbmath.org/authors/?q=ai:tao.xiangxingSummary: The purpose of this paper is to establish a necessary and sufficient condition for the boundedness of product Calderón-Zygmund singular integral operators introduced by Journé on the product Lipschitz spaces. The key idea used in this paper is to develop the Littlewood-Paley theory for the product spaces which includes the characterization of a special product Besov space and a density argument for the product Lipschitz spaces in the weak sense.On the composition of rough singular integral operators.https://zbmath.org/1460.420172021-06-15T18:09:00+00:00"Hu, Guoen"https://zbmath.org/authors/?q=ai:hu.guoen"Lai, Xudong"https://zbmath.org/authors/?q=ai:lai.xudong"Xue, Qingying"https://zbmath.org/authors/?q=ai:xue.qingyingSummary: In this paper, we investigate the behavior of the bounds of the composition for rough singular integral operators on the weighted space. More precisely, we obtain the quantitative weighted bounds of the composite operator for two singular integral operators with rough homogeneous kernels on \(L^p(\mathbb{R}^d,w)\), \(p\in(1,\infty)\), which is smaller than the product of the quantitative weighted bounds for these two rough singular integral operators. Moreover, at the endpoint \(p=1\), the \(L\log L\) weighted weak-type bound is also obtained, which has interests of its own in the theory of rough singular integral even in the unweighted case.On a \(k\)-fold beta integral formula.https://zbmath.org/1460.420222021-06-15T18:09:00+00:00"Wu, Di"https://zbmath.org/authors/?q=ai:wu.di"Shi, Zuoshunhua"https://zbmath.org/authors/?q=ai:shi.zuoshunhua"Nie, Xudong"https://zbmath.org/authors/?q=ai:nie.xudong"Yan, Dunyan"https://zbmath.org/authors/?q=ai:yan.dunyanSummary: In this paper, we investigate some necessary and sufficient conditions which ensure validity of a \(k\)-fold Beta integral formula
\[
\int_{\mathbb{R}^n}\prod\limits^k_{i=1}|x^i-t|^{-d_i}dt=C_{d_1,\dots,d_k}\prod\limits_{1\leq i<j\leq k}\biggl|x^i-x^j\biggr|^{-\alpha_{ij}}\tag{0.1}
\]
for any \(x^i\in\mathbb{R}^n\) and some nonzero real numbers \(d_i\) with \(i=1,2,\dots,k\). We establish that Eq. (0.1) holds if and only if
\[
\max\{d_1,d_2\}<n<d_1+d_2\text{ when }k=2\tag{0.2}
\]
and
\[
\max\{d_1,d_2,d_3\}<n,d_1+d_2+d_3=2n\text{ when }k=3.\tag{0.3}
\]
This yields a complete answer to the question raised by \textit{L. Grafakos} and \textit{C. Morpurgo} [Pac. J. Math. 191, No. 1, 85--94 (1999; Zbl )1006.42017)]. In addition, it turns out that formula (0.1) does not hold if \(k\geq 4\). For those \(k\), \(d_1, d_2, \ldots , d_k\) not satisfying (0.2) or (0.3), we prove that the real-valued integral \(\int_{\mathbb{R}^n}\prod\limits^k_{i=1}|x^i-t|^{-d_i}dt\) can be represented as a function of distances of consecutive differences of the sequence \(x^1, x^2,\dots, x^k\).BMO solvability and absolute continuity of caloric measure.https://zbmath.org/1460.420252021-06-15T18:09:00+00:00"Genschaw, Alyssa"https://zbmath.org/authors/?q=ai:genschaw.alyssa"Hofmann, Steve"https://zbmath.org/authors/?q=ai:hofmann.steveSummary: We show that BMO-solvability implies scale invariant quantitative absolute continuity (specifically, the weak-\(A_{\infty }\) property) of caloric measure with respect to surface measure, for an open set \(\Omega\subset\mathbb{R}^{n+1}\), assuming as a background hypothesis only that the essential boundary of \(\Omega\) satisfies an appropriate parabolic version of Ahlfors-David regularity, entailing some backwards in time thickness. Since the weak-\(A_{\infty}\) property of the caloric measure is equivalent to \(L^p\) solvability of the initial-Dirichlet problem, we may then deduce that BMO-solvability implies \(L^p\) solvability for some finite \(p\).Areas spanned by point configurations in the plane.https://zbmath.org/1460.520182021-06-15T18:09:00+00:00"Mcdonald, Alex"https://zbmath.org/authors/?q=ai:mcdonald.alexSummary: We consider an over-determined Falconer type problem on \((k+1)\)-point configurations in the plane using the group action framework introduced in [\textit{A. Greenleaf} et al., Rev. Mat. Iberoam. 