Commutators of the fractional maximal function on variable exponent Lebesgue spaces

Let \({M_\beta }\) be the fractional maximal function. The commutator generated by \({M_\beta }\) and a suitable function b is defined by \([{M_\beta },b]f = {M_\beta }(bf) - b{M_\beta }(f)\) . Denote by P(? ⁿ ) the set of all measurable functions p(·): ? ⁿ → [1,∞) such that $1 < p_ - : = \mathop {es\sin fp(x)}\limits_{x \in \mathbb{R}^n } andp_ + : = \mathop {es\operatorname{s} \sup p(x) < \infty }\limits_{x \in \mathbb{R}^n } ,$ and by B(? ⁿ ) the set of all p(·) ∈ P(? ⁿ ) such that the Hardy-Littlewood maximal function M is bounded on L ^p(·)(? ⁿ ). In this paper, the authors give some characterizations of b for which \([{M_\beta },b]\) is bounded from L ^p(·)(? ⁿ ) into L ^q(·)(? ⁿ ), when p(·) ∈ P(? ⁿ ), 0 < β < n/p ₊ and 1/q(·) = 1/p(·) ? β/n with q(·)(n ? β)/n ∈ B(? ⁿ ).     

Regular Submanifolds in Conformal Space ${\mathbb Q}^n_p$

The authors study the regular submanifolds in the conformal space Q_p~n and introduce the submanifold theory in the conformal space Q_p~n.The first variation formula of the Willmore volume functional of pseudo-Riemannian submanifolds in the conformal spaceQ_p~n is given.Finally,the conformal isotropic submanifolds in the conformal space Q_p~n are classified.     

On the Fourier transform of the characteristic functions of domains with C 1 boundary

We consider domains D ? ? ⁿ with C ¹ boundary and study the following question: For what domains D does the Fourier transform $\widehat {{1_D}}$ of the characteristic function 1 _D belong to L ^p (? ⁿ )?     

On the diophantine equation x 2 − Dy 2 = n

We propose a method to determine the solvability of the diophantine equation x2-Dy2=n for the following two cases:(1) D = pq,where p,q ≡ 1 mod 4 are distinct primes with(q/p)=1 and(p/q)4(q/p)4=-1.(2) D=2p1p2 ··· pm,where pi ≡ 1 mod 8,1≤i≤m are distinct primes and D=r2+s2 with r,s ≡±3 mod 8.     

Comparison Inequalities for One Sided Normal Probabilities

Several sharp upper and lower bounds for the ratio of two normal probabilities $\mathbb{P}\Biggl(\,\bigcap_{i=1}^{n}\bigl\{\xi^{(1)}_i\leq \mu_i\bigr\}\Biggr)\Big/\mathbb{P}\Biggl(\,\bigcap_{i=1}^{n}\bigl\{\xi^{(0)}_i\leq \mu_i\bigr\}\Biggr)$ are given in this paper for various cases, where (ξ ₁ ⁽⁰⁾ ,ξ ₂ ⁽⁰⁾ ,…,ξ _n ⁽⁰⁾ ) and (ξ ₁ ⁽¹⁾ ,ξ ₂ ⁽¹⁾ , …,ξ _n ⁽¹⁾ ) are standard normal random variables with covariance matrices R ⁰=(r _ij ⁰ ) and R ¹=(r _ij ¹ ), respectively.     

On the chromatic numbers of spheres in ℝ n

Let χ(S _r ^n?1 )) be the minimum number of colours needed to colour the points of a sphere S _r ^n?1 of radius $r \geqslant \tfrac{1} {2}$ in ? ⁿ so that any two points at the distance 1 apart receive different colours. In 1981 P. Erd?s conjectured that χ(S _r ^n?1 )→∞ for all $r \geqslant \tfrac{1} {2}$ . This conjecture was proved in 1983 by L. Lovász who showed in [11] that χ(S _r ^n?1 ) ≥ n. In the same paper, Lovász claimed that if $r < \sqrt {\frac{n} {{2n + 2}}}$ , then χ(S _r ^n?1 ) ≤ n+1, and he conjectured that χ(S _r ^n?1 ) grows exponentially, provided $r \geqslant \sqrt {\frac{n} {{2n + 2}}}$ . In this paper, we show that Lovász’ claim is wrong and his conjecture is true: actually we prove that the quantity χ(S _r ^n?1 ) grows exponentially for any $r > \tfrac{1} {2}$ .     

Absolute convergence of Walsh-Fourier series and related results

We consider the Walsh orthonormal system on the interval [0, 1) in the Paley enumeration and the Walsh-Fourier coefficients $ \hat f $ (n), n ∈ ?, of functions f ∈ L ^p for some 1 < p ≤ 2. Our aim is to find best possible sufficient conditions for the finiteness of the series Σ _n=1 ^∞ a _n | $ \hat f $ (n)| ^r , where {a _n } is a given sequence of nonnegative real numbers satisfying a mild assumption and 0 < r < 2. These sufficient conditions are in terms of (either global or local) dyadic moduli of continuity of f. The sufficient conditions presented in the monograph [2] are special cases of our ones.     

