Zygmund-type Integral Inequalities Research Articles

Generalizations of some Zygmund-type integral inequalities for polar derivatives of a complex polynomial

We prove some results for algebraic polynomials in the complex plane that relate the L-norm of the polar derivative of a complex polynomial and the polynomial under some conditions. The obtained results include several interesting generalizations of some Zygmund-type integral inequalities for polynomials and derive polar derivative analogues of some classical Bernsteintype inequalities for the sup-norms on the unit disk as well.

Generalizations of Zygmund-type integral inequalities for the polar derivative of a complex polynomial

Some Zygmund-type integral inequalities for the polar derivatives of complex polynomials, inspired by the classical Bernstein-type inequalities that relate the uniform norms of polynomials and their derivatives on the unit circle, are investigated. The obtained results sharpen as well as generalize some already known L^{delta }-estimates between polynomials and their polar derivatives.

Boundedness of the potential operators and their commutators in the local “complementary” generalized variable exponent Morrey spaces on unbounded sets

In this paper we prove a Sobolev–Spanne type $${\,^{^{\complement }}\!\mathcal M}_{\{x_0\}}^{p(\cdot ),\omega } (\varOmega )\rightarrow {\,^{^{\complement }}\!\mathcal M}_{\{x_0\}}^{q(\cdot ),\omega } (\varOmega )$$-theorem for the potential operators $$I^{\alpha }$$, where $${\,^{^{\complement }}\!\mathcal M}_{\{x_0\}}^{p(\cdot ),\omega }(\varOmega )$$ is local “complementary” generalized Morrey spaces with variable exponent p(x), $$\omega (r)$$ is a general function defining the Morrey-type norm and $$\varOmega $$ is an open unbounded subset of $${{\mathbb {R}}^n}$$. In addition, we prove the boundedness of the commutator of potential operators $$[b,I^{\alpha }]$$ in these spaces. In all cases the conditions for the boundedness are given in terms of Zygmund-type integral inequalities on $$\omega (x,r)$$, which do not assume any assumption on monotonicity of $$\omega (x,r)$$ in r.

Boundedness of some sublinear operators and their commutators on generalized local Morrey spaces

ABSTRACTIn this paper we prove the boundedness of certain sublinear operators , , generated by fractional integral operators with rough kernels , , from one generalized local Morrey space to another , , , and from the space to the weak space , , . In the case b belongs to the local Campanato space and is a linear operator, we find the sufficient conditions on the pair which ensures the boundedness of the commutator operators from to , , , , . In all cases the conditions for the boundedness of are given in terms of Zygmund-type integral inequalities on , which do not assume any assumption on monotonicity of in r.

Intrinsic Square Functions on Vanishing Generalized Orlicz-Morrey Spaces

We study the boundedness of intrinsic square functions and their commutators on vanishing generalized Orlicz-Morrey spaces. In all the cases the conditions for the boundedness are given in terms of Zygmund-type integral inequalities without assuming any monotonicity property.

Boundedness of Fractional Oscillatory Integral Operators and Their Commutators in Vanishing Generalized Weighted Morrey Spaces

In this article, we give the boundedness conditions in terms of Zygmund-type integral inequalities for oscillatory integral operators and fractional oscillatory integral operators on the vanishing generalized weighted Morrey spaces. Moreover, we investigate corresponding commutators.

Boundedness of rough fractional multilinear integral operators on generalized Morrey spaces

We consider the boundedness of fractional multilinear integral operators with rough kernels $T_{\Omega,\alpha}^{A,m}$ on the generalized Morrey spaces $M_{p,\varphi}$ . We find the sufficient conditions on the pair $(\varphi_{1},\varphi _{2})$ , which ensures the boundedness of the operators $T_{\Omega,\alpha}^{A,m}$ from $M_{p,\varphi_{1}}$ to $M_{p,\varphi_{2}}$ for $1< p<\infty$ . In all cases the conditions for the boundedness of the operator $T_{\Omega,\alpha}^{A,m}$ is given in terms of Zygmund-type integral inequalities on $(\varphi_{1},\varphi_{2})$ , which do not make any assumption on the monotonicity of $\varphi_{1}(x,r)$ , $\varphi_{2}(x,r)$ in r.

