We give analytic description for the completion of C?0 (R+) in Dirichletspace D1,p(R+, ?) := {u : R+ -> R : u is locally absolutely continuous on R+ and ||u? ||_Lp(R+,?) < ?}, for given continuouspositive weight ? defined on R+, where 1 < p < ?. The conditions are described in terms of the modified variants of the Bpconditions due to Kufner and Opic from 1984, which in our approach are focusing on integrability of ?^-p/(p-1) near zero or near infinity. Moreover, we propose applications of our results to: obtaining newvariants of Hardy inequality, interpretation of boundary value problems in ODE's defined on the halpfline with solutions in D1,p(R+, ?),new results from complex interpolation theory dealing with interpolation spaces between weighted Dirichlet spaces, and to derivationof new Morrey type embedding theorems for our Dirichlet space.

We study the regularity properties of the weak solutions u : Ω ⊆ Rn → R to elliptic problems −div a(x,Du) + b(x)φ′(|u|) u |u| = f in Ω , u = 0 on ∂Ω , with Ω ⊂ Rn a bounded open set and where the function a(x, ξ) satisfies growth conditions with respect to the second variable expressed through an N-function φ. We prove that, under a suitable interplay between the lower order terms and the datum f, which is assumed only to belong to L1(Ω), the solutions are bounded in Ω. Next, if a(x, ξ) depends on x through a H ̈older continuous function we take advantage from the boundedness of the solution u to prove the higher differentiability and the higher integrability of its gradient, under mild assumptions on the data.

Let 1 < p < ∞, ε0 ∈]0, p − 1], Ω ⊂ Rn be a Lebesgue measurable set of positive, finite measure, and let δ : (0, p − 1] → (0, ∞) be such that δb(·):= δ(·) p−·1 is nondecreasing and bounded. We show that the linear set of functions 5 f Lebesgue measurable on Ω: 0<ε sup ≤ε0(δ(ε) k − |f(x)|p−εdx ) p−1 ε < ∞ 5 Ω does not depend on small values of ε0 if and only if δb ∈ ∆2(0+) (i.e., δb(2ε) ≤ cδb(ε) for ε small, for some c > 1), which is equivalent to say that δ ∈ ∆2(0+). This means that in the case δb ∈/ ∆2(0+), the parameter ε0 plays a crucial role in the definition of a generalized grand Lebesgue space, namely, different values of ε0 define different Banach function spaces.

If I? R is a bounded interval, we prove the boundedness of Calderón singular operator and of Hardy-Littlewood Maximal operator in the generalized weighted Grand Lebesgue spaces Lpp),?(I), 1 < p< ?.

In this paper we establish an higher integrability result for second derivatives of the local solution of elliptic equation div(A(x,Du))=0in?where ? ? R, n>= 2 and A(x, ?) has linear growth with respect to ? variable. Concerning the dependence on the x-variable, we shall assume that, for the map x-> A(x, ?) , there exists a non negative function k(x), such that |DxA(x,?)|?k(x)(1+|?|)for every ?? R and a.e. x? ?. It is well known that there exists a relationship between this condition and the regularity of the solutions of the equation. Our pourpose is to establish an higher integrability result for second derivatives of the local solution, by assuming k(x) in a suitable Zygmund class.

In this paper we establish the boundedness and the higher differentiability of solutions to the {div(A(x,Du))+b(x)|u(x)|u(x)=fin ?u=0on ?? under a Sobolev assumption on the partial map x->A(x,?). The novelty here is that we deal with degenerate elliptic operator A(x,?) with p-growth, p>=2, with respect to the gradient variable, in presence of lower order terms. The interplay between b(x) and f(x), introduced in ([1]), gives a regularizing effect also in the degenerate elliptic setting.

In this paper we establish the higher differentiability of solutions to the Dirichlet problem {div(A(x,Du))+b(x)u(x)=fin?u=0on??under a Sobolev assumption on the partial map x-> A(x, ?). The novelty here is that we take advantage from the regularizing effect of the lower order term to deal with bounded solutions.

In this note we prove a modular variable Orlicz inequality for the local maximal operator. This result generalizes several Orlicz and variable exponent modular inequalities that have appeared previously in the literature.

integral(R) A(vertical bar f' (x)vertical bar h(f(x))) dx <= C-1 integral(R) A(C-2 (p) root vertical bar Mf"(x)T-h,T-p (f,x)vertical bar. h(f(x))dx, Given a N-function A and a continuous function h satisfying certain assumptions, we derive the inequality [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], with constants [C.sub.1], [C.sub.2] independent of f, where f [greater than or equal to] 0 belongs locally to the Sobolev space [W.sup.2,1] (R), f' has compact support, p 1 is smaller than the lower Boyd index of A, [T.sub.h,p] (*) is certain nonlinear transform depending of h but not of A and M denotes the Hardy-Littlewood maximal function. Moreover, we show that when h [equivalent to] 1, then Mf" can be improved by f". This inequality generalizes a previous result by the third author and Peszek, which was dealing with p = 2.

We provide regularity properties for the inverse map f^-1 under suitable assumptions on q-distorsion function of f, in bounded domains of R^2.

We introduce and investigate the grand Orlicz spaces and the grand Lorentz-Orlicz spaces. An application to the problem of global integrability of the Jacobian of orientation preserving mappings is given.

We prove that if the exponent function p((.)) satisfies log-Holder continuity conditions locally and at infinity, then the fractional maximal operator M(alpha), 0 < alpha < n, maps L(p(.)) to L(q(.)), where 1/p(x) - 1/q(x) = alpha/n. We also prove a weak-type inequality corresponding to the weak (1, n/(n - a)) inequality for M(alpha). We build upon earlier work on the Hardy-Littlewood maximal operator by Cruz-Uribe, Fiorenza and Neugebauer [3]. As a consequence of these results for M(alpha), we show that the fractional integral operator I(alpha) satisfies the same norm inequalities. These in turn yield a generalization of the Sobolev embedding theorem to variable L(p) spaces.

Yano's extrapolation theorem dated back to 1951 establishes boundedness properties of a subadditive operator acting continuously in for close to and/or taking into as and/or with norms blowing up at speed and/or , . Here we give answers in terms of Zygmund, Lorentz-Zygmund and small Lebesgue spaces to what happens if as . The study has been motivated by current investigations of convolution maximal functions in stochastic analysis, where the problem occurs for . We also touch the problem of comparison of results in various scales of spaces.

We consider a generalized version of the small Lebesgue spaces, introduced by Fiorenza, as the associate spaces of the grand Lebesgue spaces. We find a simplified expression for the norm, prove relevant properties, compute the fundamental function and discuss the comparison with the Orlicz spaces.

A quasiharmonic field is a pair $\Cal{F} = [B,E]$ of vector fields verifying $div B=0$, $curl E=0$, and coupled by a distorsion inequality. For a given $\Cal {F}$, we construct a matrix field $\Cal{A}=\Cal{A}[B,E]$ such that $\Cal{A} E=B$. This remark in particular shows that the theory of quasiharmonic fields is equivalent (at least locally) to that of elliptic PDEs. Here we stress some properties of our operator $\Cal {A}[B,E]$ and find applications of them to the study of regularity of solutions to elliptic PDEs, and to some questions of G-convergence.

Elementary interpolation was used in nonlinear partial differential equations (PDE) to study higher integrability via Riesz transforms. The analysis was conducted in Dirichlet space of locally integrable functions in RN. The uniqueness statement for solutions of the nonlinear PDE was also proved.