On Certain Integral Operators of Fractional Type

In this paper we study integral operators of the form $$T\,f\left( x \right) = \int {k_1 \left( {x - a_1 y} \right)k_2 \left( {x - a_2 y} \right)...k_m \left( {x - a_m y} \right)f\left( y \right)dy} ,$$ $$k_i \left( y \right) = \sum\limits_{j \in Z} {2^{\frac{{jn}}{{q_i }}} } \varphi _{i,j} \left( {2^j y} \right),\,1 \leqq q_i < \infty ,\frac{1}{{q_1 }} + \frac{1}{{q_2 }} + ... + \frac{1}{{q_m }} = 1 - r,$$ $0 \leqq r < 1$ , and $\varphi _{i,j}$ satisfying suitable regularity conditions. We obtain the boundedness of $T:L^p \left( {R^n } \right) \to T:L^q \left( {R^n } \right)$ for $1 < p < \frac{1}{r}$ and $\frac{1}{q} = \frac{1}{p} - r$ .     

Fractional Type Integral Operators on Variable Hardy Spaces

Given certain n × n invertible matrices A ₁, . . . , A _m and 0 ≦ α < n, we obtain the \({H^{p(.)}(\mathbb{R}^n) \to L^{q(.)}(\mathbb{R}^n)}\) boundedness of the integral operator with kernel \({k(x, y) = |x - A_1y|^{-\alpha_1} . . . |x - A_my|^{-\alpha_m}}\) , where α ₁ +  . . . + α _m = n ? α and p(.), q(.) are exponent functions satisfying log-Hölder continuity conditions locally and at infinity related by \({\frac{1}{q(.)} = \frac{1}{p(.)} - \frac{\alpha}{n}}\) . We also obtain the \({H^{p(.)}(\mathbb{R}^n) \to H^{q(.)}(\mathbb{R}^n)}\) boundedness of the Riesz potential operator.     

Nonexistence of Positive Solutions for an Integral Equation Related to the Hardy-Sobolev Inequality

Let α and s be real numbers satisfying 0<s<α<n. We are concerned with the integral equation $$u(x)=\int_{R^n}\frac{u^p(y)}{|x-y|^{n-\alpha}|y|^s}dy, $$ where \(\frac{n-s}{n-\alpha}< p< \alpha^{*}(s)-1\) with \(\alpha^{*}(s)=\frac{2(n-s)}{n-\alpha}\) . We prove the nonexistence of positive solutions for the equation and establish the equivalence between the above integral equation and the following partial differential equation $$\begin{aligned} (-\Delta)^{\frac{\alpha}{2}}u(x)=|x|^{-s}u^p. \end{aligned}$$      

Strong approximation of empirical process with independent but non-identically distributed random variables

Let{Y_t,t=1,2,…} be independent random variables with continuous distribution functionsF_i(y).For any y,dencte s=F_t(y)=1/t sum from i=1 to t F_i(y).The empirical process is defind by t~(-1/2)R(s,t) whereR(s,t)=t(1/t sum from i=1 to t I_((?)_t(Y_i)≤s)-s)=sum from i=1 to t I_(?)-ts=sum from i=1 to t I_(?)-(?)_t(y)=sum from i=1 to t I_(Y_(?)≤y)-sum from i=1 to t F_i(y).The purpose of this paper is to investigate the asymptotic properties of the empirical processR(s,t).We shall prove that for some integer sequence {t_k},there is a (?)-process (?)(s,t) such that(?)|R(s,t_k)-(?)(s,t_k)|=O(t_k~(1/2)(log t_k)~(-1/4)(log log t_k)~(1/2))a.s.where (?)(s,t) is a two-parameter Gaussian process defined in §1.     

