We introduce a new type of variable exponent function spaces $M\dot K^{\alpha (\cdot ),\lambda }_{q,p(\cdot )}(\mathbb R^n)$ of Morrey-Herz type where the two main indices are variable exponents, and give some propositions of the introduced spaces. Under the assumption that the exponents $\alpha $ and $p$ are subject to the log-decay continuity both at the origin and at infinity, we prove the boundedness of a wide class of sublinear operators satisfying a proper size condition which include maximal, potential and Calderón-Zygmund operators and their commutators of BMO function on these Morrey-Herz type spaces by applying the properties of variable exponent and BMO norms.
Let \Omega \in L^{s}\left ( S^{n-1} \right ) for s\geqslant 1 be a homogeneous function of degree zero and b a BMO function. The commutator generated by the Marcinkiewicz integral μΩ and b is defined by \left[ {b,{\mu _\Omega }} \right](f)(x) = {\left( {\int_0^\infty {{{\left| {\int_{\left| {x - y} \right| \leqslant t} {\frac{{\Omega (x - y)}}{{{{\left| {x - y} \right|}^{n - 1}}}}\left[ {b(x) - b(y)} \right]f(y){\text{d}}y} } \right|}^2}\frac{{{\text{d}}t}}{{{t^3}}}} } \right)^{1/2}}. In this paper, the author proves the \left (L^{p\left ( \cdot \right )}\left ( \mathbb{R}^{n} \right ),L^{p\left ( \cdot \right )}\left ( \mathbb{R}^{n} \right ) \right )-boundedness of the Marcinkiewicz integral operator μΩ and its commutator [b, μΩ ] when p(·) satisfies some conditions. Moreover, the author obtains the corresponding result about μΩ and [b, μΩ ] on Herz spaces with variable exponent., Hongbin Wang., and Obsahuje seznam literatury
In this paper, the boundedness of a large class of sublinear commutator operators $T_{b}$ generated by a Calderón-Zygmund type operator on a generalized weighted Morrey spaces $M_{p,\varphi }(w)$ with the weight function $w$ belonging to Muckenhoupt's class $A_{p}$ is studied. When $1<p<\infty $ and $b \in {\rm BMO}$, sufficient conditions on the pair $(\varphi _1,\varphi _2)$ which ensure the boundedness of the operator $T_{b}$ from $M_{p,\varphi _1}(w)$ to $M_{p,\varphi _2}(w)$ are found. In all cases the conditions for the boundedness of $T_{b}$ are given in terms of Zygmund-type integral inequalities on $(\varphi _1,\varphi _2)$, which do not require any assumption on monotonicity of $\varphi _1(x,r)$, $\varphi _2(x,r)$ in $r$. Then these results are applied to several particular operators such as the pseudo-differential operators, Littlewood-Paley operator, Marcinkiewicz operator and Bochner-Riesz operator.
Let $M_{\beta }$ be the fractional maximal function. The commutator generated by $M_{\beta }$ and a suitable function $b$ is defined by $[M_{\beta },b]f = M_{\beta }(bf)-bM_{\beta }(f)$. Denote by $\mathscr {P}(\mathbb R^{n})$ the set of all measurable functions $p(\cdot )\colon \mathbb R^{n}\to [1,\infty )$ such that $$ 1< p_{-}:=\mathop {\rm ess inf}_{x\in \mathbb R^n}p(x) \quad \text {and}\quad p_{+}:=\mathop {\rm ess sup}_{x\in \mathbb R^n}p(x)<\infty , $$ and by $\mathscr {B}(\mathbb R^{n})$ the set of all $p(\cdot ) \in \mathscr {P}(\mathbb R^{n})$ such that the Hardy-Littlewood maximal function $M$ is bounded on $L^{p(\cdot )}(\mathbb R^{n})$. In this paper, the authors give some characterizations of $b$ for which $[M_{\beta },b]$ is bounded from $L^{p(\cdot )}(\mathbb R ^{n})$ into $L^{q(\cdot )}(\mathbb R^{n})$, when $p(\cdot )\in \mathscr {P}(\mathbb R^{n})$, $0<{\beta }<n/p_{+}$ and $1/q(\cdot )=1/p(\cdot )-\beta /n$ with $q(\cdot )(n-\beta )/n \in \mathscr {B}(\mathbb R^{n})$.
