Invariant Function Spaces Research Articles

Annihilator Methods in Discrete Spectral Synthesis

Spectral synthesis deals with the description of translation invariant function spaces on groups. On commutative Abelian groups the basic building bricks of spectral synthesis are the exponential monomials. In this paper we exhibit some methods which can be used to characterize exponential monomials and related function classes using ring-theoretical tools, like modified differences and annihilators.

Optimal couples of rearrangement invariant spaces for the Riesz potential on the bounded domain

Abstract We prove continuity of the Riesz potential operator in optimal couples of rearrangement invariant function spaces defined in R n with the Lebesgue measure. MSC:46E30, 46E35.

Approximation by Trigonometric Polynomials in Rearrangement Invariant Quasi Banach Function Spaces

We investigate the approximation properties of trigonometric polynomials in rearrangement invariant quasi Banach function spaces. Relations between moduli of smoothness of the derivatives of a functions and those of the functions itself are studied. We prove a theorem on the relationship between the modulus of smoothness and the best approximation in rearrangement invariant quasi Banach function spaces.

Quasilinear Schrödinger equations, II: Small data and cubic nonlinearities

In part I of this project we examined low regularity local well-posedness for generic quasilinear Schr\"odinger equations with small data. This improved, in the small data regime, the preceding results of Kenig, Ponce, and Vega as well as Kenig, Ponce, Rolvung, and Vega. In the setting of quadratic interactions, the (translation invariant) function spaces which were utilized incorporated an $l^1$ summability over cubes in order to account for Mizohata's integrability condition, which is a necessary condition for the $L^2$ well-posedness for the linearized equation. For cubic interactions, this integrability condition meshes better with the inherent $L^2$ nature of the Schr\"odinger equation, and such summability is not required. Thus we are able to prove small data well-posedness in $H^s$ spaces.

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 .

Local Khintchine inequality in rearrangement invariant spaces

We prove that the local version of Khintchine inequality holds in an rearrangement invariant function space \(X\) on [0,1] if and only if the lower dilation index of the fundamental function of \(X\) is positive. A further characterization is given, based on the uniform behavior in \(X\) of the dilations of the logarithmic function. For this, a study of the space of functions acting as multiplication operators in \(X\) for the tails of Rademacher series is carried out.

Pointwise products of some Banach function spaces and factorization

The well-known factorization theorem of Lozanovskiĭ may be written in the form L1≡E⊙E′, where ⊙ means the pointwise product of Banach ideal spaces. A natural generalization of this problem would be the question when one can factorize F through E, i.e., when F≡E⊙M(E,F), where M(E,F) is the space of pointwise multipliers from E to F. Properties of M(E,F) were investigated in our earlier paper [41] and here we collect and prove some properties of the construction E⊙F. The formulas for pointwise product of Calderón–Lozanovskiĭ Eφ-spaces, Lorentz spaces and Marcinkiewicz spaces are proved. These results are then used to prove factorization theorems for such spaces. Finally, it is proved in Theorem 11 that under some natural assumptions, a rearrangement invariant Banach function space may be factorized through a Marcinkiewicz space.

Operator-valued martingale transforms in rearrangement invariant spaces and applications

In this paper the operator-valued martingale transform inequalities in rearrangement invariant function spaces are proved. Some well-known results are generalized and unified. Applications are given to classical operators such as the maximal operator and the p-variation operator of vector-valued martingales, then we can very easily obtain some new vector-valued martingale inequalities in rearrangement invariant function spaces. These inequalities are closely related to both the geometrical properties of the underlying Banach spaces and the Boyd indices of the rearrangement invariant function spaces. Finally we give an equivalent characterization of UMD Banach lattices, and also prove the Fefferman-Stein theorem in the rearrangement invariant function spaces setting.

Global wave-front sets of Banach, Fréchet and modulation space types, and pseudo-differential operators

We introduce global wave-front sets WFB(f), f∈S′(Rd), with respect to suitable Banach or Fréchet spaces B. An important special case is given by the modulation spaces B=M(ω,B), where ω is an appropriate weight function and B is a translation invariant Banach function space. We show that the standard properties for known notions of wave-front set extend to WFB(f). In particular, we prove that micro-locality and micro-ellipticity hold for a class of globally defined pseudo-differential operators Opt(a), acting continuously on the involved spaces.

A Note on Generalized Hardy-Sobolev Inequalities

We are concerned with finding a class of weight functions g so that the following generalized Hardy-Sobolev inequality holds: ∫Ωgu2≤C∫Ω|∇u|2, u∈H01(Ω), for some C>0, where Ω is a bounded domain in ℝ2. By making use of Muckenhoupt condition for the one-dimensional weighted Hardy inequalities, we identify a rearrangement invariant Banach function space so that the previous integral inequality holds for all weight functions in it. For weights in a subspace of this space, we show that the best constant in the previous inequality is attained. Our method gives an alternate way of proving the Moser-Trudinger embedding and its refinement due to Hansson.

