Weak type inequalities for the ℓ 1-summability of higher dimensional Fourier transforms

Under some weak conditions on θ, it was verified in [21, 17] that the maximal operator of the ? ₁-θ-means of a tempered distribution is bounded from H _p (? ^d ) to L _p (? ^d ) for all d/(d + α) < p ≤ ∞, where 0 < α ≤ 1 depends only on θ. In this paper, we prove that the maximal operator is bounded from H _d/(d+α)(? ^d ) to the weak L _d/(d+α)(? ^d ) space. The analogous result is given for Fourier series, as well. Some special cases of the ? ₁-θ-summation are considered, such as the Weierstrass, Picard, Bessel, Fejér, de La Vallée-Poussin, Rogosinski and Riesz summations.     

Commutators of Marcinkiewicz integral with rough kernels on Sobolev spaces

In this paper, the authors give the boundedness of the commutator [b, ??_??,?? ] from the homogeneous Sobolev space $\dot L_\gamma ^p \left( {\mathbb{R}^n } \right)$ to the Lebesgue space L ^p (? ⁿ ) for 1 < p < ??, where ??_??,?? denotes the Marcinkiewicz integral with rough hypersingular kernel defined by $\mu _{\Omega ,\gamma } f\left( x \right) = \left( {\int_0^\infty {\left| {\int_{\left| {x - y} \right| \leqslant t} {\frac{{\Omega \left( {x - y} \right)}} {{\left| {x - y} \right|^{n - 1} }}f\left( y \right)dy} } \right|^2 \frac{{dt}} {{t^{3 + 2\gamma } }}} } \right)^{\frac{1} {2}} ,$ , with ?? ?? L ¹(S ^n?1) for $0 < \gamma < min\left\{ {\frac{n} {2},\frac{n} {p}} \right\}$ or ?? ?? L(log⁺ L) ^?? (S ^n?1) for $\left| {1 - \frac{2} {p}} \right| < \beta < 1\left( {0 < \gamma < \frac{n} {2}} \right)$ , respectively.     

Θ-summability of Fourier series

A general summability method of orthogonal series is given with the help of an integrable function Θ. Under some conditions on Θ we show that if the maximal Fejér operator is bounded from a Banach space X to Y, then the maximal Θ-operator is also bounded. As special cases the trigonometric Fourier, Walsh, Walsh--Kaczmarz, Vilenkin and Ciesielski--Fourier series and the Fourier transforms are considered. It is proved that the maximal operator of the Θ-means of these Fourier series is bounded from H _p to L _p (1/2<p≤; ∞) and is of weak type (1,1). In the endpoint case p=1/2 a weak type inequality is derived. As a consequence we obtain that the Θ-means of a function f∈L ₁ converge a.e. to f. Some special cases of the Θ-summation are considered, such as the Weierstrass, Picar, Bessel, Riesz, de la Vallée-Poussin, Rogosinski and Riemann summations. Similar results are verified for several-dimensional Fourier series and Hardy spaces.     

On the Behaviour of the spectral characteristic of Feigenbaum’s map

Let T _g : [?1, 1] ?? [?1, 1] be the Feigenbaum map. It is well known that T _g has a Cantor-type attractor F and a unique invariant measure ??₀ supported on F. The corresponding unitary operator (U _g ??)(x) = ??(g(x)) has pure point spectrum consisting of eigenvalues ?? _n,r , n ?? 1, 0 ?? r ?? 2 ^n?1 ? 1 with eigenfunctions e _r ⁽ⁿ⁾ (x). Suppose that f ?? C ¹([?1, 1]), f?? is absolutely continuous on [?1, 1] and f?? ?? L ^p ([?1, 1], d??₀), p > 1. Consider the sum of the amplitudes of the spectral measure of f: $$ Sn(f): = \sum\limits_{r = 0}^{2^n - 1} {|\rho _r^{(n)} |^2 ,\rho _r^{(n)} = \int\limits_{ - 1}^1 {f(x)\overline {e_r^{(n)} (x)} d\mu _o } } (x). $$ Using the thermodynamic formalism for T _g we prove that S _n (f) ?? 2^?n q ⁿ , as n ?? ??, where the constant q ?? (0, 1) does not depend on f.     

