Weighted Lebesgue Spaces Research Articles

Nonlinear gradient estimates for double phase elliptic problems with irregular double obstacles

An elliptic double phase problem with irregular double obstacles is investigated to establish a Calderón-Zygmund type estimate in the setting of Lebesgue spaces and weighted Lebesgue spaces. We prove that the gradient of a solution to such a highly nonlinear problem is as integrable as both the nonhomogeneous term in divergence form and the gradient of the associated double obstacles under minimal regularity requirements on the given nonlinear elliptic operator.

Strongly singular bilinear Calderón–Zygmund operators and a class of bilinear pseudodifferential operators

Motivated by the study of kernels of bilinear pseudodifferential operators with symbols in a Hörmander class of critical order, we investigate boundedness properties of strongly singular Calderón–Zygmund operators in the bilinear setting. For such operators, whose kernels satisfy integral-type conditions, we establish boundedness properties in the setting of Lebesgue spaces as well as endpoint mappings involving the space of functions of bounded mean oscillations and the Hardy space. Assuming pointwise-type conditions on the kernels, we show that strongly singular bilinear Calderón–Zygmund operators satisfy pointwise estimates in terms of maximal operators, which imply their boundedness in weighted Lebesgue spaces.

Compactness of the commutators of intrinsic square functions on weighted Lebesgue spaces

Compactness of the commutators of intrinsic square functions on weighted Lebesgue spaces

Radial solutions of a biharmonic equation with vanishing or singular radial potentials

Given three measurable functions Vr≥0, Kr>0 and Qr≥0, r>0, we consider the bilaplacian equation Δ2u+V(|x|)u=K(|x|)f(u)+Q(|x|)inRNand we find radial solutions thanks to compact embeddings of radial spaces of Sobolev functions into sum of weighted Lebesgue spaces.

Compactness for the commutators of multilinear singular integral operators with non-smooth kernels

In this paper, the behavior for commutators of a class of bilinear singular integral operator associated with non-smooth kernels on the products of weighted Lebesgue spaces is considered. By some new maximal functions to control the commutators of bilinear singular integral operators and CMO functions, compactness of the commutators is proved.

A Bohr-Nikol’skii Inequality for Weighted Lebesgue Spaces

In this paper, we give a new inequality for weighted Lebesgue spaces called Bohr-Nikol’skii inequality, which combines the inequality of Bohr-Favard and the Nikol’skii idea of inequality for functions in different metrics.

On the generalized Hardy-Rellich inequalities

AbstractIn this paper, we look for the weight functions (sayg) that admit the following generalized Hardy-Rellich type inequality:$$\int_\Omega g (x)u^2 dx \les C\int_\Omega \vert \Delta u \vert ^2 dx,\quad \forall u\in {\rm {\cal D}}_0^{2,2} (\Omega ),$$for some constantC> 0, whereΩis an open set in ℝNwithN⩾ 1. We find various classes of such weight functions, depending on the dimensionNand the geometry of Ω. Firstly, we use the Muckenhoupt condition for the one-dimensional weighted Hardy inequalities and a symmetrization inequality to obtain admissible weights in certain Lorentz-Zygmund spaces. Secondly, using the fundamental theorem of integration we obtain the weight functions in certain weighted Lebesgue spaces. As a consequence of our results, we obtain simple proofs for the embeddings of${\cal D}_0^{2,2} $into certain Lorentz-Zygmund spaces proved by Hansson and later by Brezis and Wainger.

Kernel operators and their boundedness from weighted Sobolev space to weighted Lebesgue space

In this paper, for a wide class of integral operators, we consider the problem of their boundedness from a weighted Sobolev space to a weighted Lebesgue space. The crucial step in the proof of the main result is to use the equivalence of the basic inequality and certain Hardy-type inequality, so we first state and prove this equivalence.

Compactness of Commutators of One-Sided Singular Integrals in Weighted Lebesgue Spaces

AbstractWe prove that if p > 1, $w\in A_p^ +$, b ∈ CMO and $C_b^ + $ is the commutator with symbol b of a Calderón–Zygmund convolution singular integral with kernel supported on (−∞, 0), then $C_b^ + $ is compact from Lp(w) into itself.