Criteria for existence of Riesz bases consisting of root functions of Hill and 1D Dirac operators

We study the system of root functions (SRF) of Hill operator Ly=−y″+vy with a singular (complex-valued) potential v∈Hper−1 and the SRF of 1D Dirac operator Ly=i(100−1)dydx+vy with matrix L2-potential v=(0PQ0), subject to periodic or anti-periodic boundary conditions. Series of necessary and sufficient conditions (in terms of Fourier coefficients of the potentials and related spectral gaps and deviations) for SRF to contain a Riesz basis are proven. Equiconvergence theorems are used to explain basis property of SRF in Lp-spaces and other rearrangement invariant function spaces.

Potential estimates and gradient boundedness for nonlinear parabolic systems

We consider a class of parabolic systems and equations in divergence form modeled by the evolutionary p -Laplacean system u_t - \operatorname{div} (|Du|^{p-2}Du)=V(x,t) , and provide L^\infty -bounds for the spatial gradient of solutions Du via nonlinear potentials of the right hand side datum V . Such estimates are related to those obtained by Kilpeläinen and Malý [22] in the elliptic case. In turn, the potential estimates found imply optimal conditions for the boundedness of Du in terms of borderline rearrangement invariant function spaces of Lorentz type. In particular, we prove that if V\in L(n+2,1) then Du \in L^\infty_{\mathrm{loc}} , where n is the space dimension, and this gives the borderline case of a result of DiBenedetto [5]; a significant point is that the condition V \in L(n+2,1) is independent of p . Moreover, we find explicit forms of local a priori estimates extending those from [5] valid for the homogeneous case V \equiv 0 .

Approximation by polynomials in rearrangement invariant quasi Banach function spaces

In the present work we deal with the approximation properties of certain linear polynomial operators in rearrangement invariant quasi Ba- nach function spaces. We obtain some Jackson type direct theorem and sharp converse theorem of trigonometric approximation with respect to fractional positive order moduli of smoothness in these spaces.

Polynomial functions and spectral synthesis on Abelian groups

Spectral synthesis deals with the description of translation invariant function spaces. It turns out that the basic building blocks of this description are the exponential monomials, which are built up from exponential functions and polynomial functions. The author collaborated with Laczkovich [Math. Proc. Cambridge Philos. Soc. 143 (2007), no. 1, 103--120] proved that spectral synthesis holds on an Abelian group if and only if the torsion free rank of the group is finite. The author [Aequationes Math. 70 (2005), no. 1-2, 122--130] showed that the torsion free rank of an Abelian group is strongly related to the properties of polynomial functions on the group. Here we show that spectral synthesis holds on an Abelian group if and only if the ring of polynomial functions on the group is Noetherian.

Strictly singular and power-compact operators on Banach lattices

Compactness of the iterates of strictly singular operators on Banach lattices is analyzed. We provide suitable conditions on the behavior of disjoint sequences in a Banach lattice, for strictly singular operators to be Dunford-Pettis, compact or have compact square. Special emphasis is given to the class of rearrangement invariant function spaces (in particular, Orlicz and Lorentz spaces). Moreover, examples of rearrangement invariant function spaces of fixed arbitrary indices with strictly singular non power-compact operators are also presented.

The Riesz Potential Operator in Optimal Couples of Rearrangement Invariant Spaces

We prove continuity of the Riesz potential operator in optimal couples of rearrangement invariant function spaces defined in {\mathbf R}^n with the Lebesgue measure. An application is given to the Hardy–Littlewood maximal operator.

Association between temperate distributions and analytical functions in the context of wave-front sets

Let B be a translation invariant Banach function space (BF-space). In this paper we prove that every temperate distribution f can be associated with a function F analytic in the convex tube Ω = {z ...

Optimal embeddings of generalized inhomogeneous Sobolev spaces on R^n

We prove optimal embeddings in the subcritical case of the inhomogeneous Sobolev spaces built-up over function spaces in Rn with K−monotone and rearrangement invariant norm into another rearrangement invariant function spaces. The investigation is based on pointwise and integral estimates of the rearrangement or the oscillation of the rearrangement of f in terms of the rearrangement of the derivatives of f . Mathematics subject classification (2010): 46E35.

Optimal embeddings of generalized homogeneous Sobolev spaces

We prove optimal embeddings of homogeneous Sobolev spaces built over function spaces in ${\mathbb R}^n$ with $K$-monotone and rearrangement invariant norm into other rearrangement invariant function spaces. The investigation is based on pointwise and in

Translation Invariant Spaces and Asymptotic Properties of Variational Equations

We present a new perspective concerning the study of the asymptotic behavior of variational equations by employing function spaces techniques. We give a complete description of the dichotomous behaviors of the most general case of skew‐product flows, without any assumption concerning the flow, the cocycle or the splitting of the state space, our study being based only on the solvability of some associated control systems between certain function spaces. The main results do not only point out new necessary and sufficient conditions for the existence of uniform and exponential dichotomy of skew‐product flows, but also provide a clear chart of the connections between the classes of translation invariant function spaces that play the role of the input or output classes with respect to certain control systems. Finally, we emphasize the significance of each underlying hypothesis by illustrative examples and present several interesting applications.