We present an integral inequality connecting a function space (quasi-)norm of the gradient of a function to an integral of the corresponding capacity of the conductor between two level surfaces of the function, which extends the estimates obtained by V. Maz'ya and S. Costea, and sharp capacitary inequalities due to V. Maz'ya in the case of the Sobolev norm. The inequality, obtained under appropriate convexity conditions on the function space, gives a characterization of Sobolev type inequalities involving two measures, necessary and sufficient conditions for Sobolev isocapacitary type inequalities, and self-improvements for integrability of Lipschitz functions.

We prove a Riesz-Herz estimate for the maximal function $M_C f(x)=\sup_{Q\ni x}C(Q)^{-1}\int_Q |f|$ associated to a capacity $C$ which extends the equivalence between the non-increasing rearrangement of the usual Hardy-Littlewood maximal function and the average function. The proof is based on an extension of the Wiener-Stein estimates for the distribution function of the maximal function, obtained using a convenient family of modified dyadic cubes. As a byproduct we obtain a description of the norm of an interpolation space between the corresponding Morrey space and the space of integrable functions.

If C is a capacity on a measurable space, we prove that the restriction of the K-functional for (Lp(C), Loo(C)) to quasicontinuous functions, f in QC, is equivalent to the K-functional for (Lp(C), Loo(C)) restricted to QC. We apply this result to identify the interpolation space for couples of capacitary Lorentz of quasicontinuous functions.

This paper is devoted to the analysis on a capacity space, with capacities as substitutes of measures in the study of function spaces. The goal is to extend to the associated function lattices some aspects of the theory of Banach function spaces, to show how the general theory can be applied to classical function spaces such as Lorentz spaces, and to complete the real interpolation theory for these spaces included in previous work.

The aim of this note is to show that many of the basic properties of classical Lorentz spaces still hold for similar spaces associated to a capacity, including real interpolation results, and to prove that under suitable conditions they appear as interpolation spaces of two extremal ones, as in the case of rearrangement invariant spaces.

We show how all the commutator estimates of two recent papers, by M. Cwikel, N. Kalton, M. Milman and R. Rochberg and by N. Krugljak and M. Milman, can be considered as special cases of the method of couples of interpolators introduced by M.J. Carro, J. Cerdà and J. Soria, and also how the distance between orbits and the "Benson norm" considered by Krugljak and Milman can be extended, respectively, as a distance between the interpolators that appear in the general construction and as a constant that is finite when the interpolators satisfy the necessary cancellation property.

We consider interpolation of operators acting on functions that belong to a given cone $Q$ with the so--called decomposition property. The set of all positive functions whose level sets are the level sets of a fixed function is the main example, and the cone of all decreasing functions is a particular case. As applications, we obtain conditions for the identity $(E_0\cap Q,E_1\cap Q)_{\theta,p} =(E_0,E_1)_{\theta,p}\cap Q$ and interpolation results for operators which are bounded when restricted to a given family of characteristic funcions.

If $T_\mu$ is a Fourier multiplier such that $\mu$ is any (possibly unbounded) symbol with uniformly bounded $q$-variation on dyadic coronas, we prove that the commutator $[T,T_\mu]=T T_\mu - T_\mu T$ is bounded on the Besov space $B_p^{\s,r}(\R^n)$, if $T$ is any bounded linear operator on a couple of Besov spaces $B_p^{\s_j,r_j}(\R^n)$ ($j=0,1$, and $0<\s_1<\s<\s_0$).

This means an improvement of our previous work in J. Approx. Theory

We associate to every (quasi--)Banach function space, and to every entropy function $E$, a scale of spaces $\Lambda^{p,q}(E)$ similar to the classical Lorentz spaces $L^{p,q}$. Necessary and sufficient conditions for they to be normed spaces are proved and their role in real interpolation theory is analyzed. The idea is that it is enough to restrict arguments to characteristic functions. We use Sparr's interpolation method for triples to describe $(\Lambda^{p_0,q_0}(E_0),\Lambda^{p_1,q_1}(E_1))_{\theta,q}$. As applications we identify the interpolation space $(B_q,L^\infty)_{\theta,p}$, where $B_q$ is the block space introduced by M. Taibleson and G. Weiss to refine some aspects of the theory of entropy spaces. Some results about interpolation of classical Lorentz spaces are also obtained, and we characterize all the pairs $(E,L^\infty)$ that are universal right Calderón couples.

We deal with Besov spaces on $R^n$. It is proved that, under some conditions on the symbol $\mu$, much weaker than those of Marcinkiewicz Multiplier Theorem, the Fourier multiplier operator $T_\mu$ satisfies on the Besov spaces $B_p^{\s,q}$ the following commutator theorem:

If $T$ is any bounded linear operator on Besov spaces $B_p^{\s_j,q_j}(\R^n)$ ($j=0,1$, and $0<\s_1<\s<\s_0$), then the commutator $[T,T_\mu]=T T_\mu - T_\mu T$ is bounded on $B_p^{\s,q}(\R^n)$, if $T_\mu$ is a Fourier multiplier such that $\mu$ is any (possibly unbounded) symbol with uniformly bounded variation on dyadic coronas.

In a previous 1999 paper (with N. Krugljak) the case of periodic functions was considered, and here we give a simpler proof that applies for both cases.

We describe the $K$--functional and identify the real interpolated spaces of general quasi-Banach couples of classical Lorentz spaces. Applications are given which include interpolation of spaces of Lorentz-Zygmund type

We present some sufficient conditions under which pairs of Banach function lattices are Calderón couples, with special attention to the case of classical Lorentz spaces.