Die metrisierbaren linearen Teilräume des Raumes\mathfrak{O}_M von L. Schwartz

In Schwartz' terminology, a real or complex valued functionf, defined and infinitely differentiable on ? _n , belongs to \(\mathfrak{O}_M \) iff, as well as any of its derivatives, is at most of polynomial growth. The topology of \(\mathfrak{O}_M \) is defined by the seminorms sup{∣?(x)D ^p f(x)∣;x∈? ⁿ }, where ? belongs to \(\mathfrak{S}\) andD ^p is any derivative. It is well-known that \(\mathfrak{O}_M \) is non-metrisable. For any μ: ? ⁿ →?, let \(\mathfrak{B}_\mu \) be the space of all infinitely differentiable functionsf satisfying, for eachp, sup{∣(1+∣x∣²)^?μ(p) D ^p f(x)∣;x∈? ⁿ }<∞, with the obvious topology. These spaces, which are of very little use elsewhere in the theory of distributions, can be conveniently applied to characterise the metrisable linear subspaces of \(\mathfrak{O}_M \) : A linear subspace of \(\mathfrak{O}_M \) is metrisable if and only if it is, algebraically and topologically, a subspace of some \(\mathfrak{B}_\mu \) .     

The “Full Müntz Theorem” inL p[0, 1] for 0<p<∞

Denote by span {f ₁,f ₂, …} the collection of all finite linear combinations of the functionsf ₁,f ₂, … over ?. The principal result of the paper is the following. Theorem (Full Müntz Theorem in L_p(A) for p ∈ (0, ∞) and for compact sets A ? [0, 1] with positive lower density at 0). Let A ? [0, 1] be a compact set with positive lower density at 0. Let p ∈ (0, ∞). Suppose (λ _j ) _j=1 ^∞ is a sequence of distinct real numbers greater than ?(1/p). Then span {x ^λ1,x ^λ2,…} is dense in L_p(A) if and only if $\sum\limits_{j = 1}^\infty {\frac{{\lambda _j + \left( {1/p} \right)}}{{\left( {\lambda _j + \left( {1/p} \right)} \right)^2 + 1}} = \infty } $ . Moreover, if $\sum\limits_{j = 1}^\infty {\frac{{\lambda _j + \left( {1/p} \right)}}{{\left( {\lambda _j + \left( {1/p} \right)} \right)^2 + 1}} = \infty } $ , then every function from the L_p(A) closure of {x ^λ1,x ^λ2,…} can be represented as an analytic function on {z ∈ ? \ (?∞,0] : |z| < r_A} restricted to A ∩ (0, r_A) where $r_A : = \sup \left\{ {y \in \mathbb{R}:\backslash ( - \infty ,0]:\left| z \right|< r_A } \right\}$ (m(·) denotes the one-dimensional Lebesgue measure). This improves and extends earlier results of Müntz, Szász, Clarkson, Erdös, P. Borwein, Erdélyi, and Operstein. Related issues about the denseness of {x ^λ1,x ^λ2,…} are also considered.     

On quasinormal,subnormal, and hyponormal Toeplitz operators

Let ? be a non-constant function inL ^∞(D) such thatφ=φ ₁+φ ₂, whereφ ₁ is an element in the Bergman spaceL _a ² (D), and \(\phi _2 \in \overline {L_a^2 (D)} \) , the space of all complex conjugates of functions inL _a ² (D). In this paper, it is shown that if 1 is an element in the closure of the range of the self-commutator ofT _?, \(T_{\bar \phi } T_\phi - T_\phi T\phi \) , then the Toeplitz operatorT _? defined ofL _a ² (D) is not quasinormal. Moreover, if \(\phi = \psi + \lambda \bar \psi \) , whereψ∈ H ^∞(D), and λεC, it is proved that ifT _? is quasinormal, thenT _? is normal. Also, the spectrum of a class of pure hyponormal Toeplitz operators is determined.     

Wavelet bases in the Lebesgue spaces on the field of p-adic numbers

In this paper we prove that p-adic wavelets form an unconditional basis in the space L ^r (? _p ⁿ ) and give the characterization of the space L ^r (? _p ⁿ ) in terms of Fourier coefficients of p-adic wavelets.Moreover, the Greedy bases in the Lebesgue spaces on the field of p-adic numbers are also established.     