Maximal Operator in Variable Exponent Generalized Morrey Spaces on Quasi-metric Measure Space

We consider generalized Morrey spaces \({\mathcal{L}^{p(\cdot),\varphi(\cdot)}( X )}\) on quasi-metric measure spaces \({X,d,\mu}\), in general unbounded, with variable exponent p(x) and a general function \({\varphi(x,r)}\) defining the Morrey-type norm. No linear structure of the underlying space X is assumed. The admission of unbounded X generates problems known in variable exponent analysis. We prove the boundedness results for maximal operator known earlier only for the case of bounded sets X. The conditions for the boundedness are given in terms of the so called supremal inequalities imposed on the function \({\varphi(x,r)}\), which are weaker than Zygmund-type integral inequalities often used for characterization of admissible functions \({\varphi}\). Our conditions do not suppose any assumption on monotonicity of \({\varphi(x,r)}\) in r.

Sublinear operators with rough kernel generated by Calderón-Zygmund operators and their commutators on generalized local Morrey spaces

In this paper, we will study the boundedness of a large class of sublinear operators with rough kernel $T_{\Omega}$ on the generalized local Morrey spaces $LM_{p,\varphi}^{\{x_{0}\}}$ , for $s' \le p$ , $p \neq1$ or $p < s$ , where $\Omega\in L_{s}(S^{n-1})$ with $s>1$ are homogeneous of degree zero. In the case when $b \in LC_{p,\lambda}^{\{x_{0}\}}$ is a local Campanato spaces, $1< p<\infty$ , and $T_{\Omega,b}$ be is a sublinear commutator operator, we find the sufficient conditions on the pair $(\varphi_{1},\varphi_{2})$ which ensures the boundedness of the operator $T_{\Omega,b}$ from one generalized local Morrey space $LM_{p,\varphi_{1}}^{\{x_{0}\}}$ to another $LM_{p,\varphi_{2}}^{\{x_{0}\}}$ . In all cases the conditions for the boundedness of $T_{\Omega}$ are given in terms of Zygmund-type integral inequalities on $(\varphi_{1},\varphi_{2})$ , which do not make any assumptions on the monotonicity of $\varphi_{1}$ , $\varphi_{2}$ in r. Conditions of these theorems are satisfied by many important operators in analysis, in particular pseudo-differential operators, Littlewood-Paley operators, Marcinkiewicz operators, and Bochner-Riesz operators.

$$(\Phi ,\Psi )$$ ( Φ , Ψ ) -admissible potential operators and their commutators on vanishing Orlicz-Morrey spaces

We study the boundedness of $$(\Phi ,\Psi )$$ -admissible potential operators and their commutators on vanishing generalized Orlicz-Morrey spaces $$VM_{\Phi ,\varphi }(\mathbb {R}^n)$$ including their weak versions. These conditions are satisfied by most of the operators in harmonic analysis, such as the Riesz potential, fractional maximal operator and so on. In all the cases the conditions for the boundedness are given in terms of Zygmund-type integral inequalities involving the Young functions $$\Phi $$ , $$\Psi $$ and the function $$\varphi (x,r)$$ defining the space, without assuming any monotonicity property of $$\varphi (x,r)$$ on $$r$$ .

Generalized Fractional Integral Operators on Generalized Local Morrey Spaces

We study the continuity properties of the generalized fractional integral operatorIρon the generalized local Morrey spacesLMp,φ{x0}and generalized Morrey spacesMp,φ. We find conditions on the triple(φ1,φ2,ρ)which ensure the Spanne-type boundedness ofIρfrom one generalized local Morrey spaceLMp,φ1{x0}to anotherLMq,φ2{x0},1&lt;p&lt;q&lt;∞, and fromLM1,φ1{x0}to the weak spaceWLMq,φ2{x0},1&lt;q&lt;∞. We also find conditions on the pair(φ,ρ)which ensure the Adams-type boundedness ofIρfromMp,φ1/ptoMq,φ1/qfor1&lt;p&lt;q&lt;∞and fromM1,φtoWMq,φ1/qfor1&lt;q&lt;∞. In all cases the conditions for the boundedness ofIρare given in terms of Zygmund-type integral inequalities on(φ1,φ2,ρ)and(φ,ρ), which do not assume any assumption on monotonicity ofφ1(x,r),φ2(x,r), andφ(x,r)inr.

