Sharp continuity results for the short-time Fourier transform and for localization operators

We completely characterize the boundedness on Wiener amalgam spaces of the short-time Fourier transform (STFT), and on both L ^p and Wiener amalgam spaces of a special class of pseudodifferential operators, called localization operators. Precisely, sufficient conditions for the STFT to be bounded on the Wiener amalgam spaces W(L ^p , L ^q ) are given and their sharpness is shown. Localization operators are treated similarly: using different techniques from those employed in the literature, we relax the known sufficient boundedness conditions for these operators to be bounded on L ^p spaces and prove the optimality of our results. Next, we exhibit sufficient and necessary conditions for such operators to be bounded on Wiener amalgam spaces.     

Conical square function estimates and functional calculi for perturbed Hodge-Dirac operators in <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">P</Emphasis></Superscript>

Perturbed Hodge-Dirac operators and their holomorphic functional calculi, as investigated in the papers by Axelsson, Keith and the second author, provided insight into the solution of the Kato square-root problem for elliptic operators in L² spaces and allowed for an extension of these estimates to other systems with applications to non-smooth boundary value problems. In this paper, we determine conditions under which such operators satisfy conical square function estimates in a range of L ^p spaces, thus allowing us to apply the theory of Hardy spaces associated with an operator to prove that they have a bounded holomorphic functional calculus in those L ^p spaces. We also obtain functional calculus results for restrictions to certain subspaces, for a larger range of p. This provides a framework for obtaining L ^p results on perturbed Hodge Laplacians, generalising known Riesz transform bounds for an elliptic operator L with bounded measurable coefficients, one Sobolev exponent below the Hodge exponent, and L ^p bounds on the square-root of L by the gradient, two Sobolev exponents below the Hodge exponent. Our proof shows that the heart of the harmonic analysis in L² extends to L ^p for all p ∈ (1,∞), while the restrictions in p come from the operator-theoretic part of the L² proof. In the course of our work, we obtain some results of independent interest about singular integral operators on tent spaces and about the relationship between conical and vertical square functions.     

Singular operators and Fourier multipliers in weighted Lebesgue spaces with variable index

Mikhlin’s ideas and results related to the theory of spaces L _ρ ^p(·) with nonstandard growth are developed. These spaces are called Lebesgue spaces with variable index; they are used in mechanics, the theory of differential equations, and variational problems. The boundedness of Fourier multipliers and singular operators on the spaces L _ρ ^p(·) are considered. All theorems are derived from an extrapolation theorem due to Rubio de Francia. The considerations essentially use theorems on the boundedness of operators and maximal Hardy-Littlewood functions on Lebesgue spaces with constant index.     

Characterization of Lipschitz Functions via the Commutators of Singular and Fractional Integral Operators in Variable Lebesgue Spaces

We obtain characterizations of a variable version of Lipschitz spaces in terms of the boundedness of commutators of Calderón-Zygmund and fractional type operators in the context of the variable exponent Lebesgue spaces L ^p(?), where the symbols of the commutators belong to the Lipschitz spaces. A useful tool is a pointwise estimate involving the sharp maximal operator of the commutator and certain associated maximal operators, which is new even in the classical context. Some boundedness properties of the commutators between Lebesgue and Lipschitz spaces in the variable context are also proved.     

Interpolation of operators of weak type (ϕ, ϕ)

Considering the measurable and nonnegative functions ? on the half-axis [0, ∞) such that ?(0) = 0 and ?(t) → ∞ as t → ∞, we study the operators of weak type (?, ?) that map the classes of ?-Lebesgue integrable functions to the space of Lebesgue measurable real functions on ?ⁿ. We prove interpolation theorems for the subadditive operators of weak type (?₀, ?₀) bounded in L _∞(?ⁿ) and subadditive operators of weak types (?₀, ?₀) and (?₁, ?₁) in L _?(? ⁿ ) under some assumptions on the nonnegative and increasing functions ?(x) on [0, ∞). We also obtain some interpolation theorems for the linear operators of weak type (?₀, ?₀) bounded from L _∞(?ⁿ) to BMO(? ⁿ). For the restrictions of these operators to the set of characteristic functions of Lebesgue measurable sets, we establish some estimates for rearrangements of moduli of their values; deriving a consequence, we obtain a theorem on the boundedness of operators in rearrangement-invariant spaces.     