On the Gagliardo-Nirenberg type inequality in the critical Sobolev-Morrey space

Our main purpose in this article is to establish a Gagliardo-Nirenberg type inequality in the critical Sobolev–Morrey space $H\mathcal{M}^{\frac{n}{p}}_{p,q}(\mathbb{R}^{n})$ with n∈? and 1<q≤p<∞, which coincides with the usual critical Sobolev space $H^{\frac{n}{p}}_{p}(\mathbb{R}^{n})$ in the case of q=p. Indeed, we shall show the following interpolation inequality. If q<p, there exists a positive constant C _p,q depending only on p and q such that GN $$ \|f\|_{{\mathcal{M}}_{r,\frac{q}{p}r}} \leq C_{p,q}r\|f \|_{{\mathcal{M}}_{p,q}}^{\frac{p}{r}}\bigl\|(-\Delta)^{\frac{n}{2p}} f\bigr\|_{{\mathcal{M}}_{p,q}}^{1-\frac{p}{r}} $$ for all $u\in H\mathcal{M}^{\frac{n}{p}}_{p,q}( \mathbb{R}^{n})$ and for all p≤r<∞. In the case of q=p, that is, the case of the critical Sobolev space $H^{\frac{n}{p}}_{p}(\mathbb{R}^{n})$ , the corresponding inequality was obtained in Ogawa (Nonlinear Anal. 14:765–769, 1990), Ogawa-Ozawa (J. Math. Anal. Appl. 155:531–540, 1991) and Ozawa (J. Func. Anal. 127:259–269, 1995) with the growth order $r^{1-\frac{1}{p}}$ as r→∞. The inequality (GN) implies that the growth order as r→∞ is linear, which might look worse compared to the case of the critical Sobolev space. However, we investigate the optimality of the growth order and prove that this linear order is best-possible. Furthermore, as several applications of the inequality (GN), we shall obtain a Trudinger-Moser type inequality and a Brézis-Gallouët-Wainger type inequality in the critical Sobolev-Morrey space.     

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 \(\lim _{n \to \infty } (1/\sqrt n )V(\chi _0^n ) = \phi (p)\) .     

Characterization of the Bernoulli-polynomials,exp andΨ by Nörlund's multiplication formula

The system of functional equations $$\forall p\varepsilon N_ + \forall (x,y)\varepsilon D:f(x,y) = \frac{1}{p}\sum\limits_{k = 0}^{p - 1} {f(x + ky,py)}$$ is suited to characterize the functions $$(x,y) \mapsto y^m B_m \left( {\frac{x}{y}} \right),m\varepsilon N,$$ B _m means them-th Bernoulli-polynomial, $$(x,y) \mapsto \exp (x)y(\exp (y) - 1)^{ - 1}$$ (for these functionsD =R ×R ₊) and $$(x,y) \mapsto \log y + \Psi \left( {\frac{x}{y}} \right)(D = R_ + \times R_ + )$$ as those continuous solutions of this system which allow a certain separation of variables and take on some prescribed function values.     

Optimality estimations for approximately midconvex functions

Let V be a convex subset of a normed space and let a nondecreasing function α : [0, ∞) → [0, ∞) be given. A function ${f : V \rightarrow \mathbb{R}}$ is called α-midconvex if $$f\left(\frac{x+y}{2} \right)\leq \frac{f(x)+f(y)}{2}+\alpha(\|x-y\|) \quad \,{\rm for}\, x,y\in V.$$ It is known (Tabor in Control Cybern., 38/3:656–669, 2009) that if ${f : V \rightarrow \mathbb{R}}$ is α-midconvex, locally bounded above at every point of V then $$f(tx+(1-t)y)\leq tf(x)+(1-t)f(y)+P_\alpha(\|x-y\|) \quad \,{\rm for}\, x, y \in V,t \in [0,1],$$ where ${P_\alpha(r):=\sum_{k=0}^\infty \frac{1}{2^k} \alpha(2{\rm dist}(2^kr, \mathbb{Z}))}$ for ${r \in \mathbb{R}}$ . We show that under some additional assumptions the above estimation cannot be improved.     

