Boundedness of the fractional maximal operator in generalized Morrey space on the Heisenberg group

In this paper we study the fractional maximal operator M _α , 0 ≤ α < Q on the Heisenberg group ? _n in the generalized Morrey spaces M _{p, ?}(? _n ), where Q = 2n + 2 is the homogeneous dimension of ? _n . We find the conditions on the pair (? ₁, ? ₂) which ensures the boundedness of the operator M _α from one generalized Morrey space M _{p, ?1}(? _n ) to another M _{q, ?2}(? _n ), 1 < p < q < ∞, 1/p?1/q = α/Q, and from the space M _{1, ?1}(? _n ) to the weak space WM _{q, ?2}(? _n ), 1 < q < ∞, 1 ? 1/q = α/Q. We also find conditions on the φ which ensure the Adams type boundedness of M _α from $M_{p,\phi ^{\tfrac{1} {p}} } \left( {\mathbb{H}_n } \right)$ to $M_{q,\phi ^{\tfrac{1} {q}} } \left( {\mathbb{H}_n } \right)$ for 1 < p < q < ∞ and from M _{1, ?}(? _n ) to $WM_{q,\phi ^{\tfrac{1} {q}} } \left( {\mathbb{H}_n } \right)$ for 1 < q < ∞. As applications we establish the boundedness of some Schrödinger type operators on generalized Morrey spaces related to certain nonnegative potentials V belonging to the reverse Hölder class B _∞(” _n ).     

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.     

Weighted inequalities for fractional type operators with some homogeneous kernels

In this paper, we study integral operators of the form Tαf(x)=∫Rn|x-A1y|-α1 ··· |x-Amy|-αmf(y)dy,where Ai are certain invertible matrices, αi 0, 1 ≤ i ≤ m, α1 + ··· + αm = n-α, 0 ≤α n. For 1/q = 1/p-α/n , we obtain the Lp (Rn, wp)-Lq(Rn, wq) boundedness for weights w in A(p, q) satisfying that there exists c 0 such that w(Aix) ≤ cw(x), a.e. x ∈ Rn , 1 ≤ i ≤ m.Moreover, we obtain theappropriate weighted BMO and weak type estimates for certain weights satisfying the above inequality. We also give a Coifman type estimate for these operators.     

Estimates for the commutator of bilinear Fourier multiplier

Let b ₁, b ₂ ∈ BMO(? ⁿ ) and T _σ be a bilinear Fourier multiplier operator with associated multiplier σ satisfying the Sobolev regularity that $\mathop {\sup }\limits_{\kappa \in \mathbb{Z}} \left\| {\sigma _\kappa } \right\|_{W^{s_1 ,s_2 } \left( {\mathbb{R}^{2n} } \right)} < \infty$ for some s ₁, s ₂ ∈ (n/2, n]. In this paper, the behavior on $L^{p_1 } \left( {\mathbb{R}^n } \right) \times L^{p_2 } \left( {\mathbb{R}^n } \right)\left( {p_1 ,p_2 \in \left( {1,\infty } \right)} \right)$ , on H ¹(? ⁿ ) × L ^p2 (? ⁿ ) (p ₂ ∈ [2,∞)), and on H ¹(? ⁿ ) × H ¹(? ⁿ ), is considered for the commutator $T_{\sigma ,\vec b}$ defined by $${T_{\sigma ,\vec b}}({f_1},{f_2})(x) = {b_1}(x){T_\sigma }({f_1},{f_2})(x) - {T_\sigma }({b_1}{f_1},{f_2})(x) + {b_2}(x){T_\sigma }({f_1},{f_2})(x) - {T_\sigma }({f_1},{b_2}{f_2})(x)$$ . By kernel estimates of the bilinear Fourier multiplier operators and employing some techniques in the theory of bilinear singular integral operators, it is proved that these mapping properties are very similar to those of the bilinear Fourier multiplier operator which were established by Miyachi and Tomita.     

