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.
In the paper we find conditions on the pair (ω1, ω2) which ensure the boundedness of the maximal operator and the Calderón-Zygmund singular integral operators from one generalized Morrey space Mp,ω1 to another Mp,ω2 , 1 < p < ∞, and from the space M1,ω1 to the weak space WM1,ω2 . As applications, we get some estimates for uniformly elliptic operators on generalized Morrey spaces.
In this paper we prove that the maximal operator $$\tilde {\sigma }^{\kappa ,*}f:=\sup _{n\in {\mathbb P}}\frac {|{\sigma }_n^\kappa f|}{\log ^{2}(n+1)},$$ where ${\sigma }_n^\kappa f$ is the $n$-th Fejér mean of the Walsh-Kaczmarz-Fourier series, is bounded from the Hardy space $H_{1/2}( G) $ to the space $L_{1/2}( G).$.
We prove and discuss some new (Hp,Lp)-type inequalities of weighted maximal operators of Vilenkin-Nörlund means with non-increasing coefficients {q_{k}:k\geqslant 0}. These results are the best possible in a special sense. As applications, some well-known as well as new results are pointed out in the theory of strong convergence of such Vilenkin-Nörlund means. To fulfil our main aims we also prove some new estimates of independent interest for the kernels of these summability results. In the special cases of general Nörlund means tn with non-increasing coefficients analogous results can be obtained for Fejér and Cesàro means by choosing the generating sequence {q_{k}:k\geqslant 0} in an appropriate way., István Blahota, Lars-Erik Persson, Giorgi Tephnadze., and Obsahuje seznam literatury
Our aim in this paper is to deal with the boundedness of the Hardy-Littlewood maximal operator on grand Morrey spaces of variable exponents over non-doubling measure spaces. As an application of the boundedness of the maximal operator, we establish Sobolev's inequality for Riesz potentials of functions in grand Morrey spaces of variable exponents over non-doubling measure spaces. We are also concerned with Trudinger's inequality and the continuity for Riesz potentials.
The classical Hardy-Littlewood maximal operator is bounded not only on the classical Lebesgue spaces Lp(Rd) (in the case p > 1), but (in the case when 1/p(·) is log-Hölder continuous and p- = inf{p(x): x\in Rd > 1) on the variable Lebesgue spaces Lp(·)(Rd), too. Furthermore, the classical Hardy-Littlewood maximal operator is of weak-type (1, 1). In the present note we generalize Besicovitch’s covering theorem for the so-called γ-rectangles. We introduce a general maximal operator Msγδ, and with the help of generalized Φ-functions, the strong- and weak-type inequalities will be proved for this maximal operator. Namely, if the exponent function 1/p(·) is log-Hölder continuous and p- ≥ s, where 1 ≤ s ≤ ∞ is arbitrary (or
p- ≥ s), then the maximal operator Msγδ is bounded on the space Lp(·)(Rd) (or the maximal operator is of weak-type (p(·), p(·)))., Kristóf Szarvas, Ferenc Weisz., and Obsahuje seznam literatury