Let $b_1, b_2 \in {\rm BMO}(\mathbb {R}^n)$ and $T_{\sigma }$ be a bilinear Fourier multiplier operator with associated multiplier $\sigma $ satisfying the Sobolev regularity that $\sup _{\kappa \in \mathbb {Z}} \|\sigma _{\kappa }\| _{W^{s_1,s_2}(\mathbb {R}^{2n})}<\infty $ for some $s_1,s_2\in (n/2,n]$. In this paper, the behavior on $L^{p_1}(\mathbb {R}^n)\times L^{p_2}(\mathbb {R}^n)$ $(p_1,p_2\in (1,\infty ))$, on $H^1(\mathbb {R}^n)\times L^{p_2}(\mathbb {R}^n)$ $(p_2\in [2,\infty ))$, and on $H^1(\mathbb {R}^n)\times H^1(\mathbb {R}^n)$, is considered for the commutator $T_{{\sigma }, \vec {b}} $ defined by $$ \begin {aligned} T_{\sigma ,\vec {b}} (f_1,f_2) (x)=&b_1(x)T_{\sigma }(f_1, f_2)(x)-T_{\sigma }(b_1f_1, f_2)(x) &+ b_2(x)T_{\sigma }(f_1, f_2)(x)-T_{\sigma }(f_1, b_2f_2)(x) . \end {aligned} $$ By kernel estimates of the bilinear Fourier multiplier operators and employing some techniques in the theory of bilinear singular integral operators, it is proved that these mapping properties are very similar to those of the bilinear Fourier multiplier operator which were established by Miyachi and Tomita.
Let L1 = −Δ + V be a Schrödinger operator and let L2 = (−Δ)2 + V2 be a Schrödinger type operator on \mathbb{R}^{n}\left ( n\geqslant 5 \right ) where V≠ 0 is a nonnegative potential belonging to certain reverse Hölder class Bs for s\geqslant n/2. The Hardy type space H_{L2}^{1} is defined in terms of the maximal function with respect to the semigroup \left\{ {{e^{ - t{L_2}}}} \right\} and it is identical to the Hardy space H_{L2}^{1} established by Dziubański and Zienkiewicz. In this article, we prove the Lp-boundedness of the commutator Rb = bRf - R(bf) generated by the Riesz transform R = {\nabla ^2}L_2^{ - 1/2} , where b \in BM{O_\theta }(\varrho ) , which is larger than the space BMO\left (\mathbb{R}^{n} \right ). Moreover, we prove that Rb is bounded from the Hardy space H_{L2}^{1} into weak L_{weak}^1 (\mathbb{R}^n )., Yu Liu, Jing Zhang, Jie-Lai Sheng, Li-Juan Wang., and Obsahuje seznam literatury
We establish several inequalities for the spectral radius of a positive commutator of positive operators in a Banach space ordered by a normal and generating cone. The main purpose of this paper is to show that in order to prove the quasi-nilpotency of the commutator we do not have to impose any compactness condition on the operators under consideration. In this way we give a partial answer to the open problem posed in the paper by J. Bračič, R. Drnovšek, Y. B. Farforovskaya, E. L. Rabkin, J. Zemánek (2010). Inequalities involving an arbitrary commutator and a generalized commutator are also discussed.
We give a necessary and a sufficient condition for a subset S of a locally convex Waelbroeck algebra A to have a non-void left joint spectrum σl (S). In particular, for a Lie subalgebra L ⊂ A we have σl (L) ≠ ∅ if and only if [L, L] generates in A a proper left ideal. We also obtain a version of the spectral mapping formula for a modified left joint spectrum. Analogous theorems for the right joint spectrum and the Harte spectrum are also valid.
The boundednees of multilinear commutators of Calderón-Zygmund singular integrals on Lebesgue spaces with variable exponent is obtained. The multilinear commutators of generalized Hardy-Littlewood maximal operator are also considered.
Let $m$ be a positive integer, $0<\alpha <mn$, $\vec {b}=(b_{1},\cdots ,b_{m})\in {\rm BMO}^m$. We give sufficient conditions on weights for the commutators of multilinear fractional integral operators $\Cal {I}^{\vec {b}}_{\alpha }$ to satisfy a weighted endpoint inequality which extends the result in D. Cruz-Uribe, A. Fiorenza: Weighted endpoint estimates for commutators of fractional integrals, Czech. Math. J. 57 (2007), 153–160. We also give a weighted strong type inequality which improves the result in X. Chen, Q. Xue: Weighted estimates for a class of multilinear fractional type operators, J. Math. Anal. Appl., 362, (2010), 355–373.