Existence and Concentration of Bound States of a Class of Nonlinear SchrSdinger Equations in R2 with Potential Tending to Zero at Infinity

In this paper, we establish the existence and concentration of solutions of a class of nonlinear Schr?dinger equation $$- \varepsilon ^2 \Delta u_\varepsilon + V\left( x \right)u_\varepsilon = K\left( x \right)\left| {u_\varepsilon } \right|^{p - 2} u_\varepsilon e^{\alpha _0 \left| {u_\varepsilon } \right|^\gamma } , u_\varepsilon > 0, u_\varepsilon \in H^1 \left( {\mathbb{R}^2 } \right),$$ where 2 < p < ∞, α ₀ > 0, 0 < γ < 2. When the potential function V (x) decays at infinity like (1 + |x|)^?α with 0 < α ≤ 2 and K(x) > 0 are permitted to be unbounded under some necessary restrictions, we will show that a positive H ¹(?²)-solution u _? exists if it is assumed that the corresponding ground energy function G(ξ) of nonlinear Schr?dinger equation $- \Delta u + V\left( \xi \right)u = K\left( \xi \right)\left| u \right|^{p - 2} ue^{\alpha _0 \left| u \right|^\gamma }$ has local minimum points. Furthermore, the concentration property of u _? is also established as ? tends to zero.     

Sharp constants in the doubly weighted Hardy-Littlewood-Sobolev inequality

It is well known that the doubly weighted Hardy-Littlewood-Sobolev inequality is as follows,Z Rn Z Rn f(x)g(y)|x||x.y||y|dxdy6 B(p,q,,,,n)kfkLp(Rn)kgkLq(Rn).The main purpose of this paper is to give the sharp constants B(p,q,,,,n)for the above inequality for three cases:(i)p=1 and q=1;(ii)p=1 and 1q 6∞,or 1p 6∞and q=1;(iii)1p,q∞and 1p+1q=1.In addition,the explicit bounds can be obtained for the case 1p,q∞and 1p+1q1.     

Оценки отклонений оп тимальных сеток вL p -норме и теория квад ратурных формул

Let E^s=[0, 1]^s be then-dimensional unit cube, 1<p<∞, anda=(a ₁, ...,a _s ) some set of natural numbers. Denote byL _p ^(a) , (E ^s ) the class of functionsf: E ^s → C for which $$\left\| {\frac{{\partial ^{b_1 + \cdots + b_s } f}}{{\partial x_1^{b_1 } \cdots \partial x_s^{b_s } }}} \right\|_p \leqslant 1,$$ where $$0< b_1< a_1 , ..., 0< b_s< a_s .$$ Set $$R_p^{\left( a \right)} \left( N \right) = \mathop {\inf }\limits_{card \mathfrak{S} = N} R_\mathfrak{S} \left( {L_p^{\left( a \right)} \left( {E^s } \right)} \right),$$ where $R_\mathfrak{S} \left( {L_p^{\left( a \right)} \left( {E^s } \right)} \right)$ is the error of the quadrature formulas on the mesh $\mathfrak{S}$ (for the classL _p ^(a) (E ^s )), consisting of N nodes and weights, and the infimum is taken with respect to all possibleN nodes and weights. In this paper, the two-sided estimate $$\frac{{\left( {\log N} \right)^{{{\left( {l - 1} \right)} \mathord{\left/ {\vphantom {{\left( {l - 1} \right)} 2}} \right. \kern-\nulldelimiterspace} 2}} }}{{N^d }} \ll _{p, a} R^{\left( a \right)} \left( N \right) \ll _{p, a} \frac{{\left( {\log N} \right)^{{{\left( {l - 1} \right)} \mathord{\left/ {\vphantom {{\left( {l - 1} \right)} 2}} \right. \kern-\nulldelimiterspace} 2}} }}{{N^d }}$$ is proved for every natural numberN > 1, whered=min{a ₁, ...,a _s }, whilel is the number of those components of a which coincide withd. An analogous result is proved for theL _p -norm of the deviation of meshes.     