Weak-strong uniqueness criterion for the \beta -generalized surface quasi-geostrophic equation

We prove that weak-strong uniqueness holds for the $\beta $ -generalized surface quasi-geostrophic equation in the regular class $\nabla \theta \in L^{q}(0,T; L^{p}(\mathbb{R }^{2}))$ with $\frac{\alpha }{q}+\frac{2}{p}=\alpha +\beta -1$ , where $\alpha \in (0,1], \beta \in [1,2)$ and $\frac{2}{\alpha +\beta -1}<p<\infty $ .     

Riesz Means on Graphs and Discrete Groups

Suppose that Γ is a weighted graph or a discrete group. Let $m_{\alpha,R}(\lambda )=\big(1-\big|\frac{\lambda}{R}\big|\big)_{+}^{\alpha}$ be the Riesz means and let Δ be the discrete Laplacian on Γ. We prove that if D is the homogeneous dimension of Γ then the operator m _α,R (Δ) is bounded on L ^p , provided that $\alpha>D|\frac{1}{p}-\frac{1}{2}|$ .     

Well-posedness of a semilinear heat equation with weak initial data

This article mainly consists of two parts. In the first part the initial value problem (IVP) of the semilinear heat equation $$\begin{gathered} \partial _t u - \Delta u = \left| u \right|^{k - 1} u, on \mathbb{R}^n x(0,\infty ), k \geqslant 2 \hfill \\ u(x,0) = u_0 (x), x \in \mathbb{R}^n \hfill \\ \end{gathered} $$ with initial data in $\dot L_{r,p} $ is studied. We prove the well-posedness when $$1< p< \infty , \frac{2}{{k(k - 1)}}< \frac{n}{p} \leqslant \frac{2}{{k - 1}}, and r =< \frac{n}{p} - \frac{2}{{k - 1}}( \leqslant 0)$$ and construct non-unique solutions for $$1< p< \frac{{n(k - 1)}}{2}< k + 1, and r< \frac{n}{p} - \frac{2}{{k - 1}}.$$ In the second part the well-posedness of the avove IVP for k=2 with μ₀?H ^s (? ⁿ ) is proved if $$ - 1< s, for n = 1, \frac{n}{2} - 2< s, for n \geqslant 2.$$ and this result is then extended for more general nonlinear terms and initial data. By taking special values of r, p, s, and u₀, these well-posedness results reduce to some of those previously obtained by other authors [4, 14].     

A sharp estimate for the rate of convergence in mean of Birkhoff sums for some classes of periodic differentiable functions

For a badly approximable vector α, we obtain a sharp estimate for the rate of convergence in the space L _p (0 < p < ∞) of the Birkhoff means $\frac{1}{n}\sum\nolimits_{s = 0}^{n = 1} {f(x + s\alpha )} $ for an absolutely continuous periodic function f and for functions in spaces of Bessel potentials.     

A sharp equivalence between H ∞ functional calculus and square function estimates

Let (T _t ) _{t?≥ 0} be a bounded analytic semigroup on L ^p (Ω), with 1?<?p?<?∞. Let ?A denote its infinitesimal generator. It is known that if A and A ^* both satisfy square function estimates ${\bigl\|\bigl(\int_{0}^{\infty} \vert A^{\frac{1}{2}} T_t(x)\vert^2 {\rm d}t \bigr)^{\frac{1}{2}}\bigr\|_{L^p} \lesssim \|x\|_{L^p}}$ and ${\bigl\|\bigl(\int_{0}^{\infty} \vert A^{*\frac{1}{2}} T_t^*(y) \vert^2 {\rm d}t \bigr)^{\frac{1}{2}}\bigr\|_{L^{p^\prime}} \lesssim \|y\|_{L^{p^\prime}}}$ for ${x\in L^p(\Omega)}$ and ${y\in L^{p^\prime}(\Omega)}$ , then A admits a bounded ${H^{\infty}(\Sigma_\theta)}$ functional calculus for any ${\theta>\frac{\pi}{2}}$ . We show that this actually holds true for some ${\theta<\frac{\pi}{2}}$ .     