31, No. 3, 799--810 (2015; Zbl 1329.52015)]. We define the area type of a \((k+1)\)-point configuration in the plane to be the vector in \(\mathbb{R}^{\binom{k+1}{2}}\) with entries given by the areas of parallelograms spanned by each pair of points in the configuration. We show that the space of all area types is \(2k-1\) dimensional, and prove that a compact set \(E\subset\mathbb{R}^d\) of sufficiently large Hausdorff dimension determines a positve measure set of area types.Higher-order Riesz transforms of Hermite operators on new Besov and Triebel-Lizorkin spaces.https://zbmath.org/1460.420342021-06-15T18:09:00+00:00"Bui, The Anh"https://zbmath.org/authors/?q=ai:bui.the-anh"Duong, Xuan Thinh"https://zbmath.org/authors/?q=ai:duong.xuan-thinhSummary: Consider the Hermite operator \(H=-\Delta +|x|^2\) on the Euclidean space \(\mathbb{R}^n\). The aim of this article is to prove the boundedness of higher-order Riesz transforms on appropriate Besov and Triebel-Lizorkin spaces. As an application, we prove certain regularity estimates of second-order elliptic equations in divergence form with the oscillator perturbations.Bilinear Hilbert transforms and (sub)bilinear maximal functions along convex curves.https://zbmath.org/1460.420202021-06-15T18:09:00+00:00"Li, Junfeng"https://zbmath.org/authors/?q=ai:li.junfeng"Yu, Haixia"https://zbmath.org/authors/?q=ai:yu.haixiaSummary: We determine the \(L^p(\mathbb{R})\times L^q(\mathbb{R})\rightarrow L^r(\mathbb{R})\) boundedness of the bilinear Hilbert transform \(H_{\gamma}(f,g)\) along a convex curve \(\gamma\):
\[ H_{\gamma}(f,g)(x):=\text{p.v.}\int_{-\infty}^{\infty}f(x-t)g(x-\gamma(t)) \frac{\operatorname{d}t}{t},\]
where \(p\), \(q\), and \(r\) satisfy \(1/p+1/q =1/r\), and \(r>\frac{1}{2}\), \(p>1\), and \(q>1\). Moreover, the same \(L^p(\mathbb{R})\times L^q(\mathbb{R})\rightarrow L^r(\mathbb{R})\) boundedness property holds for the corresponding (sub)bilinear maximal function \(M_{\gamma}(f,g)\) along a convex curve
\[\gamma M_{\gamma}(f,g)(x):=\sup_{\varepsilon>0}\frac{1}{2\varepsilon}\int_{-\varepsilon}^{\varepsilon} |f(x-t)g(x-\gamma(t))|\operatorname{d}t.\]On a precise scaling to Caffarelli-Kohn-Nirenberg inequality.https://zbmath.org/1460.420122021-06-15T18:09:00+00:00"Bazan, Aldo"https://zbmath.org/authors/?q=ai:bazan.aldo"Neves, Wladimir"https://zbmath.org/authors/?q=ai:neves.wladimirSummary: We analyze the general form of Caffarelli-Kohn-Nirenberg inequality. Due to a new introduced parameter, this inequality presents two distinguishable ranges. One of them, the inequality is shown to be the interpolation between Hardy and weighted Sobolev inequalities. The other range, which is no more an interpolation, the positive constant in the inequality is not necessarily bounded for all value of the parameters. In both cases, the precise value of the constants were given.\(L^2\) boundedness for commutators of fractional differential type Marcinkiewicz integral with rough variable kernel and BMO Sobolev spaces.https://zbmath.org/1460.420152021-06-15T18:09:00+00:00"Chen, Yanping"https://zbmath.org/authors/?q=ai:chen.yanping.1"Ding, Yong"https://zbmath.org/authors/?q=ai:ding.yong"Zhu, Kai"https://zbmath.org/authors/?q=ai:zhu.kaiSummary: In this paper, for \(0< \gamma< 1\) and \(b\in I_{\gamma}(\text{BMO})\), the authors give the \(L^2(\mathbb{R}^n)\) boundedness of \(\mu_{\gamma;b} \), the commutator of a fractional differential type Marcinkiewicz integral with rough variable kernel, which is an extension of some known results.Weighted and unweighted Solyanik estimates for the multilinear strong maximal function.https://zbmath.org/1460.420292021-06-15T18:09:00+00:00"Qin, Moyan"https://zbmath.org/authors/?q=ai:qin.moyan"Xue, Qingying"https://zbmath.org/authors/?q=ai:xue.qingyingSummary: Let \(\omega\) be a weight in \(A^\ast_\infty\) and let \(\mathcal{M}^m_n(\vec f)\) be the multilinear strong maximal function of \(\vec f=(f_1,\dots,f_m)\), where \(f_1,\dots,f_m\) are functions on \(\mathbb{R}^n\). In this paper, we consider the asymptotic estimates for the distribution functions of \(\mathcal{M}^m_n\). We show that, for \(\lambda\in (0,1)\), if \(\lambda\rightarrow 1^-\), then the multilinear Tauberian constant \(\mathcal{C}^m_n\) and the weighted Tauberian constant \(\mathcal{C}^m_{n,\omega}\) associated with \(\mathcal{M}^m_n\) enjoy the properties that
\[
\mathcal{C}^m_n(\lambda)-1\simeq m(1-\lambda)^{\frac{1}{n}}\text{ and }\mathcal{C}^m_{n,\omega}(\lambda)-1\lesssim m(1-\lambda)^{\left(4n[\omega ]_{A^\ast_\infty}\right)^{-1}}.
\]