Anisotropic Hardy-Lorentz spaces and their applications

Let p ∈(0, 1], q ∈(0, ∞] and A be a general expansive matrix on Rn. We introduce the anisotropic Hardy-Lorentz space H~(p,q)_A(R~n) associated with A via the non-tangential grand maximal function and then establish its various real-variable characterizations in terms of the atomic and the molecular decompositions, the radial and the non-tangential maximal functions, and the finite atomic decompositions. All these characterizations except the ∞-atomic characterization are new even for the classical isotropic Hardy-Lorentz spaces on Rn.As applications, we first prove that Hp,q A(Rn) is an intermediate space between H~(p1,q1)_A(Rn) and H~(p2,q2)_A(R~n) with 0 p1 p p2 ∞ and q1, q, q2 ∈(0, ∞], and also between H~(p,q1)_A(Rn) and H~(p,q2)_A(R~n) with p ∈(0, ∞)and 0 q1 q q2 ∞ in the real method of interpolation. We then establish a criterion on the boundedness of sublinear operators from H~(p,q)_A(R~n) into a quasi-Banach space; moreover, we obtain the boundedness of δ-type Calder′on-Zygmund operators from H~(p,∞)_A(R~n) to the weak Lebesgue space L~(p,∞)(R~n)(or to H~p_A(R~n)) in the ln λcritical case, from H~(p,q)_A(R~n) to L~(p,q)(R~n)(or to H~(p,q)_A(R~n)) with δ∈(0,(lnλ)/(ln b)], p ∈(1/(1+,δ),1] and q ∈(0, ∞], as well as the boundedness of some Calderon-Zygmund operators from H~(p,q)_A(R~n) to L~(p,∞)(R~n), where b := | det A|,λ_:= min{|λ| : λ∈σ(A)} and σ(A) denotes the set of all eigenvalues of A.     

On the rates of the other law of the logarithm

Let X, X1 , X2 , . . . be i.i.d. random variables, and set Sn = X1 +···+Xn , Mn = maxk≤n |Sk|, n ≥1. Let an = o( (n)(1/2)/logn). By using the strong approximation, we prove that, if EX = 0, VarX = σ2 0 and E|X| 2+ε ∞ for some ε 0, then for any r 1, lim ε1/(r-1)(1/2) [ε-2-(r-1)]∞∑n=1 nr-2 P{Mn ≤εσ (π2n/(8log n))(1/2) + an } = 4/π . We also show that the widest a n is o( n(1/2)/logn).     

Weighted inequalities for some integral operators with rough kernels

In this paper we study integral operators with kernels $$K(x,y) = k_1 (x - A_1 y) \cdots k_m \left( {x - A_m y} \right),$$ $k_i \left( x \right) = {{\Omega _i \left( x \right)} \mathord{\left/ {\vphantom {{\Omega _i \left( x \right)} {\left| x \right|}}} \right. \kern-0em} {\left| x \right|}}^{{n \mathord{\left/ {\vphantom {n {q_i }}} \right. \kern-0em} {q_i }}}$ where Ω _i : ? ⁿ → ? are homogeneous functions of degree zero, satisfying a size and a Dini condition, A _i are certain invertible matrices, and n/q ₁ +…+n/q _m = n?α, 0 ≤ α < n. We obtain the appropriate weighted L ^p -L ^q estimate, the weighted BMO and weak type estimates for certain weights in A(p, q). We also give a Coifman type estimate for these operators.     