Boundedness of intrinsic square functions and their commutators on generalized weighted Orlicz--Morrey spaces

We shall investigate the boundedness of the intrinsic square functions and their commutators on generalized weighted Orlicz-Morrey spaces $M^{\Phi,\varphi}_{w}({\mathbb R}^n)$. In all the cases, the conditions for the boundedness are given in terms of Zygmund-type integral inequalities on weights $\varphi$ without assuming any monotonicity property of $\varphi(x,\cdot)$ with $x$ fixed.

Multilinear commutators of vector-valued intrinsic square functions on vector-valued generalized weighted Morrey spaces

In this paper, we will obtain the strong type and weak type estimates for vector-valued analogs of intrinsic square functions in the generalized weighted Morrey spaces M w p , φ ( l 2 ). We study the boundedness of intrinsic square functions including the Lusin area integral, the Littlewood-Paley g-function and g λ ∗ -function, and their multilinear commutators on vector-valued generalized weighted Morrey spaces M w p , φ ( l 2 ). In all the cases the conditions for the boundedness are given either in terms of Zygmund-type integral inequalities on φ(x,r) without assuming any monotonicity property of φ(x,r) on r.MSC:42B25, 42B35.

Commutators of sublinear operators generated by Calderón-Zygmund operator on generalized weighted Morrey spaces

In this paper, the boundedness of a large class of sublinear commutator operators Tb generated by a Calderon-Zygmund type operator on a generalized weighted Morrey spaces \({M_{p,\varphi }}(w)\) with the weight function w belonging to Muckenhoupt’s class Ap is studied. When 1 < p < ∞ and b ∈ BMO, sufficient conditions on the pair (φ1, φ2) which ensure the boundedness of the operator Tb from \({M_{p,\varphi 1}}(w)\) to \({M_{p,\varphi 2}}(w)\) are found. In all cases the conditions for the boundedness of Tb are given in terms of Zygmund-type integral inequalities on (φ1, φ2), which do not require any assumption on monotonicity of φ1(x, r), φ2(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.

Φ-Admissible singular operators and their commutators on vanishing generalized Orlicz-Morrey spaces

We study the boundedness of Φ-admissible singular operators and their commutators on vanishing generalized Orlicz-Morrey spaces including their weak versions. These conditions are satisfied by most of the operators in harmonic analysis, such as the Hardy-Littlewood maximal operator, the Calderon-Zygmund singular integral operator and so on. In all the cases the conditions for the boundedness are given in terms of Zygmund-type integral inequalities on weights without assuming any monotonicity property of on r. MSC:42B20, 42B25, 42B35, 46E30.

On the Riesz Potential and Its Commutators on Generalized Orlicz-Morrey Spaces

We consider generalized Orlicz-Morrey spacesMΦ,φ(ℝn)including their weak versionsWMΦ,φ(ℝn). In these spaces we prove the boundedness of the Riesz potential fromMΦ,φ1(ℝn)toMΨ,φ2(ℝn)and fromMΦ,φ1(ℝn)toWMΨ,φ2(ℝn). As applications of those results, the boundedness of the commutators of the Riesz potential on generalized Orlicz-Morrey space is also obtained. In all the cases the conditions for the boundedness are given either in terms of Zygmund-type integral inequalities on(φ1,φ2), which do not assume any assumption on monotonicity ofφ1(x,r),φ2(x,r)inr.

