Rearrangement Invariant Banach Function Spaces Research Articles

A unified approach to inequalities for K-functionals and moduli of smoothness

The paper provides a detailed study of crucial inequalities for smoothness and interpolation characteristics in rearrangement invariant Banach function spaces. We present a unified approach based on Holmstedt formulas to obtain these estimates. As examples, we derive new inequalities for moduli of smoothness and K-functionals in various Lorentz spaces.

Remark on Singular Integral Operators of Convolution Type on Rearrangement-Invariant Banach Function Spaces

We prove that nondegenerate singular integral operators of convolution type are bounded on a rearrangement-invariant Banach function space $X(\mathbb{R}^d)$ if and only if its Boyd indices are nontrivial, extending the result by David Boyd (1966) for the Hilbert transform.

On the essential norms of singular integral operators with constant coefficients and of the backward shift

Let X X be a rearrangement-invariant Banach function space on the unit circle T \mathbb {T} and let H [ X ] H[X] be the abstract Hardy space built upon X X . We prove that if the Cauchy singular integral operator ( H f ) ( t ) = 1 π i ∫ T f ( τ ) τ − t d τ (Hf)(t)=\frac {1}{\pi i}\int _{\mathbb {T}}\frac {f(\tau )}{\tau -t}\,d\tau is bounded on the space X X , then the norm, the essential norm, and the Hausdorff measure of non-compactness of the operator a I + b H aI+bH with a , b ∈ C a,b\in \mathbb {C} , acting on the space X X , coincide. We also show that similar equalities hold for the backward shift operator ( S f ) ( t ) = ( f ( t ) − f ^ ( 0 ) ) / t (Sf)(t)=(f(t)-\widehat {f}(0))/t on the abstract Hardy space H [ X ] H[X] . Our results extend those by Krupnik and Polonskiĭ [Funkcional. Anal. i Priloz̆en. 9 (1975), pp. 73-74] for the operator a I + b H aI+bH and by the second author [J. Funct. Anal. 280 (2021), p. 11] for the operator S S .

The space is primary for 1 < p < ∞

Abstract The classical Banach space $L_1(L_p)$ consists of measurable scalar functions f on the unit square for which $$ \begin{align*}\|f\| = \int_0^1\Big(\int_0^1 |f(x,y)|^p dy\Big)^{1/p}dx < \infty.\end{align*} $$ We show that $L_1(L_p) (1 < p < \infty )$ is primary, meaning that whenever $L_1(L_p) = E\oplus F$ , where E and F are closed subspaces of $L_1(L_p)$ , then either E or F is isomorphic to $L_1(L_p)$ . More generally, we show that $L_1(X)$ is primary for a large class of rearrangement-invariant Banach function spaces.

Noncompactness of Toeplitz operators between abstract Hardy spaces

In the beginning of 1960s, Brown and Halmos proved that a Toeplitz operator T(a) is compact on the Hardy space $$H^2=H[L^2]$$ over the unit circle $$\mathbb {T}$$ if and only if $$a=0$$ a.e. Recently, Leśnik [13] generalized this result to the setting of Toeplitz operators acting between abstract Hardy spaces H[X] and H[Y] built upon possibly different rearrangement-invariant Banach function spaces X and Y over $$\mathbb {T}$$ such that Y has nontrivial Boyd indices. We show that the general principle of noncompactness of nontrivial Toeplitz operators between abstract Hardy spaces H[X] and H[Y] remains true for much more general spaces X and Y. In particular, there are no nontrivial compact Toeplitz operators on the Hardy space $$H^1=H[L^1]$$ , although $$L^1$$ has trivial Boyd indices.

On solvability of Riemann problems in Banach hardy classes

This work deals with the rearrangement invariant Banach function space X and Banach Hardy classes generated by this space, which consist of analytic functions inside and outside the unit circle. In these Hardy classes we consider homogeneous and nonhomogeneous Riemann problems with piecewise continuous coefficient. We define new characteristic of the space X related to the power functions in X. Canonical solution is defined depending on the jumps of the argument of the coefficient of the problem. In terms of the above characteristic, we find a condition on the jumps of the argument depending on the Boyd indices of space X, which is sufficient for the solvability of these problems, and, in case of solvability, we construct a general solution. We also give an orthogonality condition for the solvability of nonhomogeneous problem. As X, considering specific spaces, we obtain previously known results.