Partial regularity for non autonomous functionals with non standard growth conditions

We prove a C ^1,μ partial regularity result for minimizers of a non autonomous integral funcitional of the form $$\mathcal{F}(u; \Omega):=\int_{\Omega}f(x, Du)\ dx$$ under the so-called non standard growth conditions. More precisely we assume that $$c |z|^{p}\leq f(x ,z) \leq L (1+|z|^{q}),$$ for 2 ≤ p < q and that D _z f(x, z) is α-Hölder continuous with respect to the x-variable. The regularity is obtained imposing that ${\frac{p}{q} < \frac{n+\alpha}{n}}$ but without any assumption on the growth of ${D^{2}_{z}f}$ .     

The Concentration Function of Additive Functions with Nonmultiplicative Weight

Suppose that g(n) is a real-valued additive function and τ(n) is the number of divisors of n. In this paper, we prove that there exists a constant C such that $\sup \limits_a \sum\limits_{n<N}{g(n) \in [a,a+1)} \tau(N-n) \leqslant C \frac{N \log N}{\sqrt{W(N)}},$ where $W(N) = 4 + \mathop {min}\limits_\lambda \left( {\lambda ^2 + \sum\limits_{p < N} {\frac{1}{p}} min(1,(g(p) - \lambda log p)^2 )} \right).$ . In particular, it follows from this result that $\mathop {\sup }\limits_a |\{ m,n:mn < N,g(N - mn) = a\} | \ll N\log N\left( {\sum\limits_{p < N,g\left( p \right) \ne 0} {(1/p)} } \right)^{ - 1/2} .$ The implicit constant is absolute.     

Existence results of fractional differential equations with irregular boundary conditions and p-Laplacian operator

In this paper, we discuss the existence of solutions for irregular boundary value problems of nonlinear fractional differential equations with p-Laplacian operator $$\left \{ \begin{array}{l} {\phi}_p(^cD_{0+}^{\alpha}u(t))=f(t,u(t),u'(t)), \quad 0< t<1, \ 1< \alpha \leq2, \\ u(0)+(-1)^{\theta}u'(0)+bu(1)=\lambda, \qquad u(1)+(-1)^{\theta}u'(1)=\int_0^1g(s,u(s))ds,\\ \quad \theta=0,1, \ b \neq \pm1, \end{array} \right . $$ where \(^{c}D_{0+}^{\alpha}\) is the Caputo fractional derivative, ? _p (s)=|s| ^p?2 s, p>1, \({\phi}_{p}^{-1}={\phi}_{q}\) , \(\frac {1}{p}+\frac{1}{q}=1\) and \(f: [0,1] \times\mathbb{R} \times\mathbb {R} \longrightarrow\mathbb{R}\) . Our results are based on the Schauder and Banach fixed point theorems. Furthermore, two examples are also given to illustrate the results.     

The Concentration Function of Additive Functions with Special Weight

Suppose that $${g\left( n \right)}$$ is an additive real-valued function, W(N) = 4+ $$\mathop {\min }\limits_\lambda $$ ( λ₂ + $$\sum\limits_{p < N} {\frac{1}{2}} $$ min (1, ( g(p) - λlog p)²), E(N) = 4+1 $$\sum\limits_{\mathop {p < N,}\limits_{g(p) \ne 0} } {\frac{1}{p}.} $$ In this paper, we prove the existence of constants C₁, C₂ such that the following inequalities hold: $\mathop {\sup }\limits_a \geqslant \left| {\left\{ {n, m, k: m, k \in \mathbb{Z},n \in \mathbb{N},n + m^2 + k^2 } \right.} \right. = \left. {\left. {N,{\text{ }}g(n) \in [a,a + 1)} \right\}} \right| \leqslant \frac{{C_1 N}}{{\sqrt {W\left( N \right)} }},$ $\mathop {\sup }\limits_a \geqslant \left| {\left\{ {n, m, k: m, k \in \mathbb{Z},n \in \mathbb{N},n + m^2 + k^2 } \right.} \right. = \left. {\left. {N,{\text{ }}g(n) = a} \right\}} \right| \leqslant \frac{{C_2 N}}{{\sqrt {E\left( N \right)} }},$ . The obtained estimates are order-sharp.     

