Lorentz-Zygmund Spaces Research Articles

Comparison and regularity results for degenerate elliptic equations related to Gauss measure

This article deals with a Dirichlet problem relative to a linear elliptic equation with zero-order terms, whose ellipticity condition is given in terms of the function ν(x)ϕ(x), where is the density of Gauss measure, ν(x) is a nonnegative integrable function. Using the notion of rearrangement with respect to Gauss measure, we prove a comparison result with a problem of the same type defined in a half space, with data depending only on the first variable. Also, some regularity results for the solutions in Lorentz–Zygmund spaces are obtained by us.

Solid hulls of quasi-Banach spaces of analytic functions and interpolation

Abstract We study solid hulls of abstract Hardy spaces of analytic functions on the unit disk. Interpolation of sublinear operators is used to description of solid hulls of quasi-Banach spaces generated by interpolation methods associated to classical Hardy spaces H p for 0 p 1 . We specialize our results to solid hulls of Hardy–Orlicz and Hardy–Lorentz spaces and we are able to characterize multipliers from Hardy–Lorentz spaces to any solid sequence space. We give, for example, a complete description of multipliers with the domain in Hardy–Lorentz–Zygmund spaces.

An integral representation formula for logarithmic potentials and embeddings of Bessel-potential spaces

We give an integral representation formula for logarithmic Riesz potentials. This plays an essential role in proving the sharpness of the embeddings of Bessel-potential spaces, which have logarithmic exponents both in the smoothness and in the underlying Lorentz—Zygmund spaces. These results are natural extensions of those obtained by Edmunds, Gurka, Opic and Trebels.

Brezis-Merle type inequality for a weak solution to the $N$-Laplace equation in Lorentz-Zygmund spaces

We consider a regularity estimate for a solution of the homogeneous Dirichlet problem for $N$-Laplace equations in a bounded domain $\Omega\subset{\mathbb R}^N$ with external force $f\in L^1(\Omega)$. Introducing the generalized Lorentz-Zygmund space, we show the multiple exponential integrability of the Brezis-Merle type for an entropy solution of the Dirichlet problem of the $N$-Laplace equation. We also discuss conditions on $f$ that guarantee the solutions are bounded.

On Ulyanov inequalities in Banach spaces and semigroups of linear operators

Let X , Y be Banach spaces and { T ( t ) : t ≥ 0 } be a consistent, equibounded semigroup of linear operators on X as well as on Y ; it is assumed that { T ( t ) } satisfies a Nikolskii type inequality with respect to X and Y : ∥ T ( 2 t ) f ∥ Y ≲ ϕ ( t ) ∥ T ( t ) f ∥ X . Then an abstract Ulyanov type inequality is derived between the (modified) K -functionals with respect to ( X , D X ( ( - A ) α ) ) and ( Y , D Y ( ( - A ) α ) ) , α > 0 , where A is the infinitesimal generator of { T ( t ) } . Particular choices of X , Y are Lorentz–Zygmund spaces, of { T ( t ) } are those connected with orthogonal expansions such as Fourier, spherical harmonics, Jacobi, Laguerre, Hermite series. Known characterizations of the K -functionals lead to concrete Ulyanov type inequalities. In particular, results of Ditzian and Tikhonov in the case X = L p , Y = L q , 1 ≤ p < q ≤ ∞ , are partly covered.

Sharp embeddings of Besov spaces involving only logarithmic smoothness

We use Kolyada's inequality and its converse form to prove sharp embeddings of Besov spaces B p , r 0 , β (involving the zero classical smoothness and a logarithmic smoothness with the exponent β ) into Lorentz–Zygmund spaces. We also determine growth envelopes of spaces B p , r 0 , β . In distinction to the case when the classical smoothness is positive, we show that we cannot describe all embeddings in question in terms of growth envelopes.