Limit ultraspherical series and their approximation properties

We study new series of the form $\sum\nolimits_{k = 0}^\infty {f_k^{ - 1} \hat P_k^{ - 1} (x)} $ in which the general term $f_k^{ - 1} \hat P_k^{ - 1} (x)$ , k = 0, 1, …, is obtained by passing to the limit as α→?1 from the general term $\hat f_k^\alpha \hat P_k^{\alpha ,\alpha } (x)$ of the Fourier series $\sum\nolimits_{k = 0}^\infty {f_k^\alpha \hat P_k^{\alpha ,\alpha } (x)} $ in Jacobi ultraspherical polynomials $\hat P_k^{\alpha ,\alpha } (x)$ generating, for α> ?1, an orthonormal system with weight (1 ? x ₂)α on [?1, 1]. We study the properties of the partial sums $S_n^{ - 1} (f,x) = \sum\nolimits_{k = 0}^n {f_k^{ - 1} \hat P_k^{ - 1} (x)} $ of the limit ultraspherical series $\sum\nolimits_{k = 0}^\infty {f_k^{ - 1} \hat P_k^{ - 1} (x)} $ . In particular, it is shown that the operator S _n ^?1 (f) = S _n ^?1 (f, x) is the projection onto the subspace of algebraic polynomials p _n = p _n (x) of degree at most n, i.e., S _n (p _n ) = p _n ; in addition, S _n ^?1 (f, x) coincides with f(x) at the endpoints ±1, i.e., S _n ^?1 (f,±1) = f(±1). It is proved that the Lebesgue function Λ _n (x) of the partial sums S _n ^?1 (f, x) is of the order of growth equal to O(ln n), and, more precisely, it is proved that $\Lambda _n (x) \leqslant c(1 + \ln (1 + n\sqrt {1 - x^2 } )), - 1 \leqslant x \leqslant 1$ .     

Atomic decomposition characterizations of weighted multiparameter Hardy spaces

Let w ?? A _??. In this paper, we introduce weighted-(p, q) atomic Hardy spaces H _w ^p,q (? ⁿ ×? ^m ) for 0 < p ? 1, q >q _w and show that the weighted Hardy space H _w ^p (? ⁿ × ? ^m ) defined via Littlewood-Paley square functions coincides with H _w ^p,q (? ⁿ × ? ^m ) for 0 < p ? 1, q > q _w . As applications, we get a general principle on the H _w ^p (? ⁿ × ? ^m ) to L _w ^p (? ⁿ ×? ^m ) boundedness and a boundedness criterion for two parameter singular integrals on the weighted Hardy spaces.     

On the H p -L q boundedness of some fractional integral operators

Let A ₁, …, A _m be n × n real matrices such that for each 1 ? i ? m, A _i is invertible and A _i ? A _j is invertible for i ≠ j. In this paper we study integral operators of the form $$Tf(x) = \int {{k_1}(x - {A_{1y}}){k_2}(x - {A_{2y}}) \ldots {k_m}(x - {A_{my}})f(y){\rm{d}}y}$$ ${k_i}(y) = \sum\limits_{j \in z} {{2^{jn/{q_i}}}} \varphi i,j({2^j}y),1 \le {q_i} < \infty ,1/{q_1} + 1/q + ... + 1/q = 1 - r,0 \le r < 1, and \varphi i,j$ satisfying suitable regularity conditions. We obtain the boundedness of T: H ^p (? ⁿ ) → L ^q (? ⁿ ) for 0 < p < 1/r and 1/q = 1/p-r. We also show that we can not expect the H ^p -H ^q boundedness of this kind of operators.     

The error term in Nevanlinna’s inequality

An upper bound is given for the error termS(r, |a _j |,f) in Nevanlinna’s inequality. For given positive increasing functions p and $ with ∫ ₁ ^∞ dr/p(r) = ∫ ₁ ^∞ dr/r ?(r) = ∞, setP(r) = ∫ ₁ ^r dt/p,Ψ(r) = ∫ ₁ ^r dt/t ?(t) We prove that $$S(r, \left\{ {a_j } \right\}, f) \leqslant \log \frac{{T(r, f)\phi (T(r, f))}}{{p(r)}} + O(1)$$ holds, with a small exceptional set of r, for any finite set of points |a _j | in the extended plane and any meromorphic function f such thatΨ(T(r, f)) =O(P(r)). This improves the known results of A. Hinkkanen and Y. F. Wang. The sharpness of the estimate is also considered.     