On boundedness of the Hilbert transform on Marcinkiewicz spaces

We study boundedness properties of the classical (singular) Hilbert transform (Hf)(t) = p.v.1/π \int_R f(s)/(t − s)ds acting on Marcinkiewicz spaces. The Hilbert transform is a linear operator which arises from the study of boundary values of the real and imaginary parts of analytic functions. Questions involving the H arise therefore from the utilization of complex methods in Fourier analysis, for example. In particular, the H plays the crucial role in questions of norm-convergence of Fourier series and Fourier integrals. We consider the problem of what is the least rearrangement-invariant Banach function space F(R) such that H : Mφ(R) → F(R) is bounded for a fixed Marcinkiewicz space Mφ(R). We also show the existence of optimal rearrangement-invariant Banach function range on Marcinkiewicz spaces. We shall be referring to the space F(R) as the optimal range space for the operator H restricted to the domain Mφ(R) ⊆ Λϕ0(R). Similar constructions have been studied by J.Soria and P.Tradacete for the Hardy and Hardy type operators [1]. We use their ideas to obtain analogues of their some results for the H on Marcinkiewicz spaces.

Gagliardo–Nirenberg Inequality for rearrangement-invariant Banach function spaces

The classical Gagliardo–Nirenberg interpolation inequality is a well-known estimate which gives, in particular, an estimate for the Lebesgue norm of intermediate derivatives of functions in Sobolev spaces. We present an extension of this estimate into the scale of the general rearrangement-invariant Banach function spaces with the proof based on the Maz’ya’s pointwise estimates. As corollaries, we present the Gagliardo–Nirenberg inequality for intermediate derivatives in the case of triples of Orlicz spaces and triples of Lorentz spaces. Finally, we promote the scaling argument to validate the optimality of the Gagliardo–Nirenberg inequality and show that the presented estimate in Orlicz scale is optimal.

Approximation in vanishing rearrangement-invariant Morrey spaces and applications

Morrey type space built on rearrangement-invariant Banach function space where the set of smooth functions with compact support is a dense set of this space is introduced in this paper. By using the denseness of the set of smooth functions with compact support, we obtain the oscillation and variation inequalities of Riesz transforms on this Morrey type space.

Rearrangements and Leibniz-type rules of mean oscillations

We shall prove a rearrangement inequality in probability measure spaces in order to obtain sharp Leibniz-type rules of mean oscillations in Lp-spaces and rearrangement invariant Banach function spaces.

Weighted inequalities for singular integral operators on the half-line

We prove weighted estimates for singular integral operators which operate on function spaces on a half-line. The class of admissible weights includes Muckenhoupt weights and weights satisfying Sawyer's one-sided conditions. The kernels of the operators satisfy relaxed Dini conditions. We apply the weighted estimates to extrapolation of maximal L p regularity of first order, second order and fractional order Cauchy problems into weighted rearrangement invariant Banach function spaces. In particular, we provide extensions, as well as a unification of recent results due to Auscher and Axelsson, and Chill and Fiorenza.

Extrapolation of $L^p$ maximal regularity for second order Cauchy problems

If the second order problem u'' + Bu' + Au = f has L p-maximal regularity for some p ∈ (1, ∞), then it has E_w-maximal regularity for every rearrangement invariant Banach function space E with Boyd indices p_E, q_E ∈ (1, ∞) and for every Muckenhoupt weight w ∈ A p- .

Duality of fully measurable grand Lebesgue space

In this paper, we prove a Hölder’s type inequality for fully measurable grand Lebesgue spaces, which involves the notion of fully measurable small Lebesgue spaces. It is proved that these spaces are non-reflexive rearrangement invariant Banach function spaces. Moreover, under certain continuity assumptions, along with several properties of fully measurable small Lebesgue spaces, we establish Levi’s theorem for monotone convergence and that grand and small spaces are associated to each other.