Interpolation of Bilinear Operators Between Quasi-Banach Spaces

We study interpolation, generated by an abstract method of means, of bilinear operators between quasi-Banach spaces. It is shown that under suitable conditions on the type of these spaces and the boundedness of the classical convolution operator between the corresponding quasi-Banach sequence spaces, bilinear interpolation is possible. Applications to the classical real method spaces, Calderon-Lozanovsky spaces, and Lorentz-Zygmund spaces are presented.

Sharp Embeddings of Besov Spaces with Logarithmic Smoothness

We prove sharp embeddings of Besov spaces B with the classical smoothness a and a logarithmic smoothness a into Lorentz-Zygmund spaces. Our results extend those with a = 0, which have been proved by D. E. Edmunds and H. Triebel. On page 88 of their paper (Math. Nachr. 207 (1999), 79-92) they have written: Nevertheless a direct proof, avoiding the machinery of function spaces, would be. desirable. In our paper we give such a proof even in a more general context. We cover both the sub-limiting and the limiting cases and we determine growth envelopes of Besov spaces with logarithmic smoothness.

The Banach–Saks index of rearrangement invariant spaces on [0,1

The set of all rearrangement invariant function spaces on [0,1] having the p-Banach–Saks property has a unique maximal element for all p∈(1,2]. For p=2 this is L 2, for p∈(1,2) this is L p,∞ 0. We compute the Banach–Saks index for the families of Lorentz spaces L p,q, 1<p<∞ , 1⩽ q⩽∞, and Lorentz–Zygmund spaces L( p, α), 1⩽p<∞, α∈ R , extending the classical results of Banach–Saks and Kadec–Pelczynski for L p -spaces. Our results show that the set of rearrangement invariant spaces with Banach–Saks index p∈(1,2] is not stable with respect to the real and complex interpoltaion methods. To cite this article: E.M. Semenov, F.A. Sukochev, C. R. Acad. Sci. Paris, Ser. I 337 (2003).

On an Extremal Scale of Approximation Spaces

This paper is concerned with an extremal scale of approximation spaces involving logarithmic weights. We give equivalent norms and study the behavior of this scale under reiteration and interpolation. Some applications to Lorentz-Zygmund operator ideals and Besov spaces with logarithmic weights are given.

Weak Type Interpolation Near “Endpoint” Spaces

Let T be an operator of weak types (a, b) and (p, q), where a<p and b<q. The Marcinkiewicz interpolation theorem and its generalizations due to Boyd, Krein–Semenov and others show that T maps certain rearrangement invariant spaces E which are “not too close” to La or Lp into certain spaces F. In this paper we consider analogous results for such an operator T in the case where, on the contrary, E is close to Lp in the sense that its fundamental function is t1/p. For example, E can be a Lorentz space Lp, r for 1⩽r⩽∞. The corresponding range spaces F are explicitly described and shown to be optimal. When E=Lp, r, then in some cases F is a member of the class of Lorentz–Zygmund spaces Lq, r(logL)s which were introduced and studied by Bennett and Rudnick. But in general F is strictly smaller than the corresponding Lorentz–Zygmund space and belongs to an apparently new class of r.i. spaces. Necessary and sufficient conditiones are given for the members of this new class to coincide with Lorentz–Zygmund spaces. Certain results of this paper can be applied to give an alternative proof and generalization of the optimal form of the limiting case of the Sobolev embedding theorem due to Hansson and Brezis–Wainger.

Optimal interpolation in spaces of Lorentz-Zygmund type

The Lorentz-Zygmund spaces, introduced by C. Bennett and K. Rudnick in [BR], are generalized by taking the exterior norm in arbitrary rearrangement invariant spaceE instead of onlyL r-spaces. On the spacesL p,α,E thus obtained, we study all operatorsT of two weak types (a, b) and (p, q) with 1≤a<p≤∞, 1≤b<q≤∞, and prove thatT:L p,α,E→L q,α−1,E. Moreover, for any set of parametersp, q, α, E, we construct the smallest possible spaceB q,α,E such thatT:L p,α,E→B q,α,E and the largest possible spaceA p,α,E such thatT:A p,α,E→L q,α−1,E. For spaces of all three types, we find their fundamental functions and Boyd indices, state various embeddings, equivalences and other properties.