Estimation of Kloosterman sums with primes and its application

Suppose that p is a large prime. In this paper, we prove that, for any natural number N < p the following estimate holds: $$ \left. {\mathop {\max }\limits_{\left( {a,p} \right) = 1} } \right|\left. {\sum\limits_{q \leqslant N} {e^{{{2\pi iaq*} \mathord{\left/ {\vphantom {{2\pi iaq*} p}} \right. \kern-\nulldelimiterspace} p}} } } \right| \leqslant \left( {N^{{{15} \mathord{\left/ {\vphantom {{15} {16}}} \right. \kern-\nulldelimiterspace} {16}}} + N^{{2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-\nulldelimiterspace} 3}} p^{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-\nulldelimiterspace} 4}} } \right)p^{0\left( 1 \right)} , $$ where q is a prime and q* is the least natural number satisfying the congruence qq* ≡ 1 (modp). This estimate implies the following statement: if p > N > p ^16/17+? , where ? > 0, and if we have λ ? 0 (modp), then the number J of solutions of the congruence $$ q_1 \left( {q_2 + q_3 } \right) \equiv \lambda \left( {\bmod p} \right) $$ for the primes q ₁, q ₂, q ₃ ≤ N can be expressed as $$ J = \frac{{\pi \left( N \right)^3 }} {p}\left( {1 + O\left( {p^{ - \delta } } \right)} \right), \delta = \delta \left( \varepsilon \right) > 0. $$ This statement improves a recent result of Friedlander, Kurlberg, and Shparlinski in which the condition p > N > p ^38/39+? was required.     

Some properties of elliptic Riesz means at critical index on Hp(TH)

Suppose f∈H^p(Tⁿ), 0 _r ^δ , δ=n/p?(n+1)/2. In this paper we eastablish the following inequality $$\mathop {\sup }\limits_{R > 1} \left\{ {\frac{1}{{\log R}}\int_1^R {\left\| {\sigma _r^\delta } \right\|_{H^p (T^R )}^p \frac{{dr}}{r}} } \right\}^{1/p} \leqslant C_{R,p} \left\| f \right\|_{H^p (T^R )} $$ It implies that $$\mathop {\lim }\limits_{R \to \infty } \frac{1}{{\log R}}\int_1^R {\left\| {\sigma _r^\delta - f} \right\|_{H^p (T^R )}^p \frac{{dr}}{r}} = 0$$ Moreover we obtain the same conclusion when p=1 and n=1.     

On the best polynomial approximation of entire transcendental functions in banach spaces. I

We study the behavior of the best approximationsE _n (?) _p of entire transcendental functions ?(z) of the order ρ=∞ by polynomials of at mostn th degree in the metric of the Banach space E′_p(Ω) of functions /tf(z) analytic in a bounded simply connected domain Ω with rectifiable Jordan boundary and such that $$\left\| f \right\|_{E'_p } = \left\{ {\iint_\Omega {\left| {f\left( z \right)} \right|^p }dxdy} \right\}^{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p}}< \infty $$ . In particular, we describe the relationship between the best approximationsE _n (?)_p and theq-order andq-type of the function ?(z).     

<Emphasis Type="Italic">L</Emphasis>-Approximation of a linear combination of the poisson kernel and its conjugate kernel by trigonometric polynomials

A linear combination Π _q,α = cos(απ/2)P + sin(απ/2)Q of the Poisson kernel P(t) = 1/2 + q cos t + q ² cos 2t + ... and its conjugate kernel Q(t) = q sin t + q ² sin 2t + ... is considered for α ∈ ? and |q| < 1. A new explicit formula is found for the value E _n?1(Π _q,α ) of the best approximation in the space L = L _2π of the function Π _q,α by the subspace of trigonometric polynomials of order at most n ? 1. More exactly, it is proved that \(E_{n - 1} \left( {\prod _{q,\alpha } } \right) = \left. {\frac{{\left| q \right|^n \left( {1 - q^2 } \right)}}{{1 - q^{4n} }}} \right\|\left. {\frac{{\cos \left( {nt - {{\alpha \pi } \mathord{\left/ {\vphantom {{\alpha \pi } 2}} \right. \kern-\nulldelimiterspace} 2}} \right) - q^{2n} \cos \left( {nt + {{\alpha \pi } \mathord{\left/ {\vphantom {{\alpha \pi } 2}} \right. \kern-\nulldelimiterspace} 2}} \right)}}{{1 - q^2 - 2q \cos t}}} \right\|_L\). In addition, the value E _n?1(Π _q,α ) is represented as a rapidly convergent series.     