Many problems in analysis are described as weighted norm inequalities that have given rise to different classes of weights. Our purpose is to show that ifferent classes of weights are related by means of composition with classical transforms. A typical example is the family $M_p$ of weights w for which the Hardy transform is bounded on $L_p(w)$. A $B_p$ weight of Ariño and Muckenhoupt is precisely one for which its Hardy transform is in $M_p$, and also a weight whose indefinite integral is an $A_{p+1}$ weight of Muckenhoupt.

We consider weak-type Hardy inequalities for weights on the half-line, and we prove that, when restricted to decreasing functions, the weak $L_p$ type of the conjugate Hardy transform does not depend on p and that, under a doubling condition, it is equivalent to the corresponding strong estimates.

The main idea underlying this paper is that the description of approximation spaces, such as Besov spaces, and the calculation of almost optimal approximation elements, in combination with real interpolation, are very useful in the so-called commutator theorems.

It is proved that, under some conditions, much weaker than those of Marcinkiewicz Multiplier Theorem, the multiplier operator $T_\mu$,

$$ T_{\mu}(\sum_{k}c_ke^{ikt}) := \sum_{k}\mu_kc_ke^{ikt}, $$

satisfies on the Besov spaces $B_p^{\s,q}$ the commutator theorem

$$ \Vert [T,T_{\mu}]\Vert_{B_p^{\s,q},B_p^{\s,q}}\leq c\Vert T\Vert, $$

for $\s_0>\s>\s_1>0$ and $q_0,q_1\geq 1$, where $\Vert T\Vert=\max(\Vert T\Vert_{B_p^{\s_0,q_0},B_p^{\s_0,q_0}}, \Vert T\Vert_{B_p^{\s_1,q_1},B_p^{\s_1,q_1}})$.

In previous papers, the authors have established a unified method for the study of the commutators of bounded linear operators and certain operators in interpolation theory. Here we show that new applications can be obtained by the same method. In particular, if we consider spaces of vector-valued functions, we obtain the boundedness of the commutators of BMO functions with some maximal operators and with Littlewood-Paley sums. We can also get the boundedness of commutators on Besov spaces.

For the real interpolation method, we identify the interpolated spaces of couples of classical Lorentz spaces through interpolation of the corresponding weighted Lebesgue spaces restricted to decreasing functions.

We prove certain vector-valued continuous inclusions for Calderón-Lozanovskii spaces and, by interpolation, we obtain results on the type and cotype of these spaces. We also give an extension of Kwapien's result which states that, for $1\leq p \leq 2$, every operator from $\ell_1$ into $\ell_p$ is $(r,1)$-summing, if $1/r=3/2-1/p$.

We deal with the basic convexity properties -rotundity, and uniform, local uniform and full rotundity- for symmetric spaces. A characterization of Orlicz-Lorentz spaces with the Kadec-Klee property for pointwise convergence is given. These results are applied to obtain criteria of convexity properties for Orlicz-Lorentz sequence spaces, and some new proofs of the sufficiency part of criteria for rotundity and uniform rotundity for Orlicz-Lorentz function spaces.

We prove that for a decreasing weight w on $[0,\infty)$, the conjugate Hardy transform is bounded on $L_p(w)$ ($1\leq p<\infty$) if and only if it is bounded on the cone of all decreasing functions of $L_p(w)$. This property does not depend on $p$.

In a previous paper we have shown that in many examples the interpolation for the cones of Banach lattices of decreasing functions is well-behaved. Here our aim is to see that this fact can be extended to Calderón products, and to the real interpolation of cones of increasing functions and of radial decreasing functions.

The boudedness of operators between Banach function lattices, like Lebesgue spaces with weights or Lorentz spaces on the half-line, restricted to decreasing functions recently has been considered in several contexts. This paper deals with the natural question of finding interpolation theorems for operators which are bounded for decreasing functions, and the real method is considered. There is no general way to identify the interpolated class of the pair $(X_0^d,X_1^d)$ of cones of decreasing functions for a couple $(X_0,X_1)$ of Banach function lattices on the halfline, even if the interpolation space $(X_0^d,X_1^d)_{\t,p}$ of this pair of spaces is known. It is shown that in many examples, which include all rearrangement-invariant spaces and a number of weighted function lattices, the interpolation for decreasing functions is well-behaved. In these cases the usual K-functional $K(f,t;X_0,X_1)$ is equivalent to the K-functional $K(f,t;X_0^d,X_1^d)$ associated to the cones of decreasing functions.

We present the dual construction of our previous work where we have established a unified method for the study of the commutators of bounded linear operators and certain operators in interpolation theory.

Recently, estimates for higher order commutators of interpolation theory have been obtained for the complex and for the real method. The analysis of cancellation properties allows us to obtain a new commutator theorem which extends the previous results. An application to the boundedness of higher order commutators for singular operators between weighted Lebesgue spaces is also given.

Rochbert and Weiss developed the study of commutators of bounded linear operators and certain operators, generally unbounded and nonlinear obtained by complex interpolation. A similar analysis was carried out for the real method by Jawerth, Rochberg and Weiss.

The purpose of this paper is to set up a unified method for both theories. Our method lieds to a simple approach to commutator theorems, giving the precise role the cancelation plays in the theory.

We study the relationship between Schechter's methods of complex interpolation and the so called commutator estimates. We obtain new commutator theorems and prove characterizations of the Domain and Range spaces associated with the corresponding quasilogarithmic operator. Our methods also provide a new approach to known results, including the higher order commutator theorems for the complex method recently obtained by R. Rochberg.

Convexity, monotonicity and smoothness properties of Köthe spaces of vector-valued functions are described.

Geometric properties of Calderón-Lozanovskii spaces between Banach function lattices and $L^\infty$ are considered. It is shown that under some general conditions these spaces posses certain monotonicity, rotundity and uniform nonsquareness properties. Some applications to renorming of Lorentz-Orlicz and Orlicz spaces are given.

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