Uniform approximation of Poisson integrals of functions from the class H ω by de la Vallée Poussin sums

We obtain asymptotic equalities for least upper bounds of deviations in the uniform metric of de la Vallée Poussin sums on the sets C _?? ^q H _?? of Poisson integrals of functions from the class H _?? generated by convex upwards moduli of continuity ??(t) which satisfy the condition ??(t)/t ?? ?? as t ?? 0. As an implication, a solution of the Kolmogorov-Nikol??skii problem for de la Vallée Poussin sums on the sets of Poisson integrals of functions belonging to Lipschitz classes H ^??, 0 < ?? < 1, is obtained.     

The maximal Fejér operator on the spacesH p (R×···×R)

It is shown that the maximal operator of the Fejér means of a tempered distribution is bounded from thed-dimensional Hardy spaceH _p (R×···×R) toL _p (R ^d ) (1/2<p<∞) and is of weak type (H ₁ ^?i ,L ₁) (i=1,…,d), where the Hardy spaceH ₁ ^?i is defined by a hybrid maximal function. As a consequence, we obtain that the Fejér means of a functionf ∈H ₁ ^?i ?L(logL) ^d?1 converge a.e. to the function in question. Moreover, we prove that the Fejér means are uniformly bounded onH _p (R×···×R) whenever 1/2<p<∞. Thus, in casef ∈H _p (R×···×R) the Fejér means converge tof inH _p (R×···×R) norm. The same results are proved for the conjugate Fejér means, too.     

Truncation and aliasing errors for Whittaker-Kotelnikov-Shannon sampling expansion

Let B _?? ^p , 1 ?? p < ??, be the space of all bounded functions from L _p (?) which can be extended to entire functions of exponential type ??. The uniform error bounds for truncated Whittaker-Kotelnikov-Shannon series based on local sampling are derived for functions f ?? B _?? ^p without decay assumption at infinity. Then the optimal bounds of the aliasing error and truncation error of Whittaker-Kotelnikov-Shannon expansion for non-bandlimited functions from Sobolev classes U(W _p ^r (?)) are determined up to a logarithmic factor.     

On the integral modulus of continuity of Fourier series

We study L ^p -integrability (1<p<??) of a sum ?? of trigonometric series under the assumptions that the sequence of coefficients of ?? belongs to the class $\overline{\mathrm{GM}}_{\theta}^{r}$ . Then we discuss the relations between the properties of ?? and the properties of the sequence (?? _n )??GM(??,r), and deduce an estimate for modulus of continuity of ?? in L ^p norm.     

Weighted Bounds for Variational Walsh–Fourier Series

For 1<p<?? and a weight w??A _p and a function in L ^p ([0,1],w) we show that variational sums with sufficiently large exponents of its Walsh?CFourier series are bounded in L ^p (w). This strengthens a result of Hunt?CYoung and is a weighted extension of a variation norm Carleson theorem of Oberlin?CSeeger?CTao?CThiele?CWright. The proof uses phase plane analysis and a weighted extension of a variational inequality of Lépingle.     

Characterization of the convergence of weighted averages of sequences and functions

The weighted averages of a sequence (c _k ), c _k ?? ?, with respect to the weights (p _k ), p _k ?? 0, with {fx135-1} are defined by {fx135-2} while the weighted average of a measurable function f: ?₊ ?? ? with respect to the weight function p(t) ?? 0 with {fx135-3}. Under mild assumptions on the weights, we give necessary and sufficient conditions under which the finite limit ?? _n ?? L as n ?? ?? or ??(t) ?? L as t ?? ?? exists, respectively. These characterizations may find applications in probability theory.     

Local Hardy spaces of Musielak-Orlicz type and their applications

Letφ:R n × [0,∞) → [0,∞) be a function such that φ(x,·) is an Orlicz function and (·,t) ∈ A ∞loc (Rn) (the class of local weights introduced by Rychkov).In this paper,the authors introduce a local Musielak-Orlicz Hardy space hφ(Rn) by the local grand maximal function,and a local BMO-type space bmoφ(Rn) which is further proved to be the dual space of hφ(Rn).As an application,the authors prove that the class of pointwise multipliers for the local BMO-type space bmo φ (Rn),characterized by Nakai and Yabuta,is just the dual of L 1 (Rn) + h Φ 0 (Rn),where φ is an increasing function on (0,∞) satisfying some additional growth conditions and Φ 0 a Musielak-Orlicz function induced by φ.Characterizations of hφ(Rn),including the atoms,the local vertical and the local nontangential maximal functions,are presented.Using the atomic characterization,the authors prove the existence of finite atomic decompositions achieving the norm in some dense subspaces of hφ(Rn),from which,the authors further deduce some criterions for the boundedness on hφ(Rn) of some sublinear operators.Finally,the authors show that the local Riesz transforms and some pseudo-differential operators are bounded on hφ(Rn).     