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).     

A decomposition for someU-type statistics

Define , $S_{k,n} = \Sigma _{1 \leqslant i_1< \cdot \cdot \cdot< l_k \leqslant n} X_{i_1 } \cdot \cdot \cdot X_{i_k } ,n \geqslant k \geqslant {\text{1}}$ where {X, X _n ,n≥1} are i.i.d. random variables withEX=0,EX ²=1 and letH _k (·) denote the Hermite polynomial of degreek. By establishing an LIL for products of correlated sums of i.i.d. random variables, the a.s. decomposition $$\begin{gathered} k!S_{k,n} = n^{k/2} H_k (S_{1n} /n^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} ) - \left( {\begin{array}{*{20}c} k \\ 2 \\ \end{array} } \right)S_{1.n}^{k - 2} \sum\limits_{i = 1}^n {(X_i^2 - 1)} \hfill \\ + O(n^{(k - 1)/2} (\log \log n)^{(k - 3/2} ) \hfill \\ \end{gathered} $$ valid whenEX ⁴<∞, elicits an LIL forη _k,n =k!S _k,n ?n ^k/2 H _k (S _1n /n ^1/2) under a reduced normalization. Moreover, whenE|X| ^p <∞ for somep in [2, 4], a Marcinkiewicz-Zygmund type strong law forη _k,n is obtained, likewise under a reduced normalization.     

Hardy-Littlewood type inequalities for Vilenkin-Fourier coefficients

The Hardy type inequality $\left( * \right) \left( {\sum\limits_{k = 1}^\infty {\frac{{\left| {\hat f\left( k \right)} \right|^p }}{{k^{2 - p} }}} } \right)^{I/p} \leqslant C_p \left\| f \right\|_{H_{ * * }^P } \left( {1/2< p \leqslant 2} \right)$ is proved for functionsf belonging to the Hardy spaceH _** ^p (G_m) defined by means of a maximal function. We extend (*) for 2<p<∞ when the Vilenkin-Fourier coefficients off are λ-blockwise monotone. It will be shown that under certain conditions on the Vilenkin system (in particular, for some unbounded type, too) a converse version of (*) holds also for allp>0 provided that the Vilenkin-Fourier coefficients off are monotone.     

Estimates for wave and Klein-Gordon equations on modulation spaces

We prove that the fundamental semi-group eit(m 2I+|Δ|)1/2(m = 0) of the Klein-Gordon equation is bounded on the modulation space M ps,q(Rn) for all 0 < p,q ∞ and s ∈ R.Similarly,we prove that the wave semi-group eit|Δ|1/2 is bounded on the Hardy type modulation spaces μsp,q(Rn) for all 0 < p,q ∞,and s ∈ R.All the bounds have an asymptotic factor tn|1/p 1/2| as t goes to the infinity.These results extend some known results for the case of p 1.Also,some applications for the Cauchy problems related to the semi-group eit(m2I+|Δ|)1/2 are obtained.Finally we discuss the optimum of the factor tn|1/p 1/2| and raise some unsolved problems.     

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.     

A generalization of the Beurling—Lax theorem

We obtain conditions for the completeness of the system {G(z)e ^τz , τ ≤ 0} in the space H _σ ² (?₊), 0 < σ < + ∞, of functions analytic in the right-hand half-plane for which $$\parallel f\parallel : = \mathop {\sup }\limits_{ - \pi /2 < \varphi < \pi /2} \left\{ {\int_0^{ + \infty } {|f(re^{i\varphi } )|^2 } e^{ - 2r\sigma |\sin \varphi |} dr} \right\}^{1/2} < + \infty $$ .