The Basis Property of Ultraspherical Jacobi Polynomials in a Weighted Lebesgue Space with Variable Exponent

В настоящей работе исследована задача о базисности ультрасферических полиномов Якоби в пространстве Лебега с переменным показателем. Найдены достаточные условия на переменный показатель $p(x)>1$, которые гарантируют равномерную ограниченность последовательности сумм Фурье $S_n^{\alpha,\alpha}(f)$, $n=0,1,…$, по ультрасферическим полиномам Якоби $P_k^{\alpha,\alpha}(x)$ в весовом пространстве Лебега $L_\mu^{p(x)}([-1,1])$ с весом $\mu=\mu(x)=(1-x^2)^\alpha$, где $\alpha>-1/2$. Случай $\alpha=-1/2$ рассмотрен отдельно. Показано, что для равномерной ограниченности последовательности сумм Фурье-Чебышeва $S_n^{-1/2,-1/2}(f)$, $n=0,1,…$, в пространстве $L_\mu^{p(x)}([-1,1])$ c $\mu(x)=(1-x^2)^{-1/2}$ достаточно и в определенном смысле необходимо, чтобы переменный показатель $p$ подчинялся условию Дини-Липшица вида $$ |p(x)-p(y)|\leqslant \frac{d}{-\ln|x-y|}, \qquadгде\quad |x-y|\leqslant \frac{1}{2},\quad x,y\in[-1,1],\quad d>0, $$ а также условию $p(x)>1$ для всех $x\in[-1,1]$. Библиография: 16 названий.

On well-posedness of vector-valued fractional differential-difference equations

We develop an operator-theoretical method for the analysis on well posedness of partial differential-difference equations that can be modeled in the form \begin{document}$ \begin{equation*} (*) \left\{\begin{array}{rll} \Delta^{\alpha} u(n) & = & Au(n+2) + f(n,u(n)), \quad n \in \mathbb{N}_0, \,\, 1< \alpha \leq 2; \\ u(0) & = & u_0;\\ u(1) & = & u_1, \end{array}\right. \end{equation*} $\end{document} where $ A $ is a closed linear operator defined on a Banach space $ X $. Our ideas are inspired on the Poisson distribution as a tool to sampling fractional differential operators into fractional differences. Using our abstract approach, we are able to show existence and uniqueness of solutions for the problem (*) on a distinguished class of weighted Lebesgue spaces of sequences, under mild conditions on sequences of strongly continuous families of bounded operators generated by $ A, $ and natural restrictions on the nonlinearity $ f $. Finally we present some original examples to illustrate our results.

$L^p$-Conjecture on Hypergroups

In this paper, we study $L^p$-conjecture on locally compact hypergroups and by some technical proofs we give some sufficient and necessary conditions for a weighted Lebesgue space $L^p(K,w)$ to be a convolution Banach algebra, where $1<p<infty$, $K$ is a locally compact hypergroup and $w$ is a weight function on $K$. Among the other things, we also show that if $K$ is a locally compact hypergroup and $p$ is greater than 2, $K$ is compact if and only if $m(K)$ is finite and $fast g$ exists for all $f,gin L^p(K)$, where $m$ is a left Haar measure for $K$, and in particular, if $K$ is discrete, $K$ is finite if and only if the convolution of any two elements of $L^p(K)$ exists.

Scattering for solutions of a dissipative nonlinear Schrödinger equation

In this study, we consider the Cauchy problem for a dissipative nonlinear Schrödinger equation. In particular, we show that for any data which belong to the weighted Lebesgue space FHγ with some 0<γ≤min⁡(n/2,1) the solutions to a dissipative nonlinear Schrödinger equation with the nonlinear exponent 1+4/(n+2γ)<p<1+4/n behave like the free solutions in FHγ at time positive infinity. The lower bound for p is less than the Strauss exponent for some 0<γ≤min⁡(n/2,1) and if the critical case γ=n/2 the lower bound is equal to 1+2/n. The main idea of proof is based on an a priori estimate for solutions in FHγ of fractional order.