Real interpolation with weighted rearrangement invariant Banach function spaces

Motivated by recent applications of weighted norm inequalities to maximal regularity of first and second order Cauchy problems, we study real interpolation spaces on the basis of general Banach function spaces and, in particular , weighted rearrangement invariant Banach function spaces. We show equivalence of the trace method and the K-method, identify real interpolation spaces between a Banach space and the domain of a sectorial operator, and reprove an extension of Dore's theorem on the boundedness of H ∞-functional calculus to this general setting.

Drop of the Smoothness of an Outer Function Compared to the Smoothness of its Modulus, Under Restrictions on the Size of Boundary Values

Let F be an outer function on the unit disk. It is well known that its smoothness properties can be twice worse than those of the modulus of its boundary values, but under some restrictions on log |F|, this gap becomes smaller. It is shown that the smoothness decay admits a convenient description in terms of a rearrangement invariant Banach function space containing log |F|. All the results are of pointwise nature.

Fully measurable grand Lebesgue spaces

Using variable exponents, we build a new class of rearrangement-invariant Banach function spaces, independent of the variable Lebesgue spaces, whose function norm is ρ(f)=esssupx∈(0,1)ρp(x)(δ(x)f(⋅)), where ρp(x) denotes the norm of the Lebesgue space of exponent p(x) (assumed measurable and possibly infinite) and δ is measurable, too. Such class contains some known Banach spaces of functions, among which are the classical and the grand Lebesgue spaces, and the EXPα spaces (α>0). We analyze the function norm and we prove a boundedness result for the Hardy–Littlewood maximal operator, via a Hardy type inequality.

Classical and approximate Taylor expansions of weakly differentiable functions

The pointwise behavior of Sobolev-type functions, whose weak derivatives up to a given order belong to some rearrangement-invariant Banach function space, is investigated. We introduce a notion of approximate Taylor expansion in norm for these functions, which extends the usual definition of Taylor expansion in L p -sense for standard Sobolev functions. An approximate Taylor expansion for functions in arbitrary-order Sobolev-type spaces, with sharp norm, is estab- lished. As a consequence, a characterization of those Sobolev-type spaces in which all functions admit a classical Taylor expansion is derived. In particular, this provides a higher-order version of a well-known result of Stein (27) on the differentiability of weakly differentiable functions. Applica- tions of our results to customary classes of Sobolev-type spaces are also presented.

Singular integral operators with operator-valued kernels, and extrapolation of maximal regularity into rearrangement invariant Banach function spaces

We prove two extrapolation results for singular integral operators with operator-valued kernels, and we apply these results in order to obtain the following extrapolation of L p -maximal regularity: if an autonomous Cauchy problem on a Banach space has L p -maximal regularity for some $${p \in (1,\infty )}$$ , then it has $${\mathbb{E}_w}$$ -maximal regularity for every rearrangement invariant Banach function space $${\mathbb{E}}$$ with Boyd indices $${1 < p_\mathbb{E} \leq q_\mathbb{E} < \infty}$$ and every Muckenhoupt weight $${w \in A_{p \mathbb{E}}}$$ . We prove a similar result for nonautonomous Cauchy problems on the line.

An improvement of dimension-free Sobolev imbeddings in r.i. spaces

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(⁎)‖f⁎‖Yp(0,1)≤cp‖∇f‖Lp(Bn)∀f∈W01,p(Bn),∀n>p where Yp(0,1) is a rearrangement-invariant Banach function space independent of the dimension n, Bn is the ball in Rn of measure 1 and cp 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(logL)p/2 is optimal in the framework of the Orlicz spaces satisfying (⁎).

New John–Nirenberg inequalities for martingales

Abstract A new John–Nirenberg theorem for martingales is proved in this paper. That is, B M O E are all equivalent for any rearrangement invariant Banach function space E .