Curvature estimates for submanifolds with prescribed Gauss image and mean curvature

We study that the n-graph defined by a smooth map ${f:\Omega\subset\mathbb R^{n}\to \mathbb R^{m}, m\ge 2,}$ in ${\mathbb R^{m+n}}$ of the prescribed mean curvature and the Gauss image. Under the condition $$\Delta_f=\left[\text{det}\left(\delta_{ij}+\sum_\alpha{\frac {\partial {f^\alpha}}{\partial {x^i}}}{\frac {\partial {f^\alpha}}{\partial {x^j}}}\right)\right]^{\frac{1}{2}} < 2,$$ we derive the interior curvature estimates $$\sup_{D_R(x)}|B|^2\le{\frac{C}{R^2}}$$ when 2 ≤ n ≤ 5 with constant C depending on the given geometric data. If there is no dimension limitation we obtain $$\sup_{D_R(x)}|B|^2\le CR^{-a}\sup_{D_{2R}(x)}(2-\Delta_f)^{-\left({\frac{3}{2}}+{\frac{1}{s}}\right)},\quad s=\min(m, n)$$ with a < 1. If the image under the Gauss map is contained in a geodesic ball of the radius ${{\frac{\sqrt{2}}{4}}\pi}$ in G _n,m we also derive corresponding estimates.     

Regularized sums of half-integer powers of a Sturm-Liouville operator

We investigate the question of the regularized sums of part of the eigenvalues z_n (lying along a direction) of a Sturm-Liouville operator. The first regularized sum is $$\sum\nolimits_{n = 1}^\infty {(z_n - n - \frac{{c_1 }}{n} + \frac{2}{\pi } \cdot z_n arctg \frac{1}{{z_n }} - \frac{2}{\pi }) = \frac{{B_2 }}{2} - c_1 \cdot \gamma + \int_1^\infty {\left[ {R(z) - \frac{{l_0 }}{{\sqrt z }} - \frac{{l_1 }}{z} - \frac{{l_2 }}{{z\sqrt z }}} \right]} } \sqrt z dz,$$ where the z_n are eigenvalues lying along the positive semi-axis, z _n ² =λ_n, $$l_0 = \frac{\pi }{2}, l_1 = - \frac{1}{2}, l_2 = - \frac{1}{4}\int_0^\pi {q(x) dx,} c_1 = - \frac{2}{\pi }l_2 ,$$ , B₂ is a Bernoulli number, γ is Euler's constant, and \(R(z)\) is the trace of the resolvent of a Sturm-Liouville operator.     

A conjecture of Segre on diophantine approximation

In 1945,B. Segre proved the following classical theorem: Every irrational ξ has an infinity of rational approximationsp/q such that (0) $$\frac{{ - 1}}{{q^2 \sqrt {1 + 4\tau } }}< \frac{p}{q} - \xi< \frac{\tau }{{q^2 \sqrt {1 + 4\tau } }},$$ where τ is any given non-negative real number. Segre conjectured that when τ≠0 and τ^?1 is not an integer, inequalities (0) can be improved by replacing \(\sqrt {1 + 4\tau } \) and \(\sqrt {1 + 4\tau } /\tau \) with larger numbers. In this paper we prove that these two numbers can be replaced with the larger numbers \(\sqrt {1 + 4\tau } + 0.2\tau ^2 \{ \tau ^{ - 1} \} (1 - \{ \tau ^{ - 1} \} )\) and \(\sqrt {1 + 4\tau } /\tau + 0.2\tau ^2 \{ \tau ^{ - 1} \} (1 - \{ \tau ^{ - 1} \} )\) respectively, where {τ^?1} is the fractional part of τ^?1.