Commutators of Singular Integrals with Kernels Satisfying Generalized Hörmander Conditions and Extrapolation Results to the Variable Exponent Spaces

We obtain boundedness results for the higher order commutators of singular integral operators between weighted Lebesgue spaces, including Lp-BMO and Lp-Lipschitz estimates. The kernels of such operators satisfy certain regularity condition, and the symbol of the commutator belongs to a Lipschitz class. We also deal with commutators of singular integral operators with less regular kernels satisfying a Hormander’s type inequality. Moreover, we give a characterization result involving symbols of the commutators and continuity results for extreme values of p. Finally, by extrapolation techniques, we derive different results in the variable exponent context.

$C_0$-semigroups of $2$-isometries and Dirichlet spaces

In the context of a theorem of Richter, we establish a similarity between C_0 -semigroups of analytic 2 -isometries \{T(t)\}_{t\geq0} acting on a Hilbert space \mathcal H and the multiplication operator semigroup \{M_{\phi_t}\}_{t\geq 0} induced by \phi_t(s)=\mathrm {exp}(-st) for s in the right-half plane \mathbb{C}_+ acting boundedly on weighted Dirichlet spaces on \mathbb{C}_+ . As a consequence, we derive a connection with the right shift semigroup \{S_t\}_{t\geq 0} given by S_tf(x)=\left \{ \begin{array}{ll} 0 &amp; \text { if }0\leq x\leq t, \\ f(x-t)&amp; \text{ if } x&gt;t, \end{array} \right . acting on a weighted Lebesgue space on the half line \mathbb{R}_+ and address some applications regarding the study of the invariant subspaces of C_0 -semigroups of analytic 2-isometries.

Bernstein–Walsh type inequalities in unbounded regions with piecewise asymptotically conformal curve in the weighted Lebesgue space

We have obtained the pointwise Bernstein–Walsh type estimation for algebraic polynomials in the unbounded regions with piecewise asymptotically conformal boundary, having exterior and interior zero angles, in the weighted Lebesgue space.

Sharp maximal estimates for multilinear commutators of multilinear strongly singular Calderón–Zygmund operators and applications

Abstract In this paper, we aim to establish the sharp maximal pointwise estimates for the multilinear commutators generated by multilinear strongly singular Calderón–Zygmund operators and BMO functions or Lipschitz functions, respectively. As applications, the boundedness of these multilinear commutators on product of weighted Lebesgue spaces are obtained. It is interesting to note that there is no size condition assumption for the kernel of the multilinear strongly singular Calderón–Zygmund operator. Due to the stronger singularity for the kernel of the multilinear strongly singular Calderón–Zygmund operator, we need to be more careful in estimating the mean oscillation over the small balls to get the sharp maximal function estimates.

Inhomogeneous shearlet coorbit spaces

In this paper, we establish inhomogeneous coorbit spaces related to the continuous shearlet transform and the weighted Lebesgue spaces [Formula: see text] for certain weights [Formula: see text]. We present an inhomogeneous shearlet frame for [Formula: see text] which gives rise to a reproducing kernel [Formula: see text] that is not contained in the space [Formula: see text]. We show that the inhomogeneous shearlet coorbit spaces are Banach spaces by introducing a generalization of the approach of Fornasier, Rauhut and Ullrich.

Welland's type inequalities for fractional operators of convolution with kernels satisfying a Hörmander type condition and its commutators

ABSTRACTLet μ be a non-negative Ahlfors n-dimensional measure on . In this context we shall consider convolution type operators , , where the kernels are supposed to satisfy certain size and regularity conditions. We prove Welland's type inequalities for the operator and its commutator , with , that include the case . As far as we know both estimates are new even in the case of the Lebesgue measure. We shall also give sufficient conditions on a pair of weights that guarantee the boundedness of between two different weighted Lebesgue spaces when the underlying measure is Ahlfors n-dimensional.

Represensibility of cones of monotone functions in weighted Lebesgue spaces and extrapolation of operators on these cones

It is shown that a sublinear operator is bounded on the cone of monotone functions if and only if a certain new operator related to the one mentioned above is bounded on a certain ideal space defin ...