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.

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.

The boundedness of the Hardy-Littlewood maximal operator is proved in weighted grand variable exponent Lebesgue space with power weights.

For kernels zi which are positive and integrable we show that the operator g bar right arrow J(v)g = integral(x)(0) v(x-s)g(s)ds on a finite time interval enjoys a regularizing effect when applied to Holder continuous and Lebesgue functions and a "contractive" effect when applied to Sobolev functions. For Holder continuous functions, we establish that the improvement of the regularity of the modulus of continuity is given by the integral of the kernel, namely by the factor N(x) = integral(x)(0) v(s)ds. For functions in Lebesgue spaces, we prove that an improvement always exists, and it can be expressed in terms of Orlicz integrability. Finally, for functions in Sobolev spaces, we show that the operator J. "shrinks" the norm of the argument by a factor that, as in the Holder case, depends on the function N (whereas no regularization result can be obtained). These results can be applied, for instance, to Abel kernels and to the Volterra function Z(x) = mu(x,0, -1) = integral(infinity)(0)x(s-1)/Gamma(s)ds, the latter being relevant for instance in the analysis of the Schrodinger equation with concentrated nonlinearities in R-2.

We prove a dimension-invariant imbedding estimate for Sobolev spaces of first order into a small Lebesgue space, and we establish the optimality of its fundamental function. Namely, for any 1 < p < ?, the inequality with a constant c_p, related to the imbedding of W_0^{1,p}(B_n) into Y_p(0,1), where Yp(0,1) is a rearrangement-invariant Banach function space independent of the dimension n, B_n is the ball in R^n of measure 1 and c_p is a constant independent of n, is satisfied by the small Lebesgue space L(p,p? /2 (0, 1). Moreover, we show that the smallest space Yp (0, 1) (in the sense of the continuous imbedding) such that (*) is true has the fundamental function equivalent to that of L(p,p?/2(0,1). As a byproduct of our results, we get that the space Lp (log L)p/2 is optimal in the framework of the Orlicz spaces satisfying the imbedding inequality.

The Generalized- ? -Space G? (p, m, w) contains many classical rearrangement invariant spaces. Here we shall study its associate space and we shall estimate its associate norm. In particular, we characterize all optimal functions u achieving the associate norm of Generalized ? -space G ? (p, m, w) when it is reflexive. For the purpose, we use the notion of relative rearrangement and new additional results on this concept. Moreover, we prove that the space G? (p, m, w) is reflexive under the conditions that m > 1 and p >= 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.

We apply the techniques of monotone and relative rearrangements to the non rearrangement invariant spaces Lp(·) (? ) with variable exponent. In particular, we show that the maps u ? L p( ·) (? ) -> k(t )u* ? L p * (·)(0, meas? ) and u ? L p( ·) (? ) -> u* ? Lp* (·) (0, meas? ) are locally ?-Ho?lderian (u * (resp. p* ) is the decreasing (resp. increasing) rearrangement of u (resp. p)). The pointwise relations for the relative rearrangement are applied to derive the Sobolev embedding with eventually discontinuous exponents.

We study the convergence and decay rate to equilibrium of bounded solutions of the quasilinear parabolic equation ut - div a(x , ? u) + f (x , u) = 0 on a bounded domain, subject to Dirichlet boundary and to initial conditions. The data are supposed to satisfy suitable regularity and growth conditions. Our approach to the convergence result and decay estimate is based on the ?ojasiewicz-Simon gradient inequality which in the case of the semilinear heat equation is known to give optimal decay estimates. The abstract results and their applications are discussed also in the framework of Orlicz-Sobolev spaces.