Commutator of hypersingular integral with rough kernels on Sobolev spaces

In this paper, the authors give the boundedness of the commutator of hypersingular integral T γ from the homogeneous Sobolev space Lpγ (Rn) to the Lebesgue space Lp(Rn) for 1p∞ and 0 γ min{ n/2 , n/p }.     

? 1-Summability of d-Dimensional Fourier Transforms

A general summability method of more-dimensional Fourier transforms is given with the help of a continuous function θ. Under some weak conditions on θ we show that the maximal operator of the ℓ ₁–θ-means of a tempered distribution is bounded from H _p (ℝ ^d ) to L _p (ℝ ^d ) for all d/(d+α)<p≤∞ and, consequently, is of weak type (1,1), where 0<α≤1 depends only on θ. As a consequence we obtain a generalization of the one-dimensional summability result due to Lebesgue, more exactly, the ℓ ₁–θ-means of a function f∈L ₁(ℝ ^d ) converge a.e. to f. Moreover, we prove that the ℓ ₁–θ-means are uniformly bounded on the spaces H _p (ℝ ^d ), and so they converge in norm (d/(d+α)<p<∞). Similar results are shown for conjugate functions. Some special cases of the ℓ ₁–θ-summation are considered, such as the Weierstrass, Picar, Bessel, Fejér, de La Vallée-Poussin, Rogosinski, and Riesz summations.     

Kolmogorov-type inequalities for norms of Riesz derivatives of functions of several variables with Laplacian bounded in L ∞ and related problems

Let L _∞,∞ ^Δ (? ^m ) be the space of functions f ∈ L _∞(? ^m ) such that Δf ∈ L _∞(? ^m ). We obtain new sharp Kolmogorov-type inequalities for the L _∞-norms of the Riesz derivatives D ^α f of the functions f ∈ L _∞,∞ ^Δ (? ^m ) and solve the Stechkin problem of approximating an unbounded operator D ^α by bounded operators on the class f ∈ L _∞(? ^m ) such that ‖Δf‖_∞ ≤ 1, and also the problem of the best recovery of the operator D ^α from elements of this class given with error δ.     

Centerpole sets for colorings of abelian groups

A subset C?G of a group G is called k-centerpole if for each k-coloring of G there is an infinite monochromatic subset G, which is symmetric with respect to a point c??C in the sense that S=cS ^?1 c. By c _k (G) we denote the smallest cardinality c _k (G) of a k-centerpole subset in G. We prove that c _k (G)=c _k (? ^m ) if G is an abelian group of free rank m??k. Also we prove that c ₁(? ⁿ⁺¹)=1, c ₂(? ⁿ⁺²)=3, c ₃(? ⁿ⁺³)=6, 8??c ₄(? ⁿ⁺⁴)??c ₄(?⁴)=12 for all n????, and ${\frac{1}{2}(k^{2}+3k-4)\le c_{k}(\mathbb{Z}^{n})\le2^{k}-1-\max_{s\le k-2}\binom {k-1}{s-1}}$ for all n??k??4.     

Best approximation with wavelets in weighted Orlicz spaces

Democracy functions of wavelet admissible bases are computed for weighted Orlicz Spaces L ^??(w) in terms of the fundamental function of L ^??(w). In particular, we prove that these bases are greedy in L ^??(w) if and only if L ^??(w) =?L ^p (w), 1?<?p?<???. Also, sharp embeddings for the approximation spaces are given in terms of weighted discrete Lorentz spaces. For L ^p (w) the approximation spaces are identified with weighted Besov spaces.     