The fractional Hausdorff operators on the Hardy spaces <Emphasis Type="Italic">H</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>(ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>)

In this paper, we study the high-dimensional fractional Hausdorff operators and establish their boundedness on the real Hardy spaces H ^p (? ⁿ ) for 0 < p < 1.     

On a Class of Weakly Coupled Systems of Elliptic Operators with Unbounded Coefficients

We consider a class of weakly coupled systems of elliptic operators \({\mathcal{A}}\) with unbounded coefficients defined in \({\mathbb{R}^N}\). We prove that a semigroup (T(t))t ≥ 0 of bounded linear operators can be associated with \({\mathcal{A}}\), in a natural way, in the space of all bounded and continuous functions. We prove a compactness property of the semigroup as well as some uniform estimates on the derivatives of the function T(t)f, when f belongs to some spaces of Hölder continuous functions, which are the key tools to prove some optimal Schauder estimates for the solution to some nonhomogeneous elliptic equations and Cauchy problems associated with the operator \({\mathcal{A}}\). Under suitable additional conditions, we then prove that the restriction of the semigroup to the subspace of smooth and compactly supported functions extends by a strongly continuous semigroup to L ^p -spaces over \({\mathbb{R}^N}\), related to the Lebesgue measure, when \({p \in [1,\infty)}\). We also provide sufficient conditions for this semigroup to be analytic when \({p \in [1,\infty)}\). Finally, we prove some L ^p ?L ^q -estimates.     

Bloom’s inequality: commutators in a two-weight setting

In 1985, Bloom characterized the boundedness of the commutator [b, H] as a map between a pair of weighted L^p spaces, where both weights are in A_p. The characterization is in terms of a novel BMO condition. We give a ‘modern’ proof of this result, in the case of p = 2. In a subsequent paper, this argument will be used to generalize Bloom’s result to all Calderón–Zygmund operators and dimensions.     

A predual of a predual of <Emphasis Type="Italic">B</Emphasis><Subscript><Emphasis Type="Italic">σ</Emphasis></Subscript> and its applications to commutators

A predual of B_σ-spaces is investigated. A predual of a predual of B_σ-spaces is also investigated, which can be used to investigate the boundedness property of the commutators. The relation between Herz spaces and local Morrey spaces is discussed. As an application of the duality results, one obtains the boundedness of the singular integral operators, the Hardy-Littlewood maximal operators and the fractional integral operators, as well as the commutators generated by the bounded mean oscillation (BMO) and the singular integral operators. What is new in this paper is that we do not have to depend on the specific structure of the operators. The results on the boundedness of operators are formulated in terms of ?_σ-spaces and B_σ-spaces together with the detailed comparison of the ones in Herz spaces and local Morrey spaces. Another application is the nonsmooth atomic decomposition adapted to B_σ-spaces.     

Generalized fractional integrals in <Emphasis Type="Italic">p</Emphasis>-adic morrey and Herz spaces

For Riesz potential I ^β (f) on p-adic linear space Q _p ⁿ and its modification \(\widetilde{I^\beta }(f)\) we give sufficient conditions of their boundedness from radialMorrey space to anotherMorrey or Campanato space. Also we study the boundedness of modified Riesz potential \(\widetilde{I^\beta }(f)\) from Herz space to special Campanato spaces.     

Wavelets and Bidemocratic Pairs in Weighted Norm Spaces

A complete characterization of weight functions for which the higher-rank Haar wavelets are greedy bases in weighted L^p spaces is given. The proof uses the new concept of a bidemocratic pair for a Banach space and also pairs (Φ, Φ), where Φ is an orthonormal system of bounded functions in the spaces L^p, p≠2.     

Maximal Regularity for Non-autonomous Equations with Measurable Dependence on Time

In this paper we study maximal L ^p -regularity for evolution equations with time-dependent operators A. We merely assume a measurable dependence on time. In the first part of the paper we present a new sufficient condition for the L ^p -boundedness of a class of vector-valued singular integrals which does not rely on Hörmander conditions in the time variable. This is then used to develop an abstract operator-theoretic approach to maximal regularity. The results are applied to the case of m-th order elliptic operators A with time and space-dependent coefficients. Here the highest order coefficients are assumed to be measurable in time and continuous in the space variables. This results in an L ^p (L ^q )-theory for such equations for \(p,q\in (1, \infty )\). In the final section we extend a well-posedness result for quasilinear equations to the time-dependent setting. Here we give an example of a nonlinear parabolic PDE to which the result can be applied.     