Positive Periodic Solutions of Systems of First Order Ordinary Differential Equations

Consider the n-dimensional nonautonomous system ?(t) = A(t)G(x(t)) ? B(t)F(x(t ? τ(t))) Let u = (u ₁,…,u _n ), $f^{i}_{0}={\rm lim}_{\|{\rm u}\|\rightarrow 0}{f^{i}(\rm u)\over \|u\|}$ , $f^{i}_{\infty}={\rm lim}_{\|{\rm u}\|\rightarrow \infty}{f^{i}(\rm u)\over \|u\|}$ , i = l,…,n, F = (f ¹…,f ⁿ ), ${\rm F_{0}}={\rm max}_{i=1,\ldots,n}{f^{i}_{0}}$ and ${\rm F_{\infty}}={\rm max}_{i=1,\ldots,n}{f^{i}_{\infty}}$ . Under some quite general conditions, we prove that either F₀ = 0 and F_∞ = ∞, or F₀ = ∞ and F_∞ = 0, guarantee the existence of positive periodic solutions for the system for all λ > 0. Furthermore, we show that F₀ = F_∞ = 0, or F_∞ = F_∞ = ∞ guarantee the multiplicity of positive periodic solutions for the system for sufficiently large, or small λ, respectively. We also establish the nonexistence of the system when either F₀ and F_∞ > 0, or F₀ and F_∞, < for sufficiently large, or small λ, respectively. We shall use fixed point theorems in a cone.     

Polynomial approximation inL p(S) forp>0

For a simple polytopeS inR ^d andp>0 we show that the best polynomial approximationE _n(f)_p≡E_n(f)L_p(S) satisfies $$E_n \left( f \right)_p \leqslant C\omega _S^r \left( {f,\frac{1}{n}} \right)p,$$ where ω _S ^r is a measure of smoothness off. This result is the best possible in the sense that a weak-type converse inequality is shown and a realization of ω _S ^r (f,t)_p via polynomial approximation is proved.     

A Functional Equation Of J.A. Baker Involving Integral Means

The functional equation $$f(x)={1\over 2}\int^{x+1}_{x-1}f(t)\ dt\ \ \ {\rm for}\ \ \ x\ \in\ {\rm R}$$ has the linear functions ?(x) = a + bx (a, b ∈ ?) as trivial solutions. It is shown that there are two kinds of nontrivial solutions, (i) ?(x) = eλ_i ^x (i = 1, 2, …), where the λ_i∈ ? are the fixed points of the map z ? sinh z, and (ii) C^∞-solutions ? for which the values in the interval [?1,1] can be prescribed arbitrarily, but with the provision that ?^(j)(? 1) = ?^(j)(0) = ?^(j)(1) = 0 for all j = 0, 1, 2 …     

The maximal variation of a bounded martingale

Let \(\chi _0^n = \left\{ {X_t } \right\}_0^n \) be a martingale such that 0≦X_i≦1;i=0, …,n. For 0≦p≦1 denote by ? _p ⁿ the set of all such martingales satisfying alsoE(X₀)=p. Thevariation of a martingale χ ₀ ⁿ is denoted byV ₀ ⁿ and defined by \(V(\chi _0^n ) = E\left( {\sum {_{l = 0}^{n - 1} } \left| {X_{l + 1} - X_l } \right|} \right)\) . It is proved that $$\mathop {\lim }\limits_{n \to \infty } \left\{ {\mathop {Sup}\limits_{x_0^n \in \mathcal{M}_p^n } \left[ {\frac{1}{{\sqrt n }}V(\chi _0^n )} \right]} \right\} = \phi (p)$$ , where ?(p) is the well known normal density evaluated at itsp-quantile, i.e. $$\phi (p) = \frac{1}{{\sqrt {2\pi } }}\exp ( - \frac{1}{2}\chi _p^2 ) where \int_{ - \alpha }^{x_p } {\frac{1}{{\sqrt {2\pi } }}\exp ( - \frac{1}{2}\chi ^2 )} dx = p$$ . A sequence of martingales χ ₀ ⁿ ,n=1,2, … is constructed so as to satisfy \(\lim _{n \to \infty } (1/\sqrt n )V(\chi _0^n ) = \phi (p)\) .     

Power Majorization and Majorization of Sequences

Let \(\bar x\) , \(\bar y\ \in\ R_n\) be vectors which satisfy x₁ ≥ x₂ ≥ … ≥ x_n and y₁ ≥ y₂ >- … ≥ y_n and Σx_i = Σy_i. We say that \(\bar x\) is power majorized by \(\bar y\) if Σx_i ^p ≤ Σy_i ^p for all real p ? [0, 1] and Σx_i ^p ≥ Σy_i ^p for p ∈ [0, 1]. In this paper we give a classification of functions ? (which includes all possible positive polynomials) for which \(\bar\phi(\bar x) \leq \bar\phi(\bar y)\) (see definition below) when \(\bar x\) is power majorized \(\bar y\) . We also answer a question posed by Clausing by showing that there are vectors \(\bar x\) , \(\bar y\ \in\ R^n\) of any dimension n ≥ 4 for which there is a convex function ? such that \(\bar x\) is power majorized by \(\bar y\) and \(\bar\phi(\bar x)\ >\ \bar\phi(\bar y)\) .