Strong and Weak Type Inequalities for Some Classical Operators in Orlicz Spaces

Inequalities involving classical operators of harmonic analysis, such as maximal functions, fractional integrals and singular integrals of convolutive type have been extensively investigated in various function spaces. Results on weak and strong type inequalities for operators of this kind in Lebesgue spaces are classical and can be found for example in [4, 20, 24]. Generalizations of these results to Zygmund spaces are presented in [4]. An exhaustive treatment of the problem of boundedness of such operators in Lorentz and Lorentz–Zygmund spaces is given in [3]. See also [8, 9] for further extensions in the framework of generalized Lorentz–Zygmund spaces. As far as Orlicz spaces are concerned, a characterization of Young functions A having the property that the Hardy–Littlewood maximal operator or the Hilbert and Riesz transforms are of weak or strong type from the Orlicz space LA into itself is known (see for example [13]). In [17, 23] conditions on Young functions A and B are given for the fractional integral operator to be bounded from LA into LB under some restrictions involving the growths and certain monotonicity properties of A and B. The main purpose of this paper is to find necessary and sufficient conditions on general Young functions A and B ensuring that the above-mentioned operators are of weak or strong type from LA into LB. Our results for (fractional) maximal operators are presented in Section 2, while Section 3 deals with fractional and singular integrals. In particular, we re-cover a result concerning the standard Hardy–Littlewood maximal operator which has recently been proved in [2, 11, 12]. Finally, in Section 4, the resolvent operator of some differential problems is taken into account and a priori bounds for Orlicz norms of solutions to elliptic boundary value problems in terms of Orlicz norms of the data are established. Let us mention that part of the results of the present paper were announced in [6].

Global limiting embeddings of logarithmic Bessel potential spaces

The paper is a continuation of [EGO II-IV], where it was shown that Bessel potential spaces HσY(Rn) , modelled upon appropriate generalized Lorentz-Zygmund spaces Y(Rn) may be embedded into Orlicz spaces LΦ(Ω) , where Φ(t) = exp(exp(. . . exp tν) . . .)) for large t , ν > 0 , and Ω is a subset of Rn with finite volume. Using weighted Hardy inequalities, we modify the Young function Φ near the origin so that the above embedding holds with Ω replaced by Rn . The resulting Young function dominates globally the Young function Ψ(t) = tq , t > 0 , for q sufficiently large and consequently, HσY(Rn) ↪→ Lq(Rn) . We also obtain an estimate of the norms of the last embeddings which is sharp in their dependence upon q provided that q is large enough. Mathematics subject classification (1991): 46E35, 46E30, 26D15.

On Embeddings of Logarithmic Bessel Potential Spaces

We present embedding theorems for certain logarithmic Bessel potential spaces modelled upon generalized Lorentz–Zygmund spaces and clarify the role of the logarithmic terms involved in the norms of the space mentioned. In particular, we get refinements of the Sobolev embedding theorems, Trudinger's limiting embedding as well as embeddings of Sobolev spaces into space ofλ-Hölder-continuous functions including the result of Brézis and Wainger.

An embedding theorem in Lorentz-Zygmund spaces

We study Sobolev embedding theorems in Lorentz-Zygmund spaces. Some limiting cases are considered.

Sharpness of embeddings in logarithmic Bessel-potential spaces

This paper is a continuation of [4], where embeddings of certain logarithmic Bessel-potential spaces (modelled upon generalised Lorentz-Zygmund spaces) in appropriate Orlicz spaces (with Young functions of single and double exponential type) were derived. The aim of this paper is to show that these embedding results are sharp in the sense of [8].

On Extrapolation Spaces and a.e. Convergence of Fourier Series

A new, unified approach to recent end point estimates for the maximal operator of partial sums of Fourier series is obtained through the use of extrapolation theory. The method involves characterizing certain extrapolation spaces associated with scales of Lorentz-Zygmund spaces.