Absence of Non-Constant Harmonic Functions with ℓp-gradient in some Semi-Direct Products

We study potential operators associated with Laguerre function expansions of convolution and Hermite types, and with Dunkl-Laguerre expansions. We prove qualitatively sharp estimates of the corresponding potential kernels. Then we characterize those 1 ≤ p,q ≤ 8, for which the potential operators are L ^p - L ^q bounded. These results are sharp analogues of the classical Hardy-Littlewood-Sobolev fractional integration theorem in the Laguerre and Dunkl-Laguerre settings.     

Weak- and strong-type inequality for the cone-like maximal operator in variable Lebesgue spaces

The classical Hardy-Littlewood maximal operator is bounded not only on the classical Lebesgue spaces L_p(R^d) (in the case p > 1), but (in the case when 1/p(·) is log-Hölder continuous and p- = inf{p(x): x ∈ R^d > 1) on the variable Lebesgue spaces L_p(·)(R^d), too. Furthermore, the classical Hardy-Littlewood maximal operator is of weak-type (1, 1). In the present note we generalize Besicovitch’s covering theorem for the so-called γ-rectangles. We introduce a general maximal operator M_s^γδ, and with the help of generalized Φ-functions, the strong- and weak-type inequalities will be proved for this maximal operator. Namely, if the exponent function 1/p(·) is log-Hölder continuous and p- ≥ s, where 1 ≤ s ≤ ∞ is arbitrary (or p- ≥ s), then the maximal operator M_s^γδ is bounded on the space L_p(·)(R^d) (or the maximal operator is of weak-type (p(·), p(·))).     

Generalized Absolute Convergence of Series of Fourier Coefficients With Respect to Haar Type Systems

For the orthogonal systems ofHaar type, introduced by Vilenkin in 1958, we study absolute convergence of series composed from positive powers of Fourier coefficients with multiplicators from the Gogoladze–Meskhia class. The conditions for convergence of the series are given in terms of either best approximations of functions in L ^p spaces by polynomials with respect to Haar type systems or fractional modulus of continuity for functions from the Wiener spaces V _p , p > 1. We establish the sharpness of the obtained results.     

Boundedness of Hausdorff operators on Lebesgue spaces and Hardy spaces

In this paper, we study the boundedness of the Hausdorff operator H_? on the real line R. First, we start with an easy case by establishing the boundedness of the Hausdorff operator on the Lebesgue space L~p(R)and the Hardy space H~1(R). The key idea is to reformulate H_? as a Calder′on-Zygmund convolution operator,from which its boundedness is proved by verifying the Hrmander condition of the convolution kernel. Secondly,to prove the boundedness on the Hardy space H~p(R) with 0 p 1, we rewrite the Hausdorff operator as a singular integral operator with the non-convolution kernel. This novel reformulation, in combination with the Taibleson-Weiss molecular characterization of H~p(R) spaces, enables us to obtain the desired results. Those results significantly extend the known boundedness of the Hausdorff operator on H~1(R).     

Approximate identities on some homogeneous Banach spaces

We study convergence of approximate identities on some complete semi-normed or normed spaces of locally L ^p functions where translations are isometries, namely Marcinkiewicz spaces \({\mathcal{M}^{p}}\) and Stepanoff spaces \({\mathcal{S}^p}\), 1 ≤ p < ∞, as well as others where translations are not isometric but bounded (the bounded p-mean spaces M ^p ) or even unbounded (\({M^{p}_{0}}\)). We construct a function f that belongs to these spaces and has the property that all approximate identities \({\phi_\varepsilon * f}\) converge to f pointwise but they never converge in norm.     

Approach regions for the square root of the Poisson kernel and boundary functions in certain Orlicz spaces

If the Poisson integral of the unit disc is replaced by its square root, it is known that normalized Poisson integrals of L ^p and weak L ^p boundary functions converge along approach regions wider than the ordinary nontangential cones, as proved by Rönning and the author, respectively. In this paper we characterize the approach regions for boundary functions in two general classes of Orlicz spaces. The first of these classes contains spaces L _Φ having the property L ^∞ ? L _Φ L ^p , 1 ? p > ∞. The second contains spaces L _Φ that resemble L ^p spaces.