Predual Spaces of Banach Completions of Orlicz-Hardy Spaces Associated with Operators

Let L be a linear operator in L ²(? ⁿ ) and generate an analytic semigroup {e ^?tL } _t??0 with kernels satisfying an upper bound of Poisson type, whose decay is measured by ??(L)??(0,??]. Let ?? on (0,??) be of upper type 1 and of critical lower type $\widetilde{p}_{0}(\omega)\in(n/(n+\theta(L)),1]$ and ??(t)=t ^?1/?? ^?1(t ^?1) for t??(0,??). In this paper, the authors first introduce the VMO-type space VMO _??,L (? ⁿ ) and the tent space $T^{\infty}_{\omega,\mathrm{v}}({\mathbb{R}}^{n+1}_{+})$ and characterize the space VMO _??,L (? ⁿ ) via the space $T^{\infty}_{\omega,\mathrm{v}}({{\mathbb{R}}}^{n+1}_{+})$ . Let $\widetilde{T}_{\omega}({{\mathbb{R}}}^{n+1}_{+})$ be the Banach completion of the tent space $T_{\omega}({\mathbb{R}}^{n+1}_{+})$ . The authors then prove that $\widetilde{T}_{\omega}({\mathbb{R}}^{n+1}_{+})$ is the dual space of $T^{\infty}_{\omega,\mathrm{v}}({\mathbb{R}}^{n+1}_{+})$ . As an application of this, the authors finally show that the dual space of $\mathrm{VMO}_{\rho,L^{\ast}}({\mathbb{R}}^{n})$ is the space B _??,L (? ⁿ ), where L ^* denotes the adjoint operator of L in L ²(? ⁿ ) and B _??,L (? ⁿ ) the Banach completion of the Orlicz-Hardy space H _??,L (? ⁿ ). These results generalize the known recent results by particularly taking ??(t)=t for t??(0,??).     

A lower bound in Nehari’s theorem on the polydisc

By theorems of Ferguson and Lacey (d = 2) and Lacey and Terwilleger (d > 2), Nehari??s theorem (i.e., if H _?? is a bounded Hankel form on H ²(D ^d ) with analytic symbol ??, then there is a function ?? in L ^??(T ^d ) such that ?? is the Riesz projection of g4) is known to hold on the polydisc D ^d for d > 1. A method proposed in Helson??s last paper is used to show that the constant C _d in the estimate ??????_?? ?? C _d ??H _?? ?? grows at least exponentially with d; it follows that there is no analogue of Nehari??s theorem on the infinite-dimensional polydisc.     

Inequivalent measures of noncompactness

Two homogeneous measures of noncompactness ?? and ?? on an infinite dimensional Banach space X are called ??equivalent?? if there exist positive constants b and c such that b ??(S)??? ??(S)??? c ??(S) for all bounded sets ${S\subset X}$ . If such constants do not exist, the measures of noncompactness are ??inequivalent.?? We ask a foundational question which apparently has not previously been considered: For what infinite dimensional Banach spaces do there exist inequivalent measures of noncompactness on X? We provide here the first examples of inequivalent measures of noncompactness. We prove that such inequivalent measures exist if X is a Hilbert space; or if (??, ??,???) is a general measure space, 1??? p??? ??, and X?=?L ^p (??, ??,???); or if K is a compact Hausdorff space and X?=?C(K); or if K is a compact metric space, 0?<??? ?? 1, and X?=?C ^0,??(K), the Banach space of H?lder continuous functions with H?lder exponent ??. We also prove the existence of such inequivalent measures of noncompactness if ?? is an open subset of ${\mathbb{R}^n}$ and X is the Sobolev space W ^m,p (??). Our motivation comes from questions about existence of eigenvectors of homogeneous, continuous, order-preserving cone maps f : C??C and from the closely related issue of giving the proper definition of the ??cone essential spectral radius?? of such maps. These questions are considered in the companion paper [28]; see, also, [27].     

Real analytic expansion of spectral projections and extension of the Hecke-Bochner identity

In this article, we review the Weyl correspondence of bigraded spherical harmonics and use it to extend the Hecke-Bochner identities for the spectral projections f × φ _k ^n?1 for function f ∈ L ^p (? ⁿ ) with 1 ≤ p ≤ ∞. We prove that spheres are sets of injectivity for the twisted spherical means with real analytic weight. Then, we derive a real analytic expansion for the spectral projections f × φ _k ^n?1 for function f ∈ L ²(? ⁿ ). Using this expansion we deduce that a complex cone can be a set of injectivity for the twisted spherical means.