Most Hardy type inequalities concern boundedness of the Hardy type operators in Lebesgue spaces. In this paper we prove some new multi-dimensional Hardy type inequalities in Hölder spaces.

We prove theorems on the boundedness of commutators [a,Hwα] of the weighted multidimensional Hardy operator Hwα:=wHα1w from a generalized local Morrey space L^{p , φ ; 0}(Rⁿ) to local or global space L^{q , ψ}(Rⁿ). The main impacts of these theorems are1.the use of CMO_s-class of coefficients a for the commutators;2.the general setting when the function φ defining the Morrey space and the weight w are independent of one another and the weight w is not assumed to be in A_p;3.recovering the Sobolev–Adams exponent q instead of Sobolev–Spanne type exponent in the case of classical Morrey spaces4.boundedness from local to global Morrey spaces;5.the obtained estimates contain the parameter s> 1 which may be arbitrarily chosen. Its choice regulates in fact an equilibrium between assumptions on the coefficient a and the characteristics of the space. The obtained results are new also in non-weighted case

We consider the general Hardy type operator inline image where inline image is a positive and measurable kernel. To characterize the weights u and v so that inline image is still an open problem for any parameters p and q. However, for special cases the solution is known for some parameters p and q. In this paper the current status of this problem is described and discussed mainly for the case inline image In particular, some new scales of characterizations in classical situations are described, some new proofs and results are given and open questions are raised.

We introduce a version of weighted anisotropic Morrey spaces and anisotropic Hardy operators. We find conditions for boundedness of these operators in such spaces. We also reveal the role of these operators in solving some classes of degenerate hyperbolic partial differential equations.

We study weighted generalized Hardy and fractional operators acting from generalized Morrey spaces L^p,φ,(ⁿ) into Orlicz-Morrey spaces L^Φ,φ,(ⁿ). We deal with radial quasi-monotone weights and assumptions imposed on weights are given in terms of Zigmund-type integral conditions. We find conditions on φ,Φ, the weight w and the kernel of the fractional operator, which insures such a boundedness. We prove some pointwise estimates for weighted generalized fractional operators via generalized Hardy operators, which allow to obtain the weighted boundedness for fractional operators from those for Hardy operators. We provide also some easy to check numerical inequalities to verify the obtained conditions.

We prove the boundedness of Potential operator in weighted generalized Morrey space in terms of Matuszewska-Orlicz indices of weights and apply this result to the Hemholtz equation in r³ with a free term in such a space. We also give a short overview of some typical situations when Potential type operators arise when solving PDEs.

In this second edition, all chapters in the first edition have been updated with new information. Moreover, a new chapter contains new and complementary information concerning: (a) a convexity approach to prove and explain Hardy-type inequalities; (b) sharp constants; (c) scales of inequalities to characterize Hardy-type inequalities; (d) Hardy-type inequalities in other function spaces; and (e) a number of new open questions.

We find conditions for the weighted boundedness of a general class of multidimensional singular integral operators in generalized Morrey spaces L p,φ (R n ,w), defined by a function φ(x,r) and radial type weight w(|x−x 0 |),x 0 ∈R n . These conditions are given in terms of inclusion into L p,φ (R n ,w), of a certain integral constructions defined by φ and w. In the case of φ=φ(r) we also provide easy to check sufficient conditions for that in terms of indices of φ and w.

We prove the boundedness of the Riesz fractional integration operator from a generalized Morrey space L-p,L-phi to a certain Orlicz-Morrey space L-Phi,L-phi which covers the Adams result for Morrey spaces. We also give a generalization to the case of weighted Riesz fractional integration operators for some class of weights.

We obtain two-weighted estimates for the Hardy type operators fromlocal generalized Morrey spaces Lp,ϕloc (X,w1) defined on an arbitrary underlyingquasi-metric measure space (X, μ, ) with the growth condition, toLq,ψloc (X,w2), where w1 = w1[(x, x0)], x0 ∈ X is a weight of radial type,while w2 = w2(x) may be an arbitrary weight. The proof allows to simultaneouslytreat a similar boundedness V Lp,ϕloc (X,w1) → V Lq,ψloc (X,w2) forvanishing Morrey spaces. We obtain sufficient conditions for such estimatesin terms of some integral inequalities imposed on ϕ, ψ and w1.w2. We alsospecially treat the one weight case where w2(x) is also of radial type. Wedo not impose doubling condition on the measure μ, but base our result onthe growth condition.The obtained results show the explicit dependence of the mapping propertiesof the Hardy type operators on the fractional dimension of the set(X, μ, ). An application to spherical Hardy type operators is also given.