Multilinear Commutators of Calderón-Zygmund Operator on Generalized Weighted Morrey Spaces

The boundedness of multilinear commutators of Calderón-Zygmund operatorTb→on generalized weighted Morrey spacesMp,φ(w)with the weight functionwbelonging to Muckenhoupt's classApis studied. When1&lt;p&lt;∞andb→=(b1,…,bm),bi∈BMO,i=1,…,m, the sufficient conditions on the pair(φ1,φ2)which ensure the boundedness of the operatorTb→fromMp,φ1(w)toMp,φ2(w)are found. In all cases the conditions for the boundedness ofTb→are given in terms of Zygmund-type integral inequalities on(φ1,φ2), which do not assume any assumption on monotonicity ofφ1(x,r), φ2(x,r)inr.

Riesz potential in generalized Morrey spaces on the Heisenberg group

We consider the Riesz potential operator , on the Heisenberg group $$ {{\mathbb{H}}_n} $$ in generalized Morrey spaces $$ {M_{{p,\varphi }}}\left( {{{\mathbb{H}}_n}} \right) $$ and find conditions for the boundedness of as an operator from $$ {M_{{p,\varphi 1}}}\left( {{{\mathbb{H}}_n}} \right) $$ to $$ {M_{{p,\varphi 2}}}\left( {{{\mathbb{H}}_n}} \right) $$ , 1 < p < ∞, and from $$ {M_{{1,\varphi 1}}}\left( {{{\mathbb{H}}_n}} \right) $$ to a weak Morrey space $$ W{M_{{1,\varphi 2}}}\left( {{{\mathbb{H}}_n}} \right) $$ . The boundedness conditions are formulated it terms of Zygmund type integral inequalities. Based on the properties of the fundamental solution of the sub-Laplacian on $$ {{\mathbb{H}}_n} $$ , we prove two Sobolev–Stein embedding theorems for generalized Morrey and Besov–Morrey spaces. Bibliography: 40 titles.

Weighted Hardy-Type Inequalities in Variable Exponent Morrey-Type Spaces

We study thep·→q·boundedness of weighted multidimensional Hardy-type operatorsHwα·andℋwα·of variable orderαx, with radial weightwx, from a variable exponent locally generalized Morrey spaceℒp·,φ·ℝn,wto anotherℒq·,ψ·ℝn,w. The exponents are assumed to satisfy the decay condition at the origin and infinity. We construct certain functions, defined byp,α, andφ, the belongness of which to the resulting spaceℒq·,ψ·ℝn,wis sufficient for such a boundedness. Under additional assumptions onφ/w, this condition is also necessary. We also give the boundedness conditions in terms of Zygmund-type integral inequalities for the functionsφandφ/w.

Maximal, potential and singular operators in vanishing generalized Morrey spaces

We introduce vanishing generalized Morrey spaces $${V\mathcal{L}^{p,\varphi}_\Pi (\Omega), \Omega \subseteq \mathbb{R}^n}$$ with a general function $${\varphi(x, r)}$$ defining the Morrey-type norm. Here $${\Pi \subseteq \Omega}$$ is an arbitrary subset in Ω including the extremal cases $${\Pi = \{x_0\}, x_0 \in \Omega}$$ and Â? = Ω, which allows to unify vanishing local and global Morrey spaces. In the spaces $${V\mathcal{L}^{p,\varphi}_\Pi (\mathbb{R}^n)}$$ we prove the boundedness of a class of sublinear singular operators, which includes Hardy-Littlewood maximal operator and Calderon-Zygmund singular operators with standard kernel. We also prove a Sobolev-Spanne type $${V\mathcal{L}^{p,\varphi}_\Pi (\mathbb{R}^n) \rightarrow V\mathcal{L}^{q,\varphi^\frac{q}{p}}_\Pi (\mathbb{R}^n)}$$ -theorem for the potential operator I Â? . The conditions for the boundedness are given in terms of Zygmund-type integral inequalities on $${\varphi(x, r)}$$ . No monotonicity type condition is imposed on $${\varphi(x, r)}$$ . In case $${\varphi}$$ has quasi- monotone properties, as a consequence of the main results, the conditions of the boundedness are also given in terms of the Matuszeska-Orlicz indices of the function $${\varphi}$$ . The proofs are based on pointwise estimates of the modulars defining the vanishing spaces